diff --git "a/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt" "b/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt" @@ -0,0 +1,1037 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf,len=1036 +page_content='Quantum interference in the resonance fluorescence of a J = 1/2 − J′ = 1/2 atomic system: Quantum beats, nonclassicality, and non-Gaussianity H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Castro-Beltr´an,1, ∗ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' de los Santos-S´anchez,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Guti´errez,3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Alcantar-Vidal1 1Centro de Investigaci´on en Ingenier´ıa y Ciencias Aplicadas and Instituto de Investigaci´on en Ciencias B´asicas y Aplicadas, Universidad Aut´onoma del Estado de Morelos, Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico 2Tecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias, Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Carlos Lazo 100, Santa Fe, Mexico City, Mexico, 01389 3Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, 62210 Cuernavaca, Morelos, M´exico (Dated: January 10, 2023) We study the resonance fluorescence of a system with angular momentum J = 1/2−J′ = 1/2 level structure driven by a single, linearly polarized, monochromatic laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Quantum interference among the two, antiparallel, π transitions leads to rich results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We develop the article around two broad overlapping themes: (i) the observation of quantum beats in the intensity and the dipole- dipole, intensity-intensity, and quadrature-intensity correlations, when the atom is subject to a strong laser and large Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The mean and modulation frequencies of the beats are given by the average and difference, respectively, among two close generalized Rabi frequencies related to a Mollow-like spectrum with two pairs of sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (ii) The nonclassical and non- Gaussian properties of phase-dependent fluorescence for the cases of weak to moderate excitation and in the regime of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The fluorescence in the beats regime is nonclassical, mainly from the third-order dipole fluctuations, which reveal them to be also strongly non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For weak to moderate driving laser and small detunings and Zeeman splittings the nonclassicality is an interplay of second- (squeezing) and third-order dipole noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' INTRODUCTION Recently, the properties of the resonance fluorescence of a single atomic system with angular momentum transi- tion J = 1/2−J′ = 1/2 driven by a monochromatic laser have been the subject of great interest due to the possi- bility of observing vacuum-induced coherence effects due to interference among the two antiparallel π transitions, emitting into the same frequency range of the electro- magnetic vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Here, the π transitions are incoher- ently coupled, mediated by spontaneous emision in the σ transitions and then excited by the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The antiparal- lel dipoles of the transitions makes it realistic to observe interference effects, while V and Λ three-level systems re- quire additional preparation because the transitions are perpendicular [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Particular attention has been de- voted to the spectrum [3–6], time-energy complementar- ity [4, 5], Young’s interference [7], photon correlations [8], frequency-resolved photon correlations [9], squeezing [10], phase shifts [11], and cooperative effects in photon correlations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The case of additional laser excitation of one of the σ transitions on the spectrum and squeezing has been studied in [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Quantum beats are among the more familiar mani- festations of quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' They appear in the modulation of the decay by spontaneous emission of mul- tilevel systems due to the energy difference among transi- tions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' So far, few experiments of quantum interference experiments have been performed on the J = 1/2 − J′ = ∗ hcastro@uaem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='mx 1/2, in this case observing Young-type fringes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Hence, further experiments are desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Quantum beats in the intensity are the result of the inability to tell the path of a particular photon when observed by a broadband detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The beats can also occur in two-time correla- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As a general rule, initial conditions should be a superposition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In this paper we investigate theoretically effects of quantum interference on the total intensity and two-time correlations such as dipole-dipole (to calculate spectra), intensity-intensity, intensity-amplitude correlations, and variance of the light emitted into the π transitions of the J = 1/2−J′ = 1/2 atomic system driven by a linearly po- larized laser and a magnetic field to break the degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' While we put emphasis on the regime of observation of quantum beats, the nonclassical and non-Gaussian prop- erties of the fluorescence are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' After describing the main features of the model in Section II, we discuss the basic dynamic and stationary properties of the atomic expectation values in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Here, we analyze the previously overlooked time- dependent behavior of the atomic populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Those of the excited states, for instance, although equal in the steady state, evolve with different Rabi frequencies and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is at the root of the formation of beats in the intensity and the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the regime of strong laser and magnetic fields these beats are character- ized by well-defined oscillations at the average frequency among two generalized Rabi frequencies, modulated at the difference of those frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' To observe beats in the intensity both ground state populations must be nonzero initially, ideally equal [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Similarly, for the two- time correlations, the vector of initial conditions must arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='03061v1 [quant-ph] 8 Jan 2023 2 have at least two nonzero terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Section IV we describe the scattered field intensity and quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Here, beats depend only on the inter- ference of the two upper populations in the nondegener- ate case, with both lower populations initially nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Cross terms of the oppposite π transitions represent in- terference in the steady state intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Then, In Section V, using the dressed states approach, we show that the double sideband spectrum [5] stems from a dipole-dipole correlation with beats, where the terms of addition of sin- gle π transitions dominate over those of the cross terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Section VI we study Brown-Twiss photon-photon correlations [16, 17], extending the work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [8] to the nondegenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Besides the ubiquitous antibunching effect, for weak to moderate laser drivings the interplay of parameters, together with detuning and Zeeman split- tings, can make for somewhat involved evolutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', long decays due to optical pumping in the non-degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Again, cross terms are minor contributors to the full correlation in the beats regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Section VII is devoted to a study of phase-dependent fluctuations by conditional homodyne detection (CHD) [18, 19] in both the temporal and spectral domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The CHD method is characterized by amplitude-intensity cor- relations (AIC), which are of third order in the field am- plitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' When the atomic operators are decomposed into a mean plus a noise operator the AIC is split into a second-order term which would be a measure of squeezing if the third-order one were negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' But the latter is not negligible outside the weak field regime of resonance fluorescence, which make the fluctuations non-Gaussian and also nonclassical by the violation of classical inequal- ities [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We obtain the spectra of the total, second- and third-order terms of the AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Narrow peaks in the spec- tra reveal population trapping when detunings favour the long term population or optical pumping of the ground state of the more detuned transition, which in the time domain show the above mentioned long decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The third-order terms make up most of the beats and thus they are non-Gaussian and nonclassical but not squeezed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Section VIII we consider squeezing by means of the variance of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As usual, squeezing in resonance fluorescence is small and restricted to weak or moderate Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Finally, in Section IX we provide a discussion and conclusions, and two Appendices give de- tails on solution methods, initial conditions, and optimal appearance of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' MODEL The system, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 1, consists of a two-level atom with transition J = 1/2 – J = 1/2 and states with magnetic quantum number m = ±J, |1⟩ = |J, −1/2⟩, |2⟩ = |J, 1/2⟩, |3⟩ = |J, −1/2⟩, |4⟩ = |J, 1/2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (1) The matrix elements are FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Scheme of the J = 1/2 – J = 1/2 atomic system interacting with a laser driving the |1⟩ − |3⟩ and |2⟩ − |4⟩ transitions with Rabi frequency Ω and detuning ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' There are spontaneous decay rates γ1, γ2 and γσ, vacuum-induced coherence γ12, and Zeeman frequency splittings Bℓ and Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' d1 = ⟨1|ˆd|3⟩ = − 1 √ 3Dez, d2 = ⟨2|ˆd|4⟩ = −d1, d3 = ⟨2|ˆd|3⟩ = � 2 3De−, d4 = ⟨1|ˆd|4⟩ = d∗ 3, (2) where D is the reduced dipole matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We choose the field polarization basis {ez, e−, e+} (linear, left cir- cular, right circular), where e± = ∓(ex ± iey)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The π transitions, |1⟩ − |3⟩ and |2⟩ − |4⟩ (m = m′), are coupled to linearly polarized light and have their dipole moments antiparallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' On the other hand, the σ tran- sitions, |1⟩ − |4⟩ and |2⟩ − |3⟩ (m ̸= m′), are coupled to circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This configuration can be found, for example, in 198Hg+ [3], and 40Ca+ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The level degeneracy is removed by the application of a static magnetic field Bz along the z direction, the Zee- man effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Note that the energy splittings gµBBz of the upper (u) and lower (ℓ) levels are different due to unequal Land´e g factors, gu and gℓ, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' µB is Bohr’s magneton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The difference Zeeman splitting is δ = (gu − gℓ)µBBz ¯h = gu − gℓ gℓ Bℓ, (3) where Bℓ = glµBBz/¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For 198Hg+ gu = 2/3 and gℓ = 2, so ¯hδ = −(4/3)µBBz = −(2/3)¯hBℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The atom is driven by a monochromatic laser of fre- quency ωL, linearly polarized in the z direction, propa- gating in the x direction, EL(x, t) = E0ei(ωLt−kLx)ez + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', (4) thus driving only the π transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The free atomic, H0, and interaction, V , parts of the Hamiltonian are, respectively: H0 = ¯hω13A11 + ¯h(ω24 + Bℓ)A22 + ¯hBℓA44, (5) V = ¯hΩ(A13 − A24)eiωLt + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (6) 3 where Ajk = |j⟩⟨k| are atomic operators, ω13 and ω24 = ω13 + δ are the frequencies of the |1⟩ − |3⟩ and |2⟩ − |4⟩ transitions, respectively, and Ω = E0D/ √ 3 ¯h is the Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The frequencies of the other transitions are ω23 = ω13 − δ and ω14 = ω13 − Bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Using the unitary transformation U = exp [(A11 + A22)iωLt], (7) the Hamiltonian in the frame rotating at the laser fre- quency is H = U †(H0 + V )U, = −¯h∆A11 − ¯h(∆ − δ)A22 + ¯hBℓ(A22 + A44) +¯hΩ [(A13 − A24) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='] , (8) where ∆ = ωL −ω13 is the detuning of the laser from the |1⟩ − |3⟩ resonance transition, and ∆ − δ is the detuning on the |2⟩ − |4⟩ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The excited states decay either in the π transitions emitting photons with linear polarization at rates γ1 = γ2, or in the σ transitions emitting photons of circular polarization at rate γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' There is also a cross-coupling of the excited states by the reservoir, responsible for the quantum interference we wish to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In general, the decay rates are written as γij = di · d∗ j |di||dj| √γiγj, i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (9) In particular, we have γii = γ1 = γ2 and γ13 = γ24 = γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Also, given that d1 and d2 are antiparallel, γ12 = γ21 = −√γ1γ2 = −γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The total decay rate is γ = γ1 + γσ = γ2 + γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (10) The decays for the π and σ transitions occur with the branching fractions bπ and bσ [5], respectively, γ1 = γ2 = bπγ, bπ = 1/3, (11a) γσ = bσγ, bσ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (11b) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' MASTER EQUATION The dynamics of the atom-laser-reservoir system is de- scribed by the master equation for the reduced atomic density operator, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In a frame rotating at the laser fre- quency (˜ρ = UρU †) it is given by ˙˜ρ = − i ¯h[H, ˜ρ] + Lγ ˜ρ, (12) where −(i/¯h)[H, ˜ρ] describes the coherent atom-laser in- teraction and Lγ ˜ρ describes the damping due to sponta- neous emission [5, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Defining S− 1 = A31, S− 2 = A42, S− 3 = A32, S− 4 = A41, S+ i = (S− i )†, (13) the dissipative part is written as Lγ ˜ρ = 1 2 2 � i,j=1 γij � 2S− i ˜ρS+ j − S+ i S− j ˜ρ − ˜ρS+ i S− j � +γσ 2 4 � i=3 � 2S− i ˜ρS+ i − S+ i S− i ˜ρ − ˜ρS+ i S− i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (14) We now define the Bloch vector of the system as Q ≡ (A11, A12, A13, A14, A21, A22, A23, A24, A31, A32, A33, A34, A41, A42, A43, A44)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (15) The equations for the expectation values of the atomic operators, ⟨Ajk⟩ = ˜ρkj, are the so-called Bloch equations, which we write as d dt⟨Q(t)⟩ = MB⟨Q(t)⟩, (16) where MB is a matrix of coeficients of the full master equation, and the formal solution is ⟨Q(t)⟩ = eMBt⟨Q(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (17) Since we are interested only in properties of the fluores- cence emitted in the π transitions we use the simplifying fact, already noticed in [8], that these Bloch equations can be split into two decoupled homogeneous sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Set 1 contains the equations for the populations and the coher- ences of the coherently driven π transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' these are ⟨ ˙A11⟩ = −γ⟨A11⟩ + iΩ(⟨A31⟩ − ⟨A13⟩), ⟨ ˙A13⟩ = − �γ 2 + i∆ � ⟨A13⟩ − iΩ(⟨A11⟩ − ⟨A33⟩), ⟨ ˙A22⟩ = −γ⟨A22⟩ − iΩ(⟨A42⟩ − ⟨A24⟩), ⟨ ˙A24⟩ = − �γ 2 + i(∆ − δ) � ⟨A24⟩ + iΩ(⟨A22⟩ − ⟨A44⟩), ⟨ ˙A31⟩ = − �γ 2 − i∆ � ⟨A31⟩ + iΩ(⟨A11⟩ − ⟨A33⟩), ⟨ ˙A33⟩ = γ1⟨A11⟩ + γσ⟨A22⟩ − iΩ(⟨A31⟩ − ⟨A13⟩), ⟨ ˙A42⟩ = − �γ 2 − i(∆ − δ) � ⟨A42⟩ − iΩ(⟨A22⟩ − ⟨A44⟩), ⟨ ˙A44⟩ = γσ⟨A11⟩ + γ2⟨A22⟩ + iΩ(⟨A42⟩ − ⟨A24⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (18) with Bloch vector R ≡ (A11, A13, A22, A24, A31, A33, A42, A44)T (19) and a corresponding matrix M, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Equations (18) do not depend on γ12, the vacuum-induced coupling of the upper levels, but on the applied magnetic field only through the difference of Zeeman splittings, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The steady state solutions, for which we introduce the 4 0 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 γt γt (d) ∆ = −2γ, δ = −4γ (b) ∆ = 2γ, δ = −2γ ( ) ∆ = −2γ, δ = −2γ ⟨A22 (t)⟩ ⟨A44 (t)⟩ ⟨A11 (t)⟩ ⟨A33 (t)⟩ (a) ∆ = 0, δ = 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Time-dependent populations ⟨A11(t)⟩ (solid-black), ⟨A22(t)⟩ (dashed-red), ⟨A33(t)⟩ (dots-green), and ⟨A44(t)⟩ (dashed-dots-blue), with the atom initially in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The parameters are: Ω = γ and (a) ∆ = δ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (b) ∆ = 2γ, δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (c) ∆ = δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (d) ∆ = −2γ, δ = −4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' short notation αjk = ⟨Ajk⟩st, are α11 = α22 = Ω2 2D, (20a) α33 = Ω2 + γ2/4 + ∆2 2D , (20b) α44 = Ω2 + γ2/4 + (∆ − δ)2 2D , (20c) α13 = Ω(∆ + iγ/2) 2D , (20d) α24 = Ω(δ − ∆ − iγ/2) 2D , (20e) αkj = α∗ jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' where D = 2Ω2 + γ2 + δ2 4 + � ∆ − δ 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (21) Note also that in the degenerate system (δ = 0) α33 = α44 and that α31 = −α42, where the minus sign arises from the fact that the dipole moments d1 and d2 are antiparallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Set 2 contains the equations for the coherences of the σ transitions and those among both upper and both lower levels, R2 ≡ (A12, A14, A21, A23, A32, A34, A41, A43)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (22) The equations for their expected values do depend on Bℓ and γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' These coherences vanish because the σ tran- sitions are driven incoherently (⟨{A14, A23, A32, A41}⟩), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', by spontaneous emission, or because they are medi- ated by those σ transitions (⟨{A12, A21, A34, A43}⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For completeness, we write the steady state results: α12 = α34 = α14 = α23 = 0, αkj = α∗ jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (23) 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 α11 = α22 α33 α44 (d) ∆ = −2γ, δ = −4γ (b) ∆ = 2γ, δ = −2γ (a) ∆ = 0, δ = 0 ( ) ∆ = −2γ, δ = −2γ Ω/γ Ω/γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Steady-state populations as a function of Rabi frequency: α11 = α22 (dashed-red), α33 (solid-black), and α44 (dots-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' All other parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The properties of the fluorescence of the π transitions, the subject matter of this article, do not depend on the equations for Set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Only the second- and third-order amplitude-intensity correlations and the dipole correla- tion for the spectrum of the σ transitions would require the full set of Bloch equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We gain valuable information on the nontrivial dynam- ics of the atomic system from single-time expectation val- ues, apparently ignored in the previous literature on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2 we show the populations for several particular cases, all with the atom initially in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the degenerate case, δ = 0, the upper populations reach opposite phases by the end of the first Rabi cycle, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is understandable since the electron oc- cupation of, say, state |1⟩ implies not to be in state |2⟩, and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The same occurs for the lower popula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Next, we show three situations for the nondegen- erate case with δ < 0 (as it is for 198Hg+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2(b) the laser is slightly detuned above the |1⟩−|3⟩ transition, but highly detuned from the |2⟩−|4⟩ transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the oscil- lations get out of phase and most of the population ends up in state |4⟩ by optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2(c) the laser is detuned below the |1⟩−|3⟩ transition, and the |2⟩−|4⟩ transition is now on resonance with the laser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' again, the oscillations are out of phase but most of the population ends up now in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2(d) we extend the pre- vious case but with stronger applied magnetic field, thus the non-degeneracy is more evident;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the large detuning on both transitions makes it recover the opposite phases of the degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 3 we show the steady state populations as a function of the Rabi frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the other parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For strong fields the populations tend to be equal (1/4), but arrive at that limit at dif- ferent rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' for instance, for large detunings on both transitions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 3(d), it takes larger fields, as compared to the degenerate case, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' On the other hand, for small detunings and weak-moderate fields, when one 5 transition is closer to resonance than the other, the lower state of the more detuned transition is more populated, as seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 3 (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' THE SCATTERED FIELD In this Section we present the main dynamical and stationary properties of the field scattered by the atom, with emphasis on the π transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Single-Time and Stationary Properties The positive-frequency part of the emitted field oper- ator is [5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 21] ˆE+(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' t) = ˆE+ free(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' t) + ˆE+ S (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ˆt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (24) where ˆE+ free(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' t) is the free-field part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' which does not con- tribute to normally ordered correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' hence we omit it in further calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and ˆE+ S (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' t) = −η r 4 � i=1 ω2 i ˆr × (ˆr × di)S− i (ˆt) (25) is the dipole source field operator in the far-field zone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' where ˆt = t−r/c is the retarded time and η = (4πϵ0c2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Since ωi ≫ δ, we may approximate the four transition as a single one ω0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (25, but cannot do so at the level of decay rates, Rabi frequencies, and splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Making ˆr = ey the direction of observation and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (2) we have ˆE+ S (r, ˆt) = ˆE+ π (r, ˆt) ez + ˆE+ σ (r, ˆt) ex, (26) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', the fields scattered from the π and σ transitions are polarized in the ez and ex directions, respectively, where ˆE+ π (r, ˆt) = fπ(r) � A31(ˆt) − A42(ˆt) � , (27a) ˆE+ σ (r, ˆt) = fσ(r) � A32(ˆt) − A41(ˆt) � , (27b) are the positive-frequency source field operators of the π and σ transitions, and fπ(r) = −ηω2 1D/ √ 3r, fσ(r) = √ 2fπ(r), (28) are their geometric factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The intensity in the π transitions is given by Iπ(r, ˆt) = ⟨ ˆE− π (r, ˆt) · ˆE+ π (r, ˆt)⟩ = f 2 π(r)⟨A13(ˆt)A31(ˆt) + A24(ˆt)A42(ˆt)⟩ = f 2 π(r)⟨A11(ˆt) + A22(ˆt)⟩, (29a) while in the steady state is Ist π = f 2 π(r) [α11 + α22] = Ω2 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (29b) 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='20 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='25 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='25 Iπ (r, t) �f 2 π (r) Iπ (r, t) �f 2 π (r) ⟨A11 (t)⟩ ⟨A22 (t)⟩ ⟨A11 (t)⟩ ⟨A22 (t)⟩ ⟨A11 (t)⟩ ⟨A22 (t)⟩ ⟨A11 (t)⟩ ⟨A22 (t)⟩ γt γt γt γt γt γt (d) (c) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Fluorescence intensity of the π transitions with equal initial ground state populations, ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The other parameters are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2: Ω = γ and (a) ∆ = δ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (b) ∆ = 2γ, δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (c) ∆ = δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (d) ∆ = −2γ, δ = −4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The insets show the excited states populations: ⟨A11(t)⟩ (solid), ⟨A22(t)⟩ (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Adding the excited state populations with the atom initially in the single state |3⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 29a gives simply Iπ(r, ˆt) = f 2 π(r)⟨A11(ˆt)⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', without the contribution of ⟨A22(ˆt)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' More interesting is the case where the ini- tial condition is ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 4 (see the populations ⟨A11(t)⟩ and ⟨A22(t)⟩ in the insets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The modulation in the intensity is reminiscent of the quantum beats in the spontaneous decay in the V three-level system [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' These beats are basically due to the inability to tell from which of the π transitions a photon comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is the standard Young-type interference [4, 5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The main requirement is that the initial condition for both ground states are nonzero (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' More interesting, though, is the case of strong resonant laser and magnetic fields and the laser is detuned far from the |2⟩ − |4⟩ resonance frequency, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Due to the laser detuning, the population ⟨A22(t)⟩ has larger frequency and smaller amplitude than that of ⟨A11(t)⟩, as seen in the insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Remarkably well-defined wave-packets or beats are observed due to the interference of the flu- orescence of both π transitions with close Rabi frequen- cies, with clear average and modulation frequencies (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The beats get scrambled with larger frequency and amplitude differences, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Save for the decay, these beats are more like the classic textbook ones, described by a modulation and an average frequency, unlike the beats from spontaneous emission or weak resonance fluorescence from two or more closely sep- arated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Henceforth, we reserve the moniker beats to those due to strong applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Further analyses of the beats are given in the next Sections, as they show up 6 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 Iπ (r, t) � f2 π (r) Iπ (r, t) � f2 π (r) ⟨A22 (t)⟩ ⟨A11 (t)⟩ ⟨A22 (t)⟩ ⟨A11 (t)⟩ γt γt γt γt (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Fluorescence intensity for Ω = 9γ, ∆ = 0, and (a) δ = −8γ and (b) δ = −15γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The insets show the excited state populations ⟨A11⟩ (solid line) and ⟨A22⟩ (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The initial conditions are ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, ⟨A11(0)⟩ = ⟨A22(0)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' also in two-time correlations with particular features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Similarly, for the σ transitions we have Iσ(r, ˆt) = ⟨ ˆE− σ (r, ˆt) · ˆE+ σ (r, ˆt)⟩ = f 2 σ(r)[⟨A23(ˆt)A32(ˆt) + A14(ˆt)A41(ˆt)⟩] = f 2 σ(r)[⟨A11(ˆt) + A22(ˆt)⟩], (30a) Ist σ = f 2 σ(r) [α11 + α22] , (30b) also showing beats with intensity twice that of the π tran- sitions given that f 2 σ(r) = 2f 2 π(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The field quadrature operator at any time is ˆEπ,φ(r, ˆt) = 1 2 � E− π (r, ˆt)e−iφ + E+ π (r, ˆt)eiφ� = fπ(r)(S1,φ − S2,φ), (31) where φ = 0, π/2 are the quadrature phases we consider, and S1,φ = 1 2 � A13e−iφ + A31eiφ� , (32a) S2,φ = 1 2 � A24e−iφ + A42eiφ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (32b) The mean quadrature field is given by ⟨ ˆEπ,φ⟩st = fπ(r) 2 � (α13 − α24) e−iφ + (α31 − α42) eiφ� = fπ(r)Re � (α13 − α24) e−iφ� (33) = fπ(r)Re �Ω (∆ + (iγ − δ)/2) D e−iφ � , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Intensity and Quadrature Fluctuations Here we introduce the intensity and quadratures of the field in terms of atomic fluctuation operators ∆Ajk = Ajk − ⟨Ajk⟩st, such that ⟨AklAmn⟩ = αklαmn + ⟨∆Akl∆Amn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (34) Only the π transitions have nonvanishing coherence terms (α13, α24 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The fluorescence in the σ transi- tions is fully incoherent (α14 = α23 = 0), so its intensity is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (30b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the remainder of this section we deal only with the π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The quadrature op- erators are then written as ˆEπ,φ(r, ˆt) = fπ(r)[απ,φ + ∆Sπ,φ(ˆt)], (35a) where απ,φ = 1 2(α31 − α42)eiφ + 1 2(α13 − α24)e−iφ, (35b) = Re �Ω (∆ + (iγ − δ)/2) D e−iφ � , ∆Sπ,φ = 1 2(∆A31 − ∆A42)eiφ + 1 2(∆A13 − ∆24)e−iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (35c) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (29b) and (34) we write the steady state intensity in terms of products of dipole and dipole fluc- tuation operator expectation values, Ist π (r) = f 2 π(r) � Icoh π,0 + Iinc π,0 + Icoh π,cross + Iinc π,cross � ,(36) where Icoh π,0 = |⟨A13⟩st|2 + |⟨A24⟩st|2, (37a) Iinc π,0 = ⟨∆A13∆A31⟩ + ⟨∆A24∆A42⟩, (37b) Icoh π,cross = −⟨A13⟩st⟨A42⟩st − ⟨A24⟩st⟨A31⟩st = −2Re (⟨A13⟩st⟨A42⟩st) , (37c) Iinc π,cross = −⟨∆A13∆A42⟩ − ⟨∆A24∆A31⟩ = −2Re (⟨∆A13∆A42⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (37d) Superindices coh and inc stand, respectively, for the co- herent (depending on mean dipoles) and incoherent (de- pending on noise terms) parts of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Subindex 0 stands for terms with the addition of single transition products, giving the total intensity, while subindex cross stands for terms with products of the two π transitions, the steady state interference part of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In 7 terms of atomic expectation values these intensities are: Icoh π,0 = |α13|2 + |α24|2 (38a) = Ω2 4D2 �γ2 2 + ∆2 + (δ − ∆)2 � , Iinc π,0 = α11 + α22 − |α13|2 − |α24|2 (38b) = Ω2 D2 � 2Ω2 − γ2 4 − ∆2 − δ2 � , Icoh π,cross = −2Re (α13α42) (38c) = Ω2 2D2 �γ2 4 + ∆(∆ − δ) � , Iinc π,cross = 2Re (α13α42) = −Icoh π,cross, (38d) The sum of these terms is, of course, the total intensity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (29a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As usual in resonance fluorescence, the coher- ent and incoherent intensities are similar only in the weak field regime, Ω ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Here, in particular, the term Iinc π,0 (no interference) becomes much larger than the others for strong driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Degree of Interference - Coherent Part In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [5], a measure of the effect of interference in the coherent part of the intensity was as Icoh π,0 + Icoh π,cross = Icoh π,0 (1 + C(δ)), C(δ) = Icoh π,cross Icoh π,0 = γ2/4 + ∆(∆ − δ) γ2/4 + δ2/4 + (∆ − δ/2)2 , (39) independent of the Rabi frequency and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Some special cases are found analytically: C(0) = 1, δ = 0, (40a) C(δ0) = 0, δ0 = ∆[1 + (γ/2∆)2], (40b) C(δmin) = −1 1 + γ2/2∆2 , δmin = 2∆[1 + (γ/2∆)2], (40c) C(δ± 1/2) = 1/2, δ± 1/2 = −∆ ± � 3∆2 + (γ2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (40d) In the degenerate case, C(δ = 0) = 1 means perfect constructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' That is because at δ = 0 both π transitions (and both σ transitions) share the same reservoir environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Increasing δ the reservoir overlap decreases, so is the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Negative values of C indicate destructive interference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' its minimum is given by δmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For large detunings, ∆2 ≫ γ2 we have δ0 = ∆, δmin = 2∆, δ± 1/2 = −∆ ± √ 3 |∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (40e) We have used the special cases δ = {0, δ0, δmin} as a guide to obtain many of the figures in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 20 10 0 10 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 δ/γ ∆ = −5γ K(δ) C(δ) ∆ = −2γ ∆ = 0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Relative weight of the interference terms C(δ) (a) and K(δ) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In (b) Ω = γ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For 198Hg+, δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Degree of Interference - Incoherent Part Likewise, we define a measure, K(δ), of the effect of interference in the intensity’s incoherent part, Iinc π,0 + Iinc π,cross = Iinc π,0(1 + K(δ)), K(δ) = Iinc π,cross Iinc π,0 = γ2/4 + ∆(∆ − δ) 2 [γ2/4 + δ2 + ∆2 − 2Ω2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (41) Unlike C(δ), K(δ) also depends on the Rabi frequency as Ω−2, since fluctuations increase with laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Special cases are: K(0) = γ2/4 + ∆2 2 [γ2/4 + ∆2 − 2Ω2], δ = 0, (42a) K(δ) = 0, δ = ∆ + γ2 4∆ or Ω ≫ γ, ∆, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (42b) The behavior of K(δ) with ∆ is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' It is ba- sically required that ∆ ∼ Ω in order to preserve the shape seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 6(b), in which case the minima for C(δ) and K(δ) are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' On-resonance, for ex- ample, Ω should be no larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='35γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Also, we can infer that the beats are little affected by the interference term unless ∆ >∼ Ω ≫ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' TWO-TIME DIPOLE CORRELATIONS AND POWER SPECTRUM The resonance fluorescence spectrum of the J = 1/2 → J = 1/2 atomic system was first considered in [3] and then very thoroughly in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Thus, here we only con- sider basic definitions and issues related to the observa- tion of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The stationary Wiener-Khintchine power spectrum is given by the Fourier transform of the field autocorrelation function Sπ(ω) = Re � ∞ 0 dτe−iωτ⟨ ˆE− π (0) ˆE+ π (τ)⟩, (43) 8 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='35 20 10 0 10 20 ω/γ Sinc π (ω) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' units) γτ � ∆E− π (0) ∆E+ π (τ) � � f2 π (r) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Dipole correlation function ⟨∆ ˆE− π (0)∆ ˆE+ π (τ)⟩ for Ω = 9γ, δ = −8γ, and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The inset shows the corre- sponding incoherent spectrum Sinc π (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' such that � ∞ −∞ Sπ(ω)dω = Ist π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' By writing the atomic operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (27a) as Ajk(t) = αjk + ∆Ajk(t), we separate the spectrum in two parts: a coherent one, Scoh π (ω) = Re � ∞ 0 e−iωτdτ � Icoh π,0 + Icoh π,cross � = π � Icoh π,0 + Icoh π,cross � δ(ω) = πΩ2 D2 � γ2 4 + � ∆ − δ 2 �2� δ(ω), (44) due to elastic scattering, where Icoh π,0 and Icoh π,cross are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (38) (a) and (c), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and an incoherent part, Sinc π (ω) = Re � ∞ 0 dτe−iωτ⟨∆ ˆE− π (0)∆ ˆE+ π (τ)⟩, specifically, Sinc π (ω) = Re � ∞ 0 dτe−iωτ [⟨∆A13(0)∆A31(τ)⟩ +⟨∆A24(0)∆A42(τ)⟩ − ⟨∆A13(0)∆A42(τ)⟩ −⟨∆A24(0)∆A31(τ)⟩] , (45) due to atomic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' An outline of the numerical calculation is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The dipole correlation ⟨ ˆE− π (0) ˆE+ π (τ)⟩ and the incoher- ent spectrum in the strong driving regime and strong nondegeneracy (large δ) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The spec- trum (inset) displays a central peak and two pairs of Mollow-like-sidebands [22] with peaks at the Rabi side- bands ±Ω1 and ±Ω2, while the correlation features de- caying quantum beats due to the closeness of the Rabi peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As usual in the strong-field regime, the dressed system approach allows to discern the origin of the peaks from the transitions among the dressed states, to find their positions [5], and thus find the frequencies of the beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The generalized Rabi frequencies are Ω1 = E+ 1 − E− 1 = � 4Ω2 + ∆2, (46a) Ω2 = E+ 2 − E− 2 = � 4Ω2 + (δ − ∆)2, (46b) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Eigenvalues of matrix M/γ and initial conditions of the correlations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (45) for Ω = 9γ and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Eigenvalues δ = −8γ δ = −15γ λ1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='749386 + 0i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='836531 + 0i λ2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='583099 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0094i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='583308 − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='9981i λ3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='583099 + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0094i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='583308 + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='9981i λ4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='569785 − 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6808i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5492 − 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4257i λ5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='569785 + 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6808i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5492 + 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4257i λ6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 + 0i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 + 0i λ7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='444846 + 0i −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='398452 + 0i λ8 0 + 0i 0 + 0i Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ⟨∆A13∆A31⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='20836 + 0i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='14734 + 0i ⟨∆A24∆A42⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='174014 + 0i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='086982 + 0i ⟨∆A13∆A42⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='000134 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='002146i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='000067 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='002011i ⟨∆A24∆A31⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='000134 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='002146i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='000067 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='002011i where E± 1 = −∆ 2 ± 1 2 � 4Ω2 + ∆2, (47a) E± 2 = Bℓ + δ − ∆ 2 ± 1 2 � 4Ω2 + (δ − ∆)2, (47b) are the eigenvalues of the Hamiltonian (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Due to the spontaneous decays these frequencies would have to be corrected, but they are very good in the relevant strong field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Indeed, we notice that Ω1 and Ω2 are very close to the imaginary parts of the eigenvalues λ2,3 and λ4,5, respectively, of matrix M, shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The beats are the result of the superposition of waves at the frequencies Ω1 and Ω2 of the spectral sidebands, with average frequency Ωav = Ω2 + Ω1 2 = � 4Ω2 + (δ − ∆)2 + √ 4Ω2 + ∆2 2 , (48) and beat or modulation frequency Ωbeat = Ω2 − Ω1 2 = � 4Ω2 + (δ − ∆)2 − √ 4Ω2 + ∆2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (49) Now, we can identify the origin and modulation fre- quency of the beats in the time-dependent intensity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (29a), since the excited state populations ⟨A11(t)⟩ and ⟨A22(t)⟩ oscillate at the generalized Rabi frequen- cies Ω1 and Ω2, respectively, with initial conditions given by a nonzero superposition of ground state pop- ulations at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the case of the dipole correlation ⟨ ˆE− π (0) ˆE+ π (τ)⟩, however, the initial conditions are given by products of stationary atomic expectation values, most of them the coherences α13, α24, which become very small in the regime of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Thus, as seen in Table I, the terms ⟨∆A13(0)∆A31(τ)⟩ and ⟨∆A24(0)∆A42(τ)⟩ are 9 0 3 6 9 12 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 3 6 9 12 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 0 3 6 9 12 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 γτ ∆ = −2γ, δ = −4γ ∆ = −2γ, δ = −2γ ∆ = 2γ, δ = −2γ ∆ = 0, δ = 0 ( ) Ω = γ (a) Ω = γ/4 (b) Ω = γ/2 g(2) π (τ) g(2) π (τ) g(2) π (τ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Photon correlations for (a) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='25γ, (b) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5γ and (c) Ω = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The pairs of values (∆, δ) are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' much larger than the cross terms ⟨∆A13(0)∆A42(τ)⟩ and ⟨∆A24(0)∆A31(τ)⟩, so the beats are basically due to the interference of those dominant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' PHOTON-PHOTON CORRELATIONS The standard method to investigate intensity fluctua- tions of a light source uses Brown-Twiss photon-photon correlations [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The conditional character of this type of measurement makes it nearly free of detector in- efficiencies, unlike a single-detector measurement of the photoelectron probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [8] the correlations of two photons from the π transitions were studied, albeit only for the degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In this paper we extend it to the case of nondegenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' These correlations are defined as g(2) π (τ) = G(2) π (τ) G(2) π (τ → ∞) (50) where, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (27a) for the field operators, G(2) π (τ) = ⟨ ˆE− π (0) ˆE− π (τ) ˆE+ π (τ) ˆE+ π (0)⟩ = f 4 π(r)⟨[A13(0) − A24(0)][A11(τ) + A22(τ)] ×[A31(0) − A42(0)]⟩, (51a) and G(2) π (τ → ∞) = � Ist π �2 = f 4 π(r) (α11 + α22)2 (51b) is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' G(2) π (τ) can be further re- duced, since ⟨A13Ajk(τ)A42(0)⟩ = ���A24Ajk(τ)A31(0)⟩ = 0, due to having vanishing initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Figure 8 shows g(2) π (τ) for moderate values of the Rabi frequency (near saturation) and the same sets of detun- ings ∆ and δ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As usual in resonance fluores- cence, the correlation shows antibunching, g(2) π (0) = 0, 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 g(2) π (τ) γτ γτ g(2) π (τ) (b) δ = −10γ (d) δ = −15γ ( ) δ = −12γ (a) δ = −8γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Photon-photon correlations showing beats in the strong field limit, Ω = 9γ, ∆ = 0, and large Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The horizontal line helps to see that the wave packet is slightly rised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' that is, a single atom cannot emit two photons simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Unlike the two-level atom resonance fluorescence, the correlation is not simply the normalized population of the excited state, nor it is only the sum of the cor- relations of each single π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Besides the terms ⟨A13(0)A11(τ)A31(0)⟩ and ⟨A24(0)A22(τ)A42(0)⟩, which are also out of phase, as seen from the time-dependent populations of their excited states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2), there are six cross terms in the full correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the nondegenerate case the multiple contributions cause in some cases quite irregular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For instance, as we will see in the next Section, the slow decay of the correlation when the laser drives the atom near saturation, but below the ω13 resonance transition, is related to a very narrow peak in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The case of strong driving and large nondegeneracy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 9, featuring quantum beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' There are several effects resulting from the increase of the nonde- generacy factor δ: (i) the larger number of visible wave packets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (ii) both average and beat frequencies approach one another, so the wave packets get shorter for larger photon-pair intervals τ, containing very few of the fast oscillations, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 9(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (iii) the wavepackets are initially slightly lifted above the g(2)(τ) = 1 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' QUADRATURE FLUCTUATIONS Squeezing, the reduction of noise in one quadrature below that of a coherent state at the expense of the other, is the hallmark of phase-dependent fluctuations of the electromagnetic field [cite].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' It is usually measured by balanced homodyne detection (BHD), but low quan- tum detector efficiency degrade the weak squeezing pro- duced in resonance fluorescence and cavity QED systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' One alternative our group has used is conditional homo- dyne detection (CHD) [18, 19], which correlates a quadra- ture amplitude on the cue of an intensity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' CHD measures a third-order amplitude-intensity corre- 10 lation (AIC) which, in the weak driving limit is reduced to the second-order one and that allows for measuring squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Being a conditional measurement it is nearly free of detector inefficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' While the original goal of CHD was to measure the weak squeezing in cavity QED [18, 19], it was soon re- alized that nonzero third-order fluctuations of the am- plitude provide clear evidence of non-Gaussian fluctua- tions and higher-order field nonclassicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the present work the fluctuations are mainly third-order ones, due to near and above saturation excitation, and violate classi- cal bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We thus explore the phase-dependent fluctu- ations under conditions of quantum interference following our recent work [20, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Amplitude-Intensity Correlations In CHD a quadrature’s field Eφ is measured by BHD on the cue of photon countings in a separate detector, where φ = 0, π/2 is the phase of the local oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is characterized by a correlation among the amplitude and the intensity of the field, hπ,φ(τ) = Hπ,φ(τ) Hπ,φ(τ → ∞), (52) where Hπ,φ(τ) = ⟨: ˆE− π (0) ˆE+ π (0) ˆEπ,φ(τ) :⟩, (53a) the dots :: indicating time and normal operator orderings, and Hπ,φ(τ → ∞) = Ist π ⟨Eπ,φ⟩st (53b) = f 3 π(r) [α11 + α22] Re � (α13 − α24) e−iφ� = f 3 π(r) Ω3 D2 Re � (∆ + (iγ − δ)/2) e−iφ� is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For the sake of concreteness, in this Section we limit our discussion to the out-of-phase quadrature, φ = π/2, which is the one that features squeezing when ωL = ω13, that is ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We do consider, however, squeezing in the in-phase quadrature φ = 0 in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' VIII on the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In several atom-laser systems hπ,φ(τ) has been proven to be time-asymmetric [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is not the case with the J = 1/2 → J = 1/2 system so we limit the analysis to positive intervals τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Omitting the geometrical factor f 3 π(r), which is later cancelled by normalization, we have Hπ,φ(τ) = ⟨ ˆE− π (0) ˆEπ,φ(τ) ˆE+ π (0)⟩ = Re � e−iφ⟨A13(0)[A13(τ) − A24(τ)]A31(0) +A24(0)[A13(τ) − A24(τ)]A42(0)⟩} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (54) Note that Hπ,φ(0) = 0 meaning that, like antibunching in g(2), the atom has to build a new photon wavepacket after one has been emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The AIC suggests nontrivial behavior when we take dipole fluctuations into account, that is, when the atomic operators are split into their mean plus noise, Ajk = αjk + ∆Ajk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' upon substitution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (54) we get Hπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) = Ist π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ⟩st + H(2) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) + H(3) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (55) or in normalized form as hπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) = 1 + H(2) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) Ist π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ⟩st + H(3) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) Ist π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ⟩st ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (56) where H(2) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) = 2Re � ⟨ ˆE+ π ⟩st⟨∆ ˆE− π (0)∆ ˆEπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ)⟩ � = Re � (α31 − α42) [⟨(∆A13(0) − ∆A24(0)) � ∆A13(τ) − ∆A24(τ))⟩e−iφ +⟨(∆A13(0) − ∆A24(0)) � ∆A31(τ)⟩ − ∆A42(τ))⟩eiφ�� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (57) H(3) π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ) = ⟨∆ ˆE− π (0)∆ ˆEπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='φ(τ)∆ ˆE+ π (0)⟩ = Re � eiφ⟨[∆A13(0) − ∆A24(0)] [∆A31(τ) − ∆A42(τ)] [∆A31(0) − ∆A42(0)]⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (58) The initial conditions of the correlations are given in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' From hπ,π/2(0) = 0 we can obtain analytically the ini- tial values of the second- and third-order terms, h(2) π,π/2(0) = 1 − (2∆ − δ)2 + γ2 2D , (59) h(3) π,π/2(0) = (2∆ − δ)2 + γ2 2D − 2, (60) where D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 11 Being the AIC a function of odd-order in the field am- plitude we rightly expect a richer landscape than that of the intensity correlations, more so when one considers quantum interference and the complex parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For instance, the correlation can take on not only nega- tive values but break classical bounds [18, 19]: 0 ≤ hφ(τ) − 1 ≤ 1 , (61a) |h(2) φ (τ) − 1| ≤ |h(2) φ (0) − 1| ≤ 1 , (61b) where the second line is valid only for weak fields such that h(3) φ (τ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' These classical bounds are stronger cri- teria for nonclassicality of the emitted field than squeezed light measurements, the more familiar probing of phase- dependent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' A detailed hierarchy of nonclas- sicality measures for higher-order correlation functions is presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [20] an inequality was obtained that considers the full hφ(τ) by calculating the AIC for a field in a coherent state, −1 ≤ hφ(τ) ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (62) For a meaningful violation of Poisson statistics, hφ(τ) must be outside these bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Also, hφ(τ) is a measure of non-Gaussian fluctuations, here of third-order in the field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Resonance fluorescence is a particularly strong case of non-Gaussian noise by being a highly nonlinear stationary nonequilib- rium process [20, 23, 24, 27, 28], thanks also to its small Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This makes resonance fluorescence unsuit- able to a quasiprobability distribution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Fluctuations Spectra Since quadrature fluctuations, such as squeezing, are often studied in the frequency domain we now define the spectrum of the amplitude-intensity correlations: Sπ,φ(ω) = 8γ1 � ∞ 0 dτ cos (ωτ) [hπ,φ(τ) − 1] (63) which, following Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (52) and (55), can be decomposed into terms of second- and third-order in the dipole fluc- tuations S(q) π,φ(ω) = 8γ1 � ∞ 0 dτ cos (ωτ)h(q) π,φ(τ), (64) where q = 2, 3, so that Sπ,φ(ω) = S(2) π,φ(ω) + S(3) π,φ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As mentioned above, the AIC was devised initially to measure squeezing without the issue of imperfect detec- tion efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Obviously, hπ,φ(τ) and Sπ,φ(ω) are not measures of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' They measure a third-order mo- ment in the field’s amplitude, while squeezing is a second- order one in its fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The so-called spectrum of squeezing is the one for q = 2, with the advantage of the AIC of not depending on the efficiency of detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Squeezing is signaled by frequency intervals where 0 2 4 6 8 10 12 0 2 4 6 10 5 0 5 10 0 1 2 0 2 4 6 8 10 12 0 2 4 6 10 5 0 5 10 1 0 1 2 0 2 4 6 8 10 12 0 2 4 6 10 5 0 5 10 1 0 1 2 Sπ,π/2 (ω) hπ,π/2 (τ) (b) Ω = γ/2 γτ ω/γ (a) Ω = γ/4 (d) Ω = γ/4 (e) Ω = γ/2 (f ) Ω = γ ( ) Ω = γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Amplitude-intensity correlations (left panel) and spectra (right panel) for the φ = π/2 quadrature in the weak- moderate field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Parameters and line styles are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 8: ∆ = δ = 0 (solid-black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ∆ = 2γ and δ = −2γ (dots-red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ∆ = −2γ and δ = −2γ (dashed-green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ∆ = −2γ and δ = −4γ (dot-dashed-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' S(2) π,φ(ω) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As a further note, the full incoherent spec- trum, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (45), can be obtained by adding the squeezing spectra of both quadratures [29], Sinc π (ω) = 1 8γ1 � S(2) π,0(ω) + S(2) π,π/2(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (65) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Results We now show plots of the AICs and their spectra in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10-12 for the φ = π/2 quadrature and the same sets of detunings ∆, δ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2, and weak to moderate Rabi frequencies, γ/4 < Ω < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' With the three parameters Ω, ∆, and δ, the landscape of effects is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We first notice a few general features seen in hπ,π/2(τ), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' With increasing Rabi frequencies, detunings, and Zeeman splittings we observe the clear breakdown of the classical inequalities besides the one at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Cor- respondingly, in the spectra, the extrema get displaced and broadened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Now, we want to single out the case of nondegeneracy with small detuning on the |1⟩ − |3⟩ transition but large on the |2⟩ − |4⟩ one, ∆ = −δ = 2γ (green-dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For weak field, Ω = γ/4, the AIC does not have a regular evolution for short times but it does decay very slowly, with a correponding very narrow spectral peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The slow decay is also clearly visible in the photon correlation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' As we mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' III regarding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2b, state |4⟩ ends up with a large portion of the steady state population due to optical pumping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' not quite a trapping state, so there is no electron shelv- ing per se, as argued in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This effect is washed out for larger Rabi frequencies, which allow for faster recycling of the populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' To a lesser degree, slow decay and sharp peak occur for opposite signs of ∆ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 12 0 2 4 6 8 10 2 1 0 1 2 3 4 10 -8 6 4 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 0 2 4 6 8 10 2 1 0 1 2 3 4 10 -8 6 4 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8 0 2 4 6 8 10 1 0 1 2 3 10 -8 6 4 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 (f ) Ω = γ ( ) Ω = γ (b) Ω = γ/2 (e) Ω = γ/2 (a) Ω = γ/4 (d) Ω = γ/4 h(2) π,π/2 (τ) ω/γ γτ S(2) π,π/2 (ω) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Second-order component of the AIC and spectra of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 0 2 4 6 8 10 1 0 1 2 10 -8 -6 -4 -2 0 2 4 6 8 10 0 1 2 0 2 4 6 8 10 1 0 1 2 10 -8 -6 -4 -2 0 2 4 6 8 10 1 0 1 2 0 2 4 6 8 10 1 0 1 2 3 10 -8 -6 -4 -2 0 2 4 6 8 10 2 1 0 1 (f ) Ω = γ ( ) Ω = γ (b) Ω = γ/2 (e) Ω = γ/2 (a) Ω = γ/4 (d) Ω = γ/4 h(3) π,π/2 (τ) ω/γ γτ S(3) π,π/2 (ω) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Third-order component of the AIC and spectra of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The splitting of the AIC and spectra into components of second and third order in the fluctuations, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 11, 12, helps to understand better the quadrature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For the second-order ones we have the squeezing spectra: around ω = 0 for ∆ = 0 and small Rabi frequencies, Ω < γ/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and in sidebands for larger detunings, Rabi frequencies and Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In h(2) π,π/2(τ) there is a reduction in amplitudes and nonclassicality for increasing Rabi frequencies except for the case of oppposite signs of detuning and difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Note that the sharp spectral peak in the latter case takes up most of the corresponding peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This is because both π transitions are largely detuned from the laser, keeping Ω small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Increasing the laser strength the third-order effects overcome the second-order ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For instance, regarding the size of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Also, a comparison of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 11 0 2 4 6 8 10 10 0 10 20 40 20 0 20 40 10 5 0 5 10 0 2 4 6 8 10 10 0 10 20 40 20 0 20 40 10 5 0 5 10 0 2 4 6 8 10 10 0 10 20 40 20 0 20 40 10 5 0 5 10 0 2 4 6 8 10 10 0 10 20 40 20 0 20 40 5 0 5 hπ,π/2 (τ) h (2) π,π/2 (τ) h (3) π,π/2 (τ) S (3) π,π/2 (τ) S (2) π,π/2 (τ) Sπ,π/2 (τ) γτ ω/γ (b) (f ) ( ) (g) (d) (h) (a) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' AIC and spectra for Ω = 9γ, ∆ = 0, (a,e) δ = −8γ, (b,f) δ = −10γ, (c,g) δ = −12γ, (d,h) δ = −15γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Lines are: full AIC and spectra (solid-black), second-order (dots-red), and third-order (dashed-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and 12 shows that h(3) φ (τ) is mainly responsible for the breakdown of the classical bounds when the driving field is on or above saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Moreover, we see that the slow-decay–sharp-peak is mainly a third-order effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' To close this Section, the AIC and spectra for very strong fields and large Zeeman splittings, Ω, |δ| ≫ γ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The AIC shows beats as in the photon correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Unlike those in g(2)(τ), these wavepack- ets oscillate around h(τ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Because the regime is that of strong excitation the third-order component clearly dominates, making the fluorescence notably non- Gaussian, and clearly violates the classical inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The spectral peaks are localized around the Rabi frequen- cies ±Ω1, ±Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Studies of the spectrum of squeezing for the J = 1/2 − J = 1/2 system were reported in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Those authors choose ±Ω1, ±Ω2 with a less strong laser but large detuning and large Zeeman splittings, observ- ing the double sidebands, but no mention or hint of beats was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' VARIANCE The variance is a measure of the total noise in a quadrature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' it is defined as Vφ = ⟨: (∆Eφ)2 :⟩ = Re � e−iφ⟨∆ ˆE−∆ ˆEφ⟩st � , (66) 13 and is related to the spectrum of squeezing as Vφ = 1 4πγη � ∞ −∞ dωS(2) φ (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (67) where η is the detector efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The maximum value of Vφ is 1/4, obtained when there is very strong driving, when almost all the emitted light is incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Negative values of the variance are a signature of squeezing but, unlike the quadrature spectra, the squeezing is the total one in the field, independent of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For the π transitions we have Vπ,φ = f 2 π(r) 2 Re � −(α13 − α24)2e−2iφ +(α11 + α22 − |α13 − α24|2) � , (68) = f 2 π(r) 2 Ω2 D � 1 − [(2∆ − δ) cos φ + γ sin φ]2 2D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (69) For φ = π/2 and φ = 0 we have, respectively, Vπ,π/2 = f 2 π(r) 2 Ω2 D � 1 − γ2 2D � , (70a) Vπ,0 = f 2 π(r) 2 Ω2 D � 1 − (2∆ − δ)2 2D � , (70b) where D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 14 we plot the variances of the out-of-phase φ = π/2 (left panel) and in-phase φ = 0 (right panel) quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The interplay of parameters is a complex one, but we mostly use the ones of previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For φ = π/2 and ∆ = 0, as usual in resonance fluorescence systems, squeezing is restricted to a small range of Rabi frequencies, detunings, and Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For φ = 0 nonzero laser or Zeeman detunings are necessary to pro- duce squeezing, with a strong dependence on their sign: on-resonance (not shown) there is no squeezing, as for a two-level atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 14(d) the laser is tuned be- low that transition, ∆ = −2γ, and there is no squeezing (positive variance) but the variance is reduced for large δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 14(e) the laser is tuned above the transition, ∆ = −2γ, and there is squeezing for larger Rabi frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Large values of δ tend to reduce the variance, be it positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Out-of-phase quadrature We now discuss a complementary view of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For φ = π/2 we can identify the Rabi frequency interval within which squeezing takes place, 0 < Ω < 1 2 � γ2/2 − δ2/2 − 2(∆ − δ/2)2, (71) and the Rabi frequency for maximum squeezing is ˜Ωπ/2 = 1 2 � γ4/2 − 2[(δ − ∆)2 + ∆2]2 3γ2 + 2[(δ − ∆)2 + ∆2]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (72) 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='04 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='01 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2 Vπ,π/2/f2 π(r) Vπ,0/f2 π(r) Ω/γ Ω/γ Ω/γ ∆/γ ∆/γ Ω/γ ( ) (a) (b) (d) (e) (f ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Variance of the quadratures of the fluorescence of the π transitions: left panel for φ = π/2 and right panel for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (a,b,d,e) as a function of Rabi frequency and (c,f) as a function of detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In all cases δ = 0 is given by a solid-black line, and δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='5γ by a dashed-red line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the dotted-blue line is δ = −2γ in (a,b,d,e) and δ = −γ in (c,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Additionally, (a) ∆ = 0, (b) ∆ = −2γ, (c) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='2γ, (d) ∆ = 0, (e) ∆ = 2γ, (f) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='8γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Thus, the variance at ˜Ωπ/2 is V (˜Ωπ/2) π,π/2 (∆ = 0, δ) = f 2 π(r) 16 (γ4/2 − 2δ4)(δ2 − γ2) γ2(γ2 + 2δ2)(δ2 + γ2), (73a) for ∆ = 0 and |δ/γ| < 1/ √ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' V (˜Ωπ/2) π,π/2 (∆, δ = 0) = f 2 π(r) 16 (γ4/2 − 8∆4)(4∆2 − γ2) γ2(γ2 + 4∆2)2 , (73b) for δ = 0 and |∆/γ| < 1/ √ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and the maximum total squeezing is obtained at ∆ = δ = 0, V (˜Ωπ/2) π,π/2 (0, 0) = −f 2 π(r) 32 , ˜Ωπ/2 = γ 2 √ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (73c) For φ = π/2 squeezing is limited to elliptical regions of weak driving and small detunings ∆ and δ: 2δ2 + 8Ω2 < γ2, ∆ = 0, (74a) 4∆2 + 8Ω2 < γ2, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (74b) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In-phase quadrature For φ = 0, squeezing is obtained in the Rabi frequency interval, for δ = 0, 0 < Ω < 1 √ 2 � ∆2 − γ2/4, |∆| > γ/2, (75) 14 with maximum squeezing at the Rabi frequency ˜Ω0 = 1 2 √ 2 � 16∆2 − γ2 12∆2 + γ2 , (76) requiring finite detuning from both π transitions (∆ ̸= 0) and stronger driving, Ω ∼ γ [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 14(d)-(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Thus, the variance at ˜Ω0 is V (˜Ω0) π,0 (δ) = −f 2 π(r) 128 4∆2 − γ2 ∆2(4∆2 + γ2), |∆| ≥ γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (77) This expression gets the asymptotic value lim ∆→∞ V (˜Ω0) π,0 = −f 2 π(r) 32 , (78) which is the same as that for the π/2 quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The region for squeezing obeys the relation 4∆2 − 8Ω2 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (79) So, to obtain squeezing in this quadrature it is necessary to have detunings ∆ > γ/4 for any Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS We have studied several properties of the resonance fluorescence of the π transitions in a J = 1/2 − J = 1/2 angular momentum atomic system driven by a linearly polarized laser field and a magnetic field along the π tran- sition to lift the level degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Interference among the various transition amplitudes create a rich landscape of effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Most notable among our results is the observa- tion of quantum beats when the atom is subject to large laser and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In this regime, two close Rabi frequencies interfere, giving rise to a well-defined modu- lation of the fast oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' These Rabi frequencies are the source of the two pairs of sidebands in the incoherent part of the power spectrum [5] and in the squeezing spec- trum [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We studied beats in the total intensity and two-time functions such as the dipole-dipole, intensity- intensity and intensity-amplitude correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the beats’ regime the role of vacuum-induced coherence is small because the upper levels are very separated due to very large difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Before the beats we considered the previously over- looked time-dependent populations and reviewed aspects of the known stationary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The fact that the upper state populations evolve out of phase should not be a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' This, and nonzero initial population of both ground states (in contrast to nonzero populations of ex- cited states for spontaneous emission), are major factors in the interference among the terms in the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Ex- cept for very strong laser fields, the steady state popula- tions depend strongly on the difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The AIC also permits to quantify the degree of non- Gaussianity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the fluctuations of third-order in the field quadrature amplitude due to strong atom-laser nonlin- earity dominate over the second-order ones with strong driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The beats are in the strongly non-Gaussian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The correlations show nonclassical features of the fluo- rescence light such as antibunching, g(2)(0) = 0, and vio- lation of classical inequalities in the amplitude-intensity correlations, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (61 -62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' We studied squeezing using the variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=', the total noise in a quadrature, as well as using the second-order part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' In the regime of beats there is squeezing, near the effective Rabi frequencies, but none in the total noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For a system with many parameters the interplay among them is a complex one, making the interpretation of results nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Thus, for most of our plots we chose parameters in two groups: i) where they are relatively small, Ω, ∆, δ ∼ γ, chosen to illustrate several degrees of vacuum-induced coherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' and ii) where they are large, Ω, ∆, δ ≫ γ, and quantum beats are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Overall, particular care must be taken regarding detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' On the one hand, large difference Zeeman splitting means that the excited levels would be very separated and in- teract with different frequency portions of the reservoir, hence diminishing the vacuum-induced coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' On the other, large laser-atom detunings, which might in- crease the VIC, mean reduced fluorescence rates, which may also be detrimental in measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The beats, then, would be better observed if ∆ ≤ γ and δ of just several γ in the strong field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ACKNOWLEDGMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Ricardo Rom´an-Ancheyta and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Ir´an Ramos-Prieto for useful comments at an early stage of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ADAV thanks CONACYT, Mexico, for scholarship No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 804318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' ORCID numbers: H´ector M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Castro-Beltr´an https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='org/0000-0002-3400-7652, Octavio de los Santos-S´anchez https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='org/0000-0002-4316-0114, Luis Guti´errez https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content='org/0000-0002-5144-4782, Appendix A: Time-Dependent Matrix Solutions and Spectra The two-time photon correlations under study have the general form ⟨W(τ)⟩ = ⟨O1(0)R(τ)O2(0)⟩, where R is the Bloch vector and O1,2 are system operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The same applies to correlations of fluctuation operators ∆R, ∆O1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Using the quantum regression formula [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' the correlations obey the equation ⟨ ˙W(τ)⟩ = M⟨W(τ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A1) which has the formal solution ⟨W(τ)⟩ = eMτ⟨W(0)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A2) where M is given by 15 M = � � � � � � � � � � � � � � −γ −iΩ 0 0 iΩ 0 0 0 −iΩ − � γ 2 + i∆ � 0 0 0 iΩ 0 0 0 0 −γ iΩ 0 0 −iΩ 0 0 0 iΩ − � γ 2 + i(∆ − δ) � 0 0 0 −iΩ iΩ 0 0 0 − � γ 2 − i∆ � −iΩ 0 0 γ1 iΩ γσ 0 −iΩ 0 0 0 0 0 −iΩ 0 0 0 − � γ 2 − i(∆ − δ) � iΩ γσ 0 γ2 −iΩ 0 0 iΩ 0 � � � � � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A3) Also, spectra of stationary systems can be evaluated more effectively using the above formal approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Be g(τ) = ⟨W(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Then, a spectrum is calculated as S(ω) ∝ � ∞ 0 cos ωτ g(τ) dτ = � ∞ 0 cos ωτ eMτg(0) dτ = Re � ∞ 0 e−(iω1−M)τg(0) dτ = Re � (iω1 − M)−1g(0) � , (A4) where 1 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For example, the inco- herent spectrum requires calculations of the type Sinc(ω) = Re � ∞ 0 dτe−iωτeMτ⟨∆Aij(0)∆Akl(0)⟩st = Re � (M − iω1)−1⟨∆Aij(0)∆Akl(0)⟩st � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A5) For the initial conditions of the correlations we use the following operator products and correlations in compact form: AklAmn = Aknδlm , (A6a) ⟨AklAmn⟩ = αknδlm, (A6b) AijAklAmn = Ainδjkδlm, (A6c) ⟨AijAklAmn⟩ = αinδjkδlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A6d) Hence, the relevant initial conditions are: ⟨A13R⟩ = (0, 0, 0, 0, α11, α13, 0, 0)T , (A7a) ⟨A24R⟩ = (0, 0, 0, 0, 0, 0, α22, α24)T , (A7b) ⟨A13RA31⟩ = (0, 0, 0, 0, 0, α11, 0, 0)T , (A7c) ⟨A24RA42⟩ = (0, 0, 0, 0, 0, 0, 0, α22)T , (A7d) ⟨A13RA42⟩ = ⟨A24RA31⟩ = 0, (A7e) where R = (A11, A13, A22, A24, A31, A33, A42, A44)T is the Bloch vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' For correlations with fluctuation oper- ator products, ∆Aij = Aij − αij, we have ⟨∆Akl∆Amn⟩ = αknδlm − αklαmn, (A8) ⟨∆Aij∆Akl∆Amn⟩ = αinδlmδjk − αilαmnδjk −αinαklδjm − αijαknδlm +2αijαklαmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A9) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' recalling that α12 = α14 = α23 = α34 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' we write the detailed initial conditions of the correlations (Set 1 of Bloch equations and quantum regression for- mula): ⟨∆A13∆R⟩ = � −α13α11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α2 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α13α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α13α24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' α11 − |α13|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' α13 − α13α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α13α42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α13α44 �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A10a) ⟨∆A24∆R⟩ = � −α24α11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α24α13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α24α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α2 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α24α31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' −α24α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' α22 − |α24|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' α24 − α24α44 �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A10b) ⟨∆A13∆R∆A31⟩ = � 2|α13|2α11 − α2 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α13 − 2α11α13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α22 − α11α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α24 − α11α24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α31 − 2α11α31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α33 + α11 − 2|α13|2 − α11α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α42 − 2α11α42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' 2|α13|2α44 − α11α44 �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A10c) ⟨∆A24∆R∆A42⟩ = � 2|α24|2α11 − α11α22, 2|α24|2α13 − α22α13, 2|α24|2α22 − α2 22, 2|α24|2α24 − 2α22α24, 2|α24|2α31 − α22α31, 2|α24|2α33 − α22α33, 2|α24|2α42 − 2α22α42, 2|α24|2α44 + α22 − 2|α24|2 − α22α44 �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A10d) 16 ⟨∆A13∆R∆A42⟩ = � 2α13α11α42, 2α2 13α42, 2α13α22α42, (2|α24|2 − α22)α13, (2|α13|2 − α11)α42, (2α13α33 − α13)α42, 2α13α2 42, (2α13α44 − α13)α42 �T , (A10e) ⟨∆A24∆R∆A31⟩ = � 2α24α11α31, (2|α13|2 − α11)α24, 2α24α22α31, 2α2 24α31, 2α24α2 31, (2α24α33 − α24)α31, (2|α24|2 − α22)α31, (2α24α44 − α24)α31 �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (A10f) Appendix B: Condition for Optimal Appearance of Beats in the Intensity We consider a simplified, unitary, model to estimate the optimal initial population of the ground states to make well-formed beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' First, we diagonalize the Hamil- tonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' The eigenvalues and eigenstates are E± 1 = −∆ 2 ± 1 2 � 4Ω2 + ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (B1a) E± 2 = Bℓ + δ − ∆ 2 ± 1 2 � 4Ω2 + (δ − ∆)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (B1b) and |u1⟩ = sin Θ1|1⟩ + cos Θ1|3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' |u2⟩ = − cos Θ1|1⟩ + sin Θ1|3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' |u3⟩ = sin Θ2|2⟩ + cos Θ2|4⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' |u4⟩ = − cos Θ2|2⟩ + sin Θ2|4⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (B2) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' where sin Θ1 = 2Ω �� ∆ + √ ∆2 + 4Ω2�2 + 4Ω2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' cos Θ1 = ∆ + √ ∆2 + 4Ω2 �� ∆ + √ ∆2 + 4Ω2�2 + 4Ω2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' sin Θ2 = 2Ω �� (δ − ∆) + � (δ − ∆)2 + 4Ω2 �2 + 4Ω2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' cos Θ2 = (δ − ∆) + � (δ − ∆)2 + 4Ω2 �� (δ − ∆) + � (δ − ∆)2 + 4Ω2 �2 + 4Ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (B3) It is now straightforward to obtain the excited-state populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' If the initial state of the system is ρ(0) = ⟨A33(0)⟩|3⟩⟨3| + ⟨A44(0)⟩|4⟩⟨4| we get ⟨A33(t)⟩ = 1 2⟨A33(0)⟩ sin2 (2Θ1)(1 − cos (Ω1t)), (B4a) ⟨A44(t)⟩ = 1 2⟨A44(0)⟩ sin2 (2Θ2)(1 − cos (Ω2t)), (B4b) and the intensity of the field is Iπ(r, t) f 2π(r) = ⟨A33(0)⟩ sin2 (2Θ1) + A44(0)⟩ sin2 (2Θ2) −⟨A33(0)⟩ sin2 (2Θ1) cos (Ω1t) −⟨A44(0)⟩ sin2 (2Θ2) cos (Ω2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' (B5) A necessary condition for the beating behavior to oc- cur is that the initial ground-state populations are both nonvanishing in the nondegenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Now, assuming the relation ⟨A33(0)⟩ ⟨A44(0)⟩ = sin2 (2Θ2) sin2 (2Θ1) (B6) is satisfied by chossing appropriate parameter values (Ω, δ, ∆) for given values of initial ground state popu- lations we would get Iπ(r, t) = f 2 π(r)⟨A33(0)⟩ sin2 (2Θ1) × [1 − cos (Ωbeatt) cos (Ωavt)] , (B7) where Ωbeat = (Ω2 − Ω1)/2 and Ωav = (Ω2 + Ω1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Ficek and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Swain, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' A 69, 023401 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'} +page_content=' [2] Z.' metadata={'source': 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