diff --git "a/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt" "b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt" @@ -0,0 +1,769 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf,len=768 +page_content='1 Synchronization of Josephson junctions in series array Abhijit Bhattacharyya Abstract—Multi-qubit quantum processors coupled to net- working provides the state-of-the-art quantum computing plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, each qubit has unique eigenfrequency even though fabricated in the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To continue quantum gate operations besides the detection and correction of errors it is required that the qubits must be synchronized in the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This study uses Kuramoto model which is a link between statistical mean-field technique and non-linear dynamics to synchronize the qubits applying small noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This noise could be any externally applied noise function or just noise from the difference of frequencies of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Kuramoto model tunes the coupled oscillators adjusting the coupling strength between the oscillators to evolve from the state of incoherence to the synchronized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Index Terms—Josephson junction, Kuramoto Model, synchro- nization, oscillators I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' INTRODUCTION J Osephson junction controls the flow of magnetic flux quanta through frequency and voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Modern instruments require measurement of voltage with a reproducible capability exceeding the uncertainty of realization of the SI volt (cur- rently 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 parts on 106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Before 1972, SI volt was represented by using carefully stabilised Weston cell banks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Drift and transportability problems with these electrochemical artifact standards limited the uniformity of voltage standards to about 1 part in 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' These uniformity was drastically improved by the usage of Josephson junction [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Josephson equation for supercurrent through a supercon- ducting tunnel junction, called as DC Josephson Effect, is defined as [2]–[4] I = Ic sin �4πe h � V dt � , (1) where Ic is critical current, h is Planck’s constant and e is electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' When a dc voltage is applied in equation (1), the phase will vary linearly with time and current will be sinusoidal with amplitude Ic and frequency fJ = 2eV/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The magnetic flux threading a superconducting loop or hole is quantized [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The superconducting magnetic flux quantum Φ0 = h/(2e) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0678×10−15 Wb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The inverse of flux quantum 1/Φ0 is called Josephson constant KJ defined as 2e/h has a value of 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='597 GHz/mV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' During each oscillation, a single quantum of magnetic flux h/(2e) passes through the junction which is very difficult to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, if an alternating current with frequency f is applied across the junction, there Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 094, India vega@barc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' abhihere@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='com is a range of bias current for which flow of flux quanta will phaselock to the applied frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Under this phase locked condition, the average voltage across the junction is precisely (h/2e)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This effect is known as ac Josephson effect observed as a constant voltage step at V =(h/2e)f in the I −V characteristic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This means a Josephson junction can act as a “Voltage to frequency converter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It is also possible for the junction to phaselock with the harmonics of fJ resulting in a series of steps at voltages V =nf(h/2e), where n is an integer denoting step number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This accuracy was limited to the condition that a Josephson voltage higher than 10mV was never used [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, if one obtain Josephson voltage over 100 mV , the accuracy could be remarkably improved besides the ability to vary the Jsephson voltage with the frequency and step number could be utilized as potentiometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Series array of Josephson junction [6] has been effectively used in development of a potentiometer system to produce (1- 10)V [1], [6] with uncertainty about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 × 10−9 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Larger series arrays were initially considered as impractical due to junction nonuniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The nonuniformity demanded each junction to be biased separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In 1977, Levinsen et al [7] stated the important of the parameter βc=4πeIcR2C/h in determining the characteristics of RF induced Josephson steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This βc is measure of the damping of Josephson oscillations by the junction shunting resistance R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Josephson junction is also a natural choice for sub- millimeter local oscillator [8], [9] as one may capitalize the voltage controlled oscillator property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, the disadvan- tage, in this case lies in very low power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Josephson constant clearly indicates that with dc voltage bias at 1 mV at 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='6 GHz, the junction may accept 100 µA current keeping under the limitation of Ic which limits the maximum output RF power at about 100 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This requirement indicates series array of junctions with a common current bias demands keeping all the junctions in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, the issue with series array of junctions operated with common current bias arises with nonuniformity of each junction due to fabrication processes [1], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' When junctions are connected in series, the system behaves as a coupled os- cillator and understanding the periodic solutions is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Two special types of periodic solutions exist [10], namely, in- phase state and splay state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' An in-phase state with period T is a state where all the oscillators always possess the same phase at all times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' θi(t) = θj(t), and θi(t + T) = θi(t) + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The splay-phase or anti-phase or rotating wave state with period T is a solution where the oscillators can be labeled so that θi(t) = Θ(t + jT/N) for all j for some function arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='03787v1 [quant-ph] 10 Jan 2023 2 Θ(t + T) = Θ(t) + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, this state indicates that all the oscillators have the same waveform Θ(t) except for a shift in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' As per [10], one may imagine that each oscillator “fires” when it reaches a certain angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For an in-phase solution, all the oscillators fire simultaneously at every instant T, while splay-phase state has a single oscillator firing every T/N instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, for splay-phase state, oscillators nearly coincide or coincide when ˙θ is small where as for large values, oscillators are not coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The definition of splay- phase does not imply that the phases of the oscillators are equi-spaced around he circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The oscillators bunch up for smaller ˙θ while spread out for large ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, splay-state shows non-uniformity in the distribution of oscillators as they are coherent for smaller ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It has been shown that [10], [11], the non-uniformity can be removed by determining a set of “natural” angles ϕj, so that the splay-phase solution satisfies ϕj(t) = 2πj/N + 2πt/T + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The “natural” angle based dynamical system gets locked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This provides an idea of phase- locking N oscillators, like N Josephson junctions, having eigenfrequencies with smaller spread which may get locked to some resonating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model provides an exactly solvable mean-field model of coupled nonlinear oscillators connecting a large of them having distributed natural frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This model links mean-field techniques and nonlinear dynamics together and also provides precise technique to tune the synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Section II discusses the theory of the Kuramoto model, Section III discusses on the reduction of the equations for the Josephson junctions connected in series to the Kuramoto Model framework and section IV discusses on the numerical analysis of the results for the generalised Kuramoto Model theory and Kuramoto model for Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' KURAMOTO MODEL Let us consider a system of N globally coupled differential equations with the stable limits cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Yoshiki Kuramoto de- veloped a mathematical model for coupled oscillators (n ⩾ 2) to synchronize which is known as “Kuramoto model” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In this model, each jth oscillator is represented by a phase variable θj(t), possessing its own natural frequency ωj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The dynamics of the system of coupled N oscillators becomes ˙θj(t) = ωj + N � i=1,j̸=i Kji sin (θj(t) − θi(t)) , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (2) where Kji is coupling coefficient of the jth oscillator with all other oscillators in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto assumed mean field coupling among phase oscillators such that Kji ≈ K/N ⩾ 0 where K is mean coupling strength which changes (2) as ˙θj(t) = ωj + K N N � i=1,j̸=i sin (θj(t) − θi(t)) , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (3) where, K ⩾ 0 is the coupling strength among the oscillators whose frequencies are distributed with a probability density g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may find a suitable rotating frame like θj → θj − Ωt transforming the system so that natural frequencies of the oscillators may have zero mean, where Ω is the first moment of the distribution function of natural frequencies g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therfore, one may consider the normal form calculation for the system such that one may define the system of equations as ˙θj = fj(θj) + K N N � i=1,i̸=j g (θi, θj) , θj ∈ Rd, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N, (4) where, function fj(θj) are eigenfrequencies defining the nat- ural dynamics in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here coupling parameter K has been added with coupling strength K/N, g is the phase response curve defining the interaction of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the following section, we are not discussing with the stability of the dynamical system, bifurcation etc while one may consult other references like [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the original paper [12], Kuramoto considered the proba- bility density g(ω) to be uni-modal and symmetric centered at mean frequency ω so that, without loss of generality, one can assume that the mean frequency ω = 0 after a shift leading to g(ω) = g(−ω) for the even and symmetric distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To diagnose the feasibility of synchronization, Kuramoto introduced the order parameter R(t) projecting the oscillation on unit circle where R(t) : 0 ⩽ R(t) ⩽ 1 is a measure of the coherence of oscillators as R(t)eȷψ(t) = 1 N N � i=1 eȷθi(t), (5) where R(t) = 0 for asynchronised oscillators, and R(t) > 0 for synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The quantity ψ(t) refers to average phase of all the oscillators at an instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Physically, this order parameter R(t) is the centroid of a set of N points eȷθi distributed in the unit circle in the complex plane at the instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the phases are uniformly spread in the range [−π, π], then R → 0 indicates that the oscillators are not synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' All the oscillators become synchronized with the same average phase ψ(t) for R(t) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the dynamics show stability of R(t) at 1, then the oscillators are synchronized and phaselocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (3) may be re-written by multiplying Ke−ȷθj on both sides of (5) and equating the imaginary parts of the both sides to reduce (3) to ˙θj(t) = ωj + KR(t) sin (ψ(t) − θj(t)) = vj(θ, ω, t) (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (6) Here, vj(θ, ω, t) is the angular velocity of a given oscillator with phase θ and natural frequency ω at the instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The equation (6) reveals that the interaction is set through R(t) and ψ(t) while the phases θj seem to evolve independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Also the effective coupling is proportional to the order parameter R(t) creating a feedback relation between coupling and synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the limit K → 0, (6) reduces to θj(t) ≈ ωjt + θ(0), (7) where, θj(0) denotes initial phase of the jth oscillator and (7) suggests that each oscillator oscillates with own natural frequencies in the absence of coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 3 In the limit of infinite number of oscillators having a distribution of frequency, phase over time, Kuramoto de- scribed the system by the probability density ρ (θ, ω, t) so that ρ (θ, ω, t) dθ gives the fraction of oscillators with phase between θ(t) and θ(t) + dθ(t) at the instant t for a given natural frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Since ρ is non-negative and 2π-periodic in θ satisfying the normalization condition � π −π ρ (θ, ω, t) dθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (8) The probability density function g must also obey the equation of continuity using the angular velocity v(θ, ω, t) as ∂ρ(θ, ω, t) ∂t + ∂ ∂θ {ρ(θ, ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='v} = 0, ∂ρ(θ, ω, t) ∂t + ∂ ∂θ [ρ(θ, ω, t) {ω + KR(t) sin (ψ(t) − θ(t))}] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (9) In the limit R(t) → 0, the dynamics provides stationary solution for ρ(θ, ω, t) = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the continuum limit, (5) gets re-defined by the order parameter R(t) and the average phase ψ(t) incorporating previously described frequency distribution as R(t)eȷψ(t) = � π −π � ∞ −∞ eȷθρ (θ, ω, t) g(ω)dωdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (10) In the strong coupling limit where K → ∞ indicate K ≫ Kc where Kc is critical coupling strength and (6) reduces to system having phases reduced to the average phase as θ(t) = ωt + θ(0) = ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (6), if oscillators get into phaselocked condition, vi(t) → 0 which provides ωj = KR(t) sin (θj(t) − ψ(t)) , −π 2 ⩽ (θj(t) − ψ(t)) ⩽ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (11) From (9), partially synchronized state leading to a locked system can be described as ∂ ∂t(ρ(θ, ω, t)) = 0 which also means ∂ ∂θ (ρ(θ, ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='v(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (11), in this partial synchronized state for vj(t) → 0 and ∂ ∂t (ρ(θ, ω, t)) = 0, reduces to ω KR(t) → sin(θj(t) − ψ(t)), which means ρ(θ, ω, t) = δ � θj(t) − ψ(t) − sin−1 � ω KR(t) �� H(cos θ), (12) such that |ω| ⩽ KR(t) and H(x) = 1, x > 0, 0, elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='. (13) Now, for the other condition ∂ ∂θ (ρ(θ, ω, t)v(t)) = 0 using (6), ρ(θ, ω, t)v(t) = C(say) = constant, or, ρ(θ, ω, t) = C |ω + KR(t) sin(θj(t) − ψ(t))|, |ω| � KR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (14) The constant C can be determined from (8) such that (14) reduces to ρ(θ, ω, t) = � ω2 − K2R2(t) 2π|ω − KR(t) sin(θj(t) − ψ(t))|, |ω| � KR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (15) Therefore, the constraint on the probablity density of the oscillators may be ρ(θ, ω, t) = δ � θj(t) − ψ(t) − sin−1 � ω KR(t) �� H(cos θ), for |ω| ⩽ KR(t) (16) and ρ(θ, ω, t) = � ω2 − K2R2(t) 2π|ω − KR(t) sin(θj(t) − ψ(t))|, elsewhere .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (17) Here δ is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (16) and (17) indicate that partial synchronized states are divided into two groups depending on the natural frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Oscillators having con- straint |ω| ⩽ KR(t) operate in mean-field resulting in locking in a common average phase ψ(t) = Ωt where Ω is the average frequency of the ensemble of the oscillators in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' On the other side, the second group of oscillators having constraint |ω| > KR(t) rotate incoherently which are called as drifting oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Inserting (16) and (17) in (10) we get R(t) = � π −π � ∞ −∞ eȷ(φ(t)−ψ(t)) δ � θ(t) − ψ(t) − sin−1 � ω KR(t) �� g(ω)dθdω + � π −π � |ω|⩽KR(t) � ω2 − K2R2(t)g(ω)dθdω 2π|ω − KR(t) sin(θ(t) − ��(t))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (18) Since g(ω) is even and symmetric, g(ω) = g(−ω) and ρ(θ + π, −ω) = ρ(θ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The even function condition makes the second term of (18) vanish which physically means all the incoherent oscillator solutions vanish resulting in order parameter R(t) only for coherent synchronized oscillators that reform as R(t) = � |ω|⩽KR(t) cos � sin−1 � ω KR(t) �� g(ω)dωdθ, = � π 2 − π 2 cos θg (KR(t) sin θ) KR(t) cos θdθ, = KR(t) � π 2 − π 2 cos2 θg (KR(t) sin θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (19) Here, (19) shows a trivial solution for which order parameter R(t) = 0 which actually shows incoherence as discussed earlier for ρ (θ, ω, t) = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, (19) also suggests 1 = K � π 2 − π 2 cos2 θ g (KR(t) sin θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 4 Setting R(t) = 0, considering K = Kc - the critical coupling strength we get, Kc = 2 πg(0), (20) that triggers the synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In general, expanding the right hand side of (19) in terms of powers of KR(t) and considering g′′(0) < 0 the order parameter can be written as R(t) ∼ � −8 (K − Kc) K3c g′′(0) , (21) which shows that near the transition point, the order parameter [12], [14] yields the form R(t) ∼ (K − Kc)β with β = 1/2 like second order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Kuramoto model can be generalized for a complex network including the connectivity parameter in the coupling term as ˙θj = ωj + N � i=1 KjiAji sin(θj − θi), (22) where, Kji is the coupling strength between nodes j and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Aji is the element of the adjacency matrix A (Aji = 1 if there is a connection between j and i else Aji = 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Any real system may have noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us discuss on the effect of the noise for the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The noise may arise from the variation of frequency of incoherent oscillators as they may not be identical or there may either be an external white noise or white noise inherent to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore the model (3) could be reframed as ˙θj = σωj + K N N � i=1 sin (θj(t) − θi(t)) + √ Γηj(t), : j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (23) where, both ωj and ηj(t) are Gaussian distributions having zero mean and unit variance while σ and Γ behave as am- plitudes of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here last term refers to white noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore (23) physically indicates locally coupled oscillators having natural frequencies of oscillators derived from Gaussian distribution in presence of stochastic effects like white noise due to fluctuations in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The reason for stochastic behavior may vary for different systems while any natural process exhibit stochastic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation of lim σ → 0 refers to the Kuramoto model having identical oscillators in presence of gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The system behaves as if the system is in contact with a heat source and the dynamics is evolving in the statistical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation for lim Γ → 0 indicates that the Kuramoto model has been constructed with oscillators having distributed natural frequencies in absence of gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The system behaves as nonlinear dynamical system relaxing to the non-equilibrium stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Beside this brief summary, one may also consult articles like [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Next, let us transform the Josephson equations for series array of junctions to Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' KURAMOTO MODEL FOR JOSEPHSON JUNCTION SERIES The Josephson junction array can be constructed using Kirchhoff’s laws considering each Josephson junction as a parallel circuit of two elements: an ideal resistance ρ carrying ideal current Iρ and a junction carrying critical current Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Actual Josephson junction also contains a capacitor in parallel to the nonlinear inductor which we have neglected due to its very small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let each of N junctions be connected serially and then coupled to external load having inductance L, resistance R and capacitance C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' C R L Ib ρ1 I1 ρ2 I2 ρN IN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Schematic circuit of qubits connected in series parallel to a Load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider Josephson junction in the series array, say jth junction and following Josephson equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' we express the circuit shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 as V (t) ρj + Ij sin φj + dQ dt = Ib, which can be written as, dφj dt = 2πρj Φ0 � Ib − Ij sin φj − dQ dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (24) Further, L ¨Q + R ˙Q + Q C = N � k=1 Vk, or, L ¨Q + � R + N � k=1 ρk � dQ dt + Q C = − N � k=1 Ikρk sin φk, (25) where Q is the charge on load capacitor, Φ0 = h/(2e) is magnetic flux quantum, h is Planck’s constant, e being the charge of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here, junction resistance ρk for any junction k is very small compared to the load variable Q/C such that one may consider, Q/C − � k ρkIb ≈ Q/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To understand the effect of external parameters like L, C and R on each junction, one may consider a scaled version of those parameters by choosing l = L N , r = R N , c = NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (26) 5 Here, it is to be noted that Φ0 Ij = 1 2πf 2 j Cj where fj is the frequency and Cj is the capacitance of jth junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us now consider transformation of time t and charge Q so that (24) and (25) become dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (24) Φ0 2πρjIj dφj dt + sin φj + 1 Ij dQ dt = Ib Ij = αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider the following transformation relation to transform time t to dimensionless form τ as Φ0 2πρjIj d dt ≡ d dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (27) such that we may write dφj dτ + sin φj + 2πρj Φ0 dQ dτ = αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (28) Substituting dimensionless time τ and scaled parameters as in (26) in (25) we get, L N �2πρjIj Φ0 �2 d2Q dτ 2 + (R + �N k=1 ρk) N �2πρjIj Φ0 � dQ dτ + Q NC = 1 N N � k=1 −Ikρk sin φk, or, l �2πρjIj Φ0 �2 d2Q dτ 2 + � r + �N k=1 ρk N � �2πρjIj Φ0 � dQ dτ + Q c = − 1 N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (29) Let us also consider the following transformation to transform charge Q to dimensionless form q as 2πρjIj Φ0 Q ≡ qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (30) Therefore, through (29), (25) transforms as d2qj dτ 2 + γj dqj dτ + ω2 0jqj = −δj N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (31) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (30) can be used to rewrite (28) as dφj dτ + sin φj + ϵj dqj dτ = αj, (32) where coefficients may be written as γj = � Φ0 2πρjIj � �1 l � � r + �N k=1 ρk N � , (33) ω2 0j = � Φ0 2πρjIj �2 1 lc, (34) δj = � Φ0 2πρjIj � 1 l , (35) and ϵj = 1 Ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (36) Let us write the equation (32) in the uncoupled form for ϵj → 0 or ˙Q → 0 such that we get, dφj dτ = αj − sin φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (37) As discussed in the Section I, the splay-state shows that transforming the dynamical system equations make a rigid sys- tem with coherent frequencies in weak coupling or uncoupled limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Hence, let us transform φj in (24) into ‘natural’ angle ψj such that dψj dt = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (37) can be transformed in terms of the ‘natural’ angle ψj such that dψj/dt − c, where c is constant to be determined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' transformation as φj → ψj as uniform rotation with first derivative remaining constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The constant ‘c’ may be determined with the fact that the time to complete one cycle by these two sets of coordinates must be same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, T = � T 0 dτ = � 2π 0 dψj c = � 2π 0 dψj ωj = � 2π 0 dφj (αj − sin φj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, 2π ωj = 2π �� α2 j − 1 �, for αj ⩾ 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Ib ⩾ Ij, which shows ωj = � α2 j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (38) Then the transformation to the natural angles satisfies dψj = � α2 j − 1 αj − sin φj dφj, (39) which on integration yields ψj = 2 tan−1 �� αj − 1 αj + 1 tan �φj 2 + π 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (40) At this point, one may construct a transformation function ψ(φj) to translate any angle φj to its natural angle ψj while another transformation function φ(ψj) may be used to invert as ψ (φ) = 2 tan−1 �� α − 1 α + 1 tan �φ 2 + π 4 �� , (41) φ (ψ) = 2 tan−1 �� α + 1 α − 1 tan �ψ 2 �� − π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (42) Here, we use the shorthand: ψj ≡ ψ(φj) and φj ≡ φ(ψj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (40), sin φj = 1 − αj cos ψj αj − cos ψj = αj − α2 j − 1 αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (43) Detailed derivation of (43) from (40) is shown in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, one may rewrite (32) using (39) and (43) as dψj dτ = dψj dφj dφj dτ = � α2 j − 1 αj − sin φj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � αj − sin φj − ϵj dqj dτ � , = � α2 j − 1 − ϵj � α2 j − 1 αj − sin φj dqj dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (44) 6 Let us rescale non-dimensional quantity τ as ˜τ such that τ = ˜τ � α2 j − 1 =⇒ d d˜τ ≡ 1 � α2 j − 1 d dτ =⇒ d2 dτ 2 ≡ � α2 j − 1 � d2 d˜τ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (45) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (44), using (45), transforms as dψj d˜τ = 1 − ϵj � α2 j − 1 αj − sin φj dqj d˜τ , (46) The weak-coupling solution of (44) may be written as ψj(τ) ≡ �� α2 j − 1 � τ + cj = ˜τ + ψj0, (47) where cj is the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Initially, at τ = 0, one may assume initial phase as ψj0 such that cj=ψj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The reference [11] discusses about the importance of the weak coupling condition for the Josephson junction arrays and drift in ψj may be obtained by averaging (46) over one cycle as �dψj d˜τ � = 1 − 1 2π � 2π 0 ϵj � α2 j − 1 αj − sin φj �dqj d˜τ � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (48) Similarly, one may rewrite non-dimensional charge equation (31) in terms of ˜τ as � α2 j − 1 � d2qj d˜τ 2 + γj � α2 j − 1dqj d˜τ + ω2 0jqj = −δj N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (49) It is usually convenient to write sin(φj)=sin(φ(ψj)) in terms of its Fourier series as sin φ(ψk) = ∞ � n=0 Akn cos (nψkn) = ∞ � n=0 Akn cos {n (˜τ + ck)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (50) Then (49) reduces to � α2 j − 1 � d2qj d˜τ 2 + γj � α2 j − 1dqj d˜τ + ω2 0jqj = −δj N N � k=1 ∞ � n=0 IkρkAkn cos {n (˜τ + ck)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (51) One may obtain the steady-state solution of (51) as qj(˜τ) = −δj N IkρkBkn cos {n (˜τ + ck) + βkn} ,(52) dqj(˜τ) d˜τ = δj N nIkρkBkn sin {n (˜τ + ck) + βkn} , (53) d2qj(˜τ) d˜τ 2 = δj N n2IkρkBkn cos {n (˜τ + ck) + βkn} ,(54) where B2 kn = A2 kn n2γ2 j � α2 j − 1 � + � n2 � α2 j − 1 � − ω2 0j �2 , (55) βkn = tan−1 � � nγj � α2 j − 1 n2 � α2 j − 1 � − ω2 0j � � = βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (56) Using the expression (43), one may derive Akn and obtain Ak0 = 1 π � π −π 1 − αk cos ψk αk − cos ψk dψk, (57) Akn = 1 π � π −π 1 − αk cos ψk αk − cos ψk cos �nπψk π � dψk (58) where n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Bkn denotes the amplitude of the linear damped oscillator while βkn denotes its phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, Bkn must be chosen to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Now, (48) may be re-written as �dψj d˜τ � = 1 − ϵjδj � α2 j − 1 2πN � 2π 0 � 1 αj − sin φj × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (59) Using (43), sin φj = αj − α2 j − 1 αj − cos ψj , or, αj − sin φj = α2 j − 1 αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (60) With this (59) may be modified using (60) to or, �dψj d˜τ � = 1 + Kj N N � k=1 Ak sin (cj − ck − ζj) , (61) where, Kj = ϵjδj � α2 j − 1 � γ2 j � α2 j − 1 �2 + � ω2 0j − � α2 j − 1 �2�2 , (62) AK = Ikρk � 1 − α2 k + αk � α2 k − 1 � , (63) ζj = tan−1 � � γj � α2 j − 1 α2 j − 1 − ω2 0j � � = β1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (64) Reader may check detailed description of the derivation in the appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the final step, one may replace the ‘initial values’ of phases by their slowly evolving components like ⟨ψj(˜τ)⟩ and ⟨ψk(˜τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Also one may get firstorder averaged equation by dropping the angular brackets so that (61) transforms to dψj d˜τ = 1 + Kj N N � k=1 Ak sin (ψj(˜τ) − ψk(˜τ) − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (65) 7 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (65) resembles the Kuramoto model in a generalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For the sake of mathematical formalities, it is important to note that except terms corresponding to n = 1 terms for other values of n becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To arrive at (65), it was assumed that the fabrication process may not guarantee exactly same values of parameters for each junction and hence one may consider that each junction has different internal resistance and different critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The difference may be very small for junctions prepared in the same batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the fabrication process is done in very skilled sequence (65) may turn into special form for assuming ρ1 = ρ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = ρN = ρ (say) and I1 = I2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = IN = Ic(say) so that each junction has nearly same frequency f (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This case of identical junctions has been studied extensively in may literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The transformation (27) for time leads to Φ0 2πρIc d dt ≡ d dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (66) while (30) entails 2πρIc Φ0 Q ≡ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (67) Consequently, (31) reduces to d2q dτ 2 + γ dq dτ + ω2 0q = − β N N � k=1 sin φk, (68) where γ = � Φ0 2πρIc � � 1 lρ � (r + ρ) , (69) ω2 0 = � Φ0 2πρIc �2 1 lc, (70) β = � Φ0 2πρIc � 1 l , (71) (72) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (55) and (56) become B2 n = A2 n n2γ2 (α2 − 1) + {n2 (α2 − 1) − ω2 0}2 , (73) βn = tan−1 � nγ √ α2 − 1 ω2 0 − n2 (α2 − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (74) Repeating the earlier exercise, one may obtain the final phase equation (59) as �dψj d˜τ � = 1 − β 2πN √ α2 − 1 � 2π 0 (α − cos (τ + cj)) × N � k=1 ∞ � n=0 nBn sin {n (˜τ + ck) + βn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (75) In the case of identical junctions, computation shows that only B1 exists while others are evaluated to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, (75) becomes �dψj d˜τ � = 1 − B1β 2πN √ α2 − 1 � 2π 0 (α − cos (τ + cj)) × N � k=1 ∞ � n=0 n sin {n (˜τ + ck) + βn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Integrating, we get dψj d˜τ = 1 + K N N � k=1 sin (ψj(˜τ) − ψk(˜τ) − β1) , (76) where K = πB1β 2π √ α2 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (77) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (76) exactly resembles as the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the following section, let us try to understand general characteristics of the Kuramoto model in general and in the context of Josephson junction array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' ANALYSIS A C + + code has been developed alongwith DISLIN code to analyse the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' DISLIN [16] is a freely available graph plotting routine that plots during runtime and can be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In this section, let us first investigate basic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model in arbitrary unit for 100 oscillators with K=4 showing synchronization after a certain settling time within a band of frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model as discussed in (3) including the effect of coupling strength (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If K is properly tuned, one may expect synchronization as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here we consider that the oscillators are oscillating possess- ing a frequency distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may control width of the distribution while keeping the zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' We consider Logistic and Lorentzian fuctions having width β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Oscillators tend get to be synchronized if K is equal to or more than some threshold value Kc as discussed in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' g(ω) = exp (−ω/β) β [1 + exp (−ω/β)]2 (78) The Logistic function is described as (78) which shows g(0)=1/(4β) where, β is the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Likewise, one may define the Lorentzian function as (79) g(ω) = b (ω2 + b2), (79) Kuramoto model for 1oo oscillators having K=4 without any noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 a sin 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0 2 4 6 8 10 time8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators with K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 having Logistic distribution of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators with K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 having Lorentzian distribution of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' so that one get g(0)=2/(πb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This g(0) estimates thresh- old value of the coupling strength as Kc=2/πg(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 where the latter is operating with threshold coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The synchronization for the latter shows phase space of order parameter as a dot denoting synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 and 6 also show similar observation of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This theoretical study clearly heps us to understand the sig- nificance of coupling strength and the treatment of frequency range of oscillators to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Next one may apply this understanding in the case of Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation is very much different hereas the definition of K is complex for both non-identical and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators oscillating with k=Kc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='509 with Logistic function of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators oscillating with k=Kc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 with Lorentzian function of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' identical junction arrays as evident from either (65) or (76) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider mean frequency may be around 5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 and 8 show simulated results of systems of 100 Josephson junctions in non-identical and identical configurations respectively operated for ˜τ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The interesting part is that synchronization is not pulling the oscillators to a certain unique frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Rather, oscillators tend to cool down to a narrow band of frequencies resulting in an arc in phase space diagram which resembles as if oscillators have a certain ‘viscosity’ in the combined system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For the non identical case, the spread of Ic is considered very small like 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1% while variation in ρj is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='05 % as fabrication is much better and junctions fabricated in the same substrate will not vary Phase graph of 10o oscillators forK=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100 dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001simulation time=10 (a) Phase graph of 100 oscillators (b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10 sin P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (T) time (r) (o)orderParameterof oscillatorsatT=0 (d) Order Parameter of oncillatore at T =io 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 40 40 90 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 UTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="9 8'0- L1 11 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="3 心1 2 6'0- 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 COSPhase graph of 10o oscillators forK=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100 dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001simulation time=10 (a) Phase graph of 100 oscillators (b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10 sin P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)orderParameterof oscillatorsatT=0 (d) Order Parameter of oncillatore at T =io 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 40 40 90 90 0 UTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 ++ sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="5 F'T COSPhasegraph of 100oscillatorsforK=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='509dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100simulation time=30 (a) Phase graph of 100 oscillators [b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 一 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (-) time (-) (o)Order Parameter of oscillators atT=0 (d) Order Parameter of oBcillatorB at T =90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 B0 2'0 90 90 sin UTS 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 ++++ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 L1 L1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 COSPhasegraphof100oscillatorsforK=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='400dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100simulationtime=30 (a) Phase graph of 1o0 oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 二 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (-) time (-) (o)orderParameterof oscillators atT=0 (d) Order Parameter of oBcillatore at T=go 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 COS9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 non-identical Josephson junctions operating with mean frequency of 5 GHz having mean Ic = 10 µA, mean internal resistance ρj = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ connected in series array to external load with parameters L=1 nH, C = 1 µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within a narrow band of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The final phase space is not a dot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 identical Josephson junctions operating at mean frequency of 5 GHz having mean Ic = 10 µA, internal resistance ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ connected in series array to external load with parameters L=1 nH, C = 1 µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within a narrow band of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The final phase space is not a dot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Another point to note is that the oscillators in the non-identical case tend to syncronize faster and better than the other case, possibly due to the noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It has already been discussed that Kuramoto model stands on the assumption that a large number of oscillators have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In our experimental regime, one may need to use smaller number of oscillators say 5 or 10 oscillators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 as asynchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The order parameter R is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 identical oscillators having Ic = 10 µA and ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ operating with 5 GHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' also shown to be oscillating at a lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The observation was made for ˜τ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The circuit parameters were kept same as those for 100 oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Evidently oscillators were not syhronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The case for the 5 non-identical oscillators is same as 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Now, to tune the circuit, let us select Ic as 10 µA and ρj Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 non identical Josephson junctions are partially syncronized changing Ib to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8785 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ as before as we wish to experiment with the same junctions while we change Ib - the bias current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='10, the synchronization is observed where one oscillator is out of sync while the rest 4 oscillators come closer to lie in a band very fast ˜τ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Phase graph of 100 oseillators for K=12205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='95dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001 simulation time=25 (a) Phase graph of 100 oscillators [b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="0 B'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 一 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o) Order Parameter of oscillators at T=0 (d) Order Parameter of oBcillatorB at T=z5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 + + 2'0 2'0 90 90 us 0 sin 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 L1 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 cosu cosyPhase graph of 100 oseillators for K=12205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='95dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001 simulation time=25 (a) Phase graph of 1oo oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10 TTTTT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 sin 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)orderParameterof oscillators atT=0 (d) Order Parameter of oBcillatorB at T=z5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 2'0 90 90 sin us 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 cosu cosyPhasegraph of 5oseillators forK=4307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='83 dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001 simulationtime=25 (a) Phase graph of 5 oscillators (b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)OrderParameter of oscillators atT=0 (d) Order Parameter of oBcillator at T =25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 2'0 90 90 us 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 cosyPhase graph of 5 oscillators for K=30333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='88dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='00lsimulation time=15 (a) Phase graph of 5oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 80 P 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)orderParameterof oscillators atT=0 (d) Order Parameter of oBcillatore at T =15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 ++ 20 90 90 uIs 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 identical Josephson junctions are partially syncronized changing Ib to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='877 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' CONCLUSION The exercises demonstrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='10 and 11 show the possibility of synchronization for few oscillators following Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, order parameter show in-course instability which later settles down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This study helps to understand applicability of junctions in series array and steps to control the level of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The process is easier and synchronization is performed well for larger number of junctions while partial synchronization is also possible following the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, this study does not state any conclusive equation for threshold coupling for Josephson junction as it discussed in case of general oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This aspect will be discussed in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' APPENDIX A From (40),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' tan �φj 2 + π 4 � = � αj + 1 αj − 1 tan ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � tan �φj 2 + π 4 � + 1 � � tan �φj 2 + π 4 � − 1 � = �� αj + 1 αj − 1 tan ψj 2 + 1 � �� αj + 1 αj − 1 tan ψj 2 − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � 1 + tan φj 2 1 − tan φj 2 + 1 � � 1 + tan φj 2 1 − tan φj 2 − 1 � = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � cos φj 2 + sin φj 2 cos φj 2 − sin φj 2 + 1 � � cos φj 2 + sin φj 2 cos φj 2 − sin φj 2 − 1 � = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2 cos φj 2 × 2 sin φj 2 � cos φj 2 − sin φj 2 �2 = 2 sin φj � cos φj 2 − sin φj 2 �2 = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2 sin φj 1 − sin φj = αj tan2 ψj 2 + tan2 ψj 2 − αj + 1 αj − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1 − sin φj 2 sin φj = 1 − αj cos ψj (αj − 1) cos2 ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1 2 sin φj = 1 2 + 1 − αj cos ψj (αj − 1) cos2 ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' sin φj = 1 − αj cos ψj αj − cos ψj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' sin φj ≡ sin φ(ψj) = αj − � α2 j − 1 � αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Phasegraph of 5oscillators forK=30453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='72dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='00l simulationtime=15 (a) Phase graph of 5 oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 E0 sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': 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T=Q (d) Order Parameter of oBcillatore at T =15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 20 90 90 sin uIs 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 心1 cos cosu11 APPENDIX B �dψj d˜τ � = 1 − ϵjδj � α2 j − 1 2πN � 2π 0 � αj − cos ψj α2 j − 1 × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 − ϵjδj 2πN � α2 j − 1 � 2π 0 (αj − cos (˜τ + cj)) × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 + ϵjδj N � α2 j − 1 � γ2 j � α2 j − 1 �2 + � ω2 0j − � α2 j − 1 �2�2 × N � k=1 Ikρk � 1 − α2 k + αk � α2 k − 1 � sin (cj − ck − ζj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 + Kj N N � k=1 Ikρk � 1 − α2 k + αk � α2 k − 1 � × sin (cj − ck − ζj) , or, �dψj d˜τ � = 1 + Kj N N � k=1 Ak sin (cj − ck − ζj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' ACKNOWLEDGMENT The author would like to Sudhir R Jain, for his ideas, inspi- ration and continuous support to conceptualize, understand and formulate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The author also expresses gratitude to Susmita Bhattacharyya and Tilottoma Bhattacharyya for their guidance.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='de/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'}