Non-orthogonal positive operator valued measure phase distributions of one- and two-mode electromagnetic fields

Barak, Ronen; Ben‐Aryeh, Y.

doi:10.1088/1464-4266/7/5/001

Cited by 8 publications

(7 citation statements)

References 77 publications

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“…Phase locking.-It is well-known that there is not a unique way to define the phase operator in quantum mechanics [42]. One option [43] that has been used to study quantum synchronization [26] is the phase distribution P(φ) = 1 2π φ|ρ|φ with |φ = ∞ n=0 e inφ |n , yielding 2πP(φ) − 1 = ∞ m n=0 ρ m,n e iφ(n−m) . Our perturbative steady-state solution (6) contains only terms with n − m = ±1, so P(φ) = 1 2π + η 1 cos φ + η 2 sin φ with To convert the phase distribution P(φ) into a single number characterizing the tendency to synchronize, we use the absolute value of the measure defined in Ref.…”

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confidence: 99%

Genuine Quantum Signatures in Synchronization of Anharmonic Self-Oscillators

et al. 2016

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We study the synchronization of a Van der Pol self-oscillator with Kerr anharmonicity to an external drive. We demonstrate that the anharmonic, discrete energy spectrum of the quantum oscillator leads to multiple resonances in both phase locking and frequency entrainment not present in the corresponding classical system. Strong driving close to these resonances leads to nonclassical steady-state Wigner distributions. Experimental realizations of these genuine quantum signatures can be implemented with current technology. DOI: 10.1103/PhysRevLett.117.073601 The synchronization of self-oscillators is a subject with great relevance to several natural sciences [1,2]. Its exciting frontiers include neuronal synchronization in the human brain [3,4] and stabilization of power-grid networks [5], as well as the engineering of high-precision clocks [6,7]. Recent advances in nanotechnology will enable experiments with large arrays of self-oscillators in the near future [8,9]. Whereas most research has focused on the classical domain, synchronization in the quantum regime [10] has become a very active topic. There has been much recent experimental progress with micro-and nanomechanical systems [11][12][13][14][15] [29,30].Studying a Van der Pol oscillator, the most prominent example of a self-oscillator, recent theoretical work characterized how synchronization quantitatively differs between its quantum and classical realization in phase locking [24,25] as well as in frequency entrainment [21,22]. While synchronization is hindered by quantum noise compared to the classical model [21,22], noise is less detrimental [24,25] than one would expect from a semiclassical description.In this Letter, we study self-oscillators for which both the damping and the frequency are amplitude dependent. We show that their synchronization behavior is qualitatively different in the quantum and the classical regime. Focusing on a Van der Pol oscillator with Kerr anharmonicity, we find two genuine quantum signatures. First, while the synchronization of one such oscillator to an external drive is maximal at one particular frequency classically, the corresponding quantum system shows a tendency to synchronize at multiple frequencies. Using perturbation theory in the drive strength, we demonstrate that these multiple resonances reflect the quantized anharmonic energy spectrum of the oscillator. We show that these features are observable in the phase probability distribution if the Kerr anharmonicity is large compared to the relaxation rates and the system is in the quantum regime; i.e., the limit cycle amplitudes are small. In the semiclassical limit, the energy spectrum becomes continuous, so that the resonances (and therefore the quantized energy spectrum) cannot be resolved. Using numerically exact simulations of the full quantum master equation, we find a second genuine quantum signature: For strong driving close to these resonances, the steady-state Wigner distribution exhibits areas of negative density; i.e., the steady state is nonclassica...

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Genuine Quantum Signatures in Synchronization of Anharmonic Self-Oscillators

et al. 2016

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“…But, in general, a distribution based on a quasiprobability distributions is not unique as other representations could be used [22], which would give different results. We circumvent this ambiguity by directly calculating a relative phase distribution from the density matrix using phase states [23]:…”

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Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers

et al. 2015

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We investigate quantum synchronization theoretically in a system consisting of two cold ions in microtraps. The ions' motion is damped by a standing-wave laser while also being driven by a blue-detuned laser which results in self-oscillation. Working in a nonclassical regime, where these oscillations contain only a few phonons and have a sub-Poissonian number variance, we explore how synchronization occurs when the two ions are weakly coupled using a probability distribution for the relative phase. We show that strong correlations arise between the spin and vibrational degrees of freedom within each ion and find that when two ions synchronize their spin degrees of freedom in turn become correlated. This allows one to indirectly infer the presence of synchronization by measuring the ions' internal state.Introduction. Two macroscopic self-oscillators synchronize when their relative phase locks to a fixed value [1]. Important studies of synchronization effects were carried out using lasers [2], with arrays of Josephson junctions [3] and over the last few years much attention has been devoted to exploring synchronization in micromechanical oscillators [4]. Recently, theoretical work has begun to explore synchronization in the quantum regime [5][6][7][8][9][10][11][12][13][14]: the formation of a relative phase preference between two (or more) weakly coupled quantum oscillators operating in a regime far from the classical correspondence limit. Differences between classical and quantum predictions for the synchronization of van der Pol oscillators have been identified in the case where the oscillators are only weakly excited [5]. Nevertheless, many important questions about quantum synchronization remain open, such as how it should be quantified and how it can best be probed experimentally.Cold ions in microtraps provide a natural platform for exploring synchronization in the quantum regime [5]. The generation of self-oscillations in the motional state of ions, phonon lasing, has already been observed [15]. Furthermore, precise control of trapping potentials of the individual ions can now be achieved with microtraps [16] allowing the vibrational frequencies of individual ions and the coupling between different ions to be tuned. Here, we investigate synchronization in two trapped-ion phonon lasers which are pumped in a similar way to that demonstrated in recent experiments [15].We identify a parameter regime where phonon lasing of an individual ion occurs with just a few quanta leading to a nonclassical state of the phonons and investigate the emergence of synchronization in this regime when a weak interion coupling is introduced (weak as it is the slowest time scale in the system). Our model includes two of the electronic levels of the ions used in the pumping process (which we refer to as "spin"), allowing us to uncover strong correlations which arise between the electronic and vibrational degrees of freedom of the individual ions. We study the degree of synchronization as the strength and detuning of the pumping lase...

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“…Finally I would like to point out that in another approach to phase measurements in MZI, which is different from the phase estimation approach, there is a quasi-conjugate uncertainty relation between the 'relative phase operator' and the operator representing the difference in the number of photons and the complete quantum phase distribution can be obtained (See for this approach in [47], and in the long list of References included). In the present work we have followed, however, the 'phase estimation methods' in which is a classical parameter and it has been shown that by using photon counting measurements these methods become very efficient for deriving the two-mode phase uncertainty in MZI.…”

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confidence: 99%