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Journal of Physics B Atomic Molecular and Optical Physics

Entanglement dynamics for a conditionally kicked harmonic oscillator

Eric G. Arrais, José S. Sales, N. G. de Almeida

Published Jul 29, 2016• 2 citations
Publisher

Abstract

Abstract The time evolution of the quantum kicked harmonic oscillator (KHO) is described by the Floquet operator which maps the state of the system immediately before one kick onto the state at a time immediately after the next. Quantum KHO is characterized by three parameters: the coupling strength V 0 , the so-called Lamb–Dicke parameter η whose square is proportional to the effective Planck constant ℏ eff , and the ratio T of the natural frequency of the oscillator and the kick frequency. To a given coupling strength and depending on T being a natural or irrational number, the phase space of the classical kicked oscillator can display different behaviors, as for example, stochastic webs or quasicrystal structures, thus showing a chaotic or localized behavior that is mirrored in the quantum phase space. On the other hand, the classical limit is studied letting ℏ eff become negligible. In this paper we investigate how the ratio T , considered as integer, rational or irrational, influences the entanglement dynamics of the quantum KHO and study how the entanglement dynamics behaves when varying either V 0 or ℏ eff parameters.

DOI
10.1088/0953-4075/49/16/165501

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Abstract The time evolution of the quantum kicked harmonic oscillator (KHO) is described by the Floquet operator which maps the state of the system immediately before one kick onto the state at a time immediately after the next. Quantum KHO is characterized by three parameters: the coupling strength V 0 , the so-called Lamb–Dicke parameter η whose square is proportional to the effective Planck constant ℏ eff , and the ratio T of the natural frequency of the oscillator and the kick frequency. To a given coupling strength and depending on T being a natural or irrational number, the phase space of the classical kicked oscillator can display different behaviors, as for example, stochastic webs or quasicrystal structures, thus showing a chaotic or localized behavior that is mirrored in the quantum phase space. On the other hand, the classical limit is studied letting ℏ eff become negligible. In this paper we investigate how the ratio T , considered as integer, rational or irrational, influences the entanglement dynamics of the quantum KHO and study how the entanglement dynamics behaves when varying either V 0 or ℏ eff parameters.
We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter chi identical to K Planck's (eff)(2)/4D(3/2) , which relates the effective Planck constant, Planck's (eff), to the kicking strength, K, and the diffusion constant, D. This relation between classical and quantum distributions is valid when chi<1 , a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by D. Our results extend a recent study performed in the chaotic regime.
We explore the properties of the quantum kicked harmonic oscillation governed by the Hamiltonian . The classical version of this model is known to exhibit certain type of stochastic behavior. We reduce the quantum equation of the model to quantum mapping equations. Numerical results of the mappings show that the behavior of the average energy as a function of time depends on the ration of the frequency of the external and that of the unperturbed system . The energy increases first very rapidly and then the energy growth saturates for irrational . The break time is independent of the strength of the external force, but is roughly proportional to . For a fixed rational the energy is damped oscillation for smaller T, and it is seemingly recurrent with time in the sense of Hogg and Huberman. In both cases, these behaviors of energy are independent of the strength of the external force and differ from the behavior of the quantum kicked rotator.
We discuss the emergence of the classical description from quantum mechanics for chaotic systems. We consider a specific model, which corresponds to an ion trapped in a harmonic potential and submitted to a sequence of laser pulses: the kicked harmonic oscillator, under conditions that lead to chaotic behavior for the classical system. We show that noise plays an essential role in the transition from quantum to classical behavior. This transition is described in terms of the separation between the classical phase space distribution and the Wigner function corresponding to the quantum system, obtained by integrating over the whole phase space the magnitude of the difference between the two distributions. It is shown that, in the semiclassical limit, this separation is governed by a single parameter, which depends on the diffusion coefficient, the Lamb-Dicke parameter, and the kick strength. The Lamb-Dicke parameter plays the role of a dimensioneless Planck constant. As this parameter goes to zero, we show that both distributions remain close together for all times.
We present a kicked harmonic oscillator where the impulsive driving is provided by stroboscopic measurements on an ancillary degree of freedom and not by the canonical quantization of a time-dependent Hamiltonian. The ancilla is dynamically entangled with the oscillator position, while the background Hamiltonian remains static. The dynamics of this system is determined in closed analytical form, allowing for the evaluation of a properly defined Loschmidt echo, ensemble averages, and phase-space portraits. As in the case of standard Floquet systems, we observe regimes with crystalline and quasicrystalline structures in phase space, resonances, and evidence of chaotic behavior, however, not originating from any classically chaotic system.
We investigate the quantum-classical correspondence of kicked harmonic oscillator (KHO) model by the averaged semiquantum method. Furthermore, we investigate the decoherence of N dimensional kicked harmonic oscillator with different coupling strengths and the dimensionalities. The averaged semiquantum method origins from the time-dependent variational principle (TDVP) formulation and contains nondiagonal matrix elements, so it can be applied to study dissipation, measurement and decoherence problems. When applied to the harmonic oscillator model coupled to a heat bath with different coupling strengths and dimensionalities of the bath, we find that the loss of coherence between superposed wave packets can be also predicted by the averaged semiquantum method.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.