We study the entanglement between two coupled detectors, the internal degrees of freedom of which are modeled by harmonic oscillators, interacting with a common quantum field, paying special attention to two less studied yet important features: finite separation and direct coupling. Distance dependence is essential in quantum teleportation and relativistic quantum information considerations. The presence of a quantum field as the environment accords an indirect interaction between the two oscillators at finite separation of a non-Markovian nature which competes with the direct coupling between them. The interplay between these two factors results in a rich variety of interesting entanglement behaviors at late times. We show that the entanglement behavior reported in prior work assuming no separation between the detectors can at best be a transient effect at very short times and claims that such behaviors represent late-time entanglement are misplaced. Entanglement between the detectors with direct coupling enters in the consideration of macroscopic quantum phenomena and other frontline issues. We find that with direct coupling entanglement between the two detectors can sustain over a finite distance, in contrast to the no direct coupling case reported before, where entanglement cannot survive at a separation more than a few inverse high-frequency cutoff scales. This work provides a functional platform for systematic investigations into the entanglement behavior of continuous variable quantum systems, such as used in quantum electro- and optomechanics.
We study the entanglement between two coupled detectors, the internal degrees of freedom of which are modeled by harmonic oscillators, interacting with a common quantum field, paying special attention to two less studied yet important features: finite separation and direct coupling. Distance dependence is essential in quantum teleportation and relativistic quantum information considerations. The presence of a quantum field as the environment accords an indirect interaction between the two oscillators at finite separation of a non-Markovian nature which competes with the direct coupling between them. The interplay between these two factors results in a rich variety of interesting entanglement behaviors at late times. We show that the entanglement behavior reported in prior work assuming no separation between the detectors can at best be a transient effect at very short times and claims that such behaviors represent late-time entanglement are misplaced. Entanglement between the detectors with direct coupling enters in the consideration of macroscopic quantum phenomena and other frontline issues. We find that with direct coupling entanglement between the two detectors can sustain over a finite distance, in contrast to the no direct coupling case reported before, where entanglement cannot survive at a separation more than a few inverse high-frequency cutoff scales. This work provides a functional platform for systematic investigations into the entanglement behavior of continuous variable quantum systems, such as used in quantum electro- and optomechanics.
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures – the so-called ‘hot entanglement’ – has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit ‘hot entanglement’ to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillators in a shared bath depicted by a finite-temperature scalar field . From the Langevin equations we derived for the normal modes and the entanglement measure constructed from the covariance matrix we examine the interplay between direct coupling, field-induced interaction and finite separation on the structure of late-time entanglement. We show that the coupling between oscillators plays a crucial role in sustaining entanglement at intermediate temperatures and over finite separations. In contrast, the field-induced interaction between the oscillators which is a non-Markovian effect becomes very ineffective at high temperature. We determine the critical temperature above which entanglement disappears to be bounded in the leading order by the inverse frequency of the center-of-mass mode of the reduced oscillator system, a result not unexpected, which rules out hot entanglement in such settings.
We consider the entanglement dynamics between two Unruh-DeWitt detectors at rest separated at a distance d. This simple model when analyzed properly in quantum field theory shows many interesting facets and helps to dispel some misunderstandings of entanglement dynamics. We find that there is spatial dependence of quantum entanglement in the stable regime due to the phase difference of vacuum fluctuations the two detectors experience, together with the interference of the mutual influences from the backreaction of one detector on the other. When two initially entangled detectors are still outside each other's light cone, the entanglement oscillates in time with an amplitude dependent on spatial separation d. When the two detectors begin to have causal contact, an interference pattern of the relative degree of entanglement (compared to those at spatial infinity) develops a parametric dependence on d. The detectors separated at those d with a stronger relative degree of entanglement enjoy longer disentanglement times. In the cases with weak coupling and large separation, the detectors always disentangle at late times. For sufficiently small d, the two detectors can have residual entanglement even if they initially were in a separable state, while for d a little larger, there could be transient entanglement created by mutual influences. However, we see no evidence of entanglement creation outside the light cone for initially separable states.
We study the dynamics of quantum entanglement between two Unruh-DeWitt detectors, one stationary (Alice), and another uniformly accelerating (Rob), with no direct interaction but coupled to a common quantum field in (3+1)D Minkowski space. We find that for all cases studied the initial entanglement between the detectors disappears in a finite time (``sudden death''). After the moment of total disentanglement the correlations between the two detectors remain nonzero until late times. The relation between the disentanglement time and Rob's proper acceleration is observer dependent. The larger the acceleration is, the longer the disentanglement time in Alice's coordinate, but the shorter in Rob's coordinate.
We consider the entanglement dynamics between two Unruh-DeWitt detectors at rest separated at a distance d. This simple model when analyzed properly in quantum field theory shows many interesting facets and helps to dispel some misunderstandings of entanglement dynamics. We find that there is spatial dependence of quantum entanglement in the stable regime due to the phase difference of vacuum fluctuations the two detectors experience, together with the interference of the mutual influences from the backreaction of one detector on the other. When two initially entangled detectors are still outside each other's light cone, the entanglement oscillates in time with an amplitude dependent on spatial separation d. When the two detectors begin to have causal contact, an interference pattern of the relative degree of entanglement (compared to those at spatial infinity) develops a parametric dependence on d. The detectors separated at those d with a stronger relative degree of entanglement enjoy longer disentanglement times. In the cases with weak coupling and large separation, the detectors always disentangle at late times. For sufficiently small d, the two detectors can have residual entanglement even if they initially were in a separable state, while for d a little larger, there could be transient entanglement created by mutual influences. However, we see no evidence of entanglement creation outside the light cone for initially separable states.
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−P for arbitrary excitation numbers (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 θ and ϑ2, the frequency parameter β0 and the coupling strength ε: 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=1 and strong couplings ε≈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.
We analyze the time evolution of quantum entanglement in a model consisting of two two-level atoms interacting with a two-mode electromagnetic field for a variety of initial states and interatomic separations. We study two specific atomic separations which give rise to symmetric atom-field couplings. For general atomic distances we consider a subset of initial states analytically, and then treat the more general situation numerically. We examine a variety of qualitative features such as entanglement sudden death, dynamical generation, protection, and transfer between subsystems. Our analysis shows a stark contrast in features of entanglement between the two special coupling schemes often considered; further, these features are uncharacteristic of those arising for general distances, due to the high degree of symmetry present in the special cases. The variety of behaviors in these two-mode cases suggest the importance of considering atomic separation carefully for any model where two atoms interact with a common field.