We provide a complete characterization of the evolution of entanglement between two oscillators coupled to a common environment. For initial Gaussian states we identify three phases with different qualitative long time behavior: There is a phase where entanglement undergoes a sudden death (SD). Another phase (SDR) is characterized by an infinite sequence of events of sudden death and revival of entanglement. In the third phase (NSD) there is no sudden death of entanglement, which persist for long time. The phase diagram is described and analytic expressions for the boundary between phases are obtained. Numerical simulations show the accuracy of the analytic expressions. These results are applicable to a large variety of non--Markovian environments. The case of non--resonant oscillators is also numerically investigated.
The dynamics of mixedness and entanglement is examined by solving the time-dependent Schrödinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially we introduce three quenched models. In the quenched models both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It…
We study the boson-parafermion entanglement of the parasupersymmetric coherent states of the harmonic oscillator and derive the degree of entanglement in terms of the concurrence. The conditions for obtaining the maximal entanglement is also examined, and it is shown that in the usual supersymmetry situation we can obtain maximally entangled Bell states.
We investigate nonequilibrium entanglement generation in a chain of harmonic oscillators with time-dependent linear coupling. We use optimal control theory to determine the coupling modulation that leads to maximum logarithmic negativity for a pair of opposite oscillators and show that it corresponds to a synchronization of the eigenmodes of the chain. We further analytically relate the maximum attainable entanglement to the irreversible work done to produce it, thus bridging nonequilibrium entanglement production and nonequilibrium thermodynamics.
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be…
Quantum entanglement has been actively sought for in optomechanical and electromechanical systems. The simplest such system is a mechanical oscillator interacting with a coherent beam, while the oscillator also suffers from thermal decoherence. For this system, we show that quantum entanglement is always present between the oscillator and continuous outgoing fields -- even when the environmental temperature is high and the oscillator is highly classical. Such universal entanglement is also shown to be able to survive more than one oscillation cycle if characteristic frequency of the optomechan…
We use a versatile model to evaluate the multipartite entanglement and the nonclassical light generation in optical parametric oscillators, exploring the differences between doubly and triply resonant cavity configurations. We demonstrate the entanglement of the pump mode with converted fields in both situations, and the fundamental differences of oscillators using parametric down conversion and four wave mixing processes as the intracavity amplification technique. The strong correlations involving the sidebands of the pump and converted fields gives the signatures of a rich dynamic of multipa…
We propose three criteria for identifying continuous variable entanglement between two many-particle systems with no restrictions on the quantum state of the local oscillators used in the measurements. Mistakenly asserting a coherent state for the local oscillator can lead to incorrectly identifying the presence of entanglement. We demonstrate this in simulations with 100 particles, and also find that large number fluctuations do not prevent the observation of entanglement. Our results are important for quantum information experiments with realistic Bose-Einstein condensates or in optics with …
The entanglement between a Pauli-like two-level system and a quantum harmonic oscillator enhanced by an interaction between them and a $δ$-pulse sequence is studied, with the decoherence due to their coupling with a Markovian bath. Without the Markovian bath, the entanglement is enhanced to maximum possible values. With the Markovian bath, the entanglement is enhanced up to some time and then dissipated, with the system in thermal equilibrium with the Markovian bath after a very long time. The time for achieving the maximum entanglement shows discontinuous jumps over the parameters of decohere…
Type-II Optical Parametric Oscillators are efficient sources of quadrature squeezed or polarization squeezed light, intensity correlated beams, and entangled light. We review here the different levels of quantum correlations and entanglement that are reached in this device, and present some applications.
We study the entanglement evolution between two harmonic oscillators having different free frequencies each leaking into an independent bath. We use an exact solution valid in the weak coupling limit and in the short time non-Markovian regime. The reservoirs are identical and characterized by an Ohmic spectral distribution with Lorents-Drude cut-off. This work is an extension of the case reported in [Phys. Rev. A 80, 062324 (2009)] where the oscillators have the same free frequency.
We study the effect of quantum entanglement maintained by virtual excitations in an ultrastrongly-coupled harmonic-oscillator system. Here, the quantum entanglement is caused by the counterrotating interaction terms and hence it is maintained by the virtual excitations. We obtain the analytical expression for the ground state of the system and analyze the relationship between the average excitation numbers and the ground-state entanglement. We also study the entanglement dynamics between the two oscillators in both the closed- and open-system cases. In the latter case, the quantum master equat…
We investigate a simple arrangement of coupled harmonic oscillators which brings out some interesting effects concerning creation of entanglement. It is well known that if each member in a linear chain of coupled harmonic oscillators is prepared in a ``classical state'', such as a pure coherent state or a mixed thermal state, no entanglement is created in the rotating wave approximation. On the other hand, if one of the oscillators is prepared in a nonclassical state (pure squeezed state, for instance), entanglement may be created between members of the chain. In the setup considered here, we …
A strategy for generating entanglement in two separated optomechanical oscillators is analysed, using entangled radiation produced from downconversion and stored in an initiating cavity. We show that the use of pulsed entanglement with optimally shaped temporal modes can efficiently transfer quantum entanglement into a mechanical mode, then remove it after a fixed waiting time for measurement. This protocol could provide new avenues to test for bounds on decoherence in massive systems that are spatially separated, as originally suggested by Wendell Furry [1] not long after the discussion by Ei…
In this paper, we propose a scheme for generating steady-state entanglement of remote micromechanical oscillators in unidirectionally-coupled cavities. For the system of two mechanical oscillators, we show that when two cavity modes in each cavity are driven at red- and blue-detuned sidebands, respectively, a stationary two-mode squeezed vacuum state of the two mechanical oscillators can be generated with the help of the cavity dissipation. The degree of squeezing is controllable by adjusting the relative strength of the pump lasers. Our calculations also show that the achieved mechanical enta…
The fundamental quantum dynamics of two interacting oscillator systems are studied in two different scenarios. In one case, both oscillators are assumed to be linear, whereas in the second case, one oscillator is linear and the other is a non-linear, angular-momentum oscillator; the second case is, of course, more complex in terms of energy transfer and dynamics. These two scenarios have been the subject of much interest over the years, especially in developing an understanding of modern concepts in quantum optics and quantum electronics. In this work, however, these two scenarios are utilized…
We observed continuous-variable entanglement between the bright beams emitted above threshold by an ultrastable optical parametric oscillator, classically phase-locked at a frequency difference of 161.8273240(5) MHz. The amplitude-difference squeezing is -3 dB and the phase-sum one is -1.35 dB. Besides proving entanglement in a new physical system, the phase-locked OPO, such unprecedented frequency-difference stability paves the way for transferring entanglement between different optical frequencies and densely implementing continuous-variable quantum information in the frequency domain.
We propose a scheme via three-level cascade atoms to entangle two optomechanical oscillators as well as two-mode fields. We show that two movable mirrors and two-mode fields can be entangled even for bad cavity limit. We also study entanglement of the output two-mode fields in frequency domain. The results show that the frequency of the mirror oscillation and the injected atomic coherence affect the output entanglement of the two-mode fields.
We show theoretically that multipartite entanglement is generated on a massive scale in the spectrum, or optical frequency comb, of a single optical parametric oscillator (OPO) emitting well above threshold. In this system, the quantum dynamics of the strongly depleted pump field are responsible for the onset of the entanglement by correlating the two-mode squeezed, bipartite-entangled pairs of OPO signal fields. (Such pairs are independent of one another in the undepleted, classical pump approximation.) We verify the multipartite nature of the entanglement by evaluating the van Loock-Furusawa…
We study entanglement dynamics in dispersive optomechanical systems consisting of two optical modes and a mechanical oscillator inside an optical cavity. The two optical modes interact with the mechanical oscillator, but not directly with each other. The appearance of optical entanglement witnesses non-classicality of the oscillator. We study the dependence of the entanglement dynamics with the optomechanical coupling, the mean photon number in the cavity and the oscillator temperature. An experimental realization with ultracold atomic ensembles is proposed.
We show that scalable multipartite entanglement among light fields may be generated by optical parametric oscillators (OPO). The tripartite entanglement existent among the three bright beams produced by a single OPO -- pump, signal, and idler -- is scalable to a system of many OPOs by pumping them in cascade with the same optical field. This latter serves as an entanglement distributor. The special case of two OPOs is studied, as it is shown that the resulting five bright beams share genuine multipartite entanglement. In addition, the structure of entanglement distribution among the fields can…
We introduce a special class of truncated Weyl-Heisenberg algebra and discuss the corresponding Hilbertian and analytical representations. Subsequently, we study the effect of a quantum network of beam splitting on coherent states of this nonlinear class of harmonic oscillators. We particularly focus on quantum networks involving one and two beam splitters and examine the degree of bipartite as well as tripartite entanglement using the linear entropy.
The effects of an intracavity photonic crystal in a multimode optical parametric oscillator are studied, with a special focus on quantum fluctuations. The capability to either stimulate or inhibit the spatial instability, lowering or increasing the parametric threshold, allows to control the intensity fluctuations and correlations. A significative quantum noise reduction and an increase of the range of squeezed quadratures are found above threshold where spatial Einstein-Podolsky-Rosen entanglement and inseparability are found.
We discuss a model comprised of a chain of three Kerr-like nonlinear oscillators pumped by two modes of external coherent field. We show that the system can be treated as nonlinear quantum scissors and behave as a three-qubit model. For such situation, different types of tripartite entangled states can be generated, even when damping effects are present in the system. Some amount of such entanglement can survive even in a long-time limit. The flow of bipartite entanglement between subsystems of the model and relations among first-order correlations, second-order correlations, and the entanglem…
We propose a system composed of a qubit interacting with a quartic (undriven) nonlinear oscillator (NLO) through a conditional displacement Hamiltonian. We show that even a modest nonlinear perturbation in the NLO potential can enhance and stabilize the quantum entanglement dynamically generated between the qubit and the NLO. In absence of the nonlinearity the entanglement between the qubit and the oscillator is known to periodically oscillate between 0 and 1, whereas the nonlinearity suppresses the dynamical decay of entanglement once it is generated. While the generation of entanglement is d…
We investigate the quantum entanglement dynamics of undriven anharmonic (nonlinear) oscillators with quartic potentials. We first consider the indirect interaction between two such nonlinear oscillators mediated by a third, linear oscillator and show that it leads to a time-varying entanglement of the oscillators, the entanglement being strongly influenced by the nonlinear oscillator dynamics. In the presence of dissipation, the role of nonlinearity is strongly manifested in the steady state dynamics of the indirectly coupled anharmonic oscillators. We further illustrate the effect of nonlinea…
In this work, we explore the dynamics of entanglement of an isolated quantum system consisting of two time-dependent, coupled harmonic oscillators. Through the use of a numerical method that relies on the estimation of the system's Wigner representation by a specific Gaussian function, we investigate the time evolution of the entanglement entropy after an instant quench in the inherent parameters of the system. Besides, from the comparison of the results obtained from the analytical expression for the time-dependent von Neumann entropy with the numerically computed entropy data, the effectiven…
A general and in principle exact approach for the continuous variable entanglement in a system of coupled harmonic oscillators in contact with a thermal bath is formulated. This allows a generalization to describe entanglement's existence between two sites in any system of this kind. It is employed a method of simultaneous diagonalization of two quadratic forms to obtain the uncoupled quantum Hamiltonian. Making use of the transformations that uncouple the system, the covariance matrix for two positions is constructed, and through the positive partial transpose criterion (PPT-criterion), the c…
We calculate the quantum correlations existing among the three output fields (pump, signal, and idler) of a triply resonant non-degenerate Optical Parametric Oscillator operating above threshold. By applying the standard criteria [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)], we show that strong tripartite continuous-variable entanglement is present in this well-known and simple system. Furthermore, since the entanglement is generated directly from a nonlinear process, the three entangled fields can have very different frequencies, opening the way for multicolored quantum info…
We propose a scheme to modulate the entanglement between two oscillators separated in space via the squeezing cavity field generated by the optical parametric amplifier instead of injecting the squeezing field directly with the assistance of Coulomb interaction. We show that the Coulomb interaction between the oscillators is the essential reason for the existence of entanglement. Due to the gain of the optical parametric amplifier and the phase of the pump driving the optical parametric amplifier can simultaneously modulate the squeezing cavity field, the radiation pressure interaction between…
We demonstrate, theoretically and experimentally, the generation of hexapartite modal entanglement by the optical parametric oscillator (OPO) operating above the oscillation threshold. We show that the OPO generates a rich structure of entanglement among sets of six optical sideband modes interacting through the non-linear crystal. The class of quantum states thus produced can be controlled by a single parameter, the power of the external laser that pumps the system. Our platform allows for the generation of massive entanglement among many optical modes with well defined but vastly different f…
We investigate new features, especially of entanglement dynamics, which arise in a quantum-classical hybrid. As a model, we study the coupling between two quantum mechanical two-level systems, i.e. two q-bits, and a classical harmonic oscillator. Their interaction is described by a hybrid coupling, in accordance with a recently developed quantum-classical hybrid theory. We discuss various situations in which entanglement of the q-bits does (not) evolve. Furthermore, we point out an experimental application in a hybrid cooling scheme and indicate topics for future study.
In this work we extend the discussion that was began in the Ref. [A. L. de Paula Jr., J. G. G. de Oliveira Jr., J. G. Peixoto de Faria, Dagoberto S. Freitas and M. C. Nemes, {\it Phys. Rev.} {\bf A89}, (2014) 022303] to deal with the dynamics of the concurrence of a many-body system. In that previous paper, the discussion was focused in the residual entanglement between the partitions of the system. The purpose of the present contribution is to shed some light on the dynamical properties of entanglement among the environment oscillators. We consider a system consisting of a harmonic oscillator…
I consider several interesting aspects of a new light source, a two-level atom, or N two-level atoms inside an Optical Parametric Oscillator. We find that in the weak driving limit, detection of a transmitted or fluorescent photon generates a highly entangled state of the atom and the cavity. This entanglement can be used with beamsplitters to create more complex quantum states and implement teleportation protocols. Also, one can store a single photon in the atoms, along the lines of recent slow and stopped light proposals and experiments.
We present the measurement of entanglement between twin beams generated with a doubly resonant optical parameter oscillator (OPO) based on four-wave mixing in hot $^{85}$Rb vapor above threshold. This is the first measurement of entanglement in an OPO with $χ^{(3)}$ media above threshold. We reconstruct the covariance matrix for several configurations and with a full picture of the four side band mode state, we study entanglement between all possible bi-partitions. We show a robust generation of entanglement with stronger generation for a specific pair of modes. For this system, we show that a…
Nonclassical beams in high order spatial modes have attracted much interest but they exhibit much less squeezing and entanglement than the fundamental spatial modes, limiting their applications. We experimentally demonstrate the relation between pump modes and entanglement of first-order HG modes (HG10 entangled states) in a type II OPO and show that the maximum entanglement of high order spatial modes can be obtained by optimizing the pump spatial mode. To our knowledge, this is the first time to report this. Utilizing the optimal pump mode, the HG10 mode threshold can be reached easily witho…
In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system a…
Multi-mode entanglement is investigated in the system composed of $N$ coupled identical harmonic oscillators interacting with a common environment. We treat the problem very general by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an $N$-mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wig…
We present a fully quantum mechanical treatment of the nondegenerate optical parametric oscillator both below and near threshold. This is a non-equilibrium quantum system with a critical point phase-transition, that is also known to exhibit strong yet easily observed squeezing and quantum entanglement. Our treatment makes use of the positive P-representation and goes beyond the usual linearized theory. We compare our analytical results with numerical simulations and find excellent agreement. We also carry out a detailed comparison of our results with those obtained from stochastic electrodynam…
We study the dynamics and redistribution of entanglement and coherence in three time-dependent coupled harmonic oscillators. We resolve the Schrödinger equation by using time-dependent Euler rotation together with a linear quench model to obtain the state of vacuum solution. Such state can be translated to the phase space picture to determine the Wigner distribution. We show that its Gaussian matrix $\mathbb{G}(t)$ can be used to directly cast the covariance matrix $σ(t)$. To quantify the mixedness and entanglement of the state one uses respectively linear and von Neumann entropies for three c…
The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schrödinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time-dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions we derive the analytical expressions for von Neumann and Rényi entropies. Making use of Wigner distribution function defined in phase space, we derive the time-dependence of position-momentum uncertainty relations. In order to show the dynamics of enta…
We explore the possibility of the controlled manipulation of the entanglement of two qubits with an external apparatus, the Rabi Hamiltonian. The novel results show that the initially entangled two qubits could have very high probability to stay unchanged by the coherent state of the photon with chosen parameters. If the inter-qubit coupling strength is negative, their entanglement could be kept for a much longer time by a relatively strong external quantum control with the suitable and novel choice of some parameters. Furthermore, their entanglement will not show any sudden death and revival.…
In this paper, we continue our investigation on controlling the state of a quantum harmonic oscillator, by coupling it to a reservoir composed of a sequence of qubits. Specifically, we show that sending qubits separable from each other but initialised at different states in pairs can stabilise the oscillator at squeezed states. However, only if entanglement is allowed in the reservoir qubit can we stabilise the oscillator at a wider set of squeezed states. This thus provides a proof for the necessity of involving entanglement in the reservoir qubits input to the oscillator, as regard to the st…
Localized electronic and nuclear spin qubits in the solid state constitute a promising platform for storage and manipulation of quantum information, even at room temperature. However, the development of scalable systems requires the ability to entangle distant spins, which remains a challenge today. We propose and analyze an efficient, heralded scheme that employs a parity measurement in a decoherence free subspace to enable fast and robust entanglement generation between distant spin qubits mediated by a hot mechanical oscillator. We find that high-fidelity entanglement at cryogenic and even …
We study the evolution of entanglement of a pair of coupled, non-resonant harmonic oscillators in contact with an environment. For both the cases of a common bath and of two separate baths for each of the oscillators, a full master equation is provided without rotating wave approximation. This allows us to characterize the entanglement dynamics as a function of the diversity between the oscillators frequencies and their mutual coupling. Also the correlation between the occupation numbers is considered to explore the degree of quantumness of the system. The singular effect of the resonance cond…
Remote entanglement of massive oscillators via wire-mediated Coulomb interaction
Technical University of Vienna（维也纳工业大学）；Institute for Theoretical Physics and Vienna Center for Quantum Science and Technology, Technical University of Vienna（维也纳大学理论物理研究所与维也纳量子科学与技术中心）；Institute of Science and Technology Austria（奥地利科学技术研究所）
We investigate entanglement in the above-threshold Optical Parametric Oscillator, both theoretically and experimentally, and discuss its potential applications to quantum information. The fluctuations measured in the subtraction of signal and idler amplitude quadratures are $Δ^2 \hat p_-=0.50(1)$, or $-3.01(9)$ dB, and in the sum of phase quadratures are $Δ^2 \hatq_+=0.73(1)$, or $-1.37(6)$ dB. A detailed experimental study of the noise behavior as a function of pump power is presented, and discrepancies with theory are discussed.
We address the presence of non-distillable (bound) entanglement in natural many-body systems. In particular, we consider standard harmonic and spin-1/2 chains, at thermal equilibrium and characterized by few interaction parameters. The existence of bound entanglement is addressed by calculating explicitly the negativity of entanglement for different partitions. This allows to individuate a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We discuss how the appearance of bound entanglement can be linked t…
We show that the optomechanical coupling between an optical cavity mode and the two movable cavity end mirrors is able to entangle two different macroscopic oscillation modes of the mirrors. This continuous variable entanglement is maintained by the light bouncing between the mirrors and is robust against thermal noise. In fact, it could be experimentally demonstrated using present technology.
Laser cooled ions trapped in a linear Paul trap are long-standing ideal candidates for realizing quantum simulation, especially of many-body systems. The properties that contribute to this also provide the opportunity to demonstrate unexpected quantum phenomena in few-body systems. A pair of ions interacting in such traps exchange vibrational quanta through the Coulomb interaction. This linear interaction can be anharmonically modulated by an elementary coupling to the internal two-level structure of one of the ions. Driven by thermal energy in the passively coupled oscillators, which are them…
We derive explicitly the thermal state of the two coupled harmonic oscillator system when the spring and coupling constants are arbitrarily time-dependent. In particular, we focus on the case of sudden change of frequencies. In this case we compute purity function, Rényi and von Neumann entropies, and mutual information analytically and examine their temperature-dependence. We also discuss on the thermal entanglement phase transition by making use of the negativity-like quantity. Our calculation shows that the critical temperature $T_c$ increases with increasing the difference between the init…
This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In gener…
