Macroscopic Thermal Entanglement Due to Radiation Pressure

Ferreira, Aires; Guerreiro, A.; Vedral, Vlatko

doi:10.1103/physrevlett.96.060407

Cited by 76 publications

(65 citation statements)

References 21 publications

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“…whereρ defined in (15) is the density operator for the effective total system (12). The relationship betweenρ r and ρ r is given by…”

Section: Exact Master Equationmentioning

confidence: 99%

“…From (12), it is easy to see that this evolution can be decomposed into two parts, a dissipative evolution of the center of mass system,…”

Section: B Density Matrixmentioning

confidence: 99%

“…Macroscopic quantum coherence phenomena (MQP) manifested in double slit experiments, micromechanical resonators, Bose-Einstein condensates, Josephson junction circuits, mesoscopic systems, or even mirrors (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) is a subject of both basic theoretical and practical application interest. Theoretically it focuses on issues at the intersection of two trunk lines of important inquires in physics: the relation between the microscopic and the macroscopic world on the one hand, and the relation between the quantum and the classical on the other.…”

Section: Introductionmentioning

confidence: 99%

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Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment

2008

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In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two-harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [Hu, Paz and Zhang, Phys. Rev. D 45, 2843 (1992)], the exact master equation with all its coefficients for this two harmonic oscillator model can be easily deduced from the known results of the single harmonic oscillator case. In the second part we give an influence functional treatment of this model and provide explicit expressions for the evolutionary operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show three applications of this master equation: on the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation, and a derivation of the uncertainty principle at finite temperature for a composite object, modeled by two interacting harmonic oscillators. The exact master equation for two, and its generalization to N , harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations and dissipation of mesoscopic objects towards the construction of a theoretical framework for macroscopic quantum phenomena.

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“…whereρ defined in (15) is the density operator for the effective total system (12). The relationship betweenρ r and ρ r is given by…”

Section: Exact Master Equationmentioning

confidence: 99%

“…From (12), it is easy to see that this evolution can be decomposed into two parts, a dissipative evolution of the center of mass system,…”

Section: B Density Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment

2008

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“…Recently, Ferreira et al raised a fundamental question: can entanglement and quantum behavior in physical systems survive at arbitrary high temperatures? [7] They found that the entanglement between a cavity mode and a movable mirror does occur for any finite temperature. This result sheds a new light on the question and help to understand macroscopic properties of solids.…”

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confidence: 99%

Thermal entanglement in ferrimagnetic chains

Wang

2006

Phys. Rev. A

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A formula to evaluate the entanglement in an one-dimensional ferrimagnetic system is derived. Based on the formula, we find that the thermal entanglement in a small size spin-1/2 and spins ferrimagnetic chain is rather robust against temperature, and the threshold temperature may be arbitrarily high when s is sufficiently large. This intriguing result answers unambiguously a fundamental question: "can entanglement and quantum behavior in physical systems survive at arbitrary high temperatures?" PACS numbers: 03.65.Ud, 75.10.Jm A physical system may exhibit entanglement at a finite temperature [1,2,3,4,5]. The thermal entanglement always vanishes above a threshold temperature for systems with finite Hilbert space dimension [6]. Recently, Ferreira et al. raised a fundamental question: can entanglement and quantum behavior in physical systems survive at arbitrary high temperatures? [7] They found that the entanglement between a cavity mode and a movable mirror does occur for any finite temperature. This result sheds a new light on the question and help to understand macroscopic properties of solids.In this report, we derive a formula to evaluate the entanglement in an one-dimensional ferrimagnetic system. Intriguingly, we find that the entanglement is rather robust against temperature in this kind of system, with the Hamiltonianwhere s n and S n are spin-1/2 and spin-s operators, respectively. The antiferromagnetic exchange interactions exist only between nearest neighbors, and they are of the same strength which is set to unity (J = 1). Physically, the system contains two kinds of spins, spin 1 2 and s, alternating on a ring (or a chain with the periodic boundary condition).Let us now study the entanglement of states of the system at thermal equilibrium described by the density operator ρ(T ) = exp(−βH)/Z, where β = 1/k B T , k B is the Boltzmann's constant, which is assume to be 1, and Z = Tr{exp(−βH)} is the partition function. The entanglement in the thermal state is referred to as the thermal entanglement.To study quantum entanglement in the ferrimagnetic system, we need a good entanglement measure. One possible way is to use the negativity [8] based on the partial transpose method [9]. In the cases of two spin halves and the (1/2,1) mixed spins, a positive partial transpose (PPT) (or the non-zero negativity) is necessary and sufficient for separability (entanglement). Although the present ferrimagnetic system is a kind of (1/2,s) system, fortunately, it was shown that due to the SU(2) symmetry in the model Hamiltonian (1), the non-zero negativity is still a necessary and sufficient condition for entanglement between a spin half and spin s [10]. This result allows us to exactly investigate entanglement features of our mixed spin systems.The negativity of a state ρ is defined aswhere µ i is the negative eigenvalue of ρ T2 , and T 2 denotes the partial transpose with respect to the second system. The negativity N is related to the trace norm of ρ T2 viawhere the trace norm of ρ T2 is equal to the sum of the abso...

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“…It has been demonstrated that a large radiation pressure can be generated in the cavity that in return may lead to entanglement between different components of the system. In particular, stationary entanglement has been predicted between the cavity mode and a vibrating mirror [5][6][7][8][9], between an atomic ensemble or a Bose-Einstein condensate located inside an optical cavity and the vibrating mirror of the cavity [10][11][12][13], between two vibrating mirrors of a ring cavity [14], between two dielectric membranes suspended inside a cavity [15], and between a membrane and a trapped atom both located inside a cavity [16][17][18]. Further studies have addressed interesting problems of entangling mechanical oscillators [19], entangling optical and microwave cavity modes [20], and the creation of a photon by a vibrating mirror [21].…”

Section: Introductionmentioning

confidence: 99%

First-order coherence versus entanglement in a nanomechanical cavity

Sun

Ficek

2012

Phys. Rev. A

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The coherence and correlation properties of effective bosonic modes of a nano-mechanical cavity composed of an oscillating mirror and containing an optical lattice of regularly trapped atoms are studied. The system is modelled as a three-mode system, two orthogonal polariton modes representing the coupled optical lattice and the cavity mode, and one mechanical mode representing the oscillating mirror. We examine separately the cases of two-mode and three-mode interactions which are distinguished by a suitable tuning of the mechanical mode to the polariton mode frequencies. In the two-mode case, we find that the occurrence of entanglement in the system is highly sensitive to the presence of the first-order coherence between the modes. In particular, the creation of the first-order coherence among the polariton and mechanical modes is achieved at the expense of entanglement between them. In the three-mode case, we show that no entanglement is created between the independent polariton modes if both modes are coupled to the mechanical mode by the parametric interaction. There is no entanglement between the polaritons even if the oscillating mirror is damped by a squeezed vacuum field. The interaction creates the first-order coherence between the polaritons and the degree of coherence can, in principle, be as large as unity. This demonstrates that the oscillating mirror can establish the first-order coherence between two independent thermal modes. A further analysis shows that two independent thermal modes can be made entangled in the system only when one of the modes is coupled to the intermediate mode by a parametric interaction and the other is coupled by a linear-mixing interaction.

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Macroscopic Thermal Entanglement Due to Radiation Pressure

Cited by 76 publications

References 21 publications

Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment

Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment

Thermal entanglement in ferrimagnetic chains

First-order coherence versus entanglement in a nanomechanical cavity

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