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Journal of Physics A Mathematical and General

Propagation of a quantum state in a continuum of coupled oscillators and applications to entanglement

Apostol Vourdas

Published Oct 26, 2005• 0 citations
Publisher

Abstract

The exponential Hilbert space approach is used to link quantum mechanics in a small system with d -dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H . In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes ( d → ∞ ) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.

DOI
10.1088/0305-4470/38/45/009

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The exponential Hilbert space approach is used to link quantum mechanics in a small system with d -dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H . In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes ( d → ∞ ) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.
A continuum of coupled oscillators is considered, described by a continuous tensor product of Hilbert spaces. The mode position Ux and the mode momentum Up are operators which act collectively on all oscillators. They obey equations of motion which are very similar to those of a harmonic oscillator. Q‐functionals are used to introduce entropic quantities that describe correlations among the oscillators. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement; and the speed with which the entanglement propagates.
A ddd-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in Zd{Z}_{d}Zd​ and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the ddd modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the ddd observers; and they also change the momentum with which the entanglement propagates.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
We apply the continuous variable approach to study entangled dynamics of coupled harmonic oscillators interacting with a thermal reservoir and to a deterministic creation of entanglement in an atomic ensemble located inside a high-Q ring cavity. In the case of harmonic oscillators, we show that a suitable unitary transformation of the position and momentum operators transforms the system to a set of independent harmonic oscillators with only one of them coupled to the reservoir. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form of smaller size matrices. This simple property allows to treat the problem to some extend analytically. We analyze the time evolution of an initial entanglement and find that the entanglement can persists for long times due to presence of constants of motion for the covariance matrix elements. In the case of an atomic ensemble located inside the cavity, the attention is focused on creation of one and two-mode continuous variable entangled states from the vacuum by applying laser pulses of a suitably adjusted amplitudes and phases. The pulses together with the cavity dissipation prepare the collective modes of the atomic ensemble in a desired entangled state.
The time evolution of a quantum particle’s product of uncertainties in position and momentum is calculated when it is coupled with an external source. We have used a simple toy model where the particle is subject to a harmonic potential and coupled with an equivalent harmonic oscillator via a linear term. It is found that the long-time-averaged product is an increasing function of the coupling strength. It diverges when one of the eigenmodes of the coupled system goes soft, with the singular term twice of that for the stationary state. Generally, there is a jump of finite size for this quantity when a small coupling is turned on, compared to the uncoupled case. Similar behaviors have also been found for the von Neumann entanglement entropy, which is calculated exactly using a covariance matrix formalism. We find that the mode-interference plays an important role in the main features of this work.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.