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arXiv:math-ph/0212014 · Journal of Physics A Mathematical and Theoretical

A semiclassically entangled puzzle

Pedro de M Rios

Published Nov 21, 2007• 3 citations
arXivHTMLPDF

Abstract

For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle.

DOI
10.1088/1751-8113/40/49/f02

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For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle.
Abstract. For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle. 1. introduction When Einstein, Podolsky and Rosen [6] explored the “strange ” behaviour of entangled states, their main goal was to set forth an argument concerning the ontology of quantum mechanics. Only quite later, did Bell [2] show that some of the discussion had a definite empirical bearing, namely, on matters related to the nonlocal character of quantum mechanics. After the empirical confirmation of this latter [1], emphasis has been placed on the “peaceful coexistence ” [15] of this kind of nonlocality and the partial order of spacetime events derived from the principle of relativity [5]. In terms of the Wigner [17] function W, the cartesian Weyl [16] representation of the density operator ˆρ as a real function on euclidean phase space, this question is addressed in the following way: let x1 and x2 denote the phase space points of two subsystems whose total system is in an entangled eigenstate of the hamiltonian ˆ H1 + ˆ H2, with corresponding Wigner function W0(x1, x2).
We study quantum van der Pol oscillator in which single-photon pumping competes with a nonlinear two-photon dissipative channel to produce a quantum limit cycle, and the question of how two such oscillators with non-identical intrinsic frequencies synchronize in the sense of phase locking when coupled dissipatively to one another. For a single oscillator, starting from a Lindblad description of the dynamics, we derive the exact Wigner-function evolution equation, and show that its Fokker--Planck approximation admits in the appropriate regime an equivalent classical Itô stochastic description that reproduces remarkably well the full Lindblad dynamics. Notably, the stochastic approach provides an efficient means of capturing system dynamics in the semiclassical regime, where a large number of energy eigenstates contribute and direct numerical simulations become computationally demanding. Restricting to the three lowest Fock states and using a parity symmetry, we obtain in closed form the stationary-state density matrix and its Wigner-function representation that displays pronounced nonclassical features. For two non-identical oscillators, we study the full Lindblad dynamics numerically and characterize synchronization through the stationary-state order parameter as a function of frequency mismatch and dissipative coupling. We find that increasing the effective nonlinear dissipative strength suppresses the synchronized region. Restricting the two oscillators to the three-level qutrit subspace, a parity symmetry reduces the stationary-state problem to an exact 414141-dimensional linear system, allowing to compute the synchronization order parameter. In this reduced description, results in the limit of large dissipation agree closely with those from the full dynamics, while exhibiting in the limit of small dissipation a double-peak synchronization structure accompanied by a phase reversal.
In this work, we study the quantum Rabi model, a simple two-state system coupled to a harmonic oscillator. With increasing interaction strength between the subsystems, the model undergoes a quantum phase transition (QPT), a phenomenon explained by a change in the shape of the potential in the semiclassical limit. We numerically cal- culate the Wigner quasiprobability distribution of the oscillator subsystem for various eigenstates and coupling strengths, demonstrate its nontrivial structural change as the QPT is crossed, and compare it with the Husimi Q function. Finally, we employ the Wigner negativity to quantify the nonclassicality of the states and observe that the neg- ativity is enhanced near the QPT. This work can contribute to the understanding of the Wigner distribution and Wigner negativity in a system relevant to quantum technology and computing.