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arXiv:1203.6755 · Physical Review A

Critical behavior in ultrastrong-coupled oscillators

Vivishek Sudhir, Marco G. Genoni, Jinhyoung Lee, M. S. Kim

Published Jul 16, 2012• 28 citations
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Abstract

We investigate the strong-coupling regime of a linear $x$-$x$ coupled harmonic-oscillator system by performing a direct diagonalization of the Hamiltonian. It is shown that the $x$-$x$ coupled Hamiltonian can be equivalently described by a Mach-Zehnder-type interferometer with a quadratic unitary operation in each of its arms. We show a sharp transition of the unitary operation from an elliptical phase rotator to an elliptical squeezer as the coupling gets stronger, leading to the continuous generation of entanglement, even for a significantly thermal state in the ultrastrong-coupled regime. It is also shown that this critical regime cannot be achieved by a classical Hookian coupling. Finally, the effect of a finite-temperature environment is analyzed, showing that entanglement can still be generated from a thermal state in the ultrastrong-coupled regime but is destroyed rapidly.

DOI
10.1103/physreva.86.012316

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We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong-coupling regime with additional single-mode squeezing on both oscillators, as well as higher-order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second-order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher-order interactions.
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In the present paper we study the entanglement properties of thermal (a.k.a.~\emph{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 and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.
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