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Physical Review Letters

Manifestation of Classical Bifurcation in the Spectrum of the Integrable Quantum Dimer

S. Aubry, Sergej Flach A, K. Kladko, Eckehard Olbrich

Published Mar 4, 1996• 88 citations
Publisher

Abstract

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.

DOI
10.1103/physrevlett.76.1607

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We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.
We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.
In this Brief Report the properties of both dynamical and static entanglement in the integrable quantum dimer are studied in terms of the reduced-density linear entropy and von Neumann entropy with various coupling parameters, total boson numbers, and initial states. The mean entanglement, which is defined to be averaged over time, is used to describe the influence of the classical separatrix on the behavior of entanglement. It is shown that the mean entanglement exhibits a maximum near the position of the corresponding classical separatrix energy and that the static entanglement of the state with the largest eigenvalue of the quantum spectrum displays a maximum near the bifurcation point. For weak coupling and larger total boson number the maximum entanglement state is exactly at the position of the classical separatrix and bifurcation. In strong coupling all initial states have nearly the same mean entanglement.
Abstract We consider the open two-site Bose–Hubbard dimer, a well-known quantum mechanical model that has been realised recently for photons in two coupled photonic crystal nanocavities. The system is described by a Lindblad master equation which, for large numbers of photons, gives rise to a limiting semiclassical model in the form of a four-dimensional vector field. From the situation where both sites trap the same amount of photons under symmetric pumping, one encounters a transition that involves symmetry breaking, the creation of periodic oscillations and multistability as the pump strength is increased. We show that the associated one-parameter bifurcation diagram of the semiclassical model captures the essence of statistical properties of computed quantum trajectories as the pump strength is increased. Even for small numbers of photons, the fingerprint of the semiclassical bifurcations can be recognised reliably in observables of quantum trajectories.
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Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on "quantumtorus" and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.
We study the connection between the phase behavior of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a nonstochastic ``tilted'' operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behavior of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling---supplemented by trajectory umbrella sampling---we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behavior of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalize to other problems.