The transition from classical to quantum behavior for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative size ofh is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effectiveh is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is non-monotonic inh. Open nonlinear quantum systems are critical in understanding the foundations of quantum behavior, particularly the transition from quantum to classical mechanics. For example, density matrix formulations have been used to argue that quantum systems decohere rapidly when the classical counterpart is chaotic, with the decoherence rate determined by the classical Lyapunov exponents of the system[1]. This applies to entanglement and fidelity issues as well [2,3,4], since decoherence amounts to entanglement with the environment.A powerful alternative way of studying open quantum systems is the Quantum State Diffusion (QSD) approach [5]. This approach enabled the resolution of an important paradox, namely that in the absence of a QSD-like formulation, classical chaos cannot be recovered from quantum mechanics, indicating that theh → 0 limit is singular. Early QSD work[6] studied the convergence towards classical trajectories for a chaotic system, considering quantum Poincaré sections of the quantities x and p . It showed that the classical chaotic attractor is recovered when the system parameters were such thath was small relative to the system's characteristic action. As the relativeh increased, the attractor disappeared gradually, suggesting a persistence of chaos into the quantum region, consistent with later, more quantitative analyses [8,9]. Related work[10] studied a quantum system that is being continuously weakly measured, which leads to similar equations as those for QSD [11]. This also showed that chaos is recovered in the classical limit, and that it persists, albeit reduced, substantially into the quantum regime. Another related study [12] of coupled Duffing oscillators, showed that quantum effects, specifically entanglement, persist in a quantum system even when the system is classical enough to be chaotic.While the quantum persistence of chaos is interesting, it is still consistent with the understanding that chaos is a classical phenomenon that is suppressed quantum mechanically. Do quantum effects always decrease chaos, however? A closed Hamiltonian quantum system studied within a gaussian wavepacket approximation [13] manifested chaos absent in its classical version. This has been understood to be an artifact of the approximation, since the full quantum system is not chaotic. Follow-up work with an open system [14] also manifested quantum chaos, but it is not clear if this was not due to the approximatio...
We study the largest Lyapunov exponents λ and dynamical complexity for an open quantum driven double-well oscillator, mapping its dependence on coupling to the environment Γ as well as effective Planck’s constant β2. We show that in general λ increases with effective Hilbert space size (as β decreases, or the system becomes larger and closer to the classical limit). However, if the classical limit is regular, there is always a quantum system with λ greater than the classical λ, with several examples where the quantum system is chaotic even though the classical system is regular. While the quantum chaotic attractors are generally of the same family as the classical attractors, we also find quantum attractors with no classical counterpart. Contrary to the standard wisdom, the correspondence limit can thus be the most difficult to achieve for certain classically chaotic systems. These phenomena occur in experimentally accessible regimes.
Kingsbury et al. Reply: The preceding Comment [1] by Finn, Jacobs, and Sundaram further investigating our system is indeed relevant. Their results, particularly the complete absence of chaos for the À ¼ 0:3 case, are somewhat surprising and inconsistent with our results.Our published results reported analysis on the inprinciple experimentally accessible time series. For À ¼ 0:125 Poincaré sections indicate a chaotic attractor at ¼ 0:01, which is altered but persists for ¼ 0:3 and disappears for ¼ 1. For À ¼ 0:3, Poincaré sections show no chaos at ¼ 0:01, an attractor for ¼ 0:3, which disap-
The quantized 0(1,2)/0(2) X Z 2 sigma model has no continuum limit in four dimensions. 11.A lattice formulation of the 0( 1,2 )/0( 2) X Z 2 sigma model is developed, based on the continuum theory presented in the preceding paper. Special attention is given to choosing a lattice action (the "geodesic" action) that is appropriate for fields having noncompact curved configuration spaces. A consistent continuum limit of the model exists only if the renormalized scale constant PR vanishes for some value of the bare scale constant p. The geodesic action has a special form that allows direct access to the small-P limit. In this limit half of the degrees of freedom can be integrated out exactly. The remaining degrees of freedom are those of a compact model having a P-independent action which is noteworthy in being unbounded from below yet yielding integrable averages. Both the exact action and the Pindependent action are used to obtain PR from Monte Carlo computations of field-field averages (twopoint functions) and current-current averages. Many consistency cross-checks are performed. It is found that there is no value of for which PR vanishes. This means that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because these fields have neither the geometry nor the symmetries of the original model we conclude that the 0( 1,2)/0(2)XZ2 model has no continuum limit.PACS number(s): 1 l.lO.Lm, 04.60. + n, ll.lO.Gh, 12.25. +e
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