Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.
Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, nonunitary evolution, and thermalization, but it still lacks a precise definition. The Grobe-Haake-Sommers conjecture, which links Ginibre level repulsion to classical chaotic dynamics, was recently shown to fail [Phys. Rev. Lett. 133, 240404 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.240404]. We properly restore the quantum-classical correspondence through a dynamical approach based on the von Neumann entropy (VNE) and out-of-time-order correlators (OTOCs), which reveal signatures of chaos beyond spectral statistics. Focusing on the open anisotropic Dicke model, we identify two distinct regimes: transient chaos, marked by rapid early-time growth of VNE and OTOCs followed by low saturation values, and steady-state chaos, characterized by high long-time values. We introduce a random matrix toy model and show that Ginibre spectral statistics signals short-time chaos rather than steady-state chaos. Our results establish VNE dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales.
We investigate the connection between classical and quantum chaos in a dissipative system of two coupled collective spins. Although the classical dynamics possesses a stable fixed-point attractor, we find pronounced signatures of chaos in the transient dynamics preceding relaxation. In the quantum model, such transient chaos is accompanied by a transition to random-matrix statistics in the relevant part of the Liouvillian spectrum and by a qualitative increase in the number of Liouvillian modes contributing to expectation values of physical observables. The transition occurs before the bifurcation at which the stable fixed point disappears and a chaotic attractor emerges. Our results show that Liouvillian quantum chaos can be associated with transient classical chaos, rather than requiring a chaotic asymptotic attractor, and provide a refinement of the Grobe-Haake-Sommers conjecture.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.
How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.