Classical small systems coupled to finite baths

Toppling Pencils-Macroscopic Randomness from Microscopic Fluctuations

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol'd-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

Quantum Chaos and Quantum Randomness-Paradigms of Entropy Production on the Smallest Scales

Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs' paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the "quantum death" of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.

Quantum Metrology beyond the Classical Limit under the Effect of Dephasing

Quantum sensors have the potential to outperform their classical counterparts. For classical sensing, the uncertainty of the estimation of the target fields scales inversely with the square root of the measurement time T. On the other hand, by using quantum resources, we can reduce this scaling of the uncertainty with time to 1/T. However, as quantum states are susceptible to dephasing, it has not been clear whether we can achieve sensitivities with a scaling of 1/T for a measurement time longer than the coherence time. Here, we propose a scheme that estimates the amplitude of globally applied fields with the uncertainty of 1/T for an arbitrary time scale under the effect of dephasing. We use one-way quantum-computing-based teleportation between qubits to prevent any increase in the correlation between the quantum state and its local environment from building up and have shown that such a teleportation protocol can suppress the local dephasing while the information from the target fields keeps growing. Our method has the potential to realize a quantum sensor with a sensitivity far beyond that of any classical sensor.

Correlations of correlations: Secondary autocorrelations in finite harmonic systems

The momentum or velocity autocorrelation function C(t) for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of C(t) ("a correlation of the correlation"), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation C(t) (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations show nonrandom long-time scale pattern, and higher-order correlations converge to the lowest normal mode.

Noncanonical statistics of a finite quantum system with non-negligible system-bath coupling

The canonical statistics describes the statistical properties of an open system by assuming its coupling with the heat bath is infinitesimal in comparison with the total energy in thermodynamic limit. In this paper, we generally derive a noncanonical density matrix for the open system with a finite coupling to the heat bath, which deforms the energy shell to effectively modify the conventional canonical way. The obtained noncanonical distribution reflects the back action of system on the bath and thus depicts the statistical correlations between two subsystems by the mutual information as a result of energy conservation.

Finite kicked environments and the fluctuation-dissipation relation

In this work we derive a generalized map for a system coupled to a kicked environment composed of a finite number N of uncoupled harmonic oscillators. Dissipation is introduced via the interaction between system and environment which is switched on and off simultaneously (kicks) at regular time intervals. It is shown that kicked environments naturally generate a non-Markovian rotated dynamics, describe more complicated system-environment couplings which involve position and momentum, and satisfy an unusual fluctuation-dissipation relation. As an example, the motion of a kicked Brownian particle is discussed.

Tsallis power laws and finite baths with negative heat capacity

It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for classical Hamiltonian systems. By using well-known fundamental statistical mechanics expressions, we rigorously show that Tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity, while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium. We comment on the implications of the finite bath for the recent attempts towards a q-generalized central limit theorem.

Quantum bath refrigeration towards absolute zero: challenging the unattainability principle

A minimal model of a quantum refrigerator, i.e., a periodically phase-flipped two-level system permanently coupled to a finite-capacity bath (cold bath) and an infinite heat dump (hot bath), is introduced and used to investigate the cooling of the cold bath towards absolute zero (T=0). Remarkably, the temperature scaling of the cold-bath cooling rate reveals that it does not vanish as T→0 for certain realistic quantized baths, e.g., phonons in strongly disordered media (fractons) or quantized spin waves in ferromagnets (magnons). This result challenges Nernst's third-law formulation known as the unattainability principle.

Responses to applied forces and the Jarzynski equality in classical oscillator systems coupled to finite baths: an exactly solvable nondissipative nonergodic model

Responses of small open oscillator systems to applied external forces have been studied with the use of an exactly solvable classical Caldeira-Leggett model in which a harmonic oscillator (system) is coupled to finite N-body oscillators (bath) with an identical frequency (ω(n) = ω(o) for n = 1 to N). We have derived exact expressions for positions, momenta, and energy of the system in nonequilibrium states and for work performed by applied forces. A detailed study has been made on an analytical method for canonical averages of physical quantities over the initial equilibrium state, which is much superior to numerical averages commonly adopted in simulations of small systems. The calculated energy of the system which is strongly coupled to a finite bath is fluctuating but nondissipative. It has been shown that the Jarzynski equality is valid in nondissipative nonergodic open oscillator systems regardless of the rate of applied ramp force.