Transient chaos in quantum and classical mechanics

Bogolubov's classical example of statistical relaxation in a many-dimensional linear oscillator is discussed. The relation of the discovered relaxation mechanism to quantum dynamics as well as to some new problems in classical mechanics is considered.     

Stable quantum computation of unstable classical chaos

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.     

Generic quantum ratchet accelerator with full classical chaos

A simple model of quantum ratchet transport that can generate unbounded linear acceleration of the quantum ratchet current is proposed, with the underlying classical dynamics fully chaotic. The results demonstrate that generic quantum ratchet transport can occur with any type of classical phase space structure. The quantum ratchet transport with full classical chaos is also shown to be very robust to noise due to the large linear acceleration afforded by the quantum dynamics. One possible experiment allowing observation of these predictions is suggested.     

Interaction between classical and quantum systems and the measurement of quantum observables

Quantum mechanics presumes classical measuring instruments with which they interact. The problem of measurement interaction between classical and quantum systems is posed and solved. The restriction to compatible measurements comes about naturally as the condition for the integrity of the classical system. A technical device is the perspective on classical mechanics as quantum mechanics with essentially hidden dynamical variables. Work supported in part by U.S. Atomic Energy Commission, ERDA.     

Emergence of chaos in quantum systems far from the classical limit

The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation--as all experimental systems must be--their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.     

The quantum measurement problem and selection of classical states

The problem of selection of preferred basis during passage from quantum to classical systems is treated with the help of a simple example of a 2-state system like the sugar molecule. A simple principle leading to this selection is stated and demonstrated in case of the chosen example. The principle, stated simply is that the preferred basis is the one in which the system environment interaction hamiltonian is diagonal. Talk given at the International Symposium on Theoretical Physics, Bangalore, November 1984.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Continuous quantum measurement with independent detector cross correlations

We investigate the advantages of using two independent, linear detectors for continuous quantum measurement. For single-shot measurement, the detection process may be quantum limited if the detectors are twins. For weak continuous measurement, cross correlations allow a violation of the Korotkov-Averin bound for the detector's signal-to-noise ratio. The joint weak measurement of noncommuting observables is also investigated, and we find the cross correlation changes sign as a function of frequency, reflecting a crossover from incoherent relaxation to coherent, out of phase oscillations. Our results are applied to a double quantum-dot charge qubit, simultaneously measured by two quantum point contacts.     

Interplay between classical and quantum signals: partial trace and measurement channels

We continue the study of similarities between quantum information theory and theory of classical Gaussian signals. The possibility of using quantum entropy for classical Gaussian signals was explored a long time ago. Recently we demonstrated that some basic quantum channels can be represented as linear transforms of classical Gaussian signals. Here we consider bipartite quantum systems and show that an important quantum channel given by the partial trace operation has a simple classical representation, namely, a coordinate projection of a classical “prequantum signal.” We also consider the classical signal realization of quantum channels corresponding to state transforms in the process of measurement. The latter induces a difficult interpretational problem — the output signal corresponding to one system depends on a measurement that has been done on the second system. This situation might be interpreted as a sign of quantum nonlocality, action at a distance. Although we do not exclude such a possibility, i.e., that, in complete accordance with Bell, the creation of a realistic prequantum model is impossible without action at a distance, we found another interpretation of this situation that is not related to quantum nonlocality.     

Relevance of classical chaos in quantum mechanics: The hydrogen atom in a monochromatic field

We present analytical and numerical results on the mechanism of excitation and ionization of hydrogen atoms under microwave fields. In particular we predict the existence of a critical value of the microwave field, the quantum delocalization border, above which the quantum packet delocalizes and strong excitation and ionization takes place. Below the quantum border, the packet is localized even though the corresponding classical system can be chaotic and obeys a diffusion equation.Our studies reveal some other unexpected new features of quantum dynamics which also could be observed in laboratory experiments and provides a quantum theory for subthreshold ionization.     

Continuous quantum measurement: inelastic tunneling and lack of current oscillations

We study the dynamics of a charge qubit, consisting of a single electron in a double well potential coupled to a point-contact (PC) electrometer, using the quantum trajectories formalism. Contrary to previous predictions, we show formally that, in the sub-Zeno limit, coherent oscillations in the detector output are suppressed, and the dynamics is dominated by inelastic processes in the PC. Furthermore, these reduce the detector efficiency and induce relaxation even when the source-drain bias is zero. This is of practical significance since it means the detector will act as a source of decoherence. Finally, we show that the sub-Zeno dynamics is divided into two regimes: low and high bias in which the PC current power spectra show markedly different behavior.     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

Quantum instantons and quantum chaos

We suggest a closed form expression for the path integral of quantum transition amplitudes to construct a quantum action. Based on this we propose rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential.