The famous Niels Bohr’s quantum–classical correspondence principle states that classical mechanics is a limiting case of the more general quantum mechanics. The quantum laws show up in two different, although not entirely independent ways: Discrete spectrum of finite motion; and Interference phenomena. The specific features of quantum dynamics of classically chaotic systems seem to be in striking contrast with those of genuine classical chaos. The plain two-dimensional areas with closed irregular borders called billiards are pet systems often used to illustrate characteristic features of the classical dynamical chaos. Response of an evolving in time quantum system to a weak external perturbation is of prime interest in the context of the problem of stability and reversibility of quantum motion. The chapter discusses the decoherence phenomenon in the electron transport through an open ballistic quantum dot as the problem is seen from the point of view of the general resonance-scattering theory.
The famous Niels Bohr’s quantum–classical correspondence principle states that classical mechanics is a limiting case of the more general quantum mechanics. The quantum laws show up in two different, although not entirely independent ways: Discrete spectrum of finite motion; and Interference phenomena. The specific features of quantum dynamics of classically chaotic systems seem to be in striking contrast with those of genuine classical chaos. The plain two-dimensional areas with closed irregular borders called billiards are pet systems often used to illustrate characteristic features of the classical dynamical chaos. Response of an evolving in time quantum system to a weak external perturbation is of prime interest in the context of the problem of stability and reversibility of quantum motion. The chapter discusses the decoherence phenomenon in the electron transport through an open ballistic quantum dot as the problem is seen from the point of view of the general resonance-scattering theory.
The famous Nils Bohr's quantum-classical correspondence principle states that the classical mechanics is a limiting case of the more general quantum mechanics. This implies that ``under certain conditions" quantum laws of motion become equivalent to classical laws. One of the conditions is fairly obvious: the corresponding classical action should be very large as compared with the Planck's constant. But this condition is not sufficient. Suppression of effects of quantum interference ("decoherence") is the phenomenon of primary importance for understanding of the Bohr's principle. Being, in essence, of quite general nature, this problem takes on special significance in the case of non-linear classically chaotic quantum systems. Whereas the rapid decay of phase correlations is an underlying feature of the classical dynamical chaos, the "quantum chaos" by itself is not capable of destroying the quantum phase coherence. Strictly speaking, any initially pure quantum state remains pure during an arbitrary long unitary evolution. Basically, formation of incoherent mixed states (decoherence) results only from the process of preparation of a mixed initial state or interaction with a noisy environment. Just the decoherence brigdes the quantum and classical worlds. Peculiarities of the time behavior of Peres fidelity, state purity, Shanon and von Neumann entropies are analyzed in detail. We demonstrate the ways the decoherence shows up in periodically driven systems that can be associated with Ramsey-type interferomentry experiments with ion traps. Finally, decoherence in ballistic electron quantum transport caused by interaction with a disordered environment is considered.
We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
The study of the quantum–classical correspondence has been focused on the quantum measurement problem. However, most of the discussion in the preceding chapters is motivated by a broader question: Why do we perceive our quantum Universe as classical? Therefore, emergence of the classical phase space and Newtonian dynamics from the quantum Hilbert space must be addressed. Chapter 6 starts by re-deriving decoherence rate for non-local superpositions using the Wigner representation of quantum states. We then discuss the circumstances that, in some situations, make classical points a useful idealization of the quantum states of many-body systems. This classical structure of phase space emerges along with the (at least approximately reversible) Newtonian equations of motion. Approximate reversibility is a non-trivial desideratum given that the quantum evolution of the corresponding open system is typically irreversible. We show when such approximately reversible evolution is possible. We also discuss quantum counterparts of classically chaotic systems and show that, as a consequence of decoherence, their evolution tends to be fundamentally irreversible: They produce entropy at the rate determined by the Lyapunov exponents that characterize classical chaos. Thus, quantum decoherence provides a rigorous rationale for the approximations that led to Boltzmann’s H-theorem.
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo).We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution.The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process.Open problems related to the simple operational characterization of the decoherence strength are discussed.