Sub-Planck structure in phase space and its relevance for quantum decoherence

Abstract

Heisenberg's principle 1 states that the product of uncertainties of position and momentum should be no less than Planck's constanth. This is usually taken to imply that phase space structures associated with sub-Planck (≪h) scales do not exist, or, at the very least, that they do not matter. I show that this deeply ingrained prejudice is false: Non-local "Schrödinger cat" states of quantum systems confined to phase space volume characterized by 'the classical action' A ≫h develop spotty structure on scales corresponding to sub-Planck a =h 2 /A ≪h. Such structures arise especially quickly in quantum versions of classically chaotic systems (such as gases, modelled by chaotic scattering of molecules), that are driven into nonlocal Schrödinger cat -like superpositions by the quantum manifestations of the exponential sensitivity to perturbations 2 . Most importantly, these sub-Planck scales are physically significant: a determines sensitivity of a quantum system (or of a quantum environment) to perturbations. Therefore sub-Planck a controls the effectiveness of decoherence and einselection caused by the environment 3−8 . It may also be relevant in setting limits on sensitivity of Schrödinger cats used as detectors.One of the characteristic features of classical chaos is the evolution of the small scale structure in phase space probability distributions: As a consequence of the exponential sensitivity to initial conditions, an initially regular "patch" in phase space with a characteristic size ∆ will, after a time t, develop structure on the scales:where Λ is, in effect, the Lyapunov exponent 9,10 . This is a consequence of the exponential stretching and of the conservation of phase space volume in reversible evolutions. What happens with small scale structure in quantum versions of classically chaotic systems? We shall investigate this question using Wigner function W (x, p) -the closest quantum analogue of the classical phase space distribution 11 :Above, ρ is the density operator. When the quantum state is a pure |ψ >, the integrand becomes exp(ipy/h)ψ * (x + y/2)ψ(x − y/2). In a quantum system the smallest spatial structures in the wave function ψ(x) will be set by the highest phase space frequencies available. Given a finite energy, ψ(x), and, hence, W (x, p) cannot reach scales as small as the exponential squeezing of Eq. (1) would eventually imply. Thus, in a quantum system, there must be a scale below which structure should not appear. That some limit must exist was apparent already some time ago: Berry and Balazs (who, as we shall see below in connection with Eq. (13)) have an enviable record of correctly anticipating various aspects of quantum chaos) have conjectured 12,13 that structure saturates on scales given by Planck constant. If this were the case, W should be smooth on scales small compared to 2πh.I shall show that copious structure appears in the Wigner function W on much smaller sub-Planck scales associated with the action of the order of:and explain its origin. Most importantly, I shall demonstra...

Sub-Planck-scale structures in the Pöschl-Teller potential and their sensitivity to perturbations

“…As shown by Ref. [1], the latter produces sub-Planck scale structures in the phase space Wigner distribution. The behavior at fractional revival times can be derived in a straightforward way from Eq.…”

“…2(b)) and six interference terms, among which two diagonal partners (W 13 and W 24 ) overlap at the center of the phase space and generate a smaller chess-board interference pattern. This structure is the signature of sub-Planck scale structures [1,10] for a compass-like state. The central interference pattern can also be seen as the superposition of interferences of two orthogonally situated cat states, and it is formed by small "tiles" much smaller than the individual CS peaks.…”

“…Non-local superposition states show interference and the presence of phase space regions where W (x, p) is negative is an unambiguous signature of quantum behavior. The oscillatory structures resulting from quantum interference can have a dimension much smaller than the Planck's constant , and these sub-Planck scale structures were first found by Zurek in quantum chaotic systems [1]. Sub-Planck scale structures usually appear as alternate small 'tiles' of maxima and minima.…”

“…The phase space area of these 'tiles' are given by 2 divided by the effective phase-space area A globally occupied by the state, which can be significantly larger than . These structures are very sensitive to environmental decoherence [1,2,3,4,5,6] and may have important application in Heisenberg-limited measurements and quantum parameter estimation [7,8]. A classical analogue of sub-Planck scale structures, regarded as sub-Fourier sensitivity, has been studied in Ref.…”

Mesoscopic quantum superposition of the generalized cat state: A diffraction limit

“…Cat states and their generalizations are known to achieve Heisenberg limited sensitivity in estimation of parameters like coordinate/momentum displacements and phase space rotations [1]. A criterion to distinguish quantum states without classical counterparts, from those without the same, are studied in [2,3].…”