We study the behavior of an open quantum system, with an $N$--dimensional space of states, whose density matrix evolves according to a non--unitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's and Harper's map are the two examples we consider). A non--unitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large $N$) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analitic tools. The existence of a regime where the entropy grows with a rate which is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.
In the pursuit of speculative new particles, forces, and dimensions with vanishingly small influence on normal matter, understanding the ultimate physical limits of experimental sensitivity is essential. Here, I show that quantum decoherence offers a window into otherwise inaccessible realms. There is a standard quantum limit for diffusion that restricts some entanglement-generating phenomena, like soft collisions with new particle species, from having appreciable classical influence on normal matter. Such phenomena are classically undetectable but can be revealed by the anomalous decoherence …
We study the decoherence process for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system--environment coupling strength). For longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is inv…
Some hypothetical particles are considered essentially undetectable because they are far too light and slow-moving to transfer appreciable energy or momentum to the normal matter that composes a detector. I propose instead directly detecting such feeble particles, like sub-MeV dark matter or even gravitons, through their uniquely distinguishable decoherent effects on quantum devices like matter interferometers. More generally, decoherence can reveal phenomena that have arbitrarily little classical influence on normal matter, giving new motivation for the pursuit of macroscopic superpositions.
We investigate the sensitivity of quantum systems that are chaotic in a classical limit, to small perturbations of their equations of motion. This sensitivity, originally studied in the context of defining quantum chaos, is relevant to decoherence in situations when the environment has a chaotic classical counterpart.
The effects of quantum and thermal corrections on the dynamics of a damped nonlinearly kicked harmonic oscillator are studied. This is done via the Quantum Langevin Equation formalism working on a truncated moment expansion of the density matrix of the system. We find that the type of bifurcations present in the system change upon quantization and that chaotic behavior appears for values of the nonlinear parameter that are far below the chaotic threshold for the classical model. Upon increase of temperature or Planck's constant, bifurcation points and chaotic thresholds are shifted towards low…
Although various pieces of indirect evidence about the nature of dark matter have been collected, its direct detection has eluded experimental searches despite extensive effort. If the mass of dark matter is below 1 MeV, it is essentially imperceptible to conventional detection methods because negligible energy is transferred to nuclei during collisions. Here I propose directly detecting dark matter through the quantum decoherence it causes rather than its classical effects such as recoil or ionization. I show that quantum spatial superpositions are sensitive to low-mass dark matter that is in…
We numerically investigate decoherence of a two-spin system (central system) by a bath of many spins 1/2. By carefully adjusting parameters, the dynamical regime of the bath has been varied from quantum chaos to regular, while all other dynamical characteristics have been kept practically intact. We explicitly demonstrate that for a many-body quantum bath, the onset of quantum chaos leads to significantly faster and stronger decoherence compared to an equivalent non-chaotic bath. Moreover, the non-diagonal elements of the system's density matrix decay differently for chaotic and non-chaotic ba…
Using the Calogero-Moser model and the Nakamura equations for a multi-partite quantum system, we prove an inequality between the mean bi-partite entanglement rate of change under the variation of a critical parameter and the level-curvature. This provides an upper bound for the rate of production or distraction of entanglement induced dynamically. We then investigate the dependence of the upper bound on the degree of chaos of the system, which in turn, through the inequality, gives a measure of the stability of the entangled state. Our analytical results are supported with extensive numerical …
Violation of correspondence principle may occur for very macroscopic byt isolated quantum systems on rather short timescales as illustrated by the case of Hyperion, the chaotically tumbling moon of Saturn, for which quantum and classical predictions are expected to diverge on a timescale of approximately 20 years. Motivated by Hyperion, we review salient features of ``quantum chaos'' and show that decoherence is the essential ingredient of the classical limit, as it enables one to solve the apparent paradox caused by the breakdown of the correspondence principle for classically chaotic systems…
Chaotic evolutions exhibit exponential sensitivity to initial conditions. This suggests that even very small perturbations resulting from weak coupling of a quantum chaotic environment to the position of a system whose state is a non-local superposition will lead to rapid decoherence. However, it is also known that quantum counterparts of classically chaotic systems lose exponential sensitivity to initial conditions, so this expectation of enhanced decoherence is by no means obvious. We analyze decoherence due to a "toy" quantum environment that is analytically solvable, yet displays the cruci…
We study the differences between the process of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models wich contain both regular and chaotic environments. In all cases the system of interest is a "quantum walker", i.e. a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment wich has a D dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) timescale t*, wich scales with the dimensionality of t…
We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system--environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.
In a recent Phys. Rev. E Rapid Communication, the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time, $τ$, for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: 1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called `logtime', $τ\propto - \ln (\hbar)$, differing only by a numerical coefficient; and 2) the limitation of the semiclassical approximation as expressed in the authors' Eq. (8) is not limited by their argument leading to…
We show that some classically chaotic quantum systems uncoupled from noisy environments may generate intrinsic decoherence with all its associated effects. In particular, we have observed time irreversibility and high sensitivity to small perturbations in the initial conditions in a quasiperiodic version of the kicked rotor. The existence of simple quantum systems with intrinsic decoherence clarifies the quantum--classical correspondence in chaotic systems.
We provide a path integral treatment of the decoherence process induced by a heat bath on a single particle whose dynamics is classically chaotic and show that the decoherence rate is given by the Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculated semiclassically within diagonal approximation, when the particle initial state is a single gaussian wave packet. The calculation is performed for weak dissipation and in the high temperature limit. This situation allows us to simplify the heat bath description to a single random potential. Although the dissipa…
We prove that any spectral sequence obeying a certain growth law is the quantum spectrum of an equivalence class of classically integrable non-linear oscillators. This implies that exceptions to the Berry-Tabor rule for the distribution of quantum energy gaps of classically integrable systems, are far more numerous than previously believed. In particular we show that for each finite dimension $k$, there are an infinite number of classically integrable $k$-dimensional non-linear oscillators whose quantum spectrum reproduces the imaginary part of zeros on the critical line of the Riemann zeta fu…
We study the effects of dissipation and decoherence induced on a harmonic oscillator by the coupling to a chaotic system with two degrees of freedom. Using the Feynman-Vernon approach and treating the chaotic system semiclassically we show that the effects of the low dimensional chaotic environment are in many ways similar to those produced by thermal baths. The classical correlation and response functions play important roles in both classical and quantum formulations. Our results are qualitatively similar to the high temperature regime of the Caldeira-Leggett model.
We study the decoherence of a one-particle system, whose classical correpondent is chaotic, when it evolves coupled to a weak quenched environment. This is done by analytical evaluation of the Loschmidt Echo, (i.e. the revival of a localized density excitation upon reversal of its time evolution), in presence of the perturbation. We predict an exponential decay for the Loschmidt Echo with a (decoherence) rate which is asymptotically given by the mean Lyapunov exponent of the classical system, and therefore independent of the perturbation strength, within a given range of strengths. Our results…
We discuss the behavior of fidelity for a classically chaotic quantum system in the metallic regime. We show the existence of a critical value of the perturbation below which the exponential decay of fidelity is determined by the width of the Breit-Wigner distribution and above which the quantum decay follows the classical one which is ruled by the Lyapunov exponent. The independence of the decay {\it rate} from the perturbation strength derives from the similarity of the quantum and classical relaxation process inside the Heisenberg time scale.
We describe the quantum mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs. We find that the time scale $τ$ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension $τ=\fr{7}{6}λ^{-1}\ln(A/\hbar)$, with $λ$ the Lyapunov exponent and $A$ a typical classical action.
We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave ne…
We show that quantum interference can be classically interpreted in terms of a phase invariant quantity, not unlike the Berry's phase. Under this interpretation, closed loops in time become fundamental quantum entities, and all quantum states become periodic. Decoherence is then seen to occur naturally as a consequence. This formalism, although counterintuitive, provides a useful way of assigning ``classical'' meaning to quantum probabilities.
A direct classical analog of quantum decoherence is introduced. Similarities and differences between decoherence dynamics examined quantum mechanically and classically are exposed via a second-order perturbative treatment and via a strong decoherence theory, showing a strong dependence on the nature of the system-environment coupling. For example, for the traditionally assumed linear coupling, the classical and quantum results are shown to be in exact agreement.
We propose a strategy to suppress decoherence of a solid-state qubit coupled to non-Markovian noises by attaching the qubit to a chaotic setup with the broad power distribution in particular in the high-frequency domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-frequency components of our control are generated by the chaotic setup driven by a low-frequency field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide…
The population decay due to a small opening in an otherwise closed cavity supporting chaotic classical dynamics displays a quantum correction on top of the classical exponential form, a pure manifestation of quantum coherence that acquires a universal form and can be explained by path interference. Being coherent, such enhancement is prone to decoherence effects due to the coupling of the system to an external environment. We study this interplay between incoherent and coherent quantum corrections to decay by evaluating, within a Caldeira-Legget scenario, off-diagonal contributions to the deco…
The following statements belonging to the folklore of the theory of environmental decoherence are shown to be incorrect: 1) linear coupling to harmonic oscillator bath is a universal model of decoherence, 2) chaotic environments are more efficient decoherers.
The effect of chaotic bath dynamics on the decoherence of a quantum system is examined for the vibrational degrees of freedom of a diatomic molecule in a realistic, constant temperature collisional bath. As an example, the specific case of I$_2$ in liquid xenon is examined as a function of temperature, and the results compared with an integrable xenon bath. A crossover in behavior is found: the integrable bath induces more decoherence at low bath temperatures than does the chaotic bath, whereas the opposite is the case at the higher bath temperatures. These results, verifying a conjecture due …
Using recent results in the field of quantum chaos we derive explicit expressions for the time scale of decoherence induced by the system-environment entanglement. For a generic system-environment interaction and for a generic quantum chaotic system as environment, conditions are derived for energy eigenstates to be preferred states in the weak coupling regime. A simple model is introduced to numerically confirm our predictions. The results presented here may also help understanding the dynamics of quantum entanglement generation in chaotic quantum systems.
The out-of-time-order correlator (OTOC), recently analyzed in several physical contexts, is studied for low-dimensional chaotic systems through semiclassical expansions and numerical simulations. The semiclassical expansion for the OTOC yields a leading-order contribution in $\hbar^2$ that is exponentially increasing with time within an intermediate, temperature-dependent, time-window. The growth-rate in such a regime is governed by the Lyapunov exponent of the underlying classical system and scales with the square-root of the temperature.
The dynamics of non-polar diatomic molecules interacting with a far-detuned narrow-band laser field, that only may drive rotational transitions, is studied. The rotation of the molecule is considered both classically and quantum mechanically, providing links to features known from the heavy symmetric top. In particular, quantum decoherence in the molecular rotation, being induced by spontaneous Raman processes, is addressed. It is shown how this decoherence modifies the rotational dynamics in phase space.
We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int delta(w-W(x)) dx, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of W(x), starting from a coherent state. We have shown that for a classically chaotic quantum counterpart the distribution P(w) in the semi-classical limit becomes a Gaussian distribution that is fully de…
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of th…
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).
The decay of the overlap between a wave packet evolved with a Hamiltonian H and the same state evolved with H}+$Σ$ serves as a measure of the decoherence time $τ_ϕ$. Recent experimental and analytical evidence on classically chaotic systems suggest that, under certain conditions, $τ_ϕ$ depends on H but not on $Σ$. By solving numerically a Hamiltonian model we find evidence of that property provided that the system shows a Wigner-Dyson spectrum (which defines quantum chaos) and the perturbation exceeds a crytical value defined by the parametric correlations of the spectra.
Close to a spacelike singularity, pure gravity and supergravity in four to eleven spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possib…
We analyze the interplay of chaos, entanglement and decoherence in a system of qubits whose collective behaviour is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement…
We investigate decoherence in quantum systems coupled via dephasing-type interactions to an arbitrary environment with chaotic underlying classical dynamics. The coherences of the reduced state of the central system written in the preferential energy eigenbasis are quantum Loschmidt echoes, which in the strong coupling regime are characterized at long times scales by fluctuations around a constant mean value. We show that due to the chaotic dynamics of the environment, the mean value and the width of the Loschmidt echo fluctuations are inversely proportional to the quantity we define as the ef…
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem we aim to distinguish effects of the two types of dynamics from those depending on the choice of the wave packet. To isolate the former we introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find at intermediate times a weak effect of spectral correlations that is reminiscent of the correlation hole.
This paper is motivated by the suggestion [W. Zurek, Physica Scripta, T76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become non-classical within 20 years, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as hbar^{2/3} for ch…
A popular model of decoherence based on the linear coupling to harmonic oscillator heat baths is analized and shown to be inappropriate in the regime where decoherence dominates over energy dissipation, called pure decoherence regime. The similar mechanism essentially related to the energy conservation implies that, on the contrary to the recent conjectures, chaotic environments can be less efficient decoherers than regular ones. Finally, the elastic scattering mechanism is advocated as the simplest source of pure decoherence.
The effect of introducing measuring devices in a ``quantum pinball'' system is shown to lead to a chaotic evolution for the particle position as defined in Bohm's approach to Quantum Mechanics.
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 ess…
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well-known that quantum coherence can obliterate many chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact many classical chaotic behavior. How effective decoherence works to sust…
It has been long recognized that the task of semiclassical evaluation of quantum spectra for the classically nonintegrable systems is fundamentally more complex than for the classically integrable ones. Below it is argued that the quantum spectra of the chaotic systems can differ among themselves by level of their complexity.
We study environmentally induced decoherence of an electromagnetic field in a homogeneous, linear, dielectric medium. We derive an independent oscillator model for such an environment, which is sufficiently realistic to encompass essentially all of linear physical optics. Applying the ``predictability sieve'' to the quantum field, and introducing the concept of a ``quantum halo'', we recover the familiar dichotomy between background field configurations and photon excitations around them. We are then able to explain why a typical linear environment for the electromagnetic field will effectivel…
In a recent paper [Nature 412, 712 (2001)], Zurek has argued that (1) time evolution typically causes chaotic quantum systems to generate structure that varies on the scale of phase-space volume elements of size $(\hbar^2/A)^d$, where A is a classical action characteristic of the state and d is the number of degrees of freedom, and that (2) this structure implies that a small change in a phase-space coordinate X by an amount $δX \sim \hbar X/A$ generically results in an orthogonal state. While we agree with (1), we argue that (2) is not correct if the number of degrees of freedom is small. Our…
We study the open system dynamics of a circuit QED model operating in the ultrastrong coupling regime. If the resonator is pumped periodically in time the underlying classical system is chaotic. Indeed, the periodically driven Jaynes-Cummings model in the Born-Oppenheimer approximation resembles a Duffing oscillator which in the classical limit is a well-known example of a chaotic system. Detection of the field quadrature of the output field acts as an effective position measurement of the oscillator. We address how such detection affects the quantum chaotic evolution in this bipartite system.…
Quantum systems in mixed states can be unentangled and yet still correlated in a way that is not possible for classical systems. These correlations can be quantified by the quantum discord and might provide a resource for certain mixed-state quantum information processing tasks. Here we report on the generation of discordant states of two trapped atomic ions via Markovian decoherence processes. While entanglement is strictly non-increasing under such operations, discord can be generated in various forms. Firstly we show that, starting from two classically correlated qubits, it is possible to g…
We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equa…
We study the dynamics of entanglement in spin gases. A spin gas consists of a (large) number of interacting particles whose random motion is described classically while their internal degrees of freedom are described quantum-mechanically. We determine the entanglement that occurs naturally in such systems for specific types of quantum interactions. At the same time, these systems provide microscopic models for non--Markovian decoherence: the interaction of a group of particles with other particles belonging to a background gas are treated exactly, and differences between collective and non--co…
The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in- depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the wi…
We study the decoherence of two coupled spins that interact with a chaotic spin-bath environment. It is shown that connectivity of spins in the bath is of crucial importance for the decoherence of the central system. The previously found phenomenon of two-step decoherence (Phys. Rev. Lett. {\bf 90}, 210401 (2003)) turns out to be typical for the bath with a slow enough dynamics or no dynamics. For a generic random system with chaotic dynamics a conventional exponential relaxation to the pointer states takes place. Our results confirm a conjecture of Paz and Zurek (Phys. Rev. Lett. {\bf 82}, 51…
To suppress decoherence of solid-state qubits which are coupled to the non-Markovian noises, we propose a strategy to couple the qubit with a chaotic device, of which the broad power distribution in the high-frequency domain can be used to freeze the noises just like the dynamical decoupling control (DDC) method. Compared with the DDC, high-frequency components can be generated by the chaotic device even driven by a low-frequency field and we do not need to optimize the control fields to generate complex control pulses. As an application to superconducting circuits, we find that various noises…
We study the dynamics of a quantum spin ensemble controlled by trains of ultrashort pulses. To model disturbances of the kicks, we consider that the spins are submitted to different kick trains which follow regular, random, stochastic or chaotic dynamical processes. These are described by dynamical systems on a torus. We study the quantum decoherence and the population relaxation of the spin ensemble induced by these classical dynamical processes disturbing the kick trains. For chaotic kick trains we show that the decoherence and the relaxation processes exhibit a signature of chaos directly r…
We study the ultimate limits to the decoherence rate associated with dephasing processes. Fluctuating chaotic quantum systems are shown to exhibit extreme decoherence, with a rate that scales exponentially with the particle number, thus exceeding the polynomial dependence of systems with fluctuating $k$-body interactions. Our findings suggest the use of quantum chaotic systems as a natural test-bed for spontaneous wave function collapse models. We further discuss the implications on the decoherence of AdS/CFT black holes resulting from the unitarity loss associated with energy dephasing.
Dephasing decoherence induced by interaction of one qubit with a quantum bath can be simulated classically by random unitary evolution without the need for a bath and this random unitary evolution is equivalent to the quantum case. For a general dephasing model and a single qubit system, we explicitly construct the noise functional and completely specify the random unitary evolution. To demonstrate the technique, we applied our results to three paradigmatic models: spin-boson, central spin, and quantum impurity. For multiple qubits, we identify a class of generalized quantum dephasing models t…
The entropy production rate for an open quantum system with a classically chaotic limit has been previously argued to be independent of $\hbar$ and $D$, the parameter denoting coupling to the environment, and to be equal to the sum of generalized Lyapunov exponents, with these results applying in the near-classical regime. We present results for a specific system going well beyond earlier work, considering how these dynamics are altered for the Duffing problem by changing $\hbar,D$ and show that the entropy dynamics have a transition from classical to quantum behavior that scales, at least for…
Heisenberg's principle$^1$ states that the product of uncertainties of position and momentum should be no less than Planck's constant $\hbar$. This is usually taken to imply that phase space structures associated with sub-Planck ($\ll \hbar$) 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' $A \gg \hbar$ develop spotty structure on scales corresponding to sub-Planck $a = \hbar^2 / A \ll \hbar$. Suc…
