The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein–Podolsky–Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.     

On a model for quantum friction III. Ergodic properties of the spin-boson system

We investigate the dynamics of a 2-level atom (or spin 1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared. From a more philosophical point of view our results show that, contrary to the conventional wisdom, quantum dynamics can be simpler at positive than at zero temperature.     

Hopping Transport and Stochastic Dynamics of Electrons in Plasma Layers

We investigate the stochastic dynamics and the hopping transfer of electrons embedded into two‐dimensional atomic layers. First we formulate the quantum statistics of general atom ‐ electron systems based on the tight‐binding approximation and express ‐ following linear response transport theory ‐ the quantum‐mechanical time correlation functions and the conductivity by means of equilibrium time correlation functions. Within the relaxation time approach an expression for the effective collision frequency is derived in Born approximation, which takes into account quantum effects and dynamic effects of the atom motion through the dynamic structure factor of the lattice and the quantum dynamics of the electrons. In the last part we derive Pauli equations for the stochastic electron dynamics including nonlinear excitations of the atomic subsystem. We carry out Monte Carlo simulations and show that mean square displacements of electrons and transport properties are in a moderate to high temperature regime strongly influenced by by soliton‐type excitations and demonstrate the existence of percolation effects (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The diffusion in the quantum Smoluchowski equation

A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit.     

Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime

The study of optomechanical systems has attracted much attention, most of which are concentrated in the physics in the smallamplitude regime. While in this article, we focus on optomechanics in the extremely-large-amplitude regime and consider both classical and quantum dynamics. Firstly, we study classical dynamics in a membrane-in-the-middle optomechanical system in which a partially reflecting and flexible membrane is suspended inside an optical cavity. We show that the membrane can present self-sustained oscillations with limit cycles in the shape of sawtooth-edged ellipses and exhibit dynamical multistability. Then, we study the dynamics of the quantum fluctuations around the classical orbits. By using the logarithmic negativity, we calculate the evolution of the quantum entanglement between the optical cavity mode and the membrane during the mechanical oscillation. We show that there is some synchronism between the classical dynamical process and the evolution of the quantum entanglement.     

Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons

In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such a state are. We confirm the relaxation hypothesis through an ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which have already been observed in the recent experiment on the nonequilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita et al., Nature (London) 440, 900 (2006)10.1038/nature04693].     

Dissipative dynamics of a quantum two-state system in presence of nonequilibrium quantum noise

We analyze the real-time dynamics of a quantum two-state system in the presence ofnonequilibrium quantum fluctuations. The latter are generated by a coupling of thetwo-state system to a single electronic level of a quantum dot which carries anonequilibrium tunneling current. We restrict to the sequential tunneling regime andcalculate the dynamics of the two-state system, of the dot population, and of thenonequilibrium charge current on the basis of a diagrammatic perturbative method valid fora weak tunneling coupling. We find a nontrivial dependence of the relaxation and dephasingrates of the two-state system due to the nonequilibrium fluctuations which is directlylinked to the structure of the unperturbed central system. In addition, aHeisenberg-Langevin-equation of motion allows us to calculate the correlation function ofthe nonequilibrium fluctuations. By this, we obtain a generalized nonequilibriumfluctuation relation which includes the equilibrium fluctuation-dissipation theorem in thelimit of zero transport voltage. A straightforward extension to the case with atime-periodic ac voltage is shown.     

Level Dynamics and Universality of Spectral Fluctuations

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas–Yukawa gas.     

Stability of Equilibria with a Condensate

A quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a quantum dot, which can trap finitely many Bosons, has multiple equilibria at fixed temperature. We extend the notion of return to equilibrium to systems possessing a multitude of equilibrium states and show that the above system returns to equilibrium in a weak coupling sense: any local perturbation of an equilibrium state converges in the long time limit to an asymptotic state. The latter is, modulo an error term, an equilibrium state which depends, in an explicit way, on the initial local perturbation. The error term vanishes in the small coupling limit.We deduce this stability result from properties of structure and regularity of eigenvectors of the Liouville operator, the generator of the dynamics. Among our technical results is a virial theorem for Liouville type operators which has new applications to systems with and without a condensate.Supported by a CRM-ISM postdoctoral fellowship and by McGill University     

Fundamental aspects of quantum Brownian motion

With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.     

The second law of thermodynamics under unitary evolution and external operations

The von Neumann entropy cannot represent the thermodynamic entropy of equilibrium pure states in isolated quantum systems. The diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, is a natural generalization of the von Neumann entropy and applicable to equilibrium pure states. We show that the diagonal entropy is consistent with the second law of thermodynamics upon arbitrary external unitary operations. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that quantum trajectories under unitary evolution are restricted by the Hamiltonian dynamics and that the external operation is performed without reference to the microscopic state of the system.     

Gauss Markov process of a quantum oscillator

We give a proper definition of a quantum Gauss process. From there we derive the generator (dissipative Liouville operator) of a Gauss Markov process for a quantum oscillator without using a microscopic model. Dissipative Liouville operators derived from microscopic models are recovered as special cases. The dynamics following from the generator is investigated by studying the relaxation of the first moments and equilibrium correlations.Work supported by Deutsche Forschungsgemeinschaft     

Geometric phase of a qubit driven by a phase noise laser under non-Markovian dynamics

Robustness of the geometric phase (GP) with respect to the environmental effects is a basic condition for an effective quantum computation. Here, we study quantitatively the GP of a two-level atom system driven by a phase noise laser under non-Markovian dynamics in terms of different parameters involved in the whole system. We find that with the change of the damping coupling, the GP is very sensitive to its properties exhibiting long collapse and revival phenomena, which play a significant role in enhancing the stabilization and control of the system dynamics. Moreover, we show that the GP can be considered as a tool for testing and characterizing the nature of the qubit–environment coupling. Due to the significance of how a system is quantum correlated with its environment in the construction of a scalable quantum computer, the entanglement dynamics between the qubit with its environment under external classical noise is evaluated and investigated during the time evolution.     

Correlation Dynamics of Three-Qubit System Under a Classical Dephasing Environment

By starting from the stochastic Hamiltonian of the three correlated spins and modeling their frequency fluctuations as caused by dephasing noisy environments described by Ornstein-Uhlenbeck (OU) processes, we study the dynamics of quantum correlations, including entanglement and quantum discord. Of course, in this article, we use two definitions for the quantum discord (global quantum discord and quantum dissension). We prepared initially our open system with the Greenberger-Horne-Zeilinger (GHZ) and W states and present the exact solutions for evolution dynamics of entanglement and quantum discord between three spins under both Markovian and non-Markovian regime of this classical noise. By comparison the dynamics of entanglement with that of quantum discord we find that entanglement can be more robust than quantum discord against this noise. It is shown that by considering non-Markovian extensions the survival time of correlations prolong. Also, we compare the results of two definitions of the quantum discord and show that the quantum dissension is equal to the global quantum discord for GHZ state, but they are unequal for the W state.     

Detecting Quantum Dissonance and Discord in Exact Dynamics of qubit Systems

In this paper, we evaluate the quantum and classical correlations in exact dynamics of qubit systems interacting with a common dephasing environment. We show the existence of a sharp transition between the classical and quantum loss of correlations during the time evolution. We show that it is possible to exploit a large class of initial states in different tasks of quantum information and processing without any perturbation of the correlations from the environment noisy for large time intervals. On the other hand, we include the dynamics of a new kind of correlation so-called quantum dissonance, which contains the rest of the nonclassical correlations. We show that the quantum dissonance can be considered as an indicator to expect the behavior of the dynamics of classical and quantum correlations in composite open quantum systems.     

Quench Dynamics of Neutral Atoms in Out-Equilibrium One-Dimensional Optical Lattices

A quantum simulator is proposed for nucleation and growth dynamics using an out-of equilibrium optical lattice. We calculate the density of neutral atoms in the lattice and we establish the connection with the KolmogorovMehl-Johnson-Avrami model. Here we show that an Avrami equation can describe most of the evolution in time of the population growth in the lattice, coherence between neutral atoms leads a complex growth rate.     

Quench Dynamics of Neutral Atoms in Out-Equilibrium One-Dimensional Optical Lattices

A quantum simulator is proposed for nucleation and growth dynamics using an out-of equilibrium optical lattice. We calculate the density of neutral atoms in the lattice and we establish the connection with the KolmogorovMehl-Johnson-Avrami model. Here we show that an Avrami equation can describe most of the evolution in time of the population growth in the lattice, coherence between neutral atoms leads a complex growth rate.