arXiv subjects
Publications and source records attributed to Andrey R. Kolovsky.
This chapter gives an overview of transport problems where chaotic dynamics of the system plays a crucial role. We begin with single-particle transport problems and then come to conservative and then dissipative systems of identical particles, which follows the historical way of developing the theory of Quantum Chaos over the past 40 years. We also include brief descriptions of key laboratory experiments on the discussed transport problems.
We analyze the self-thermalization dynamics of the $M$-site Bose-Hubbard model in terms of the single-particle density matrix that is calculated by using the pseudoclassical approach. It is shown that a weak inter-particle interaction, which suffices to convert the integrable system of non-interacting bosons into a chaotic system, has a negligible effect on the thermal density matrix given by the Bose-Einstein distribution. This opens the door for equilibration where the two coupled Bose-Hubbard systems, which are initially in different thermal states, relax to the same thermal state. When we couple these two subsystems by using a lattice of the length $L\ll M$, we numerically calculate the quasi-stationary current of Bose particles across the lattice and show that its magnitude is consistent with the solution of the master equation for the boundary driven $L$-site Bose-Hubbard model.
In the absence of decoherence the current of fermionic particles across a finite lattice connecting two reservoirs (leads) with different chemical potentials is known to be ballistic. It is also known that decoherence typically suppresses this ballistic current. However, if decoherence is weak, the change in the current may be undetectable. In this work we show that the effect of a weak decoherence can be amplified by orders of magnitude if the lattice has edge states.
We analyze quantum transport of charged fermionic particles in the tight-binding lattice connecting two particle reservoirs (the leads). If the lead chemical potentials are different they create an electric field which tilts the lattice. We study the effect of this tilt on quantum transport in the presence of weak relaxation/decoherence processes in the lattice. It is shown that the Landauer ballistic transport regime for a weak tilt (small chemical potential difference) changes to the diffusive Esaki-Tsu transport regime for a strong tilt (large chemical potential difference), where the critical tilt for this crossover is determined by the condition that the Wannier-Stark localization length coincides with the lattice length.
We study effects of relaxation/decoherence processes on quantum transport of non-interacting Fermi particles across the tight-binding chain, where we distinguish between relaxation processes in the contacts (external decoherence) and those in the chain (internal decoherence). It is argued that relaxation processes in the contacts can essentially modify the resonant transmission as compared to the Landauer theory. We also address quantum transport in disordered chains. It is shown that external decoherence reduces conductance fluctuations but does not alter the Anderson localization length. This is in strong contrast with the effect of internal decoherence which is found to suppress the Anderson localization.
We revisit the problem of two-terminal transport of non-interacting Fermi particles in a mesoscopic device. First, we generalize the problem by including into consideration relaxation processes in contacts (which are characterized by the contact self-thermalization rate $γ$) and then solve it by using the master equation approach. In the limit $γ\rightarrow0$ the obtained results are shown to reproduce those of the Landauer-Büttiker theory. Thus, the presented analysis proves analytical correspondence between the Landauer-Büttiker and master-equation approaches to quantum transport, -- the problem which resisted solution for decades.
We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the single particle density matric in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance.
We revisit the problem of quantum bi- and multi-stability by considering the dissipative Double Resonance Model. For a large driving frequency, this system has a simpler phase structure than the driven dissipative nonlinear oscillator -- the paradigm model for classical and quantum bistability. This allows us to obtain an analytical estimate for the lifetime of quantum limit cycles. On the other hand, for a small driving frequency, the system is much reacher than the nonlinear oscillator. This allows us to address a novel phenomenon of dissipation- and chaos-assisted tunneling between quantum limits cycles.
We analyze Josephson's oscillation of Bose particles in the open (dissipative) Bose-Hubbard dimer. First, we excite the dimer from the vacuum state into a state suitable for observing the oscillation by using a special protocol for external driving. Next, we switch off the driving and observe the oscillation. It is shown that the main mechanism for the decay of Josephson's oscillation is the dephasing due to fluctuating number of particles in open systems. An analytical estimate for the decay time is obtained.
We revisit the phenomenon of the resonant transmission of fermionic carriers through a quantum device connected to two contacts with different chemical potentials. We show that, besides the traditional in solid-state physics Landauer-Büttiker approach, this phenomenon can be also described by the non-Markovian master equation for the reduced density matrix of the fermions in the quantum device. We identify validity regions of both approaches in the system parameter space and argue that for large relaxation rates the accuracy of the latter approach greatly exceeds the accuracy of the former.
We revisit quantum dynamics of the damped and driven nonlinear oscillator. In the classical case this system has two stationary solutions (the limit cycles) in the certain parameter region, which is the origin of the celebrated bistability phenomenon. The quantum-classical correspondence for the oscillator dynamics is discussed in details.
We revisit the Born-Markov approximation for an open quantum system by considering a microscopic model of the bath, namely, the Bose-Hubbard chain in the parameter region where it is chaotic in the sense of Quantum Chaos. It is shown that strong ergodic properties of the bath justify all approximations required for deriving the Markovian master equation from the first principles.
We theoretically analyze the depletion dynamics of an ensemble of cold atoms in a quasi one-dimensional optical lattice where atoms in one of the lattice sites are subject to decay. Unlike the previous studies of this problem in R. Labouvie, {\em et. al}, Phys. Rev. Lett. {\bf 116}, 235302 (2016) we focus on the case where the system is brought to the chaotic regime, that crucially modifies the depletion dynamics as compared to the regular case. It is shown that depletion of the affected site results in gradual depletion of the neighbouring sites according to $t^{1/3}$ scaling law. We also show that by measuring occupations of the lattice sites one can extract important information on chaotic dynamics of the original conservative system.
We analyze quantum state of fermionic carriers in a transport channel attached to a particle reservoir. The analysis is done from the first principles by considering microscopic models of the reservoir and transport channel. In the case of infinite effective temperature of the reservoir we demonstrate a full agreement between the results of straightforward numerical simulations of the system dynamics and solution of the specified master equation on the single-particle density matrix of the carriers in the channel. This allows us to predict the quantum state of carriers in the case where transport channel connects two reservoirs with different chemical potentials.
We study depletion dynamics of an open system of weakly interacting fermions with two-body random interactions. In this model fermions are escaping from the high-energy one-particle orbitals, that mimics the evaporation process used in laboratory experiments with neutral atoms to cool them to ultra-low temperatures. It is shown that due to dynamical thermalization the system instantaneously adjusts to the new chemical potential and temperature, so that occupation numbers of the one-particle orbitals always obey the Fermi-Dirac distribution. In this way we are able to describe the evaporation process which leads to a significant cooling of particles remaining inside the system. We also briefly discuss the evaporation process in the SYK black hole model that corresponds to strongly interacting fermions.
We study the current of Bose particles between two reservoirs connected by a one-dimensional channel. We analyze the problem from first principles by considering a microscopic model of conductivity in the noninteracting limit. Equations for the transient and the stationary current are derived analytically. The asymptotic current has a form similar to the Landauer-Büttiker equation for electronic current in mesoscopic devices.
We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When the tilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change of the eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratory experiment by observing Bloch oscillations of cold atoms.
We analyze the spectrum and eigenstates of a quantum particle in a bipartite two-dimensional tight-binding dice network with short range hopping under the action of a dc bias. We find that the energy spectrum consists of a periodic repetition of one-dimensional energy band multiplets, with one member in the multiplet being strictly flat. The corresponding macroscopic degeneracy invokes eigenstates localized exponentially perpendicular to the dc field direction, and super-exponentially along the dc field direction. We also show that the band multiplet is characterized by a topological winding number (Zak phase), which changes abruptly if we vary the dc field strength. These changes are induced by gap closings between the flat and dispersive bands, and reflect the number of these closings.