Measuring Loschmidt echo via Floquet engineering in superconducting circuits

The Loschmidt echo is a useful diagnostic for the perfection of quantum time-reversal process and the sensitivity of quantum evolution to small perturbations. The main challenge for measuring the Loschmidt echo is the time reversal of a quantum evolution. In this work, we demonstrate the measurement of the Loschmidt echo in a superconducting 10-qubit system using Floquet engineering and discuss the imperfection of an initial Bell-state recovery arising from the next-nearest-neighbor (NNN) coupling present in the qubit device. Our results show that the Loschmidt echo is very sensitive to small perturbations during quantum-state evolution, in contrast to the quantities like qubit population that is often considered in the time-reversal experiment. These properties may be employed for the investigation of multiqubit system concerning many-body decoherence and entanglement, etc., especially when devices with reduced or vanishing NNN coupling are used.     

Decoherence and the Loschmidt echo

Decoherence causes entropy increase that can be quantified using, e.g., the purity sigma=Trrho(2). When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo M(t). It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of sigma(t) and the average Mmacr;(t). In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of sigma and Mmacr; has a regime where it is dominated by the Lyapunov exponents.     

Classical decays in decoherent quantum maps

The linear entropy and the Loschmidt echo have proved to be of interest recently in the context of quantum information and of the quantum to classical transitions. We study the asymptotic long-time behavior of these quantities for open quantum maps and relate the decays to the eigenvalues of a coarse-grained superoperator. In specific ranges of coarse graining, and for chaotic maps, these decay rates are given by the Ruelle-Pollicott resonances of the classical map.     

Quantum dynamics of Gaudin magnets

Quantum dynamics of many-body systems is a fascinating and significant subject for both theory and experiment. The question of how an isolated many-body system evolves to its steady state after a sudden perturbation or quench still remains challenging. In this paper, using the Bethe ansatz wave function, we study the quantum dynamics of an inhomogeneous Gaudin magnet. We derive explicit analytical expressions for various local dynamic quantities with an arbitrary number of flipped bath spins, such as: the spin distribution function, the spin–spin correlation function, and the Loschmidt echo. We also numerically study the relaxation behavior of these dynamic properties, gaining considerable insight into coherence and entanglement between the central spin and the bath. In particular, we find that the spin–spin correlations relax to their steady value via a nearly logarithmic scaling, whereas the Loschmidt echo shows an exponential relaxation to its steady value. Our results advance the understanding of relaxation dynamics and quantum correlations of long-range interacting models of the Gaudin type.     

Sudden transition and sudden change from open spin environments

We investigate the necessary conditions for the existence of sudden transition or sudden change phenomenon for appropriate initial states under dephasing. As illustrative examples, we study the behaviors of quantum correlation dynamics of two noninteracting qubits in independent and common open spin environments, respectively. For the independent environments case, we find that the quantum correlation dynamics is closely related to the Loschmidt echo and the dynamics exhibits a sudden transition from classical to quantum correlation decay. It is also shown that the sudden change phenomenon may occur for the common environment case and stationary quantum discord is found at the high temperature region of the environment. Finally, we investigate the quantum criticality of the open spin environment by exploring the probability distribution of the Loschmidt echo and the scaling transformation behavior of quantum discord, respectively.     

New mechanism for non-trivial intra-molecular vibrational dynamics

We investigate the time evolution process of one selected (initially prepared by optical pumping) vibrational molecular state S, coupled to all other intra-molecular vibrational states R of the same molecule, and also to its environment Q. Molecular states forming the first reservoir R are characterized by a discrete dense spectrum, whereas the environment reservoir Q states form a continuous spectrum. Assuming the equidistant reservoir R states we find the exact analytical solution of the quantum dynamic equations. S-Q and R-Q couplings yield to spontaneous decay of the S and R states, whereas S-R exchange leads to recurrence cycles and Loschmidt echo at frequencies of S-R transitions and double resonances at the interlevel reservoir R transitions. Due to these couplings the system S time evolution is not reduced to a simple exponential relaxation. We predict various regimes of the system S dynamics, ranging from exponential decay to irregular damped oscillations. Namely, we show that there are possible four dynamic regimes of the evolution: (i) independent of the environment Q exponential decay suppressing backward R - S transitions, (ii) Loschmidt echo regime, (iii) incoherent dynamics with multicomponent Loschmidt echo, when the system state is exchanged its energy with many states of the reservoir, (iv) cycle mixing regime, when long time system dynamics looks as a random-like. We suggest applications of our results for interpretation of femtosecond vibration spectra of large molecules and nano-systems.     

Relatively-long time decay of Loschmidt echo of a Bose-Einstein condensate in a double-well potential

We study the long-time decay of quantum Loschmidt echo (LE) of a Bose-Einstein condensate (BEC) in a double-well potential. In the tunneling and self-trapping phases of the BEC, the LE has exponential and Gaussian decays, respectively, for relatively-long times. In the crossover region, the LE behaves differently from both the tunneling and the self-trapping phases. These results indicate that relatively-long time decay of the LE is suitable for characterizing the dynamical phase transition of the BEC.     

Statistics of the work done on a quantum critical system by quenching a control parameter

We study the statistics of the work done on a quantum critical system by quenching a control parameter in the Hamiltonian. We elucidate the relation between the probability distribution of the work and the Loschmidt echo, a quantity emerging usually in the context of dephasing. Using this connection we characterize the statistics of the work done on a quantum Ising chain by quenching locally or globally the transverse field. We show that for local quenches starting at criticality the probability distribution of the work displays an interesting edge singularity.     

Interaction quench in the Hubbard model

Motivated by recent experiments in ultracold atomic gases that explore the nonequilibrium dynamics of interacting quantum many-body systems, we investigate the opposite limit of Landau's Fermi-liquid paradigm: We study a Hubbard model with a sudden interaction quench, that is, the interaction is switched on at time t=0. Using the flow equation method, we are able to study the real time dynamics for weak interaction U in a systematic expansion and find three clearly separated time regimes: (i) An initial buildup of correlations where the quasiparticles are formed. (ii) An intermediate quasi-steady regime resembling a zero temperature Fermi liquid with a nonequilibrium quasiparticle distribution function. (iii) The long-time limit described by a quantum Boltzmann equation leading to thermalization of the momentum distribution function with a temperature T proportional, variantU.     

Loschmidt echo in a system of interacting electrons

We study the Loschmidt echo for a system of electrons interacting through mean-field Coulomb forces. The electron gas is modeled by a self-consistent set of hydrodynamic equations. It is observed that the quantum fidelity drops abruptly after a time that is proportional to the logarithm of the perturbation amplitude. The fidelity drop is related to the breakdown of the symmetry properties of the wave function.     

Ehrenfest ‘phase-space’ trajectories and quantum chaos in He atom under strong,oscillating magnetic fields: an application of time-dependent quantum fluid density functional theory (TDQFDFT)

Using a dynamical signature proposed earlier from our laboratory, quantum chaos in He atom interacting with strong, oscillating magnetic fields has been studied through a comparison between the nonlinear divergence of two neighbouring Ehrenfest ‘phase-space’ (EPS) trajectories differing slightly in initial conditions and the Loschmidt echo. The dynamical EPS signature can detect quantum chaos independently of the Loschmidt echo and in agreement with the latter, even for low-lying states, in the same spirit as that of classical chaos. This time-dependent signature extends the concept of quantum chaos to systems which have no classical counterparts and brings the concept of quantum chaos closer to that of classical chaos.     

Faster than Lyapunov decays of the classical Loschmidt echo

We show that in the classical interaction picture the echo dynamics, namely, the composition of perturbed forward and unperturbed backward Hamiltonian evolution, can be treated as a time-dependent Hamiltonian system. For strongly chaotic (Anosov) systems we derive a cascade of exponential decays for the classical Loschmidt echo, starting with the leading Lyapunov exponent, followed by a sum of the two largest exponents, etc. In the loxodromic case a decay starts with the rate given as twice the largest Lyapunov exponent. For a class of perturbations of symplectic maps the echo dynamics exhibits a drift resulting in a superexponential decay of the Loschmidt echo.     

Decay of Loschmidt echo enhanced by quantum criticality

We study the transition of a quantum system from a pure state to a mixed one, which is induced by the quantum criticality of the surrounding system E coupled to it. To characterize this transition quantitatively, we carefully examine the behavior of the Loschmidt echo (LE) of E modeled as an Ising model in a transverse field, which behaves as a measuring apparatus in quantum measurement. It is found that the quantum critical behavior of E strongly affects its capability of enhancing the decay of LE: near the critical value of the transverse field entailing the happening of quantum phase transition, the off-diagonal elements of the reduced density matrix describing S vanish sharply.     

Loschmidt Echo and Fidelity Decay Near an Exceptional Point

Non‐Hermitian classical and open quantum systems near an exceptional point (EP) are known to undergo strong deviations in their dynamical behavior under small perturbations or slow cycling of parameters as compared to Hermitian systems. Such a strong sensitivity is at the heart of many interesting phenomena and applications, such as the asymmetric breakdown of the adiabatic theorem, enhanced sensing, non‐Hermitian dynamical quantum phase transitions, and photonic catastrophe. Like for Hermitian systems, the sensitivity to perturbations on the dynamical evolution can be captured by Loschmidt echo and fidelity after imperfect time reversal or quench dynamics. Here, a rather counterintuitive phenomenon in certain non‐Hermitian systems near an EP is disclosed, namely the deceleration (rather than acceleration) of the fidelity decay and improved Loschmidt echo as compared to their Hermitian counterparts, despite large (non‐perturbative) deformation of the energy spectrum introduced by the perturbations. This behavior is illustrated by considering the fidelity decay and Loschmidt echo for the single‐particle hopping dynamics on a tight‐binding lattice under an imaginary gauge field.     

Revivals in Caldeira–Leggett Hamiltonian dynamics

We reconsider the problem of quantum system interacting with a complex environment discussed by Caldeira and Leggett (CL), and generalize their results for a quantum oscillator coupled to a reservoir R with dense discrete spectrum of oscillators with close to ${\omega }_s$ frequencies. Dynamics consists of recurrence cycles with partial revivals of the initial state. This revival or Loschmidt echo appears in each cycle. Width and number of the Loschmidt echo components increase with the recurrence cycle number leading to irregular, stochastic-like time evolution.     

Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.     

Decoherence in a two-level system coupled to a fermion environment with phase transitions

The decoherence process characterized by Loschmidt echo (LE) in a two-level system dephasingly coupled to a fermion environment with phase transitions is studied in this Letter. The results show that the LE of the two-level system may act as a witness of the environment's phase transitions, which is similar to the relation between quantum phase transitions and the LE.     

Edge state equilibration in a two-subband system

In order to honor Jörg Kotthaus, I present unpublished experimental results which were obtained in 1994 when I was a postdoc in Munich.The scattering between edge states in the quantum Hall regime is strongly reduced compared to scattering in the bulk of a two-dimensional electron gas. For edge states with different Landau quantum numbers an equilibration length as long as 100μm has been determined. In the case of Landau levels with different spin quantum numbers this length may reach values of 1 mm. Here we set out to explore the equilibration between edge states with different subband quantum numbers. Using parabolic quantum wells as a tunable multi-subband system we find that intersubband scattering can reduce the equilibration length to values below 5μm.