Non-equilibrium dynamics from few- to many-body systems : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: (1) full quantum dynamics and (2) semiclassical phase-space method. We rely on arguments based on the eigenstate thermalization hypothesis (ETH), quantum chaos as seen from the distribution of level spacings, and the concept of chaotic eigenstates in demonstrating equilibration dynamics of local observables in the system after an integrability-breaking quench. The same issue on quantum thermalization can be viewed from a different perspective using semiclassical phase-space methods. In particular, we employ the truncated Wigner approximation (TWA) to simulate the quantum dynamics. In this case, we show that the marginal distributions of the individual trajectories which sample the initial Wigner distribution are in good agreement with the corresponding microcanonical distribution.
We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: (1) full quantum dynamics and (2) semiclassical phase-space method. We rely on arguments based on the eigenstate thermalization hypothesis (ETH), quantum chaos as seen from the distribution of level spacings, and the concept of chaotic eigenstates in demonstrating equilibration dynamics of local observables in the system after an integrability-breaking quench. The same issue on quantum thermalization can be viewed from a different perspective using semiclassical phase-space methods. In particular, we employ the truncated Wigner approximation (TWA) to simulate the quantum dynamics. In this case, we show that the marginal distributions of the individual trajectories which sample the initial Wigner distribution are in good agreement with the corresponding microcanonical distribution.
The interplay of particle and resonant wave scattering including nonlinear effects \ncreates systems of diverse and interesting quantum many-body physics. A better \nunderstanding of the physics in these systems could lead to new and exiting \napplication exploiting their quantum nature. \nAs an example, in this thesis we investigate the scattering of bright matter-wave \nsolitons in ultracold gases on a square well in one spatial dimension. For this, solutions \nof the mean-field Gross-Pitaevskii approximation and a full quantum manybody \nmethod, the so-called multiconfigurational time-dependent Hartree approach \n(MCTDH), are compared. \nThe MCTDH method is based on a finite basis set expansion, which naturally \nleads to errors in system properties, such as energies and densities, when compared \nto exact results. In this thesis, we propose an efficient solution to this problem \nby rescaling the interaction strength between the particles. Even for very large \ninteractions in the Tonks-Girardeau limit, the rescaling leads to significant improvements. \nThis is validated by successfully applying the rescaling to problems \nin ring systems as well as external confinements, such as a harmonic well and a \ndouble-well. \nThe MCTDH method is then applied to the soliton scattering problem and \ncompared to results from mean-field calculations. The latter verify that solitons, \nwhen scattered on a well, show quantum effects, such as reflection. For the first \ntime, we show that a soliton can be additionally permanently trapped by the well \ndue to resonances with bound states. \nFor this thesis, to extend these results to a full many-body approach, we developed \nQiwiB. It is a program package implementing the MCTDHB method, \nwhich is a derivative of the MCTDH method, but optimised for bosonic systems. \nLimits for the validity of the MCTDHB approach are addressed by convergence \nstudies on the soliton scattering problem. Furthermore, we demonstrate that the \nscattering on the well enables the creation of macroscopic binary quantum superposition \nstates, i.e. NOON states. Novel NOON states corresponding to a \nsuperposition of a reflected soliton and a trapped soliton are observed. These \nstates are shown to exist for a large range of initial conditions, and a possible \nexperimental realisation is discussed.
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general W- and general GHZ-state, coupled to a global spin-1/2 bath or several independent spin-1/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.
Ultracold atoms in optical lattices constitute a versatile many-body platform with highly tunable parameters, allowing us to emulate a multitude of complex quantum systems, in the laboratory, even those eluding analytical treatment. In particular, the nonequilibrium dynamics of strongly correlated many-body systems represents one of the most challenging problems of modern quantum physics, with applications ranging from thermalization dynamics and transport properties to the management of correlations and the control of the dynamics. Understanding nonequilibrium phenomena of strongly correlated systems is a formidable task. A very promising route to gain insight into such systems is to examine few-body setups which contain only a few relevant degrees of freedom, yet incorporating the quantum correlations between the particles.
The present dissertation contributes to the understanding of the nonequilibrium dynamics of strongly-correlated quantum many-body systems by exploring systems of few-bosons − of one or two species − trapped in optical lattices. The systems are driven out-of-equilibrium either by performing a quench of a Hamiltonian parameter or by considering a time-periodic modulation of the external confinement. In the course of several consecutive works, we showcase different ways to couple the nonequilibrium modes, while unveiling their correlated nature and microscopic origin. To simulate the nonequilibrium dynamics, a sophisticated, highly flexible ab-initio method for numerically solving the time-dependent many-body Schrodinger equation is utilized, namely the Multi-Layer Multi-Configuration Time-Dependent Hartree Method for Atomic Mixtures (ML-MCTDHX).
Within the first part we study in six consecutive works the correlated nonequilibrium dynamics of few-boson systems in one-dimensional finite lattices. Starting from weak interactions, it is shown that a sudden increase of the interaction strength generates a global density-wave tunneling dynamics as well as intrawell breathing and cradle-like excited-band processes. The cradle process is a dipole-like process generated by the quench-induced over-barrier transport and it is one of the central results of the present thesis. Remarkably enough, the interaction quenches couple the density-wave and cradle modes, inducing resonance phenomena between the inter and intrawell dynamics. We further show that the cradle mode is inherently related to the initial delocalization and, following a quench from strong-to-weak interactions, can be excited only for incommensurate setups with filling larger than unity. Alternatively, a sudden ramping down of the lattice depth which favors the spatial delocalization is employed to access the cradle mode for setups with filling smaller than unity. Following a multiple interaction quench protocol, we observe the rise of several lowest-band tunneling modes as well as the cradle and the breathing mode. Besides the cradle mode, all other excited modes are highly tunable possessing different frequencies during and in between the quenches. In the excitation dynamics a monotonic behavior with increasing quench amplitude and a non-linear dependence on the duration of the application of the quenched interaction strength is revealed. Additionally, a periodic population transfer between momenta for quenches of increasing interaction is observed, with a power-law frequency dependence on the quench amplitude. Linear interaction quenches from a superfluid to a Mott-insulator state excite various inter- and intraband tunneling modes. The competition between the quench rate and the interparticle repulsion leads to a resonant dynamical response, at moderate ramp times, being related to avoided-crossings in the many-body eigenspectrum.
The resultant higher-band excitation dynamics is shown to obey an exponential decay possessing two distinct time scales with varying ramp time. Inspecting the crossover from shallow to deep lattices we find that for a diabatic quench the excited-band fraction decreases, while approaching the adiabatic limit it exhibits a nonlinear behavior for increasing height of the potential barrier. Quenching from strong-to-weak interactions leads to a melting of the Mott-insulator and negligible higher-band excitations.
Performing quenches either on the wavevector or the phase of a spatially dependent interaction profile triggers various tunneling channels and a rich excitation dynamics which is amplified for increasing inhomogeneity amplitude. Most importantly, the phase quench is shown to induce a directional transport enabling us to discern, otherwise, energetically degenerate tunneling pathways.
Finally, a periodic population transfer between distinct momenta for quenches of increasing wavevector and a directed occupation of higher momenta following a phase quench is observed. Employing a quench of an additional harmonic trap from strong-to-weak confinement, we find that the competition between the initial localization and the repulsive interaction leads to a resonant response of the system related to avoided-crossings in the many-body eigenspectrum with varying final trap frequency. Furthermore, we show that these avoided-crossings can be utilized to prepare the system in a desired state.
The second part comprises two efforts and is devoted to the study of the nonequilibrium dynamics of finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices. For a shaken lattice, a wide range of driving frequencies is covered and a resonant behavior of the intrawell dynamics is revealed and found to be related to a rich intraband excitation spectrum.
Moreover, it is shown that for increasing repulsion a strong suppression of the interwell tunneling and an enhanced excitation dynamics occurs. For a vibrating lattice, an additional interaction quench gives rise to admixtures of different excitations in the outer wells, an enhanced breathing in the center and an amplification of the emerging tunneling dynamics. The occurence of multiple resonances between the inter- and intrawell dynamics at different quench amplitudes is revealed, with the position of the resonances being tunable via the driving frequency and thus allowing for further control of the mode coupling in optical lattices.
In the third and final part of this thesis we unravel the quench dynamics of a Bose-Bose and a Bose-Fermi mixture. Utilizing an interspecies interaction quench we cross the miscibility-immiscibility threshold in a harmonically confined Bose-Bose mixture. We show that increasing the interspecies repulsion coefficient results in a filamentation of the density of each species, with the spontaneously generated filaments being strongly correlated and exhibiting domain-wall structures. Strikingly, by following the reverse quench protocol, i.e., upon decreasing the interspecies interaction strength, the formation of multiple dark-antidark solitary waves is observed. These solitary wave structures are found to decay into the many-body environment, soon after their generation in sharp contrast to the predictions of the mean-field approximation. To relate our findings with possible experimental realizations, we simulate, for the first time for binary mixtures, single-shot images showcasing that the growth rate of the variance of a sample of single-shots probes the degree of entanglement inherent in the system. As a next step we investigate the expansion dynamics of a mass balanced Bose-Fermi mixture confined in a one-dimensional optical lattice upon quenching an imposed harmonic trap from strong-to-weak confinement. Tuning the interspecies interaction strength we realize the immiscible and miscible correlated ground state phases. We further show that the system's dynamical response crucially depends on the initial phase and consists of an expansion of each cloud and an interwell tunneling dynamics. Varying the quench amplitude and referring to a fixed phase a multitude of response regimes is unveiled, being richer within the immiscible phase, which are described by distinct expansion strengths and tunneling channels. Finally, in the expansion dynamics a two-body anti-correlated behavior between the predominantly occupied wells is unveiled.
This thesis addresses different aspects of the dynamics of isolated quantum many-body systems far from equilibrium, ranging from transient phenomena at short times to questions about relaxation and equilibration at late times. The first part deals with nonequilibrium critical phenomena. Dynamical quantum phase transitions are characterized in a two-dimensional spin system by studying the distribution of Fisher zeros in the complex time plane. Moreover, a new topologically driven transition in the nonequilibrium steady state of quenched Chern insulators is investigated, resulting in the identification of universal nonanalytic behavior of the Hall conductance as a function of the quench parameter. The second part of the thesis comprises a study of transient dynamics of the transverse-field Ising model in two and three dimensions based on a perturbative derivation of classical networks to encode the time-evolved wave function; this construction is also applicable to represent the state as an artificial neural network. In the last part of the thesis a possible definition of the irreversibility of quantum many-body dynamics is introduced, which is based on the echo dynamics of observables under imperfect effective time reversal. Echo dynamics is studied in generic and integrable quantum systems as well as in the semiclassical limit. The relation to out-of-time-order correlators is explored, which were recently introduced to probe the quantum butterfly effect.
This cumulative doctoral thesis, based on the works [P1–P5], addresses various topical concepts as well as foundational questions in the nonequilibrium physics of quantum many-body systems. Publication [P1] concerns itself with the question of time’s arrow in quantum mechanics. Based on the recursion method, we attribute autocorrelation functions of few-body observables at infinite temperature (almost) monotonously decreasing functions, constructed from temporal derivatives of the correlation function that indicate a directed approach to equilibrium. We numerically find that in chaotic models, these constructions comprise a notion of simplicity of the underlying dynamics. Addressing the ubiquity of simple relaxation dynamics in otherwise complex chaotic systems, in Ref. [P3] we introduce a scheme based on Lanczos coefficients to approximate correlation functions in the thermodynamic limit. Examining several pertinent condensed-matter type models and benchmarking with state-of-the-art numerics, we find the method to yield good results already for a low number of included Lanczos coefficients, provided that the latter exhibit a smooth and increasing structure. We underpin these findings with analytical arguments on the method’s applicability. Ref. [P2] provides an estimate on the equilibration time of such infinite-temperature correlation functions in the thermodynamic limit. Similarly to Ref. [P3], we find the estimate to capture the equilibration times very reasonably, given that the corresponding Lanczos coefficients exhibit a smooth growth. Substantiated by analytical considerations, the convergence of the estimate implies that equilibration timescales of the relaxation processes are finite and conceivably short. In Ref. [P5] we scrutinize the interpretative power of the Lanczos coefficients concerning the system’s integrability or chaoticity, operator growth and the indicativeness of physical behavior. To this end, we analytically deter- mine the Lanczos coefficients for a Majorana observable in a strictly quadratic model and illustrate that within this model any structure of the Lanczos coefficients may be realised. Different to the other works, Ref. [P4] is not cast in the language of the recursion method. Focusing on the full eigenstate thermalization hypothesis, it explores the connection between thermalization dynamics of multi-point correlation functions and the scaling of thermal free cumulants in setups mimicking a system-bath situation.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.