Session W31: Non-equilibrium Quantum Dynamics in Atomic Systems

Quantum critical states and phase transitions in the presence of non equilibrium noise

Ultracold atomic, molecular or trapped ion systems, offer unique possibilities to realize interesting quantum phases and phase transitions. On the other hand they are easily driven out of equilibrium by external (classical) noise sources. It is natural to expect that noise will destroy the subtle correlations underlying quantum critical behavior. This is indeed the case for thermal noise. Surprisingly we find that the $1/f$ noise, ubiquitous in such systems, does preserve the critical behavior. The emergent states show intriguing interplay of intrinsic quantum-critical and external noise-driven fluctuations. We demonstrate this general phenomenon with several specific examples in solid state and ultracold atomic systems. Our approach shows that genuine quantum phase transitions can be well defined even for systems driven out of equilibrium.   

Quenching across a quantum critical point: non-trivial power laws in different topological sectors

We study what happens when the Hamiltonian of a system with different topological sectors is quenched at a finite rate across a quantum critical point. We show that the quenching leads to a residual energy which scales as a power of the quenching rate, where the power depends on the topological sector in a non-trivial way. This generalises the idea of a Landau-Zener transition in two important ways: depending on the sector, the analysis may involve more than two states, and the effective Hamiltonian in the low-energy subspace may involve non-linear quenching even when the quenching of the original Hamiltonian is linear in time. We also discuss how one can dynamically evolve from one topological sector to another. As a specific example, we discuss all these ideas in the context of the Kitaev model defined on a two-leg ladder which can be studied by introducing Majorana fermions.   

Anomalous non-ergodic scaling in adiabatic multicritical quantum quenches

We investigate non-equilibrium dynamical scaling in adiabatic quench processes across quantum multicritical points. Our analysis shows that the resulting power-law scaling depends sensitively on the control path, and that anomalous critical exponents may emerge depending on the universality class. We argue that the observed anomalous behavior originates in the fact that the dynamical excitation process takes place asymmetrically with respect to the static multicritical point, and that non-critical energy modes may play a dominant role. As a consequence, dynamical scaling requires introducing new non-static exponents.   

Thermalization in Quenched Spinor Condensates

Motivated by recent experiments, we consider the dynamics of spin-one spinor condensates after a quantum quench from the polar to ferromagnetic state. We apply the truncated Wigner approximation to the spinor system with all spatial and spin degrees of freedom. This involves propagating the Gross-Pitaevskii equation averaged over an initial noise distribution provided by the Wigner function. For short times, we find agreement with the linearized Bogoliubov analysis. For this, the we show that the longitudinal magnetization grows with twice the gain exponent of the transverse magnetization. For long times (where the linearized theory fails) we provide evidence of thermalization. We interpret the results for large quenches, as a dynamical Berezinsky-Kosterlitz-Thouless transition resulting from the unbinding of vortices in the spin and charge degrees of freedom.   

Quench dynamics near a quantum critical point

We study the dynamical response of a system to a sudden small change of the tuning parameter starting at the quantum critical point. In particular we find the scaling laws of different physical quantities (the excitation probability, number of excited quasiparticles, heat and entropy) with the quench amplitude and the system size. We extend the analysis to quenches with arbitrary power law dependence on time, showing the close connection of these scaling laws with the scaling behavior of the fidelity susceptibility and other generalized susceptibilities. We illustrate the relevance of these results for experiments with cold atoms.   

Quantum quenches and thermalization in one-dimensional systems

We use quantum quenches to study the dynamics and thermalization of hardcore bosons and fermions in finite one-dimensional lattices. We perform exact diagonalizations and find that, far away from integrability, few-body observables thermalize. We then study the breakdown of thermalization as one approaches an integrable point. This is found to be a smooth process in which the predictions of standard statistical mechanics continuously worsen as the system moves toward integrability. We establish a direct connection between the presence or absence of thermalization and the validity or failure of the eigenstate thermalization hypothesis, respectively.\\ {\bf References}\\ \noindent M. Rigol, Phys. Rev. Lett. {\bf 103}, 100403 (2009); Phys. Rev. A {\bf 80}, 053607 (2009).   

Numerical verification of the instanton method on macroscopic quantum tunneling: phase slip dynamics

Instanton methods have been widely used for studying quantum tunneling in various contexts. Nevertheless, how accurate instanton methods are for the problems of macroscopic quantum tunneling (MQT) still remains unclear because of lack of their direct comparison with exact time evolution of the many-body Schroedinger equation. In this talk, we show numerical verification of instanton methods applied to coherent MQT. By specifically applying the quasi-exact numerical method of time-evolving block decimation to the system of bosons in a ring lattice, the real-time quantum dynamics of supercurrents is simulated. There we see a coherent oscillation between two macroscopically distinct current states occurs due to MQT. The tunneling rate extracted from the coherent oscillation is compared with that given by the instanton method. We show that the error is within 10 percent when the effective Planck's constant is sufficiently small. We also discuss phase-slip dynamics associated with the coherent oscillations.   

Non-Equilibrium Enhancement of Superfluidity in a Bose-Hubbard Model

The existence of superfluidity depends on the energy distribution of excitations in a system.~However, the distribution at thermal equilibrium is rarely optimal for the production of a superfluid state. It has been shown that simultaneously pushing a system out of equilibrium while balancing the induced heat through dissipation can be an effective way to enhance the system's superfluid properties. To that end, we consider how exciting a nonequilibrium site-density distribution in the Bose-Hubbard model can increase the superfluid region in the finite-temperature phase diagram. This work is supported by DARPA.   

Quantum distillation: dynamical generation of low-entropy states of strongly correlated fermions in an optical lattice

Correlations between particles can lead to subtle and sometimes counterintuitive phenomena. We analyze one such case, occurring during the sudden expansion of fermions in a lattice when the initial state has a strong admixture of double occupancies. We promote the notion of quantum distillation: during the expansion, and in the presence of strongly repulsive interactions, doublons group together, forming a nearly ideal band insulator, which is metastable with a low entropy. We propose that this effect could be used for cooling purposes in experiments with two-component Fermi gases.   

Emergent ``super-solitons'' following an interaction strength quantum quench across a Luttinger liquid-Mott insulating phase boundary

Rapid progress in cold atom experiments has motivated the study of non-equilibrium many-body dynamics following a sudden deformation of the system Hamiltonian (a ``quantum quench''). Here, we consider the dynamics of localized excitations produced via a quench across a quantum phase boundary separating critical Luttinger liquid and gapped Mott insulating states. Our initial liquid ground state is labeled by a Luttinger interaction parameter $K$, and subject to a density-inhomogeneity forming external potential. For the Mott insulator, we employ the quantum Sine Gordon model at the Luther-Emery (LE) point. We find that over a wide range of initial $K$ values, the quench induces the production of relativistic, non-dispersive traveling density waves, which we dub ``super-solitons.'' The super-solitons are generated from generic antecedent localized density lumps, and appear to be a robust feature of the post-quench dynamics. An isolated exception occurs for the case of $K = K_{LE}$; here, the density dynamics are generically dispersive, and depend sensitively upon the shape of the initial inhomogeneity. We show that the super-solitons do not interact, and we demonstrate that an inhomogeneous Luttinger parameter $K$ can be used to produce super-solitons with different characteristics in the same system.   

Quantum Quenches in an XXZ Spin Chain with an Inhomogeneous Initial State

We present results for the non-equilibrium dynamics of a quantum \textit{XXZ} spin chain, whose spins are initially arranged in a domain wall profile, via the application of a spatially-varying magnetic field in the $z$-direction. The system is driven out of equilibrium by rapidly turning off the magnetic field, leaving the system in a highly excited state. We study the time-evolution of the domain wall profile and various two-point correlation functions. The results are obtained both numerically and analytically via a bosonization approach. For the case of the \textit{XX} chain, which maps onto a model of non-interacting fermions, we find an interesting even-odd dichotomy in the long-time behavior of the transverse correlation function for sites spaced by $n$ lattice points. For the \textit{XXZ} chain, we highlight how the domain wall dynamics depend on whether the system is in the gapless \textit{XX} phase or in the gapped, Ising phase.   

Nonequilibrium dynamics of weakly and strongly paired superconductors

We study small oscillations of the order parameter in weakly and strongly paired superconductors driven slightly out of equilibrium, in the collisionless approximation. While it was known for quite some time that the amplitude of the oscillations in a weakly paired superconductor decays as $1/t^{1/2}$, we show that in a superconductor sufficiently strongly paired so that its fermions form bound states usually referred to as molecules, these oscillations decay as $1/t^{3/2}$. The transition between these two regimes happens when the chemical potential of the superconductor vanishes, thus the behavior of the oscillations can be used to distinguish weakly and strongly paired superconductors. These results are obtained in the mean field approximation which may not be reliable in the crossover region between the strong and weak pairing, so we also obtain identical results within the two-channel model, which can be tuned to be reliable throughout the entire crossover, although it then describes a special type of interactions between the fermions which may be difficult to observe experimentally. Finally, we interpret the result in the strongly paired superconductor as the probability of the molecular decay as a function of time.   

Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization

By means of exact diagonalization, we study level statistics and the structure of the eigenvectors of one-dimensional gapless bosonic and fermionic systems across the transition from integrability to quantum chaos. These systems are integrable in the presence of only nearest-neighbor terms, whereas the addition of next-nearest neighbor hopping and interaction may lead to the onset of chaos. We show that the strength of the next-nearest neighbor terms required to observe clear signatures of nonintegrability is inversely proportional to the system size. The transition to chaos is also seen to depend on particle statistics, bosons responding first to the integrability breaking terms. In addition, we discuss the use of delocalization measures as main indicators for the crossover from integrability to chaos. The analysis and findings described in this work \footnote{L. F. Santos and M. Rigol, arXiv:0910.2985} are intimately reflected by studies of thermalization.   

Signatures of the random singlet phase after a bond-breaking

We study the time evolution of one-dimensional hardcore bosons, initially prepared in a state with random nearest-neighbor hopping, after the severing of one nearest-neighbor bond. The initial system is equivalent to an xy spin chain with random nearest-neighbor bonds and displays a random singlet phase. Analytic results are demonstrated with exact numerical time-evolution of finite-sized systems. The correlations between pairs of sites in the time-evolving system after a single bond is broken display signatures of the critical phase as well as evidence of the ``light cone'' effect as information about the broken bond moves through the system.   

Non-adiabatic processes below the gap energy in $p$-wave superfluids

We argue that non-adiabatic transitions to subgap states on vortices in $p$-wave superfluids can significantly complicate the use of their non-abelian statistics for quantum information processing. To avoid these difficulties, we suggest to work in the regime of a small number of subgap states, which can be achieved in atomic superfluids. In this regime, the semiclassical approximation to the Bogoluibov-deGennes equations by Kopnin and Salomaa is not applicable. Based on numerical calculations in the spherical geometry, we calculate the system parameters for which the energy gap to the subgap states is maximized for $p$-wave superfluids obtained either by Feshbach resonances or by using microwave induced precessing dipole interactions of dipolar molecules.