Post-quench dynamics and pre-thermalization in a resonant Bose gas
We explore the dynamics of a resonant Bose gas following its quench to a strongly interacting regime near a Feshbach resonance. For such deep quenches, we utilize a self-consistent dynamic field approximation and find that after an initial regime of many-body Rabi-like oscillations between the condensate and finite-momentum quasiparticle pairs, at long times, the gas reaches a pre-thermalized nonequilibrium steady state. We explore the resulting state through its broad stationary momentum distribution function, that exhibits a power-law high momentum tail. We study the dynamics and steady-state form of the associated enhanced depletion, quench-rate dependent excitation energy, Tan’s contact, structure function and radio frequency spectroscopy. We find these predictions to be in a qualitative agreement with recent experiments.
i.1 Background and motivation
Degenerate atomic gases have radically expanded the scope of quantum many-body physics beyond the traditional solid-state counter part, offering opportunity to study highly coherent, strongly interacting, and well-characterized, defects-free systems. Atomic field-tuned Feshbach resonances (FRs) FRrmp (); BlochRMP (); ZweirleinReview (); GurarieRadzihovskyAOP () have become a powerful experimental tool that has been extensively utilized to explore strong resonant interactions in these systems. Feshbach resonances have thus led to a seminal realization of paired -wave fermionic superfluidity, with the associated BCS-to-Bose-Einstein condensate (BEC) crossover GrimmBCS-BEC (); JinBCS-BEC (); ZweirleinReview (); GurarieRadzihovskyAOP () through a universal unitary regime JasonHo (); VeillettePRA (); PowellSachdevPRA (), and phase transitions driven by species imbalance PartridgePolarized (); RSpolarized () and by Mott-insulating physics in an optical lattice GreinerMI (); Doniach (); JackishZoller (); refsFermionsFR_opticallattice (); ZhaochuanPRL (). Numerous other promising many-body states and phase transitions, such a -wave fermionic superfluidity pwaveGR (); pwaveYip (); pwaveJin () and Stoner ferromagnetism magnetismKetterle () have been proposed and continue to be explored.
Unmatched by their extreme coherence and high tunability of system parameters, such as FR interactions and single-particle (trap and lattice) potentials, atomic gases have also enabled numerous experimental realizations of highly nonequilibrium, strongly-interacting many-body states and associated phase transitions GreinerMI (); JinBCS-BEC (); BlochRMP ().
This has motivated extensive theoretical studies PolkovnikovRMP2011 (); CazalillaRMP2011 (); Schmiedmayer (), with a particular focus on nonequilibrium dynamics following a quench of Hamiltonian parameters, . In addition to studies of specific physical systems, experiments on these closed and highly coherent systems have driven theory to address fundamental questions in quantum statistical mechanics. These include the conditions for and nature of thermalization under unitary time evolution of a closed quantum system vis-á-vis eigenstate thermalization hypothesis Srednicki (); Rigola (), role of conservation laws and obstruction to full equilibration of integrable models argued to instead be characterized by a generalized Gibb’s ensemble (GGE), emergence of statistical mechanics under unitary time evolution for equilibrated and nonequilibrium stationary states KinoshitaWengerWeissNature2006 (); RigolOlshanniNaturePRL07 (). These questions of post-quench dynamics have been extensively explored in a large number of systems AlmanVishwanath06 (); Barankov06 (); AGR06 (); Yuzbashan07 (); MitraGiamarchi13 (); GurarieIsing13 (); CalabreseCardy07 (); SotiriadisCardy10 (); Sondhi13 (); NatuMueller13 (); ChinGurarie13 (); YinLR14 (); BacsiBalazs13 (); MitraSpinChain14 (); Essler14 (); NessiCazalilla14 ()
Early studies of a Feshbach-resonant Fermi gas predicted persistent coherent post-quench oscillations Barankov (); AGR06 () and, more recently found topological nonequilibrium steady states and phase transitions Yuzbashan (); Foster ().
Resonant Bose gas quenched dynamics studies date back to seminal experiments on Rb Donley (); Claussen (), that demonstrated coherent Rabi-like oscillations between atomic and molecular condensates HollandKokkelmans (), enabling a measurement of the molecular binding energy. More recently, oscillations in the dynamic structure function have also been observed in quasi-2D bosonic Cs ChinGurarie13 () and studied theoretically NatuMueller13 (); Ranon13 () for shallow quenches between weakly-repulsive interactions (small gas parameter where is the s-wave scattering length).
Such resonant bosonic gases were also predicted to exhibit distinct atomic and molecular superfluid phases, separated by a quantum Ising phase transition (rather than just a fermionic smooth BCS-BEC crossover) and other rich phenomenology RPWmolecularBECprl04 (); StoofMolecularBEC04 (); RPWmolecularBECpra08 (); ZhouPRA08 (); BonnesWessePRB12 (), thereby providing additional motivation for their studies.
Important recent developments are experiments by Makotyn, et al, Makotyn (), that explored dynamics of Rb following a deep quench to the vacinity of the unitary point on the molecular (positive scattering length, ) side of the Feshbach resonance. It was discovered that even near the unitary point, where a Bose gas is expected to be unstable PetrovShlypnikov (), the three-body decay rate (on the order of an inverse milli-second) appears to be more than an order of magnitude slower than the two-body equilibration rate (both measured to be proportional to Fermi energy, as expected FeiZhou (); LRunpublished (). This thereby opened a window of time scales from a microsecond (a scale of the quench) to a milli-second for observation of a metastable strongly-interacting nonequilibrium dynamics.
Stimulated by these fascinating experimental developments and motivated by the aforementioned earlier work, in a recent brief publication YinLR14 () we reported on results for the upper-branch repulsive dynamics of a resonant Bose gas following a deep-detuning quench close to the unitary point on the molecular side () of the FR Makotyn (). Taking the aforementioned slowness of as an empirical fact, consistent with experimental observations we predicted a fast evolution to a pre-thermalized strongly-interacting stationary state, characterized by a broad, power-law steady-state momentum distribution function, , with a time scale for the pre-thermalization of momenta set by the inverse of the excitation spectrum, . The associated condensate depletion was found to exhibit a monotonic growth to a nonequilibrium value exceeding that of the corresponding ground state. In the current manuscript we present the details of the analyses that led to these results as well as a large number of other predictions.
The rest of the paper is organized as follows. We conclude the Introduction with a summary of our key results. In Section II, starting with a one-channel model of a Feshbach-resonant Bose gas, we develop its approximate Bogoluibov and self-consistent dynamic field forms. In Section III, as a warmup we analyze the equilibrium self-consistent model for the strongly interacting case and compare its predictions to that of the Bogoluibov approximation. In Section IV we utilize the Bogoluibov model to study the nonequilibrium dynamics following a shallow-quench, computing the momentum distribution function probed in the time-of-flight, the radio-frequency (RF) spectroscopy signal, , and the structure function probed via Bragg spectroscopy. Then in Section V we generalize the quench to a more experimentally realistic case of a finite-rate ramp and study the effect of ramp rate. In Section VI we employ the self-consistent dynamic field theory to study these and a number of other observables for deep quenches in a strongly interacting regime relevant to JILA experiments Makotyn (). In Section VII we study excitation energy, an important measure of long time nonequilibrium stationary state, for both sudden quench and finite ramp-rate cases, and discuss its dependence on quench depth and ramp rate. We generalize Tan’s Contact to nonequilibrium process and study its long time behavior in Section VIII. Finally in Section IX we conclude with a discussion of our predictions for experiments and of the future directions for this work. We relegate the details of most calculations to Appendices.
i.3 Summary of results
Before turning to the derivation and analysis, we briefly summarize the key predictions of our work. Working within the upper-branch of a single-channel model of a resonantly interacting Bose gas we studied an array of nonequilibrium observables following its Feshbach resonance quench toward the unitary point. One central quantity extensively studied in recent time of flight measurements ChinGurarie13 (); Makotyn () is the momentum distribution function, at time after a quench from a ground state of an initial Hamiltonian to a final Hamiltonian . Motivated by experiments we take to be a superfluid BEC ground state in the upper branch of the repulsive Bose gas footnote (). For a shallow quench in the scattering length , away from the immediate vicinity of the unitary point, the calculation is controlled by an expansion in a small interaction parameter, . Within the lowest, Bogoluibov approximation the momentum distribution function is given by (choosing units where and throughout) NatuMueller13 ()
where characterizes the “depth” of the quench, and we have rescaled the momentum and time with the coherence length and pre-thermalization timescale , as and , respectively. We start the system in a weakly interacting state, characterized by a short positive scattering length and quench it to (). Following coherent oscillations, the gas then exhibits pre-thermalization dynamics, where after a dephasing time , set by the inverse of the excitation spectrum consistent with experiments Makotyn (), the initial narrow Bogoluibov momentum distribution evolves to a stationary state, characterized by a broadened distribution function
where we defined as the nonequilibrium analog of Tan’s contact for the nonequilibrium steady state, given by
Within above approximation the quasi-particles do not scatter, precluding full thermalization, and the above final state remains nonequilibrium, completely determined by the depth-quench parameter , with the associated diagonal density matrix ensemble.
The associated condensate depletion is then straightforwardly computed and monotonically pre-thermalizes to
a value exceeding that for the ground state of the final scattering length and greater than the initial ground state depletion at scattering length .
With the goal of understanding deep quenches of a strongly interacting Bose gas Makotyn (); YinLR14 (); SykesPRA14 () near a Feshbach resonance, we developed a self-consistent dynamic field theory of coupled Gross-Petaevskii equation for the condensate and a Heisenberg equation for atoms excited out of the condensate. It accounts for strong time-dependent depletion of the condensate, with feedback on dynamics of excitations. Within this nonpertubative (but uncontrolled) approximation this amounts to solving for a Heisenberg evolution of with a time-dependent Bogoluibov-like Hamiltonian, parameterized by a condensate density . The latter is self-consistently determined by the atom-number constraint equation, AGR06 (); YinLR14 (). Our treatment here is closely related to the analysis of post-quench quantum coarsenning dynamics of the Sondhi13 () and Ising SotiriadisCardy10 () models. The resulting momentum distribution function, (projected column density measured in experiments Makotyn ()) and the corresponding depletion are illustrated in Figs. 1,3.
We also studied the excitation energy after a constant ramp rate between and scattering lengths. As illustrated in Fig. 4, we found that it displays a form
for a ramp-rate below the microscopic energy cutoff .
To further characterize the post-quench evolution and the resulting pre-thermalized steady-state we have also computed a time dependent structure function , a Fourier transform of the density-density connected correlation function. For the weakly interacting, shallow-quench regime, at temperature it is given by
Utilizing our self-consistent dynamic field theory we extended above calculation of to deep quenches of strongly interacting resonant condensates. The resulting time-dependent structure function and its steady-state form are illustrated in Fig. 5.
We also computed the RF spectroscopy signal JinRF (); Wild (), that measures the transition rate of atoms from two resonantly interacting hyperfine states into a third weakly interacting hyperfine state, for the quench process. Within the Bogoluibov approximation the response is given by
as measured experimentally, with the amplitude proportional to Tan’s contact, that in the simplest Bogoluibov approximation is given by .
We next turn to a single-channel Feshbach resonant model, followed by its detailed analysis that led to above and other results.
Ii Model of a resonant superfluid
A resonant gas of bosonic atoms can be modeled by a single-channel grand-canonical Hamiltonian, (defining )
where is a bosonic atom field operator, is a single-particle Hamiltonian, is the chemical potential, and the pseudo-potential characterizes the atomic two-body interaction on the scale longer than its microscopic range , typically on the order of ten angstroms. For simplicy, we have set .
As discussed in detail in Ref. GurarieRadzihovskyAOP () and references therein, near a Feshbach resonance the magnetic field-dependent coupling controls the s-wave scattering length through the renormalized coupling (-matrix) ,
related to the scattering length via . As illustrated in Fig. 6, for a sufficiently strong attractive interaction, in a vacuum, the two-atom scattering length diverges at , as the two-body bound state forms for and turns positive on the so-called “BEC” side of the Feshbach resonance. is the range of the potential and is the corresponding momentum cutoff. It is this scattering-length tunability that enables studies of phase transitions in resonant Bose RPWmolecularBECprl04 (); StoofMolecularBEC04 (); RPWmolecularBECpra08 (); ZhouPRA08 (); BonnesWessePRB12 () (and BCS-BEC crossover in Fermi GrimmBCS-BEC (); JinBCS-BEC (); ZweirleinReview (); FRrmp (); BlochRMP (); ZweirleinReview (); GurarieRadzihovskyAOP ()) gases and quenched dynamics ChinGurarie13 (); Makotyn (); YinLR14 (); SykesPRA14 () that is our focus here.
Expressing the Hamiltonian, (9) in terms of the operator , it decomposes into
is the lowest order mean-field ground-state energy, and
are the operator components organized by respective orders in the excitation .
ii.1 Bogoluibov approximation for weakly interacting bosons
We set the stage for the study of dynamics following a shallow quench ChinGurarie13 () and of a self-consistent dynamic field treatment YinLR14 () of a deep quench Makotyn () by first briefly summarizing the results for the ground state and excitations in the Bogoluibov approximation Bogoliubov (); Fetter ().
In the weakly interacting limit the atomic gas is characterized by a small gas parameter , well-approximated by the Bogoluibov quadratic Hamiltonian, neglecting the nonlinear components of . Focusing on the uniform (bulk) condensate and eliminating the chemical potential in favor the condensate density by requiring the vanishing of the component (equivalent to a minimization of over ), , neglecting the difference between the condensate density, and total atom density, , the grand-canonical Hamiltonian reduces to ,
where the quadratic Hamiltonian was straightforwardly diagonalized in terms of the Bogoluibov quasi-particles , related to the atomic Nambu spinor by a pseudo-unitary transformation,
satisfies a pseudo eigenvalue equation and preserves the canonical commutation relation, , corresponding to , defined by
with and the third Pauli matrix.
With in (15), the Bogoluibov spectrum is given by a well-known gapless form,
that interpolates between the low-momentum zeroth-sound with velocity (a Goldstone mode of the symmetry breaking) and the high-momentum quadratic dispersion, with crossover scale set by the correlation length . The corresponding coherence factors defining are given by
The ground state is a vacuum of Bogoluibov quasi-particles, , with the energy density given by
where the momentum distribution function
with Tan’s contact and
The interaction-driven condensate depletion, is given by
and provides an important measure of the validity of the Bogoluibov approximation that neglects the difference between and .
Clearly, for a large gas parameter, the depletion is substantial and must be accounted for. Although there is no currently available systematic analysis in this nonperturbative limit, as we will show in subsequent sections, an uncontrolled self-consistent method, akin to a spherical, large- model ZinnJustin (); ChaikinLubensky (); SotiriadisCardy10 (); Sondhi13 (); LRunpublished () captures important qualitative physics in this resonantly interacting regime.
ii.2 Generalization for large scattering length
To extend the analysis to a large we need to account (even if approximately) for the nonlinear components of the Hamiltonian, neglected in the Bogoluibov model. To this end, in the spirit of variational theory or a spherical model ChaikinLubensky (), we replace these nonlinear operators by their “best” approximation in terms of operators up to a quadratic order in fluctuation field . Using Wick’s theorem, we have
where we kept the depletion density and neglected “anomalous” averages (e.g., ) and high order correlators (e.g., ) that we expect to be subdominant.
With these we approximate and by a linear and quadratic forms
The grand-canonical Hamiltonians now take the following forms: , where
Above, for simplicity, we have defined and and in Eqs. (29a),(29b),(29c) have discarded the ”anomalous average” term to satisfy the constraint of Goldstone theorem, which requires a gapless excitation spectrum. This amounts to the widely used Popov approximation HFB_Popov ().
Following what was done in the last subsection, we fix the chemical potential by requiring
For a uniform system, this gives
Thus we obtain the grand-canonical Hamiltonian
where . It exhibits the standard Bogoluibov form with gapless spectrum, but also approximately accounts for a potentially strong depletion through the condensate density replacing the full density as the self-consistently determined parameter of the Hamiltonian.
Iii self-consistent analysis for strongly interacting ground state
Before turning to our main focus of nonequilibrium post-quench dynamics, we examine the ground state properties of a strongly interacting resonant Bose gas, characterized by a large scattering length and gas parameter . This regime lies beyond the scope of the standard Bogoluibov theory. Nevertheless we expect to be able to treat it qualitatively correctly (even if quantitatively uncontrolled) by taking into account the large depletion through the Hamiltonian (32) and the self-consistency condition through the atom number conservation
where in the second line we calculated the depletion by diagonalizing (32) as in Sec. II.1 of the conventional Bogoluibov theory, but with replacing . Such treatment is quite close in spirit to the self-consistent Hartree-Fock approximations, and the BCS and other mean-field gap equations.
In the dimensionless form for , the self-consistency equation reduces to
where , with the mometum scale set by the boson density .
The solution to Eq. (34) is illustrated in Fig. 2. We find that the self-consistency constraint suppresses condensate depletion, leading to a higher condensate fraction than the Bogoluibov approximation for the same strength of the interaction parameter . We also observe that, as expected the correction to Bogoluibov theory from the self-consistency condition grows (from zero) with increasing gas parameter , thereby avoiding the spurious transition to a vanishing condensate state appearing in the Bogoluibov theory.
Iv Dynamics for shallow quench
We now turn to nonequilibrium dynamics following a change in the scattering length from its initial value to the final value , as can be realized experimentally in a Feshbach resonant Bose gas by a change in the external magnetic field Makotyn (). Here we assume the change is instantaneous (sudden quench), allowing analytical analysis. In this section, we focus on shallow quenches characterized by both and , so that the Bogoluibov approximation remains rigorously valid.
For shallow quenches, the system is well approximated by Hamiltonian (15) with and for the initial and final Hamiltonians, respectively, with corresponding Bogoluibov quasi-particle bases prior to the quench and post the quench. Focussing on a sudden quench, the two sets of bases are related to the atomic basis via a pseudo-unitary transformations
define Bogoluibov transformations for Hamiltonians (with interaction ) before and (with interaction ) after the quench, respectively. The corresponding excitation spectra are
and the two quasi-particle bases are related by
We take the initial state to be the ground state of the pre-quenched Hamiltonian footnote (), and thus a vacuum of quasi-particles, . At finite temperature this generalizes to Bose-Einstein distribution of quasi-particle occupation,
Because experiments probe physical observables expressed in terms of atomic operators, we need to compute time evolution of . Using free post-quench evolution of quasi-particles
the relation (39), together with the simplicity of matrix elements of quasi-particles in the pre-quench ground state (vacuum of ), we find
where the matrix
evolves the initial Bogoluibov spinor