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arXiv:2102.07051 · Physical Review Letters

Maximum Energy Growth Rate in Dilute Quantum Gases

Ran Qi, Zhe-Yu Shi, Hui Zhai

Published Jun 16, 2021• 11 citations
arXivHTMLPDF

Abstract

In this Letter we study how fast the energy density of a quantum gas can increase in time, when the interatomic interaction characterized by the s-wave scattering length a_{s} is increased from zero with arbitrary time dependence. We show that, at short time, the energy density can at most increase as sqrt[t], which can be achieved when the time dependence of a_{s} is also proportional to sqrt[t], and especially, a universal maximum energy growth rate can be reached when a_{s} varies as 2sqrt[ℏt/(πm)]. If a_{s} varies faster or slower than sqrt[t], it is, respectively, proximate to the quench process and the adiabatic process, and both result in a slower energy growth rate. These results are obtained by analyzing the short time dynamics of the short-range behavior of the many-body wave function characterized by the contact, and are also confirmed by numerically solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomic gases.

DOI
10.1103/physrevlett.126.240401

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In this Letter we study how fast the energy density of a quantum gas can increase in time, when the interatomic interaction characterized by the s-wave scattering length a_{s} is increased from zero with arbitrary time dependence. We show that, at short time, the energy density can at most increase as sqrt[t], which can be achieved when the time dependence of a_{s} is also proportional to sqrt[t], and especially, a universal maximum energy growth rate can be reached when a_{s} varies as 2sqrt[ℏt/(πm)]. If a_{s} varies faster or slower than sqrt[t], it is, respectively, proximate to the quench process and the adiabatic process, and both result in a slower energy growth rate. These results are obtained by analyzing the short time dynamics of the short-range behavior of the many-body wave function characterized by the contact, and are also confirmed by numerically solving an example of interacting bosons with time-dependent Bogoliubov theory. These results can also be verified experimentally in ultracold atomic gases.
Recent advances in the tunability of ultracold atomic gases have created opportunities for studying interesting quantum many-body systems. Fano-Feshbach resonances, in particular, allow experimenters to freely adjust the scattering of atoms by controlling an external magnetic field. By rapidly changing this field near a resonance, it is possible to drive systems out of equilibrium towards novel quantum states where correlations between atoms change dynamically. In this thesis, we take a wave-function-based approach to theoretically examine the response of several interesting systems to suddenly-switched, or "quenched", interactions.\nWe first calculate the time evolution of a Bose-Einstein condensate that is quenched to the unitarity regime, where the scattering length 'a' diverges. Working within the time-dependent variational formalism, we find that the condensate does not deplete as quickly as the usual Bogoliubov theory would suggest. We also make a quantitative prediction for the dynamics of short-range pair correlations, encoded in Tan's contact. We then consider the dynamics of these correlations for quenches to small 'a', and we find that bound states can cause high-contrast oscillations of the contact. These dynamics can be modelled quantitatively at short times by using a properly-chosen two-body model. Finally, we characterize the nonlocal correlation waves that are generated by an interaction quench in arbitrary dimensionality. Our analysis demonstrates that the large-momentum limit of the post-quench momentum distribution can sometimes include contributions from both the short range and the long range, depending on the quench protocol.
A fundamental problem of out-of-equilibrium physics is the speed at which the order parameter grows upon crossing a phase transition. Here, we investigate the ordering dynamics in a Fermi gas undergoing a density-wave phase transition induced by quenching infinite-range, cavity-mediated interactions. We observe, in real time, the exponential rise of the order parameter and track its growth over several orders of magnitude. Remarkably, the growth rate can exceed the Fermi energy by an order of magnitude, consistent with predictions from a linearized instability analysis. This case contrasts with the ordering process driven by short-range interactions. We then generalize our results to linear interaction ramps, where deviations from the adiabatic behavior are captured by a simple dynamical ansatz. Our study offers a paradigmatic example of the interplay between strong short- and long-range interactions in quantum nonequilibrium dynamics.
The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.
We study the many-body dynamics of weakly interacting Bose gases with two-particle losses. We show that both the two-body interactions and losses in atomic gases may be tuned by controlling the inelastic scattering process between atoms by an optical Feshbach resonance. Interestingly, the low-energy behavior of the scattering amplitude is governed by a single parameter, i.e. the complex s s -wave scattering length a_c a c . The many-body dynamics are thus described by a Lindblad master equation with complex scattering length. We solve this equation by applying the Bogoliubov approximation in analogy to the closed systems. Various peculiar dynamical properties are discovered, some of them may be regarded as the dissipative counterparts of the celebrated results in closed Bose gases. For example, we show that the next-order correction to the mean-field particle decay rate is to the order of |n a_c^{3}|^{1/2} | n a c 3 | 1 / 2 , which is an analogy of the Lee-Huang-Yang correction of Bose gases. It is also found that there exists a dynamical symmetry of symplectic group Sp (4,\mathbb{C}) ( 4 , ℂ ) in the quadratic Bogoliubov master equation, which is an analogy of the SU(1,1) dynamical symmetry of the corresponding closed system. We further confirmed the validity of the Bogoliubov approximation by comparing its results with a full numerical calculation in a double-well toy model. Generalizations of other alternative approaches such as the dissipative version of the Gross-Pitaevskii equation and hydrodynamic theory are also discussed in the last.
We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].