Entropy production and statistical relaxation of dipolar bosons and fermions in interaction quench dynamics

TL;DR

This work investigates how dipolar many-body quantum systems relax after an interaction quench, contrasting bosonic and fermionic dipoles in a 1D harmonic trap. By solving the full time-dependent Schrödinger equation with variational MCTDH-X/F methods and tracking multiple many-body entropy measures, the authors identify GOE-like statistical relaxation for strong quenches in dipolar bosons, accompanied by Hilbert-space delocalization and orbital fragmentation, while dipolar fermions remain nonrelaxed under the same protocol. Relaxation in bosons occurs when all entropy measures saturate to GOE-predicted values, signaling chaotic spreading in Fock space and a relaxation time set by $1/M$-type fragmentation. The study highlights a strong statistics-dependent (nonuniversal) relaxation behavior in long-range interacting systems and suggests experimental exploration with ultracold dipolar gases, with extensions to $1/r^{\alpha}$ interactions.

Abstract

We study the out-of-equilibrium dynamics of dipolar bosons and fermions after a sudden change in the interaction strength from zero to a finite repulsive value. We simulate the interaction quench on the initial state which is the ground state of harmonic potential with noninteracting bosons and fermions. We solve the time-dependent many-boson Schrödinger equation exactly using numerical methods. To understand the many-body dynamics we analyze several measures of many-body information entropy, monitoring their time evolution and assessing their dependence on interaction strength. We establish that for weak interaction quench the dynamics is statistics independent, both dipolar bosons and fermions do not relax. Whereas it is significantly different for dipolar bosons from that of dipolar fermions in the stronger interaction quench. When dipolar bosons exhibit concurrent signature of relaxation in all entropy measures, dipolar fermions fail to relax. For dipolar bosons and for larger interaction quench, the many-body information entropy measures dynamically approach the value predicted for the Gaussian orthogonal ensemble of random matrices, implying statistical relaxation. The relaxation time is uniquely determined when the orbital fragmentation exhibits a $1/M$ population in each orbital ($M$ is the number of orbitals) and all entropy measures saturate to the maximum entropy values. The relaxation time also becomes independent of the strength of dipolar interaction. Whereas, for the same quench protocol, dipolar fermions exhibit modulated oscillations in all entropy dynamics. Our study is also complemented by the measures of delocalization in Hilbert space, clearly establishing the onset of chaos for strongly interacting dipolar bosons. It highlights the importance of many-body effects with a possible exploration in quantum simulation with ultracold atoms.

Figures (8)

• Figure 1: Time evolution of four different entropy measures for $N=4$ quenched dipolar bosons in the weak interaction limit, $g_d=0.002,0.02,0.2$. a) coefficient entropy $S_C(t)$, b) occupation entropy $S_n(t)$, c) coefficient inverse participation ratio $I_C(t)$, d) many-body coefficient entropy $S_C^{N}(t)$. The overall observation is same for $g_d=0.002$ and $0.02$, i.e, the many-body state can be described in mean-field theory. For $g_d=0.2$, large fluctuations emerge in all quantities due to absence of strong correlations between the particles. The thin dashed lines are provided to guide the eye and to estimate the magnitude of fluctuations. See the text for details.
• Figure 2: Time evolution of four different entropy measures for $N=4$ quenched dipolar bosons in the strong interaction limit, $g_d=1.0, 1.5, 2.0$. a) coefficient entropy $S_C(t)$, the inset exhibits linear increase in entropy in short time, b) occupation entropy $S_n(t)$, c) coefficient inverse participation ratio $I_C(t)$, d) many-body coefficient entropy $S_C^{N}(t)$. Statistical relaxation happens as all the entropy measures manifest the convergence to GOE predictions. See the text for details.
• Figure 3: Configuration space population at time $t=100$, the distribution of the magnitude of the coefficients $\{ |C_{\vec{n}}(t)|^{2}\}$ as a function of index $n$ for different interaction quench. The index $n$ is computed from vector $\vec{n}$ using the mapping described in Ref. Streltsov:2010. Figs (a)-(d) represent the magnitude of the coefficients for (a) $g_d=0.002$, (b) $g_d=0.02$, (c) $g_d=0.2$, (d) $g_d=2$. The state stays rather localized for $g_d=0.002$ and $0.02$. With increasing in the interaction strength $g_d$, more coefficients in the expansion become significant. The state rapidly becomes delocalized.
• Figure 4: Orbital fragmentation dynamics for $N=4$ interacting dipolar bosons quenched to $g_d=2$ and computation is done with different orbitals. (a), $M=10$, in the inset, population in ten natural orbitals up to time $t=1$, (b) $M=9$, (c) $M=8$, (d) $M=7$. For all cases approximately at time $t \approx 20$, the population of the orbitals coalesces around the value $1/M$. See the text for details.
• Figure 5: Time evolution of four different entropy measures for $N=4$ quenched dipolar fermions in the weak interaction limit, $g_d=0.002,0.02,0.2$. a) coefficient entropy $S_C(t)$, b) occupation entropy $S_n(t)$, c) coefficient inverse participation ratio $I_C(t)$, d) many-body coefficient entropy $S_C^{N}(t)$. Possibility of relxation is ruled out. See the text for details.