Relaxation towards negative temperatures in bosonic systems: Generalized Gibbs ensembles and beyond integrability

Motivated by the recent experimental observation of negative absolute temperature states in systems of ultracold atomic gases in optical lattices [Braun et al., Science 339, 52 (2013)], we investigate theoretically the formation of these states. More specifically, we consider the relaxation after a sudden inversion of the external parabolic confining potential in the one-dimensional inhomogeneous Bose-Hubbard model. First, we focus on the integrable hard-core boson limit, which allows us to treat large systems and arbitrarily long times, providing convincing numerical evidence for relaxation to a generalized Gibbs ensemble at negative temperature $T<0$, a notion we define in this context. Second, going beyond one dimension, we demonstrate that the emergence of negative temperature states can be understood in a dual way in terms of positive temperatures, which relies on a dynamic symmetry of the Hubbard model. We complement the study by exact diagonalization simulations at finite values of the on-site interaction.

Figure 1

Graphical illustration of the quench dynamics studied in this article. Starting from the ground state of the system at $J=0$ (a product state), we study the quenches towards $J=1$ combined with either $V\to V$ or $V\to -V$. Both quenches are dual to each other: while $V>0$ results in a final state at $T>0$ (top), negative absolute temperatures emerge for $V<0$ (bottom).

Figure 2

Time evolution of the particle density for short times (upper plot) and long times (lower plot), showing $N=200$ bosons on $L=800$ lattice sites and $\left|V\right|=50/L^2$. Due to Eq. (6), the dynamics of the particle density for $+V$ and $-V$ is the same and, apart from that, also identical for hard-core bosons and spinless fermions. While the motion for short times appears to be coherent to a large extend, it becomes incoherent in the long-time limit, where a quasistationary distribution emerges (for a comparison with thermal reference ensembles, see Fig. 5).

Figure 3

Momentum distribution functions as defined in Sec. 2 after the quench from $J=0$ to $J=1$ and $V\to V$ (upper plot) and $V\to -V$ (lower plot) for $N=200$ hard-core bosons on $L=800$ lattice sites and $\left|V\right|=50/L^2$. Starting from a constant momentum distribution in the initial state (infinite temperature), a sharp peak around momentum zero or $\pi$, respectively, builds up on a short time scale, which broadens at longer times. An enhancement of momentum states centered around zero ($\pi$) indicates positive (negative) absolute temperatures in the long time limit (see Fig. 5 for the a comparison with equilibrium ensembles).

Figure 4

Dynamic symmetry and its breakdown: scaling plot of the total kinetic energy per particle as a function of time over particle number for different initial conditions: $J_{\mathrm{init}}=0$ (blue circles), $J_{\mathrm{init}}=0.2$ (red x symbols), and $J_{\mathrm{init}}=0.4$ (green “+” symbols). The lower curves correspond to the quench to $J=1,V\to V$ and show approximately equilibration to a state which can be associated to positive temperatures in the GGE in the sense discussed in the text (dashed lines: reference values independently obtained from the GGE). The upper curves correspond to $J=1,V\to -V$ and (approximately) equilibrate in the same sense to negative temperature GGEs. Different system sizes $N=50,100$ are shown, where the compression ${\left(4N\right)}^2\left|V\right|=50$ is kept constant. At short times, the data collapse upon rescaling to a surprising degree.

Figure 5

Momentum distributions of $N=50$ hard-core bosons on $L=200$ lattice sites in the final state after the quench $J=0\to J=1$ and $V\to V>0$ (left) or $V\to V<0$ (right), $\left|V\right|=50/L^2$. We compare the momentum distribution of the final state, averaged over times $1000J^{-1}<t<2000J^{-1}$ (black solid curve) with the Gibbs ensemble (GE, blue circles), where $\beta J=\pm 0.85$ and $\mu =\pm 0.80J$ are calculated from energy and particle number conservation. We also compare the distributions to the generalized Gibbs ensemble (GGE, red x symbols). As predicted by the dynamic symmetry, the final momentum distributions for $V\to \pm V$ coincide upon shifting all momenta by $\pi$. The discrepancies between the GGE and the long-time average have been previously observed in Ref. [20] and were attributed to finite size effects.

Figure 6

Density profile of the long-time limit after the quench from $J=0$ to $J=1$ and $V\to \pm V$, showing the same parameters and ensembles as in Fig. 5. The particle densities for $\pm V$ have been simulated and found to be identical as expected from the dynamic symmetry. The long time average (black solid line) coincides with the GGE prediction (red x symbols) to great accuracy, while deviations from the thermal Gibbs ensemble (blue circles) are pronounced.

Figure 7

Lagrange multipliers ${\lambda }_n$ as a function of their index $n$ of the the generalized Gibbs ensemble (GGE). The blue and rising (red and sloping) curves correspond to positive (negative) temperature states, reached after the quench $J:0\to 1$ and $V\to V$ ($V\to -V$). In order to compare the system to the Gibbs ensemble (GE) we also plot $(E_n-\mu )/T$ as a function of $n$. The vector $(\vec{E}-\mu \mathbf{1})/T$ can be understood as a projection of $\vec{\lambda }$ on the subspace spanned by $\vec{E}$ and $\mathbf{1}$, as discussed in the main text.

Figure 8

Momentum distribution functions for $N=4$ particles on $L=16$ lattice sites (trapping potential $\left|V\right|=50/L^2$). The rows correspond to various initial interaction strengths $U/J=40,20,10,4$, whereas the columns correspond to the quenches $J:0\to 1$ combined with quenches from $(U,V)$ to $(U,V)$ (left), $(-U,-V)$ (middle), and $(U,-V)$ (right). While the first two quenches are characterized by peaks around zero and $\pi$, the latter case is expected to be unstable and shows a washed-out momentum distribution. Note that when $\left|U\right|$ is much larger than all other energy scales, differences between the last two columns become smaller, i.e., both quenches approach the hard-core limit for $\left|U\right|\gg J$.