Violent relaxation in quantum fluids with long-range interactions

Violent relaxation is a process that occurs in systems with long-range interactions. It has the peculiar feature of dramatically amplifying small perturbations, and rather than driving the system to equilibrium, it instead leads to slowly evolving configurations known as quasistationary states that fall outside the standard paradigm of statistical mechanics. Violent relaxation was originally identified in gravity-driven stellar dynamics; here, we extend the theory into the quantum regime by developing a quantum version of the Hamiltonian mean field (HMF) model which exemplifies many of the generic properties of long-range interacting systems. The HMF model can either be viewed as describing particles interacting via a cosine potential, or equivalently as the kinetic $XY$ model with infinite-range interactions, and its quantum fluid dynamics can be obtained from a generalized Gross-Pitaevskii equation. We show that singular caustics that form during violent relaxation are regulated by interference effects in a universal way described by Thom's catastrophe theory applied to waves and this leads to emergent length scales and timescales not present in the classical problem. In the deep quantum regime we find that violent relaxation is suppressed altogether by quantum zero-point motion. Our results are relevant to laboratory studies of self-organization in cold atomic gases with long-range interactions.

Figure 1

Nonequilibrium phase diagram for the quantum HMF model with repulsive interactions ($\epsilon >0$). Violent relaxation leads to quasistationary states which are very slowly evolving nonequilibrium configurations; for the HMF model these are the bicluster states. On this semilog plot, the vertical axis measures the initial deviation from equilibrium quantified by $v_0=A{\omega }_{\text{pl}}cos\theta$ which is the initial velocity field due to small plasma oscillations [see Eq. (D1)]. The horizontal axis captures quantum effects via the effective Planck constant $\chi =\hslash /\sqrt{\left|\epsilon \right|m{R}^2}$. When $\chi \ll 1$ and $\chi \lesssim 2A{\omega }_{\text{pl}}$ the system forms a bicluster at late times [see Fig. 5]. By contrast, for $\chi \lesssim 1$, the $k=\pm 1$ modes dominate and the system continues to undergo plasma oscillations without relaxing [see Eq. (11) and Fig. 5]. Finally, for $\chi \gtrsim 1$ all Fourier modes experience a free Schrödinger-type dispersion relation and violent relaxation is suppressed by quantum zero-point motion [see Fig. 5]. The gray region denotes initial conditions that invalidate the short-time linear response procedure detailed in Eqs. (14) and (15).

Figure 2

Long-range interactions tend to amplify initial perturbations. We illustrate this feature here with the Newtonian trajectories of 52 particles obeying the classical HMF model with (a) attractive ($\epsilon <0$) and (b) repulsive ($\epsilon >0$) interactions. At $t=0$, the particles are spaced evenly around the ring with very slightly varying initial velocities $v_i(\theta ,0)=0.005cos\theta$. In both cases, the LRI cause the particles to cluster and this behavior repeats such that the envelopes of the trajectories form a quasiperiodic series of cusp-shaped caustics (or “chevrons”[75, 76]). However, there are key differences: First, the attractive interactions give rise to a single cluster point (monocluster) around the ring whereas repulsive interactions produce two cluster points (bicluster), and second there are very different timescales associated with the two cases with the repulsive case being much slower (note the different scales on the time axes).

Figure 3

Quantum Jeans instability: dynamics of the density profile $\rho (\theta ,\tau )$ for attractive interactions ($\epsilon <0$) with initial conditions ${\rho }_0={(1+0.01cos\theta )}^2$ and $v_0=0$ with $\chi ={10}^{-3}$. Interference tempers the (singular) classical caustic and replaces it with a characteristic interference pattern. The dynamical timescale is set by ${\omega }_{\text{pl}}\approx {\omega }_1=\text{Im}(i/\sqrt{2})\sqrt{1-{\chi }^2/2}$ [Eq. (11)].

Figure 4

Trajectories of test particles, each computed by solving Newton's equations for a particle moving in a given external potential $V(\theta ,\tau )=-M_{\chi }\left[\rho \left(\tau \right)\right]cos\theta$. This is exactly the mean field potential $\mathrm{\Phi }(\theta ,\tau )$ computed numerically in the quantum dynamics shown in Fig. 3, where $M_{\chi }\left[\rho \left(\tau \right)\right]$ is the self-consistent magnetization.

Figure 5

Dynamics of the density profile $\rho (\theta ,\tau )$ for $\epsilon >0$ in the $log\chi \text{−}A{\omega }_{\text{pl}}$ plane in correspondence with the different regimes plotted in Fig. 1. The initial conditions in all panels are ${\rho }_0=1/2\pi$ and $v_0=A{\omega }_{\text{pl}}cos\theta$ and time has been rescaled by $T_{\mathrm{bc}}$. For $\chi \gg 1$ no bicluster develops (a). For $A{\omega }_{\text{pl}}<\chi \ll 1$ (b) classical plasma oscillations occur, and a very weak $\pi$ periodic focusing occurs, but is dispersed by the quantum pressure before the classical focusing time $T_{\text{bc}}$. In (c)–(f) $\chi$ is held fixed while the plasma amplitude $A{\omega }_{\text{pl}}$ is tuned. This has the effect of a deeper effective potential $V_{\text{eff}}$ [see Eq. (15)], and consequently a more classical-like pattern emerges. This change of behavior is captured by the nonequilibrium phase diagram shown in Fig. 1, and is discussed at length in Sec. 6. (a) $\chi =5.0\times {10}^1$, $A{\omega }_{\text{pl}}=5.0\times {10}^{-3}$. (b) $\chi =5.0\times {10}^{-1}$, $A{\omega }_{\text{pl}}=5.0\times {10}^{-3}$. (c) $\chi =2.5\times {10}^{-3}$, $A{\omega }_{\text{pl}}=5.0\times {10}^{-3}$. (d) $\chi =2.5\times {10}^{-3}$, $A{\omega }_{\text{pl}}=1.3\times {10}^{-2}$. (e) $\chi =2.5\times {10}^{-3}$, $A{\omega }_{\text{pl}}=2.5\times {10}^{-2}$. (f) $\chi =2.5\times {10}^{-3}$, $A{\omega }_{\text{pl}}=5.0\times {10}^{-2}$.

Figure 6

The formation of a cusp catastrophe: illustration of how a line of initial data in phase space (i.e., a vanishingly thin water-bag distribution) is folded over by the dynamics. Projecting down onto the $\theta \text{−}t$ plane produces a cusp-shaped envelope on which the density of trajectories diverges. This cusp catastrophe is structurally stable against perturbations and hence occurs generically without the need for special initial conditions. In the case of the bicluster, this folding occurs simultaneously at two symmetric points around the ring.

Figure 7

Classical trajectories in two dimensions will generically form cusp-shaped caustics where the density of trajectories diverges, as shown in (a). In the wave or quantum theory interference removes the singularity and replaces it with a universal wave function, the Pearcey function $\mathrm{Pe}(a,b)$, which is valid in the immediate locale of the caustic. In (b) we plot $\left|\mathrm{Pe}(a,b)\right|$. Note that this function contains interesting subwavelength features such as vortices at its nodes. (a) Cusp catastrophe. (b) Pearcey function.

Figure 8

Quantum biclusters: dynamics of the density profile $\rho (\theta ,\tau )$ for repulsive interactions ($\epsilon >0$) with initial conditions ${\rho }_0=1/2\pi$ and $v_0=A{\omega }_{\text{pl}}cos\theta$. This simulation used $A{\omega }_{\text{pl}}=0.01$, $\chi =0.005$, and included Fourier components up to $k_{\text{max}}=58$. The timescale is in units of the classical bicluster formation time $T_{\text{bc}}=\mathcal{O}\left({\left[A{\omega }_{\text{pl}}\right]}^{-1}\right)$ which is two orders of magnitude larger than the inverse plasma frequency ${\omega }_{\text{pl}}^{-1}$ relevant in the attractive case.

Figure 9

Schematic depiction of noncommutativity of the thermodynamic $N\to \infty$, and classical $\hslash \to 0$ limits for the finite time dynamics of pure quantum states. The sequence $\hslash \to 0$ followed by $N\to \infty$ takes us from the quantum to classical many-body descriptions and then to the Vlasov equation which gives an exact description of the classical dynamics in the thermodynamic limit. Conversely, if the GGPE captures the leading order dynamical behavior in the thermodynamic limit, then the sequence $N\to \infty$ followed by $\hslash \to 0$ leads to the classical Euler equations, which emerge from the Vlasov equation in the zero-temperature approximation.