Coherent Backscattering in Fock Space: A Signature of Quantum Many-Body Interference in Interacting Bosonic Systems

We predict a generic signature of quantum interference in many-body bosonic systems resulting in a coherent enhancement of the average return probability in Fock space. This enhancement is robust with respect to variations of external parameters even though it represents a dynamical manifestation of the delicate superposition principle in Fock space. It is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. Using a semiclassical approach based on interfering paths in Fock space, we calculate the magnitude of the backscattering peak and its dependence on gauge fields that break time-reversal invariance. We confirm our predictions by comparing them to exact quantum evolution probabilities in Bose-Hubbard models, and discuss their relevance in the context of many-body thermalization.

Figure 1

Illustration of coherent backscattering in Fock space. A many-body system, represented here by a Bose-Hubbard chain with $L=5$ sites and $N=6$ particles, is prepared in a well-defined initial Fock state $n^{(\mathrm{i})}=(n_1^{(\mathrm{i})},\dots ,n_L^{(\mathrm{i})})$ (upper part). After a given evolution time $t$, the final populations are found to be $n^{(\mathrm{f})}$ (lower parts). A solution $\gamma$ of the Gross-Pitaevskii equation joining $n^{(\mathrm{i})}$ with $n^{(\mathrm{f})}$ contributes with an amplitude $K_{\gamma }\simeq A_{\gamma }{e}^{i{R}_{\gamma }}$ to this process, where $R_{\gamma }$ is a classical action. Under averaging, only pairs of identical Fock space trajectories yield a systematically nonvanishing contribution to the probability $P=|\sum _{\gamma }{K}_{\gamma }{|}^2$ when $n^{(\mathrm{f})}\ne n^{(\mathrm{i})}$ (left column). For $n^{(\mathrm{f})}=n^{(\mathrm{i})}$ (right column), however, constructive interference additionally arises if trajectories are paired with their time-reversed counterparts (dot-dashed green line). This gives rise to a coherent enhancement of the probability to detect the system in the initial Fock state after the evolution, as compared to other states with comparable distributions of the population.

Figure 2

Average quantum (black diamonds) and classical (red crosses) evolution probabilities in Fock space for a Bose-Hubbard chain ($L=5$, $N=14$), at different evolution times $t/\tau =1.5\text{(a)}$, $2.5\text{(b)}$, $5\text{(c)}$, $10\text{(d)}$, $20\text{(e)}$, $50\text{(f)}$ with $\tau \equiv \hslash /J$. The average was performed over an ensemble of ${10}^3$ realizations of on-site energies ${\epsilon }_{\alpha }\in [0,10J]$, with interaction strength $U=4J$. The probabilities are displayed for the set of Fock states ${\mathbf{n}}^{\text{( f)}}$ having the same total interaction energy as ${\mathbf{n}}^{(\mathrm{i})}=(3,2,3,2,4)$ (marked by the vertical dashed line). While quantum-classical correspondence holds for $t\sim \tau$, quantum interference sets in and stabilizes for $t\gtrsim 4\tau$ (see the Supplemental Material [20]) yielding, in accordance with Eq. (14), a systematic enhancement of the quantum backscattering probability to ${\mathbf{n}}^{(\mathrm{f})}={\mathbf{n}}^{(\mathrm{i})}$ by about a factor 2 as compared to other final states ${\mathbf{n}}^{(\mathrm{f})}\ne {\mathbf{n}}^{(\mathrm{i})}$ and to its classical prediction.

Figure 3

Evolution probability $\overline{P}({\mathbf{n}}^{(\mathrm{f})},{\mathbf{n}}^{(\mathrm{i})},t)$ in Fock space averaged over random on-site energies ${\epsilon }_{\alpha }\in [0,W]$ for a Bose-Hubbard ring of $L=6$ sites (see inset). We show results for evolution times $t=10\tau$, hopping phases $\varphi =0$, $\pi /8$, $\pi /4$, $\pi /2$ (black diamonds, red upper triangles, green lower triangles, blue squares), and initial states ${\mathbf{n}}^{(\mathrm{i})}$ indicated by a vertical line. In (a) we have $N=17$ particles, with interaction strength $U=4J$ and $W=10J$. The lower (more “quantum”) panel (b) has $N=7$, $U=J$, and $W=2J$. In both cases, the breaking of time-reversal invariance for $\varphi =\pi /8$, $\pi /4$ destroys the coherent enhancement of the backscattering probability to the initial state. In the semiclassical regime (a) the evolution probabilities globally agree with the classical prediction for $\varphi =0$ (red crosses), while they significantly exceed the latter in the quantum regime (b).