Full counting statistics of time-of-flight images

Inspired by recent advances in cold atomic systems and nonequilibrium physics, we introduce a characterization scheme, the time-of-flight full counting statistics. We benchmark this method on an interacting one-dimensional Bose gas and show that there the time-of-flight image displays several universal regimes. Finite momentum fluctuations are observed at larger distances, where a crossover from exponential to Gamma distribution occurs upon decreasing momentum resolution. Zero-momentum particles, on the other hand, obey a Gumbel distribution in the weakly interacting limit, characterizing the quantum fluctuations of the former quasicondensate. Time-of-flight full counting statistics is demonstrated to capture (pre-)thermalization processes after a quantum quench and can be useful for characterizing exotic quantum states such as many-body localized systems or models of holography.

Figure 1

Sketch of ToF experiment with quasi-one-dimensional Bose gas. At $t=0$ the atoms, initially confined to a tube of length $L$, are released form the trap. Interactions between the particles are typically short ranged and become quickly negligible due to the rapid expansion in transverse direction (not shown here). After propagation time $t$, the density profile of the expanded cloud is investigated by taking an absorption image at position $R$ with a laser beam of waist $\mathrm{\Delta }R$. Atoms expand freely after release from the trap, and the distribution of the measured intensity provides direct information on the structure of the initial quantum state.

Figure 2

Distribution of normalized intensity $\tilde{I}$ (symbols) at $T=0$ temperature, plotted for different momentum resolutions, $\mathrm{\Delta }\tilde{p}$. We used $K=10,\tilde{p}=15\times 2\pi$, and ${\xi }_h/L=0.002$. Solid lines are fits with the Gamma distribution from Eq. (10). The distribution smoothly evolves from exponential to Gamma as $\mathrm{\Delta }p$ increases, and reflects the two-mode squeezed structure of the Bogoliubov ground state in momenta $p$ and $-p$. Inset shows the parameter of the fitted Gamma distribution $\alpha$ as a function of $\mathrm{\Delta }\tilde{p}$.

Figure 3

PDF of the normalized variable $(\widehat{I}-\langle \widehat{I}\rangle )/\delta I$ for $p=mR/t=0$, plotted for different Luttinger parameters $K$, with $\delta I$ referring to standard deviation. We used periodic boundary conditions to compare with analytical results. For weak interactions ($K\gg 1$) the PDF converges to the Gumbel distribution (12) (solid line), also predicted by a particle-number-preserving Bogoliubov approach. We used $\mathrm{\Delta }\tilde{p}=0.1\times 2\pi$ and ${\xi }_h/L=0.002$.

Figure 4

Joint PDF of signals ${\tilde{I}}_0$ and ${\tilde{I}}_1$, evaluated for dimensionless momenta ${\tilde{p}}_0=0$ and ${\tilde{p}}_1=2\pi$, for strong ($K=2$) and weak ($K=10$) interactions at $T=0$ temperature. The anticorrelation, observable for any interaction strength, reflects particle-number conservation. Particles with nonzero wave numbers $p_1$, leave holes behind in the quasicondensate.

Figure 5

Finite-temperature distribution of the normalized zero-momentum intensity ${\tilde{I}}_{p=0}$, for different dimensionless temperatures $\tilde{T}=k_{B}T/\left(K\mathrm{\Delta }\right)$. The PDF crosses over from the zero-temperature Gumbel distribution, Eq. (12), to an exponential distribution, as a signature of the thermal depletion of the quasicondensate by the thermally populated $p\ne 0$ modes. The experimentally accessible temperature range, $T\sim 30–120\mathrm{nK}$ [58], corresponds to $\tilde{T}\sim 0.12–0.46$. We used $N=3500$ and $L=39\mu \mathrm{m}$ (density $\rho =90\mu {\mathrm{m}}^{-1}$), and a chemical potential $\mu /h=1.6\mathrm{kHz}$, implying $K\approx 77,c\approx 2,7\mathrm{mm}/\mathrm{s}$ and ${\xi }_h/L\approx 0.007$ for ${}^{87}\mathrm{Rb}$ atoms. We assumed $\mathrm{\Delta }\tilde{p}/\left(2\pi \right)=0.1$, corresponding to a time of flight $t=1\mathrm{s}$, and a real-space resolution $\mathrm{\Delta }R=12\mu \mathrm{m}$, but shorter times of flight can also be applied by using a focusing method, yielding similar images.

Figure 6

Distribution of the normalized zero-momentum intensity ${\tilde{I}}_{p=0}$ after an interaction quench, for different holding times, ${\tau }_h=1.3–177\mu \mathrm{s}$, measured in dimensionless units, ${\tilde{\tau }}_h={\tau }_h{c}_f/L$. Distributions are plotted for two different quenches of durations $\tau =0.71\mu \mathrm{s}$ (rapid), and $\tau =71\mu \mathrm{s}$ (slow). We have used $N=3684,L=39\mu \mathrm{m}$, and a chemical potential $\mu /h=1.6\mathrm{kHz}$, corresponding to $K_0=80,c_0=2.7\mathrm{mm}/\mathrm{s}$, and ${\xi }_h^0=0.27\mu \mathrm{m}$, and assumed an interaction quench to $K_f=7$, yielding $c_f=c_0{K}_0/K_f=30.8\mathrm{mm}/\mathrm{s}$. We assumed a modest momentum resolution, $\mathrm{\Delta }\tilde{p}/\left(2\pi \right)=0.1$. Similar to the finite-temperature thermalization plotted in Fig. 5, the PDF after a rapid quench crosses over from the equilibrium Gumbel distribution, Eq. (12), to an exponential distribution as ${\tilde{\tau }}_h$ increases, even though the number of excitations in the system remains constant after the quench. For a slower quench, the PDF for short holding times ${\tau }_h$ is much wider than the Gumbel distribution, showing that increasing interactions have time to deplete the quasicondensate during the quench protocol, resulting in larger particle-number fluctuations. As in the case of rapid quench, for larger holding times this PDF crosses over to a thermal distribution.

Figure 7

Expectation value $\langle {\widehat{I}}_{R,\mathrm{\Delta }R}\left(t\right)\rangle /N$ plotted as a function of dimensionless wave number $\tilde{p}/\left(2\pi \right)$ for different dimensionless temperatures $\tilde{T}$, with parameters $K=10,{\xi }_h/L=0.002$, and $\mathrm{\Delta }\tilde{p}=0.1\times 2\pi$, using logarithmic scales on both axis. As $T$ increases, the expectation value develops a wide flat region for small momenta $p\ll \hslash /{\xi }_T$. For larger momenta $\hslash /{\xi }_T\ll p\ll \hslash /{\lambda }_T$, the intensity shows a power-law decay $\sim 1/p^2$. This behavior can be explained by the exponential decay of two-point correlations in finite-temperature Luttinger liquids, Eq. (15), with correlation length ${\xi }_T$. For even larger momenta $p\gg \hslash /{\lambda }_T$, we get back the zero-temperature results, resulting in a crossover to the different power-law behavior $\langle {\widehat{I}}_{R,\mathrm{\Delta }R}\left(t\right)\rangle \sim 1/p$.

Figure 8

Distribution of normalized intensity $\tilde{I}$ (symbols) at $T=0$ for stronger interactions, and fits with the Gamma distribution from Eq. (10) (solid lines), plotted for different momentum resolutions $\mathrm{\Delta }\tilde{p}$. We used $K=2,\tilde{p}=15\times 2\pi$, and ${\xi }_h/L=0.002$. Similarly to the limit of weak interactions, the distribution smoothly evolves from exponential to Gamma as $\mathrm{\Delta }p$ increases. Inset shows the parameter of the fitted Gamma distribution $\alpha$ as a function of $\mathrm{\Delta }\tilde{p}$, increasing approximately linearly with the same slope as in the weakly interacting limit.

Figure 9

Finite-temperature distribution of the normalized zero-momentum intensity ${\tilde{I}}_{p=0}$ for strong interactions, using different dimensionless temperatures $\tilde{T}=k_{B}T/\left(K\mathrm{\Delta }\right)$. The PDF crosses over from the zero-temperature limit (deviating from Gumbel distribution due to strong interactions) to an exponential distribution, as a signature of the depletion of the zero mode by the thermally populated $p\ne 0$ modes. As in the limit of weak interactions, the crossover is governed by the dimensionless temperature $\tilde{T}$. We used $K=1.5,\mathrm{\Delta }\tilde{p}/\left(2\pi \right)=0.1$, and ${\xi }_h/L\approx 0.002$.