Signatures of Many-Body Localization in a Controlled Open Quantum System

In the presence of disorder, an interacting closed quantum system can undergo many-body localization (MBL) and fail to thermalize. However, over long times, even weak couplings to any thermal environment will necessarily thermalize the system and erase all signatures of MBL. This presents a challenge for experimental investigations of MBL since no realistic system can ever be fully closed. In this work, we experimentally explore the thermalization dynamics of a localized system in the presence of controlled dissipation. Specifically, we find that photon scattering results in a stretched exponential decay of an initial density pattern with a rate that depends linearly on the scattering rate. We find that the resulting susceptibility increases significantly close to the phase transition point. In this regime, which is inaccessible to current numerical studies, we also find a strong dependence on interactions. Our work provides a basis for systematic studies of MBL in open systems and opens a route towards extrapolation of closed-system properties from experiments.

Popular Summary

The behavior of an isolated quantum system follows one of two distinct paradigms. It can approach a thermal equilibrium state, where any initial quantum correlations spread throughout the system, rendering the system effectively classical. Alternatively, in the presence of disorder, a system can be what is known as “many-body localized” (MBL). This regime has recently received a lot of attention because here some quantum coherences remain local up to infinite times. However, experimental investigation of this novel state is complicated by unavoidable interference from the environment, which acts as a source of fluctuations (a “bath”) that eventually thermalizes the system. We develop a method to implement a controllable bath and present a systematic study of its effects on a MBL system.

In our experiment, we illuminate a charge-density pattern in an ensemble of ultracold potassium atoms (the MBL system) with nearly resonant light, and we investigate the system’s response. Here, the light intensity controls the coupling to the bath, and the charge-density pattern decays as a stretched exponential with a linearly dependent rate. Furthermore, we find that the susceptibility of the MBL system to the photon bath strongly increases when approaching the MBL transition, which is analogous to the effects of finite temperatures in the vicinity of a quantum phase transition.

Using control over the bath, our study is a first step toward extrapolating the behavior of truly isolated MBL systems from experimental measurements. The method for implementing a controllable photon bath also has applications in a variety of experiments on cold atoms, both within and outside of the context of MBL.

Figure 1

Schematic phase diagram of an open MBL system and illustration of the effects of a single-photon scattering event. (a) Coupling a disordered system to a thermal environment destroys MBL on a time scale inversely proportional to the coupling strength $\gamma$. Nonetheless, for sufficiently weak couplings, the characteristics of both the MBL (i.e., persisting density pattern) and the ergodic Griffiths phase (i.e., power-law decay of density pattern [19, 20, 21, 22]) survive sufficiently long to enable their experimental characterization. The shaded area represents the regime of weak dissipation in which the intrinsic dynamics of the phases can be discerned. Because of the diverging time scales at the critical point, the respective signatures become increasingly difficult to observe in its proximity. A sharp transition is only expected in the closed-system limit ($\gamma =0$), while finite dissipative couplings are expected to produce a crossover regime between the two characteristic dynamics, similar to the effect of temperature in ground-state quantum phase transitions. Black arrows indicate the regime that is considered in this work. (b) Schematic of a scattering event. An atom in an initial localized superposition of Wannier states (blue) becomes localized on the length scale of the scattered photon’s wavelength $\lambda /2\pi$, which, in our system, is less than the lattice constant $d$. This dephases superpositions to incoherent mixtures of single Wannier states in the ground band (yellow) and also produces a small population in higher bands (faint yellow), which can be seen as the result of position measurements with sublattice site resolution. Band excitations can then lead to atom loss since, in most excited bands, atoms are not trapped.

Figure 2

Time evolution of a charge-density wave in the presence of photon scattering. Upper panel: An initially prepared one-dimensional charge-density wave evolves in the presence of quasiperiodic disorder of strength $\mathrm{\Delta }=4J$ at $U=2J$ under the influence of varying scattering rates $\gamma$. Higher scattering rates result in shorter lifetimes of the imbalance. The finite imbalance lifetime at $\gamma =0$ is due to residual couplings between different 1D tubes [14] and off-resonant scattering of lattice photons, which are not included in the time-evolving block decimation (TEBD) simulation ($\gamma =0$) indicated by the gray shaded region. The dashed line extrapolates the simulation’s mean value. Each experimental data point is the average of six disorder phase realizations, with error bars denoting the standard error of the mean. Solid lines are stretched exponential fits, used to extract the imbalance decay rates (see Appendix pp9). The lower panel shows the corresponding time evolution of the normalized atom number fitted by simple exponentials.

Figure 3

Noninteracting susceptibility vs disorder strength: Susceptibilities for both the rate model and the experiment. Error bars indicate the fit uncertainty. The black dashed line indicates the upper bound of $\chi \le p_{\mathrm{dp}}$. The inset shows measured imbalance decay rates ${\mathrm{\Gamma }}_{\mathcal{I}}$ as a function of the scattering rate $\gamma$ at $\mathrm{\Delta }=3J$. We observe a linear behavior, the slope of which is directly given by $\chi$. We compare the data to the predictions of a rate model [31], indicated by the brown line, which is parallel to the fit through the experimental data. Hence, experiment and theory give the same susceptibility. The offset is caused by the constant background decay ${\mathrm{\Gamma }}_{\mathrm{bg}}$ in the experiment.

Figure 4

Measured susceptibilities at different disorder strengths versus interactions $U$. At finite interaction strengths, we compare our results to numerical TEBD simulations that do not include particle loss (triangles). The theoretical values at $U=0$ are calculated from the rate model discussed earlier (squares). We observe a strong interaction dependence of the experimental susceptibilities close to the phase transition ($\mathrm{\Delta }=4J$) but only a weak effect deep in the localized phase ($\mathrm{\Delta }=6J$). Error bars indicate the fit uncertainty. The solid lines are guides to the eye, and the gray shaded region indicates the statistical uncertainty of the TEBD simulations.

Figure 5

Level scheme of ${}^{40}\mathrm{K}$: Schematics of the atomic hyperfine levels and transitions relevant to the scattering of photons. Levels are labeled using both their quantum number $m_j$ and their respective hyperfine manifold $F$. Here, $F$ labels the manifold that the state is adiabatically connected to at low magnetic fields. The quantum number $m_i$ is, to a good approximation, not coupled to optical transitions as the system is close to the Paschen-Back regime. The two spin states $|m_I=-3⟩$ and $|-4⟩$ adiabatically connect to the $|m_F=-7/2⟩$ and $|-9/2⟩$ states at low magnetic fields. The dedicated scattering light is shown as red arrows; spontaneous emission processes are indicated as wavy lines, along with their branching ratios.

Figure 6

Band-excitation probabilities. (a) As a function of the maximum possible number of individual photons considered in a scattering burst. We plot the integrated band-excitation probability for bursts with up to $n$ photons. This integral quickly converges to the average band-excitation probability of a scattering burst. (b) Excitation probability as a function of the number of scattering bursts. The probability of staying in the ground band steadily decreases, while the probability of higher bands, which get lost, increases. At intermediate numbers of scattering bursts, a finite population of atoms builds up in the first longitudinally excited band, which is trapped. Time traces used to extract the imbalance decay rate in this work usually contain up to 10 scattering bursts per particle.

Figure 7

Spatial band structure of the lowest three bands of the longitudinal ($x$) lattice along the $x$ direction at $y$, $z=0$. The ground band is illustrated in black, the first longitudinally excited band in dark blue, and the second longitudinally excited band in light blue. The structure emerges because of the Gaussian shape of the dipole and lattice beams. The red line illustrates the pure trapping potential. The dashed horizontal lines illustrate that atoms in the second excited band can tunnel out of the system, but atoms in the first excited band remain trapped. The cloud size in the ground band is indicated by $w_g$, which corresponds to the $1/e^2$ width of a Gaussian fit. For the first excited band, the indicated size $w_{1\mathrm{st}}$ is derived from the band structure calculation, which agrees well with the result of an in situ measurement.

Figure 8

Atom number loss. (a) Fraction of atoms remaining in the ground or first excited bands vs time. The dashed line shows an exponential fit to extract the time scale. (b) Noninteracting atom number susceptibilities. Error bars show the uncertainty of the extracted value, including both the fitting procedure and the error in the initial data.

Figure 9

Imbalance and atom number decay rate as a function of the scattering rate. (a) Imbalance decay rate for various interaction strengths at $\mathrm{\Delta }=4J$. Our data are consistent with a linear behavior for all parameter values. The imbalance decay rates additionally show a background decay rate at ${\gamma }_{\mathrm{dp}}=0$, which heavily depends on both the interaction and the disorder strength. (b) Atom number decay rates as a function of the band-excitation rate. Again, we find a linear scaling.