## A logarithmic signature

Some one-dimensional disordered interacting quantum systems have been theoretically predicted to display a property termed many-body localization (MBL), where the system retains the memory of its initial state and fails to thermalize. However, proving experimentally that something does not occur is tricky. Instead, physicists have proposed monitoring the entanglement entropy of the system, which should grow logarithmically with evolution time in an MBL system. Lukin *et al.* observed this characteristic logarithmic trend in a disordered chain of interacting atoms of rubidium-87. This method should be generalizable to other experimental platforms and higher dimensions.

*Science*, this issue p. 256

## Abstract

An interacting quantum system that is subject to disorder may cease to thermalize owing to localization of its constituents, thereby marking the breakdown of thermodynamics. The key to understanding this phenomenon lies in the system’s entanglement, which is experimentally challenging to measure. We realize such a many-body–localized system in a disordered Bose-Hubbard chain and characterize its entanglement properties through particle fluctuations and correlations. We observe that the particles become localized, suppressing transport and preventing the thermalization of subsystems. Notably, we measure the development of nonlocal correlations, whose evolution is consistent with a logarithmic growth of entanglement entropy, the hallmark of many-body localization. Our work experimentally establishes many-body localization as a qualitatively distinct phenomenon from localization in noninteracting, disordered systems.

Isolated quantum many-body systems maintain their initial global purity while undergoing unitary time evolution. However, the presence of interactions drives local thermalization: The coupling between any subsystem and its complement mimics the contact with a bath. This causes the subsystem’s degrees of freedom to be ultimately described by a thermal ensemble, even if the full system is in a pure state (*1*–*3*). A consequence of thermalization is that local information about the initial state of the subsystem gets scrambled and transferred into nonlocal correlations that are only accessible through global observables (*4*–*6*).

Disordered systems (*7*–*18*) can provide an exception to this paradigm of quantum thermalization. In such systems, particles can localize and transport ceases, which prevents thermalization. This phenomenon is called many-body localization (MBL) (*6*, *7*, *19*–*23*). Experimental studies have identified MBL through the persistence of the initial density distribution (*24*–*29*) and two-point correlation functions during transient dynamics (*25*). However, while particle transport is frozen, the presence of interactions gives rise to slow coherent many-body dynamics that generate nonlocal correlations, which are inaccessible to local observables (*30*–*32*). These dynamics are considered to be the hallmark of MBL and distinguish it from its noninteracting counterpart, called Anderson localization (*7*–*11*, *14*, *15*, *18*). Their observation, however, has remained elusive because it requires exquisite control over the system’s coherence.

We study these many-body dynamics by probing the entanglement properties of an MBL system with a fixed particle number (*30*–*34*). We distinguish two types of entanglement that can exist between a subsystem and its complement (Fig. 1A): (i) Number entanglement implies that the particle number in one subsystem is correlated with the particle number in the other. This type of entanglement is generated through tunneling across the boundary between the subsystems. (ii) Configurational entanglement implies that the configuration of the particles in one subsystem is correlated with the configuration of the particles in the other. It therefore requires the presence of at least one particle in each subsystem. Tunneling alone does not generate configurational entanglement, as it acts individually on each particle. Interactions, in contrast, can entangle pairs of particles. As a result, the combination of tunneling and interactions can lead to configurational entanglement at long distances.

The formation of particle and configurational entanglement changes in the presence or absence of interactions and disorder in the system (Fig. 1B). In thermal systems without disorder, interacting particles delocalize and rapidly create both types of entanglement throughout the entire system. In contrast, for Anderson localization, number entanglement builds up only locally at the boundary between the two subsystems. Here, the lack of interactions prevents the formation of a substantial amount of configurational entanglement. In MBL systems, number entanglement builds up in a similarly local way as for Anderson localization. However, notably, the presence of interactions additionally enables the slow formation of configurational entanglement throughout the entire system.

In this work, we realize an MBL system and characterize its key properties: breakdown of quantum thermalization, finite localization length of the particles, area-law scaling of the number entanglement, and slow growth of the configurational entanglement that ultimately results in a volume-law scaling. The first three properties are also present for an Anderson localized state; the slowly growing configurational entanglement qualitatively distinguishes our system from a noninteracting, localized state.

## Experimental system

In our experiments, we study MBL in the interacting Aubry-André model for bosons in one dimension (*35*, *36*), which is described by the Hamiltonian

Our experiments begin with a Mott-insulating state in the atomic limit with one ^{87}Rb atom on each site of a two-dimensional optical lattice (Fig. 2B). The system is placed in the focus of a high-resolution imaging system through which we project site-resolved optical potentials (*37*). We first isolate a single, one-dimensional chain from the Mott insulator and then add the site-resolved potential offsets *38*). This projects the many-body state onto the number basis, which consists of all possible distributions of the particles within the chain.

In some realizations, particle loss during the time evolution and imperfect readout reduce the number of detected atoms compared with the initial state, thereby injecting classical entropy into the system. We eliminate this entropy by postselecting the data on the intended atom number, thereby reaching a fidelity of

## Breakdown of thermalization

We first investigate the breakdown of thermalization in a subsystem that consists of a single lattice site. The conserved total atom number enforces a one-to-one correspondence between the particle number outcome on a single site and the number in the remainder of the system, entangling the two during tunneling dynamics. Ignoring information about the remaining system puts the subsystem into a mixed state of different number states. The associated number entropy is given by *38*). Because the atom number is the only degree of freedom of a single lattice site,

Counting the atom number on an individual lattice site in different experimental realizations allows us to obtain the probabilities

We perform measurements of *38*). For weak disorder, the measured entropy agrees with the predicted value, whereas the entropy is significantly reduced from the thermal value for strong disorder, signaling the absence of thermalization in the system. A reduced entropy implies that the subsystem does not occupy all available degrees of freedom and retains some memory of its initial conditions for arbitrarily long evolution times. We indeed find that the probability of retrieving the initial state of one atom per site increases for strong disorder (Fig. 2E, inset).

## Spatial localization

The breakdown of thermalization is expected to be a consequence of the spatial localization of the particles. Previous experiments have determined the decay length of an initially prepared density step into empty space (*27*). We measure the localization by directly probing density-density correlations within the system. These correlations are captured by

We measure the density-density correlations *38*). We eliminate the contribution coming from the disorder potential by defining *38*). We show the fitted localization length as a function of the disorder strength (Fig. 3C). For increasing disorder, the correlation length decreases from the entire system size down to around one lattice site.

Our observation of localized particles is consistent with the description of MBL in terms of local integrals of motion (*30*–*32*). It describes the global eigenstates as product states of exponentially localized orbitals. The correlation length extracted from our data is a measure of the size of these orbitals. Because the localized orbitals form a complete set of locally conserved quantities, this picture connects the breakdown of thermalization in MBL with nonthermalizing, integrable systems.

## Dynamics and spreading of entanglement

We now turn to a characterization of the entanglement properties of larger subsystems, starting with a subsystem covering half the system size. As for the case of a single lattice site, the particle number in the subsystem can become entangled with the number in the remaining system through tunneling dynamics, resulting in the number entropy *38*). An analogous relation exists for spin systems with conserved total magnetization instead of the particle number.

The dynamics of *30*–*34*).

In our experiment, we can independently probe both types of entanglement. We obtain the number entropy *39*, *40*). Here, we choose a complementary approach to probe the configurational entanglement in the system. It exploits the configurational correlations between the subsystems, quantified by the correlator (*38*)*38*). Our measurements lie within the numerically verified parameter regime. Even for large systems, one can find a regime of proportionality between C and

We study the time dynamics of *30*–*32*). The agreement of the long-term dynamics of

Considering the entropy in subsystems of different sizes gives us insights into the spatial distribution of entanglement in the system: In a one-dimensional system, locally generated entanglement results in a subsystem-size-independent entropy, whereas entanglement from nonlocal correlations causes the entropy to increase in proportion to the size of the subsystem. In reference to the subsystem’s boundary and volume, these scalings are called area law and volume law. We find almost no change in

## Conclusion

Our method, which is based on measurements of the particle number fluctuations and their configurations, can be generalized to higher dimensions and different experimental platforms, where a direct measurement of entanglement entropy remains challenging (e.g., trapped ions, neutral atoms, and superconducting circuits). In the future, experiments at larger system sizes will be of interest to shed light on the critical properties of the thermal-to-MBL phase transition, which are the subject of ongoing studies (*41*–*44*). In our system, it is experimentally feasible to increase the system size at unity filling to a numerically intractable regime. Additionally, the full control over the disorder potential on every site opens the way to studying the role of rare regions and Griffiths dynamics as well as the long-time behavior of an MBL state with a link to a thermal bath (*45*–*47*). Ultimately, these studies will further our understanding of quantum thermodynamics and whether such systems are suitable for future applications as quantum memories (*6*).

## Supplementary Materials

science.sciencemag.org/content/364/6437/256/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S12

Table S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We acknowledge discussions with I. Cirac, E. Demler, J. Eisert, C. Gross, W. W. Ho, H. Pichler, D. A. Huse, and A. Polkovnikov.

**Funding:**We are supported by grants from the National Science Foundation, the Gordon and Betty Moore Foundations EPiQS Initiative, an Air Force Office of Scientific Research MURI program, an Army Research Office MURI program, and the NSF Graduate Research Fellowship Program. J.L. acknowledges support from the Swiss National Science Foundation. V.K. acknowledges support from the Harvard Society of Fellows and the William F. Milton Fund.

**Author contributions:**A.L., M.R., R.S., M.E.T., and J.L. performed the experiment and collected and analyzed data. All authors contributed to discussion of the data, development of the theoretical concepts, and writing of the manuscript. M.G. supervised the project.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**Published data are available on the Zenodo public database (

*48*).