Abstract
When a system thermalizes it loses all memory of its initial conditions. Even within a closed quantum system, subsystems usually thermalize using the rest of the system as a heat bath. Exceptions to quantum thermalization have been observed, but typically require inherent symmetries^{1,2} or noninteracting particles in the presence of static disorder^{3,4,5,6}. However, for strong interactions and high excitation energy there are cases, known as manybody localization (MBL), where disordered quantum systems can fail to thermalize^{7,8,9,10}. We experimentally generate MBL states by applying an Ising Hamiltonian with longrange interactions and programmable random disorder to ten spins initialized far from equilibrium. Using experimental and numerical methods we observe the essential signatures of MBL: initialstate memory retention, Poissonian distributed energy level spacings, and evidence of longtime entanglement growth. Our platform can be scaled to more spins, where a detailed modelling of MBL becomes impossible.
Main
It is exceedingly rare in nature for systems to localize, or retain local information about their initial conditions at long times. In an important counterexample, Anderson demonstrated that localization can arise due to the presence of disorder, which can destructively scatter propagating waves and prevent transport of energy or particles^{3}. Although this interference effect can be applied to generic quantum systems, most experimental work has been restricted to the narrow parameter regime of low excitation energies and no interparticle interactions^{4,5,6}.
Whether such localization persists in the more general case of arbitrary excitation energy and nonzero interparticle interactions was theoretically explored by Anderson^{3}, and more recently by others^{7,8,9,10}. This MBL phase is predicted to emerge for a broad set of interaction ranges and disorder strengths, although the precise phase diagram is not well known^{11} because equilibrium statistical mechanics breaks down in the MBL phase and numerical simulations are limited to ∼20 particles^{8,9}. Very recent experiments searching for MBL have measured constrained mass transport and the breakdown of ergodicity in disordered atomic systems with interactions^{12,13}.
Here we report the direct observation of MBL in a longrange transverse field Ising model with programmable, random disorder. This is a nonintegrable model that cannot be mapped to noninteracting particles (a necessary condition for MBL^{7}) and we can easily tune the disorder strength and interaction range over a parameter space that exhibits this phenomenon. Our experiment is effectively a closed quantum system over the timescales of interest, because the system localizes approximately 60 times faster than the coupling rate to the outside environment (see Supplementary Information).
Each of the effective spin1/2 particles is encoded in the ^{2}S_{1/2} F = 0, m_{F} = 0〉 and F = 1, m_{F} = 0〉 hyperfine ‘clock’ states of a ^{171}Yb^{+} ion, denoted ↓〉_{z} and ↑〉_{z}, respectively. We confine a chain of ten ions in a linear radiofrequency Paul trap and apply optical dipole forces to generate the effective spin–spin coupling^{14} of a disordered Ising Hamiltonian:
where σ_{i}^{γ} (γ = x, z) are the Pauli matrices acting on the ith spin, J_{i, j} is the coupling strength between spins i and j, B is a uniform effective transverse field, D_{i} is a sitedependent disordered potential, and ℏ = 1 (see Supplementary Information). After the chain evolves for some time, we collect the statedependent fluorescence on an intensified chargecoupled device camera for siteresolved imaging. This, in addition to our ability to perform highfidelity rotations, allows measurement of the singlesite magnetization 〈σ_{i}^{γ}〉(γ = x, y, z) as well as arbitrary spin correlation functions along any direction.
J_{i, j} is a tunable, longrange coupling that falls off approximately algebraically as J_{i, j} ∝ J_{max}/ i − j ^{α} (ref. 15), where J_{max} is typically 2π (0.5 kHz). Here, we tune α between 0.95 and 1.81, although for most of the data α ≈ 1.13. We directly measure the complete spin–spin coupling matrix (Fig. 1a), demonstrating the longrange interactions required to exhibit MBL in this model. Moreover, it has been shown numerically that (1) exhibits MBL for the experimental parameters^{16}.
The sitespecific programmable disorder term D_{i} is sampled from a uniform random distribution with D_{i} ∈ [−W, W], where W characterizes the strength of the disorder. The disorder is generated by sitedependent laserinduced Stark shifts (see Supplementary Information), which also allow for preparation of the system into any desired product state. To ensure we observe the general behaviour of the disordered Hamiltonian, we average over 30 distinct random instances of disorder (Fig. 1b, c), which leads to a sampling error that is smaller than the features of interest (see Supplementary Information).
An important signature of the MBL phase is manifested in the spectral statistics of adjacent energy levels of the Hamiltonian. In the thermalizing phase, the energy levels are given by the eigenvalues of a random matrix, a matrix whose elements are given by a random distribution, due to level repulsion. However, in the MBL phase, this level repulsion is greatly suppressed because eigenstates typically differ by multiple spins flips. As a result, the level spacing between adjacent energy eigenvalues are Poissondistributed^{8,9}. Using our directly measured spin–spin couplings and applied realizations for the strongest experimental disorder W = 8J_{max} and B = 4J_{max}, we calculate the distribution of adjacent energy level splittings and find them to be Poissondistributed, as expected for a MBL state (Fig. 1d).
Before searching for evidence of localization in the system’s time evolution, we first find parameters that cause the measured state to thermalize in the absence of disorder. We increase the transverse field B and look for conditions that result in the singlesite magnetization along two orthogonal directions approaching and remaining at their thermal equilibrium values (see Supplementary Information).
Figure 2a shows the measured dynamics of 〈σ_{i}^{z}〉 for B = 4J_{max} and D_{i} = 0 with the spins initialized in the Néel ordered state, ↑↓↑↓↑↓↑↓↑↓〉_{z} along the z direction. This configuration has an energy equivalent to an infinite temperature thermal state, because the expectation value of the Hamiltonian is zero. At long times, each expectation value 〈σ_{i}^{z}〉 approaches zero, losing memory of the initial ordering. As the transverse field B is increased, the system appears to thermalize more quickly and the level statistics approach those of random matrices rather than Poissonians, as expected for a generic thermodynamic system (see Supplementary Information).
When B ≫ J, the Hamiltonian is effectively an XY model^{17,18} and conserves ∑ _{i}σ_{i}^{z}, because Ising processes that flip spins along the large field are energetically forbidden. Thus, being in a spin configuration with half of the spins up and half of the spins down maximizes the accessible energy states. In addition, the Néel state is never an eigenstate, even for B ≫ J and W ≫ J, because the uniform B field at each site still allows spin exchange in the zbasis.
If a system is thermal, the Eigenstate Thermalization Hypothesis (ETH) provides a general framework where observables reach the value predicted by the microcanonical ensemble^{19,20,21}. This allows us to calculate the expected thermal value of the reduced density matrix given the Hamiltonian and an initial state (see Supplementary Information). To further establish that the system is thermalizing, we measure the reduced density matrix for each spin, ρ_{i} = Tr_{{j≠i}}ρ, without applied disorder and B = 4J_{max}, as shown in Fig. 2a. In our experiment, the spins are initially prepared in a product state with high fidelity. However, at long times, the measured reduced density matrices show that each of the spins is very close to the zero magnetization mixed state, implying the system has locally thermalized.
We apply the random disordered potential, D_{i} ≠ 0, with B = 4J_{max} and observe the emergence of MBL as we increase the strength of disorder. Because the manybody eigenstates in the MBL phase are not thermal, transport of energy and spins is suppressed, and ETH fails. Thus, observables will not relax to their thermal values^{9} and memory of the initial conditions will be evident in the singlesite magnetization. When starting in the Néel ordered state, Fig. 2b–f shows the time evolution of 〈σ_{i}^{z}〉 for different disorder strengths. The frozen moments of the spins increase with increasing disorder as the emergent integrals of motion become more strongly localized^{10}.
With the maximum applied disorder, W = 8J_{max}, we measure the singlespin reduced density matrix for the initial state and the averaged matrix for J_{max}t ≥ 5. In this case, localization of the spins leads to a marked difference in the measured and thermal reduced density matrices, indicating memory of the system’s initial conditions and a breakdown of ETH.
To quantify the localization, we measure the normalized Hamming distance (HD)^{22}:
which gives the number of spin flips away from the initial state, ψ_{0}, normalized by the length of the chain, N. At long times, the HD approaches 0.5 for a thermalizing state and remains at 0 for a fully localized state. In Fig. 3a, we measure that the longtime HD is 0.5 in the absence of disorder, and becomes smaller as the disorder strength is increased and the system more strongly localizes. This time evolution agrees with exact diagonalization numerical calculations for (1) (see Supplementary Information).
Figure 3b shows that for finite but weak disorder, the timeaveraged HD for J_{max}t > 5 is essentially unchanged, indicating weak or no localization. However, once the random field is sufficiently strong we observe a crossover from a thermalizing to a localized state. Once in this regime, the system becomes more localized with increasing disorder strength.
There is great theoretical interest in mapping the MBL phase diagram with respect to interaction range and disorder strength^{11,22,23}. We have taken the first steps towards this goal by measuring a change in the timeaveraged HD for W = 8J_{max} and J_{max}t > 5 as we adjusted the interaction range, 0.95 < α < 1.81 (Fig. 3c). For shorterrange interactions, the system appears more localized, because the state approaches a fully localized Anderson insulator as α → ∞. This change in timeaveraged HD with a change in interaction range makes clear that the longrange couplings are playing a role in the observed dynamics, thus indicating the observed effect is a manybody phenomenon.
Although there are predictions of a manybody delocalization transition at α = 1.5 (ref. 23), we did not observe this effect as we tuned α across this boundary. The lack of a sharp transition, along with the presence of MBL states for α < 1, may be due to finite size effects (see Supplementary Information). As this system is scaled to many dozens of spins, it will allow better study of the phase transition and mapping of the phase boundary in a regime where numerics are intractable.
A hallmark of MBL is the characteristic growth of entanglement under coherent time evolution^{24}, although its experimental observation has been elusive so far. In Anderson insulators without manybody interactions, the entanglement production from weakly entangled initial states shows a quick saturation after a sharp transient regime. However, in MBL systems a longtime growth sets in, which is logarithmically slow for shortrange interactions^{25} and can become algebraic with powerlaw interactions^{26}.
This entanglement growth can be measured using a suitable witness operator or even full state tomography^{27}. We instead indirectly characterize the entanglement growth in this system by measuring the quantum Fisher information (QFI)^{28,29,30}. The QFI gives a lower bound on the entanglement in the system while requiring only a measurement of twobody correlators, which can be efficiently accessed with our siteresolved imaging. Importantly, the QFI is able to distinguish MBL from single particle localization through the anticipated characteristic entanglement growth (see Supplementary Information). With no applied disorder, we observe a fast initial growth of the QFI following a Lieb–Robinson bound^{17,18} as the correlations propagate through the system, but no further growth afterwards (Fig. 4a). In contrast, for the cases of applied disorder of W = 6J_{max} and W = 8J_{max}, the further growth of the QFI is consistent with a logarithmic increase of entanglement at long times in a MBL state (Fig. 4b), but absent for single particle localized systems.
Our experimental platform is well suited for studying deep and intractable questions about thermalization and localization in quantum manybody systems. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems^{24}.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank L. Duan, Z.X. Gong, T. Grover, C. Laumann, S. Wang, N. Yao and J. Zhang for helpful discussions. This work is supported by the ARO Atomic and Molecular Physics Program, the AFOSR MURI on Quantum Measurement and Verification, the IARPA MQCO Program, and the NSF Physics Frontier Center at JQI. M.H. acknowledges the Deutsche Akademie der Naturforscher Leopoldina (grant No. LPDS 201307 and LPDR 201501) and P.H. acknowledges the EU IP SIQS, the SFB FoQuS (FWF Project No. F4016N23) and the ERC synergy grant UQUAM.
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J.S., A.L., P.R., B.N., P.W.H. and C.M. performed the experimental work and P.H., M.H. and D.A.H. performed the theoretical work. All authors contributed to the preparation of this manuscript.
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Smith, J., Lee, A., Richerme, P. et al. Manybody localization in a quantum simulator with programmable random disorder. Nature Phys 12, 907–911 (2016). https://doi.org/10.1038/nphys3783
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