Research Article

Quantum thermalization through entanglement in an isolated many-body system

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Science  19 Aug 2016:
Vol. 353, Issue 6301, pp. 794-800
DOI: 10.1126/science.aaf6725
  • Fig. 1 Schematic of thermalization dynamics in closed systems.

    An isolated quantum system at zero temperature can be described by a single pure wavefunction Embedded Image. Subsystems of the full quantum state are pure, as long as the entanglement between subsystems (indicated by the gray lines between the particles) vanishes (upper panels). If suddenly perturbed, the full system evolves unitarily, developing considerable entanglement between all parts of the system (lower panels). The bar graphs show the probability of an observable before and after perturbation of the system. Although the full system remains in a pure and in this sense zero-entropy state, the entropy of entanglement causes the subsystems to equilibrate, and local thermal mixed states appear to emerge within a globally pure quantum state.

  • Fig. 2 Experimental sequence.

    (A) Using tailored optical potentials superimposed on an optical lattice, we deterministically prepared two copies of a six-site Bose-Hubbard system, where each lattice site is initialized with a single atom. We reduced the lattice depth along x (specified in units of the lattice recoil energy Er) to enable tunneling and obtained either the ground state (adiabatic melt) or a highly excited state (sudden quench) in each six-site copy. After a variable evolution time, we froze the evolution and characterized the final quantum state by either acquiring number statistics or the local and global purity. Even and odd refer to the atom number parity. (B) Site-resolved number statistics of the initial distribution (left panel, showing a strong peak at one atom with vanishing fluctuations) and the distribution at later times (middle panel), compared with the predictions of a canonical thermal ensemble (red bars) of the same average energy as the quenched quantum state [J/(2π) = 66 Hz; U/(2π) = 103 Hz]. Error bars are SEM. Measurements of the global many-body purity show that it is static and high (right panel). This is in contrast to the vanishing global purity of the canonical thermal ensemble, yet this same ensemble accurately describes the local number distribution that we observed. (C) To measure the atom number locally, we allowed the atoms to expand in half-tubes along the y direction while pinning the atoms along x. In separate experiments, we applied a many-body beam splitter by allowing the atoms in each column to tunnel in a projected double-well potential. The resulting atom number parity (even or odd) on each site encodes the global and local purity.

  • Fig. 3 Dynamics of entanglement entropy.

    Starting from a low-entanglement ground state, a global quantum quench leads to the development of large-scale entanglement between all subsystems. We quenched a six-site system from the Mott insulating product state (J/U ≪ 1) with one atom per site to the weakly interacting regime of J/U = 0.64 [J/(2π) = 66 Hz] and measured the dynamics of the entanglement entropy. Shown are the dynamics for (A) one-, (B) two-, and (C) three-site subsystems and (D) the full system. As it equilibrates, a subsystem acquires local entropy, whereas the entropy of the full system remains constant and at a value given by measurement imperfections (D). The measured dynamics are consistent with exact numerical simulations (24) with no free parameters (solid lines). Error bars are SEM. For the largest entropies encountered in the three-site subsystem shown in (C), the large number of populated microstates leads to a significant statistical uncertainty in the entropy, which is reflected in the upper error bar extending to large entropies or being unbounded (24). The inset in (A) shows the slope of the early time dynamics, extracted from (A) to (C) with a piecewise linear fit (24). The dashed line is the mean of these measurements.

  • Fig. 4 Thermalized many-body systems.

    After the quench, the many-body state reaches a thermalized regime with saturated entanglement entropy. (A) In contrast to the ground state, for which the Rényi entropy only weakly depends on subsystem size, the entanglement entropy of the saturated quenched state grows almost linearly with size. As the subsystem size becomes comparable to the full system size, the subsystem entropy bends back to near zero, reflecting the globally pure zero-entropy state. For small subsystems, the Rényi entropy in the quenched state is nearly equal to the corresponding thermal entropy from the canonical thermal ensemble density matrix. (B) The mutual information IAB = SA + SBSAB quantifies the amount of classical (statistical) and quantum correlations between subsystems A and B (gray region). For small subsystems, the thermalized quantum state has SA + SBSAB, thanks to the near-volume law scaling (red arrow), leading to vanishing mutual information. When the volume of AB approaches the system size, the mutual information will grow because SA + SB exceeds SAB. (C) Mutual information IAB versus the volume of AB for the ground state and the thermalized quenched state. For small system sizes, the quenched state exhibits smaller correlations than the adiabatically prepared ground state, and the mutual information is nearly vanishing (red arrow). When probed on a scale near the system size, the highly entangled quenched state exhibits much stronger correlations than the ground state. Throughout this figure, the entanglement entropies from the last time point in Fig. 3 are averaged over all relevant partitionings with the same subsystem volume; we have also corrected for the extensive entropy unrelated to entanglement (24). All solid lines represent numerical calculations with no free parameters (24).

  • Fig. 5 Observation of local thermalization.

    (A) After quenching to J/U = 2.6, the saturated average particle number on each site (density) is nearly equal among the sites of the system, which resembles a system at thermal equilibrium. By comparison, the ground state for the same Bose-Hubbard parameters has appreciable curvature. (B) In measuring the probabilities of observing a given particle number on a single site, we can obtain the local single-site density matrix and observe the approach to thermalization. Using two different metrics—trace distance and fidelity—we compare the observed state to the mixed state derived from the subsystem of a canonical thermal ensemble after a quench to J/U = 0.64. The trace distance provides an effective distance between the mixed states in Hilbert space, whereas the fidelity is an overlap measure for mixed states. The two metrics illustrate how the pure state subsystem approaches the thermal ensemble subsystem shortly after the quench. The starting value of these quantities is given by the overlap of the initial pure state with the thermal mixed state. Solid lines connect the data points.

  • Fig. 6 Local observables in a globally pure quenched state.

    (A) In a quench, the ground state |g>〉 of the initial Hamiltonian (represented in its eigenbasis in the first panel) is projected onto many eigenstates |n>〉 of the new Hamiltonian. The full quantum state undergoes unitary evolution according to the eigenstate amplitudes and energies, cn and En, respectively. [left angle bracket]E[right angle bracket] denotes the full system energy expectation value; ℏ is the reduced Planck’s constant. According to the ETH, the expectation value of observables at long times can be obtained from a diagonal ensemble (illustrated by the probability weights in the eigenstates of the quenched Hamiltonian), as well as from a microcanonical ensemble. (B) Along with the microcanonical ensemble, several other closely related ensembles (colored lines) are compared to the data . The dashed line indicates the expectation value of the full system energy. (C) Thermalization of local observables. For the different temperatures and subsystems shown, the measured number statistics are in excellent agreement with microcanonical and canonical thermal ensembles, verifying the thermal character of the local density matrix (24). A grand-canonical ensemble reproduces the data very well, as long as the subsystem is small compared with the full system. The error bars are the standard deviation of our observations over times between 10 and 20 ms. (D) Thermalization occurs even for global quantities such as the full-system interaction energy Embedded Image. The thermalization dynamics as calculated from our number-resolved images are in close agreement with exact numerical simulation and a canonical prediction (24). Error bars are SEM.

Supplementary Materials

  • Quantum thermalization through entanglement in an isolated many-body system

    Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Markus Greiner

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