Ergodic dynamics and thermalization in an isolated quantum system

Journal name:
Nature Physics
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Published online

Statistical mechanics is founded on the assumption that all accessible configurations of a system are equally likely. This requires dynamics that explore all states over time, known as ergodic dynamics. In isolated quantum systems, however, the occurrence of ergodic behaviour has remained an outstanding question1, 2, 3, 4. Here, we demonstrate ergodic dynamics in a small quantum system consisting of only three superconducting qubits. The qubits undergo a sequence of rotations and interactions and we measure the evolution of the density matrix. Maps of the entanglement entropy show that the full system can act like a reservoir for individual qubits, increasing their entropy through entanglement. Surprisingly, these maps bear a strong resemblance to the phase space dynamics in the classical limit; classically, chaotic motion coincides with higher entanglement entropy. We further show that in regions of high entropy the full multi-qubit system undergoes ergodic dynamics. Our work illustrates how controllable quantum systems can investigate fundamental questions in non-equilibrium thermodynamics.

At a glance


  1. Pulse sequence and the resulting quantum dynamics.
    Figure 1: Pulse sequence and the resulting quantum dynamics.

    a, Pulse sequence showing first the initial state of the three qubits (equation (4)) followed by the unitary operations for a single time step (equation (3)). These operations are repeated N times before measurement. Single-qubit rotations are generated using shaped microwave pulses in 20ns; the three-qubit interaction is generated using a tunable coupling circuit controlled using square pulses of length 5ns for κ = 0.5 and 25ns for κ = 2.5. b, The state of a single qubit is measured using state tomography and shown in a Bloch sphere. The initial state is shown in red with subsequent states shown in blue for N = 1–20.

  2. Entanglement entropy and classical chaos.
    Figure 2: Entanglement entropy and classical chaos.

    a,b, The entanglement entropy (colour) of a single qubit (see equation (5)) averaged over qubits and mapped over a 31 × 61 grid of the initial state, for various time steps N and two values of interaction strength κ = 0.5 (a) and κ = 2.5 (b). The entanglement entropy of a single qubit can range from 0 to 1. c, The entanglement entropy averaged over 20 steps for κ = 0.5 and over 10 steps for κ = 2.5; for both experiments the maximum pulse sequence is 500ns. The left/right asymmetry is the result of experimental imperfections and is not present in numerical simulations (see Supplementary Information). d, A stroboscopic map of the classical dynamics is computed numerically and shown for comparison. The map is generated by randomly choosing 5,000 initial states, propagating each state forwards using the classical equations of motion, and plotting the orientation of the state after each step as a point. We observe a clear connection between regions of chaotic behaviour (classical) and high entanglement entropy (quantum).

  3. Multi-qubit entanglement.
    Figure 3: Multi-qubit entanglement.

    We represent the three-qubit density matrix for two initial states shown inset, one where the entropy was low (top) and one where the entropy was high (bottom). In both cases, the initial state was evolved for N = 10 time steps and κ = 0.5. Each bar indicates the expectation value of one possible combination of Pauli operators on the three qubits, the corresponding operator is shown using coloured squares. The increase in multi-qubit correlations in the lower panel signifies that the contrast between high and low entropy is the result of entanglement.

  4. Ergodic dynamics.
    Figure 4: Ergodic dynamics.

    The overlap of the time-averaged three-qubit density matrix with a microcanonical ensemble (see equation (6)) versus number of time steps N, for κ = 2.5. We choose three different initial states, shown inset. A value of 1.0 indicates that the dynamics are fully ergodic.


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Author information

  1. These authors contributed equally to this work.

    • C. Neill,
    • P. Roushan,
    • M. Fang &
    • Y. Chen


  1. Department of Physics, University of California, Santa Barbara, California 93106-9530, USA

    • C. Neill,
    • M. Fang,
    • Z. Chen,
    • B. Campbell,
    • B. Chiaro,
    • A. Dunsworth,
    • P. J. J. OMalley,
    • C. Quintana,
    • A. Vainsencher,
    • J. Wenner &
    • J. M. Martinis
  2. Google Inc., Santa Barbara, California 93117, USA

    • P. Roushan,
    • Y. Chen,
    • A. Megrant,
    • R. Barends,
    • E. Jeffrey,
    • J. Kelly,
    • J. Mutus,
    • D. Sank,
    • T. C. White &
    • J. M. Martinis
  3. Department of Physics, Boston University, Boston, Massachusetts 02215, USA

    • M. Kolodrubetz &
    • A. Polkovnikov


C.N., P.R. and Y.C. designed and fabricated the sample and co-wrote the manuscript. C.N., P.R. and M.F. designed the experiment. C.N. performed the experiment and analysed the data. M.K. and A.P. provided theoretical assistance. All members of the UCSB team contributed to the experimental set-up and to the manuscript preparation.

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