Thermalization and dephasing in an isolated system of coupled qubits

AI summary

Overview

Research area: Quantum physics, specifically quantum thermodynamics and the foundations of statistical mechanics in isolated quantum systems, with a proposed implementation in superconducting circuits.

Technical level: Advanced. The paper assumes familiarity with density matrices, master equations, interaction-picture perturbation theory, and superconducting qubit physics.

Scope: A theoretical and numerical study of whether an isolated, all-to-all coupled ensemble of qubits internally thermalizes and dephases when a single qubit is treated as the subsystem and the remaining qubits as its effective bath.

What This Paper Is About

A system that evolves unitarily cannot, by definition, thermalize or dephase. The paper asks what happens if one singles out just one qubit from an isolated ensemble of coupled qubits and treats the rest as a bath: does that subsystem relax, lose coherence, and reach a thermal distribution even though the whole ensemble remains closed? The goal is to identify which coupling mechanisms produce genuine thermalization and dephasing, to predict the final temperature from the initial state alone, and to propose a superconducting circuit that could realize the physics.

Key Contributions

1. A self-consistent weak-coupling framework for internal thermalization. Each of the (N) all-to-all coupled qubits is taken in turn as the system, with the other (N-1) qubits as its bath; the same weak-coupling master equation is written for every qubit and solved consistently across the ensemble.

2. Identification of nonlinearity as the ingredient for thermalization. For three-wave mixing, the number of excitations is not conserved and non-degenerate qubits can exchange energy, producing an ergodic system that reaches a thermal distribution. For purely linear coupling, only degenerate qubits interact and no thermal distribution is reached.

3. A quantitative temperature prediction from energy conservation. The long-time temperature is fixed uniquely by the energy of the initial non-thermal state, with closed-form approximations (\tilde{\beta} \approx \pi/\sqrt{6\Delta\varepsilon}) for weakly anharmonic oscillators and (\tilde{\beta} \approx \pi/\sqrt{12\Delta\varepsilon}) for ideal two-level qubits, valid for (N, \Delta\varepsilon \gg 1).

4. A concrete experimental proposal. An all-to-all network of asymmetric SQUIDs connected by inductors, whose third-order nonlinearity (\alpha_k \varphi_k^3) generates the required three-wave mixing terms after diagonalizing the Josephson part of the Hamiltonian.

Main Findings

• Three-wave mixing thermalizes the ensemble. Numerically solving the rate equations for (N=100) weakly anharmonic oscillators with level truncation (n_k^{\max}=150), equal couplings (|M_{ijk}|^2=0.001), energies (\Omega_k = k\Delta\Omega), and the initial state (\rho_{10,10}^{(10)}(0)=1) (all other oscillators in their ground state), the populations of the lowest oscillator on levels (n_1 = 0, 1, \ldots, 9) evolve to values matching the thermal populations of Eq. (6). The numerical long-time results agree with both the exact evaluation of the energy-conservation condition and the integral approximation.

• The final temperature is set only by the initial energy. Equating the initial excitation energy to the equilibrium energy of the bosonic or two-level distribution determines (\beta). For (N=300) ideal qubits with (|M_{pqr}|^2=0.005), simulations from various initial states collapse onto the thermal curves of Eq. (11) with (\beta) from Eq. (13), shown as circle symbols overlapping the solid lines.

• Thermalization is insensitive to the energy dependence of the coupling. For ideal qubits with (N=100), three different coupling forms were compared — (|M_{pqr}|^2 = 0.001) for all combinations, (|M_{pqr}|^2 = 0.0002,p\exp(-|q-r|/N)), and (|M_{pqr}|^2 = (0.005/p)\exp(|q-r|/N)) — starting from (\rho_{gg}^{(10)}(0)=\rho_{ee}^{(10)}(0)=0.5). All three stabilize to the same (\Omega_k)-dependent long-time populations. The authors state that thermalization was found in every case considered but is not proven analytically.

• Off-diagonal elements decay exponentially. For ideal qubits under three-wave mixing, the master equation gives (\dot{\rho}{ge}^{(j)} = -\tfrac{1}{2}\Gamma\Sigma^{(j)}\rho_{ge}^{(j)}), and the numerically tracked (\rho_{ge}^{(10)}) decays exponentially for all three coupling forms. In the large-(N) linear-coupling limit, populations decay at rate (\Gamma_\Sigma^{(j)}) and coherences at (\Gamma_\Sigma^{(j)}/2), with (\Gamma_\Sigma^{(j)} = (2\pi/\hbar^2)\nu_0\langle g^2\rangle).

• The Cauchy principal value only adds a phase. A detailed derivation in the Supplemental Material shows that retaining the principal value gives (\rho_{ge}^{(j)}(t) = \rho_{ge}^{(j)}(0)e^{iP_j t}e^{-\Gamma_\Sigma^{(j)}t/2}), with (P_j \equiv (2/\hbar^2)\nu_j\langle g_j^2\rangle\ln(\Delta_2^{(j)}/\Delta_1^{(j)})). For a symmetric frequency distribution (P_j) vanishes.

• Linear coupling does not thermalize. With (\hat{\mathbb{V}}^{(j)} = \sum_{k\neq j}(g_{jk}\hat{a}_j^\dagger\hat{a}_k + \text{h.c.})), the delta functions (\delta(\Omega_j-\Omega_k)) restrict interaction to degenerate qubits. Detailed-balance populations are a steady-state solution of the master equation but are not reached. Even when level broadening is included for stronger coupling, the long-time distribution is Lorentzian, Eq. (21), and qubits far from the central frequency remain untouched. Supplemental Fig. S4 shows this for (N=10000) with (\Gamma_0 = 10^{-4}), (10^{-3}), and (10^{-2}).

• Dephasing under linear coupling requires large (N). In the limit (N\rightarrow\infty), a single qubit relaxes and dephases almost exponentially even with linear weak coupling, but the residual long-time populations are still not thermal.

• Anharmonic oscillators and ideal qubits share the qualitative behavior. Diagonal elements reach a steady state in the long-time limit in both cases; the thermal forms differ, being Eq. (6) for harmonic oscillators and Eq. (11) for two-level systems.

• The integral approximation has a finite-size validity range. For small (N=10) (Supplemental Figs. S1 and S2), the analytic expression fails quantitatively, whereas for large (N) (Supplemental Fig. S3, (N=100)) it is a good approximation.

Methodology in Plain English

The authors start from a closed set of (N) qubits coupled all-to-all, either as weakly anharmonic oscillators or as ideal two-level systems. They pick one qubit, trace out the other (N-1) qubits to form an effective bath, and derive a standard weak-coupling master equation for that single qubit. The coupling Hamiltonian is written in the interaction picture and inserted into the master equation, producing rate equations for the populations: a loss rate out of level (n_j) and gain rates from levels (n_j+1) and (n_j-1). The bath expectation values are themselves expressed through the populations of the other qubits, so the same set of equations is applied to every qubit and solved self-consistently.

For three-wave mixing, the interaction terms (M_{pqr}a_p^\dagger a_q a_r + \text{h.c.}) allow one qubit to emit into two others or absorb from two others, with energy-conserving delta functions (\delta(\Omega_j-\Omega_k-\Omega_l)) and (\delta(\Omega_k-\Omega_l-\Omega_j)). The resulting rate equations are integrated numerically, and the long-time populations are compared with thermal distributions whose temperature is obtained by matching the initial and final total energy. For ideal qubits, the same procedure is applied to the ground and excited populations and to the coherence (\rho_{ge}), which closes into a simple exponential decay equation. A separate analysis treats linear coupling in the rotating wave approximation, including the effect of level broadening beyond the strict weak-coupling limit. Finally, the authors write down a circuit Hamiltonian for asymmetric SQUIDs coupled through a central node, enforce current conservation to eliminate the central phase, diagonalize the Josephson part, and show that the remaining cubic term reproduces the three-wave-mixing perturbation.

Why This Matters

The work addresses a foundational tension in quantum technology: quantum information processing relies on unitary evolution, whereas thermalization to a Gibbs state with vanishing off-diagonal elements corresponds to a return to classical behavior and loss of potential quantum advantage. Understanding which microscopic ingredients cause an apparently closed system to behave thermally is relevant both for fundamental statistical mechanics and for engineering multi-qubit hardware.

Potential applications:

• Superconducting qubit coherence. Predicting intrinsic relaxation and dephasing rates in arrays of coupled qubits and SQUIDs from the coupling nonlinearity and frequency distribution.
• Quantum simulation of statistical mechanics. Using coupled superconducting circuits or cold atoms as analog simulators of thermalization, ergodicity, and equilibration in closed quantum systems.
• Quantum thermodynamics. Treating self-thermalization as a resource, where the final temperature is determined solely by the stored energy, relevant to on-chip thermometry and thermal machines.
• Circuit design and calibration. Anticipating steady-state populations in all-to-all coupled transmon or SQUID networks, and choosing linear versus nonlinear coupling deliberately.

Industry relevance: The proposed platform is based on superconducting quantum circuits, the leading architecture for gate-based quantum computing. The results give hardware designers a quantitative rule for when unwanted thermalization arises from qubit-qubit nonlinearity, and when a linearly coupled architecture will instead freeze into a non-thermal steady state.

Future Directions

• Experimental demonstration. Realizing the asymmetric-SQUID network with inductive all-to-all coupling and verifying the predicted thermal populations and exponential coherence decay.
• Beyond the weak-coupling limit. The analysis assumes strict delta-function energy conservation. The authors note that relaxing this requirement does not change the dynamics qualitatively, but a systematic treatment of stronger coupling and level broadening remains open.
• Thermalization versus many-body localization. The paper cites recent work ascribing localization in three-wave-mixing systems to Hilbert-space fragmentation from non-uniform energy distributions. Testing disordered or non-equidistant spectra, and the crossover to the large-(N) limit where level density exceeds the coupling-induced smearing, is a natural next step.
• Extensions of the model. Including only third-order nonlinearity here, the authors leave open the role of higher-order terms, alternative coupling modalities, and the principal-value phase shifts (P_j) in asymmetric frequency distributions.

Target Audience

Researchers in quantum thermodynamics and the foundations of quantum statistical mechanics; experimentalists working with superconducting qubits, transmons, and SQUID arrays; and graduate students familiar with open quantum systems and master-equation techniques who want a concrete, quantitatively worked example of thermalization emerging inside an isolated system.

Authors’ abstract

A system evolving unitarily does not dephase or thermalize by definition. Yet, if one considers a part of a unitary system, situation is different. Here we analyze an isolated set of coupled qubits taking one qubit at a time as a subsystem, and writing self-consistently a weak-coupling master equation for it. Weakly anharmonic oscillators and ideal qubits (two-level systems) demonstrate dephasing, i.e. the diagonal elements of the density matrix reach a steady-state in the long-time limit, and the off-diagonal elements of ideal qubits decay exponentially. For the linear coupling between the qubits, only the degenerate ones interact, and the system does not reach thermal distribution. However, non-linearity, in our analysis in form of three-wave mixing, ensures a thermal distribution in the long time limit, where the temperature is determined uniquely by the energy in the initial non-thermal state. Finally, we present a potential experimental scheme based on a superconducting quantum circuit.