Quantum Thermalization

Quantum thermalization is the emergence of thermal predictions for a declared class of local observables or subsystems from the unitary dynamics of an isolated many-body system. The global pure state does not literally become a mixed Gibbs state. Information remains in nonlocal correlations, exact conserved quantities remain fixed, and a finite system can recur.

Experiments test this idea by following local densities, correlations or a small subsystem after a preparation or quench. A stationary signal is only the first step: its value must also agree with a thermal ensemble fixed by the initial conserved quantities. Relaxation and Thermalization develops the dephasing and ensemble arguments; here the emphasis is on interpreting measurements in quantum simulators and materials.

Required background. Relaxation and Thermalization supplies the distinction between unitary evolution, equilibration and thermalization. Quantum Quenches supplies the prepared-state, final-Hamiltonian and probe protocol. Statistical Ensembles Overview supplies the constrained thermal comparison. Number Operators and Conserved Quantities supplies the charge and symmetry-sector constraints that comparison must respect.

Local thermal behavior in an isolated system

Equilibration means that the measured observables stay near reproducible values for most times in a stated late-time window. Thermalization additionally requires those values to match a justified thermal ensemble. A plateau can satisfy the first condition and fail the second. A finite isolated system generally fluctuates and recurs, so neither condition demands exact pointwise convergence forever (Gogolin and Eisert, 2016).

Begin with a specified Hamiltonian, preparation and observable. A trapped gas, a periodic chain and a driven circuit can have different conserved quantities and boundary effects even when their early density traces look similar.

For a closed system prepared in $|\psi_0\rangle$,

$$|\psi(t)\rangle = e^{-iHt/\hbar}|\psi_0\rangle, \qquad \rho_A(t) = \operatorname{Tr}_{\bar A} |\psi(t)\rangle\langle\psi(t)|.$$

The reduced state $\rho_A(t)$ can approach thermal predictions even though

$$S\!\left( |\psi(t)\rangle\langle\psi(t)| \right) = 0$$

for all $t$. Entanglement transfers locally accessible information into correlations with $\bar A$; it does not destroy global information.

Comparing measurements with a thermal prediction

Choose an operational target

For observables $O_a$ in a declared set $\mathcal O$, define a normalized residual

$$R_a(t) = \frac{ \langle O_a(t)\rangle - \langle O_a\rangle_{\mathrm{ens}} }{ \sigma_a },$$

where $\sigma_a$ includes measurement uncertainty and uncertainty in the predicted value. Small residuals are meaningful only if the errors are small enough to distinguish the proposed ensembles. A large error bar does not establish thermalization.

Use several observables and several microscopically different preparations with matched conserved densities. Separate temporal fluctuations from shot noise by repeating measurements at the same times. Correlated observables also require their covariance when assessing joint agreement; several correlated residuals are not independent tests.

Fix the ensemble independently

For a canonical candidate,

$$\rho_\beta = \frac{e^{-\beta H}}{Z(\beta)}, \qquad Z(\beta) = \operatorname{Tr}e^{-\beta H},$$

the inverse temperature should be fixed by energy matching,

$$\langle\psi_0|H|\psi_0\rangle = \operatorname{Tr}(\rho_\beta H),$$

within the symmetry sector actually prepared. More precisely, for a projector $P$ onto an invariant sector, use $\rho_{\beta,P}=P e^{-\beta H}/\operatorname{Tr}(P e^{-\beta H})$. A compatible set of commuting symmetry labels defines a sector; noncommuting charges cannot all be assigned sharp values simultaneously.

If the preparation samples sectors, preserve their weights unless a grand-canonical approximation is justified for the observables in question. The energy distribution also matters: a narrow energy-density distribution can often be represented by one temperature, whereas a broad or bimodal preparation need not be characterized by its mean energy alone. The canonical discussion explains these limits.

Fitting $\beta$ separately to every observable makes the test circular. Use one independently fixed parameter set and predict observables that were not used to fit it.

Compare the correct reduced state

For subsystem tomography, the thermal comparison is the reduction of the appropriate full-system ensemble,

$$\rho_A^{\mathrm{th}}=\operatorname{Tr}_{\bar A}\rho_{\mathrm{ens}}.$$

It is not generally the Gibbs state $e^{-\beta H_A}/Z_A$ of the isolated subsystem Hamiltonian. Couplings across the boundary affect the reduced state. Neglecting them needs a weak-coupling or suitable bulk approximation; small subsystems can have large boundary corrections. Exercise 2 makes this distinction explicit with two spins.

The subsystem size also matters. Keeping $A$ fixed while increasing the full system tests a local thermodynamic limit. Keeping a finite fraction $|A|/|AB|=f>0$ probes a different limit: exact total-energy and charge constraints can remain visible in subsystem fluctuations. Agreement for one site does not establish agreement for half the sample.

Kaufman et al. (2016) combined local measurements with subsystem Rényi entropy in an isolated atom system. A Rényi entropy, $S_2(\rho_A)=-\ln\operatorname{Tr}\rho_A^2$, tests purity; it is not the same quantity as the von Neumann entropy. Neither entropy alone determines the entire reduced density matrix.

Connecting the dynamics to eigenstates

The eigenstate thermalization hypothesis (ETH) connects the structure of energy eigenstates to thermal behavior (Rigol et al., 2008; D’Alessio et al., 2016). The practical diagonal test asks whether eigenstate expectation values of a local operator become a smooth function of energy within one exact symmetry sector:

$$O_{nn} = \langle E_n|O|E_n\rangle \approx O_{\mathrm{th}}(E_n).$$

For a meaningful comparison:

1. resolve compatible symmetry sectors and handle degeneracies before comparing levels or matrix elements;
2. use bulk energy-density windows unless the claim concerns an edge;
3. subtract the smooth energy dependence before measuring fluctuations;
4. track extreme outliers as well as typical variance.

A variance that decreases appropriately with size supports weak ETH; one small variance at one size does not establish the limiting statement. Absence of outliers in accessible sizes does not prove strong ETH in the thermodynamic limit. Conversely, one atypical scarred eigenstate does not prevent broad classes of initial states from thermalizing if they have negligible overlap with it.

ETH is sufficient under appropriate initial-state and spectral assumptions, but an experiment can establish local thermalization without reconstructing all eigenstates. Time-domain tests and eigenstate tests are complementary, not interchangeable.

Extra conserved quantities and slow constraints

An integrable model carries an extensive family of local or quasilocal conserved quantities $Q_j$. An ordinary Gibbs state constrained only by energy and particle number can therefore discard information that local observables retain. The generalized candidate is

$$\rho_{\mathrm{GGE}} = \frac{1}{Z_{\mathrm{GGE}}} \exp\!\left( - \sum_j\lambda_jQ_j \right).$$

The multipliers are fixed by initial charge values (Rigol et al., 2007). A truncated charge family should be enlarged until predictions for the observables of interest stabilize. Integrability and Generalized Gibbs Ensembles Preview develops the construction.

The quantum Newton’s cradle (Kinoshita et al., 2006) illustrates why many collisions and strong interactions do not guarantee ordinary thermalization: one-dimensional kinematics and near-integrability can greatly restrict redistribution. Deliberately breaking integrability tests whether the extra memory then decays.

Prethermal relaxation

A prethermal state is a long-lived intermediate regime controlled by approximate conservation laws, a large energy separation, weak integrability breaking, or a high-frequency drive (Mori et al., 2018). Its defining hierarchy is

$$\tau_{\mathrm{local}} \ll t \ll \tau_{\mathrm{escape}},$$

where local observables have relaxed but the slow constraint has not.

The plateau must be tested against an effective or generalized ensemble appropriate to the approximate charges. Eventual drift is not an experimental nuisance to erase; it distinguishes prethermalization from exact stationarity. Parameter scaling is especially valuable. For example, if an integrability-breaking perturbation has strength $g$, a measured escape time should be compared with a theoretically motivated $g$ dependence rather than reported at one value.

Gring et al. (2012) used full interference-contrast distributions in a split one-dimensional gas to identify a thermal-like intermediate state distinct from final equilibrium. The prethermalization discussion explains the effective constraints.

Not every two-stage trace is prethermal. Inhomogeneous dephasing, transport from distant regions, detector bandwidth, or a weak external bath can also produce separated timescales.

Internal relaxation and environmental relaxation

A bath can produce the same local values as isolated-system thermalization. Distinguishing the mechanisms requires independent information about coupling to the environment.

Observation	What else is needed?
Gibbs-like local values	Check whether the temperature follows initial energy or an external bath.
Growing subsystem entropy	Establish global purity before interpreting it as entanglement.
Exponential relaxation	Compare with bath rates; exponential decay alone does not identify a mechanism.
Initial-state independence	Match conserved quantities and check for environmental attraction.
Changing total energy	Include work or energy exchange; a time-independent isolated Hamiltonian conserves energy.

Compare the relaxation time with environmental times relevant to the same state and observable. For loss, dephasing and heating rates, a useful first check is

$$\tau_{\mathrm{rel}}\ll\tau_{\mathrm{env}}, \qquad \tau_{\mathrm{env}}=\min\bigl(\Gamma_{\mathrm{loss}}^{-1},\Gamma_\phi^{-1},\Gamma_{\mathrm{heat}}^{-1}\bigr).$$

Several weak channels can add, and a per-particle rate is not a whole-sample error rate. The relevant coupling operators determine which observables are affected. Varying bath coupling is more informative than relying on one nominal coherence time.

Markovian evolution alone guarantees neither thermalization nor even convergence to a stationary state. Protected degrees of freedom can continue oscillating; a unique attractor, when present, need not be Gibbsian. Steady States and Relaxation explains those conditions. They differ from the eigenstate mechanism in an isolated system.

Measurements in different platforms

Platform	Useful probe and principal limitation
Optical-lattice atoms	Occupations, correlations and subsystem purity; traps, loss and boundaries require control.
One-dimensional gases	Momentum and interference distributions; tube averaging and residual integrability matter.
Trapped-ion spins	Local readout and correlation spreading; long-range coupling changes transport and boundary times.
Superconducting qubits	Local tomography and randomized measurements; gate errors and decoherence constrain the time window.
Pump–probe solids	Time-resolved spectra and order parameters; heating, nonthermal occupations and depth averaging complicate interpretation.

In a solid, “electrons thermalize in $50\ \mathrm{fs}$” often means that a measured distribution is well fit by a Fermi–Dirac form over a limited energy window. A complete claim should test other observables, conservation constraints, electron–phonon energy transfer, matrix-element effects, and whether one temperature describes the whole electronic subsystem.

Two different times can govern the same experiment: a local interaction scale sets early relaxation, while conserved densities redistribute much more slowly over a large distance. Hydrodynamic tails therefore need not contradict thermalization. Varying the size, probe distance and observation window helps separate bulk relaxation from transport and boundary return.

The useful conclusion specifies which observables agree, at which conserved densities, for which preparations and over which times. Current theorem assumptions and exceptional state families are discussed in Thermalization and Eigenstate Thermalization research.

Exercises

1. Global purity and local entropy

A bipartite system starts in a pure product state and evolves unitarily into an entangled state. What happens to the global von Neumann entropy and to the entropy of subsystem $A$?

Solution

Unitary evolution preserves the eigenvalues of the global density operator, so

$$S(\rho_{AB}(t)) = 0$$

at all times. The reduced state

$$\rho_A(t) = \operatorname{Tr}_B\rho_{AB}(t)$$

can become mixed as entanglement develops, so $S(\rho_A)$ can increase. Local entropy growth therefore does not require global information loss.

2. A subsystem need not have its bare Gibbs state

Consider the equilibrium state of two Ising spins $s_A,s_B=\pm1$ with

$$H=-h_A\sigma_A^z-h_B\sigma_B^z-J\sigma_A^z\sigma_B^z.$$

Sum over $s_B$ to obtain the probability ratio $p_A(+1)/p_A(-1)$. Does it equal the bare-spin Gibbs ratio $e^{2\beta h_A}$? This is an equilibrium comparison, not a claim that this two-spin Hamiltonian thermalizes.

Solution

For fixed $s_A$, the unnormalized probability is

$$w(s_A)=2e^{\beta h_A s_A}\cosh\bigl[\beta(h_B+Js_A)\bigr].$$

Consequently,

$$\frac{p_A(+1)}{p_A(-1)} =e^{2\beta h_A}\frac{\cosh[\beta(h_B+J)]}{\cosh[\beta(h_B-J)]}.$$

The additional factor generally differs from one. It becomes one at $J=0$, and also at $h_B=0$ in this symmetric example. At $\beta h_A=\beta h_B=\beta J=1/2$, the full result gives $\langle\sigma_A^z\rangle\simeq0.615$, whereas the bare Gibbs state gives $\tanh(1/2)\simeq0.462$. Fitting a new temperature to spin $A$ can hide the missing interaction; it does not validate the full thermal model.

3. Choose the ensemble

Two initial states have equal energy and particle number but different values of an additional exactly conserved local charge $Q$. They equilibrate to different local densities. Does this refute equilibration or ETH?

Solution

It first refutes the incomplete ensemble that omitted $Q$. The appropriate candidate must retain its constraint, for example

$$\rho \propto \exp(-\beta H-\lambda_Q Q).$$

If both states equilibrate locally to predictions fixed by their respective charge values, equilibration and generalized thermalization are compatible with the observations. ETH must likewise be formulated within the correct charge data or integrable structure.

4. Closed or open?

A simulator relaxes in $\tau_{\mathrm{rel}}=12\ \mathrm{ms}$, while independent calibrations give loss, dephasing, and heating times of $2.0\ \mathrm{s}$, $0.30\ \mathrm{s}$, and $0.80\ \mathrm{s}$. Evaluate the isolation hierarchy.

Solution

The shortest environmental time is $0.30\ \mathrm{s}=300\ \mathrm{ms}$, so

$$\frac{\tau_{\mathrm{rel}}} {\min(\Gamma_{\mathrm{loss}}^{-1}, \Gamma_\phi^{-1}, \Gamma_{\mathrm{heat}}^{-1})} = \frac{12}{300} = 0.04.$$

Relaxation is much faster than the calibrated environmental processes, leaving a plausible closed-system window. This is a preliminary comparison, not an error bound. If the three calibrated rates contribute independently to the same observable, their summed exposure during relaxation is $0.012(1/2+1/0.30+1/0.80)\simeq0.061$. State dependence, the later observation window and the measured quantity still matter.

References

• L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics,” Advances in Physics 65, 239–362 (2016), doi:10.1080/00018732.2016.1198134. Reviews ETH, chaos, and numerical diagnostics.
• C. Gogolin and J. Eisert, “Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems,” Reports on Progress in Physics 79, 056001 (2016), doi:10.1088/0034-4885/79/5/056001. Reviews rigorous and operational frameworks.
• M. Gring et al., “Relaxation and Prethermalization in an Isolated Quantum System,” Science 337, 1318–1322 (2012), doi:10.1126/science.1224953. Observes a prethermal state in split one-dimensional Bose gases.
• A. M. Kaufman et al., “Quantum Thermalization through Entanglement in an Isolated Many-Body System,” Science 353, 794–800 (2016), doi:10.1126/science.aaf6725. Measures subsystem thermal behavior and entanglement in an isolated atom array.
• T. Kinoshita, T. Wenger, and D. S. Weiss, “A Quantum Newton’s Cradle,” Nature 440, 900–903 (2006), doi:10.1038/nature04693. Demonstrates striking suppression of ordinary thermalization near one-dimensional integrability.
• T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, “Thermalization and Prethermalization in Isolated Quantum Systems: A Theoretical Overview,” Journal of Physics B 51, 112001 (2018), doi:10.1088/1361-6455/aabcdf. Reviews ETH, integrability breaking, and prethermal mechanisms.
• M. Rigol, V. Dunjko, and M. Olshanii, “Thermalization and Its Mechanism for Generic Isolated Quantum Systems,” Nature 452, 854–858 (2008), doi:10.1038/nature06838. Connects ETH to quench thermalization in lattice models.
• M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, “Relaxation in a Completely Integrable Many-Body Quantum System,” Physical Review Letters 98, 050405 (2007), doi:10.1103/PhysRevLett.98.050405. Establishes generalized-ensemble logic for integrable dynamics.

Further Reading

• J. Eisert, M. Friesdorf, and C. Gogolin, “Quantum Many-Body Systems out of Equilibrium,” Nature Physics 11, 124–130 (2015), doi:10.1038/nphys3215. A concise overview of equilibration and its exceptions.
• A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, “Colloquium: Nonequilibrium Dynamics of Closed Interacting Quantum Systems,” Reviews of Modern Physics 83, 863–883 (2011), doi:10.1103/RevModPhys.83.863. A broad entry to quenches, relaxation, and nonequilibrium many-body methods.