Heating, Finite Size, and Open-System Evidence

Nonequilibrium signals are bounded by several clocks: local relaxation, prethermal drift, intrinsic heating, boundary traversal, finite-size recurrence, particle loss, bath memory, decoherence, and measurement backaction. Evidence for a dynamical regime is credible only when those clocks are measured or bounded separately and the claimed observation window lies between the relevant ones.

Required background. Floquet phases and time crystals supplies the isolated driven claims, and driven-dissipative matter supplies open steady-state and trajectory distinctions. Helpful background. Non-Markovian dynamics and memory kernels is useful when bath correlations are not short compared with system dynamics.

Build a timescale hierarchy

For a local energy scale $g$, drive period $T$, linear size $L$, characteristic propagation speed $v$, loss rate $\gamma$, and bath correlation time $\tau_B$, a useful starting record is

$$T,\quad g^{-1},\quad t_{\mathrm{rel}},\quad t_{\mathrm{trav}}\sim\frac{L}{v},\quad t_*,\quad \gamma^{-1},\quad \tau_B,$$

plus the preparation and detector-integration times. The symbols are not automatically ordered. A prethermal interpretation, for example, needs

$$t_{\mathrm{rel}}\ll t_{\mathrm{obs}}\ll \min(t_*,\gamma^{-1},t_{\mathrm{rec}}),$$

and the lower and upper inequalities must be supported by data rather than assumed.

Boundary traversal $L/v$ is not necessarily a full recurrence time, but it is when opposite boundaries can first affect a local light cone. Exact or near recurrences depend on the discrete many-body spectrum and can be much later. In small integrable or nearly commensurate devices, however, conspicuous revivals may occur near a few traversal times and imitate protected oscillations.

Heating rate and prethermal drift

For an isolated periodically driven lattice, choose a fixed reference operator—often a truncated effective Hamiltonian $H_{\mathrm{eff}}^{(n)}$—and monitor

$$e_n(t)=\frac1{N_{\mathrm s}}\langle H_{\mathrm{eff}}^{(n)}\rangle_t,$$

where $N_{\mathrm s}$ is the number of lattice sites. This page uses $\hbar=1$, consistently with the timescale $g^{-1}$ above. A plateau is not defined by a visually flat curve. Fit its drift over a declared interval, propagate uncertainty and correlations, and report the dimensionless rate

$$\Gamma_n(t)=\frac{1}{g^2}\left\lvert\frac{\mathrm d e_n}{\mathrm dt}\right\rvert,$$

where $g$ is the declared local energy scale. Repeat for successive truncation orders and for an observable not used to select the interval. At high frequency, intrinsic absorption should decrease strongly with $\Omega/g$ under the local bounded assumptions; at resonances it can increase abruptly. If the fitted rate instead tracks laser noise, loss, or calibration drift and is insensitive to frequency, the plateau is extrinsically limited.

Controlled truncation and exponentially slow absorption under bounded-local hypotheses are established by Kuwahara, Mori, and Saito 2016 and Abanin et al. 2017.

Heating toward infinite temperature also depends on the Hilbert space. In a finite spin system, local observables can approach their infinite-temperature values inside a fixed symmetry sector, as demonstrated for generic interacting Floquet lattices by D’Alessio and Rigol 2014. An unbounded bosonic continuum has no normalizable infinite-temperature state without a cutoff, so “heating to infinite temperature” must be replaced by a measured growth and loss statement over a specified energy range.

For the broader distinction between prethermal plateaus, kinetic relaxation, and true late-time thermalization, see Mori et al. 2018.

Size scaling and recurrence controls

At least three sizes are needed to distinguish a trend from a pairwise difference, and they must be compared at matched intensive parameters and preparation fidelity. Useful diagnostics include:

• collapse versus $t/L^z$ for an expected hydrodynamic or traversal scale;
• the drift of revival time with $L/v$;
• the plateau lifetime at fixed threshold;
• the minimum many-body quasienergy gap in the relevant symmetry block; and
• local observables measured far from boundaries.

Changing $L$ also changes spectral density and often device quality. A lifetime that grows with qubit number can reflect a simultaneous improvement in calibration, while a lifetime that shrinks can reflect accumulated control error rather than thermodynamic instability. Each size therefore needs its own error and noise characterization.

Open-system alternatives

A Markovian fit assumes bath correlations decay rapidly compared with the system evolution and that initial system–bath correlations are negligible. When $\tau_B$ is comparable to the measured decay time, a single Lindblad rate can absorb memory effects and give a misleading extrapolation. Varying the coupling strength and bath spectrum, or performing process-tensor or memory-kernel tests on a reduced system, can expose the mismatch.

Loss and dephasing can either destroy an intrinsic signal or create a stationary-looking one by balancing the drive. Compare unconditional averages with single-shot or trajectory statistics. A survival-conditioned sample has density operator

$$\rho_{\mathrm{cond}}(t) =\frac{M_0(t)\rho_0M_0^\dagger(t)} {\operatorname{Tr}[M_0(t)\rho_0M_0^\dagger(t)]},$$

which is nonlinear because of normalization and does not represent the trace-preserving ensemble. Reporting its long-lived oscillation without the declining survival probability overstates the unconditional evidence.

Measurement backaction needs a drive-off control with the same observation sequence. Increasing probe cadence while holding total evolution time fixed tests whether the apparent lifetime is Zeno-enhanced or probe-limited. Detector bandwidth should be applied to simulated traces before comparing oscillation amplitudes or decay constants.

A minimum comparison matrix

Candidate explanation	Positive trend	Decisive stress test
High-frequency prethermalization	Lifetime grows rapidly with $\Omega/g$ before an external ceiling	Frequency scan plus successive $H_{\mathrm{eff}}$ orders
Finite-size recurrence	Revival time follows traversal or spectral commensurability	Size and boundary-condition variation
Localization or fragmentation	Memory tied to disorder or exact Krylov sectors	Disorder/constraint-breaking perturbation
Bath-limited steady behavior	Rate follows calibrated coupling or noise spectrum	Tunable bath and unconditional-versus-conditional comparison
Preparation artifact	Signal concentrated in a narrow initial-state family	Generic, symmetry-related, and deliberately perturbed preparations
Measurement backaction	Lifetime changes with probe cadence	Matched weak-probe and drive-off sequences

The strongest conclusion is the intersection of what survives these tests. A result can rigorously establish a 100-cycle prethermal plateau in a finite open device without establishing an infinite-time phase; that bounded statement is scientifically valuable and more reproducible.

Evidence status

These evidence practices and platform boundaries were checked through 10 August 2026. No single extrapolation protocol is universal across cold atoms, solid-state spins, cavities, and quantum processors. Dated calibration records, negative results, and superseding platform evidence belong in the Quantum Matter and Emergence Research dossier; executable sweeps belong in the chapter’s reproducible verification workflow.

Exercises

1. Competing ceilings. An intrinsic prethermal lifetime is $t_*=t_0e^{c\Omega/g}$, a boundary revival occurs at $L/v$, and loss acts at $1/\gamma$. Write the maximum clean observation window and describe two scans that identify the active ceiling.

Solution

The upper edge is approximately $t_{\max}=\min(t_*,L/v,1/\gamma)$, subject to a more precise recurrence estimate. Scan $\Omega$ at fixed $L$ and $\gamma$: growth followed by saturation separates intrinsic heating from another ceiling. Then vary $L$ or $\gamma$ separately; linear motion with $L$ indicates traversal, whereas inverse motion with $\gamma$ indicates loss.

2. Conditional bias. If the unconditional signal is $S(t)=P_0(t)S_0(t)+[1-P_0(t)]S_{\mathrm{jump}}(t)$, when can a persistent no-jump signal $S_0$ have negligible influence on $S$?

Solution

When the survival probability $P_0(t)$ becomes small, its contribution is suppressed even if $S_0(t)$ remains perfectly oscillatory. The unconditional signal can then be governed by jumped trajectories. A no-jump plot must therefore be accompanied by $P_0(t)$ and by the trace-preserving average.

References

• Abanin, Dmitry A., Wojciech De Roeck, Wen Wei Ho, and François Huveneers. “Effective Hamiltonians, Prethermalization, and Slow Energy Absorption in Periodically Driven Many-Body Systems.” Physical Review B 95, 014112 (2017). DOI.
• D’Alessio, Luca, and Marcos Rigol. “Long-Time Behavior of Isolated Periodically Driven Interacting Lattice Systems.” Physical Review X 4, 041048 (2014). DOI.
• Kuwahara, Tomotaka, Takashi Mori, and Keiji Saito. “Floquet–Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems.” Annals of Physics 367, 96–124 (2016). DOI.
• Mori, Takashi, Tatsuhiko N. Ikeda, Eriko Kaminishi, and Masahito Ueda. “Thermalization and Prethermalization in Isolated Quantum Systems: A Theoretical Overview.” Journal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018). DOI.