Nonequilibrium, Driven, and Open Quantum Matter

Quenches, periodic drives, constraints, and environments give quantum matter several routes away from equilibrium. Their observable behavior depends on preparation and timescale as well as the Hamiltonian.

Enter this chapter

Begin with Quenches and Relaxation in Lattice and Continuum Matter to distinguish dephasing, local relaxation, and prethermal evolution after a quench.

Helpful background. Quenches and Relaxation in Lattice and Continuum Matter and Equilibration, Thermalization, and Dephasing develop the definitions used by the chapter’s opening pages.

Topics

1. Quenches and Relaxation in Lattice and Continuum Matter — How do quench protocol, conserved quantities, spectra, and observable choice distinguish dephasing, prethermalization, equilibration, and thermalization in a model?
2. Kibble–Zurek Scaling and Defect Production — How does a finite-rate ramp through a quantum critical region set freeze-out scales and defect production?
3. Nonthermal Fixed-Point Realizations in Quantum Matter — Which model and platform observables establish self-similar approach to a nonthermal fixed point rather than transient redistribution?
4. Floquet Engineering in Quantum Matter — How can periodic driving engineer a useful quantum-matter Hamiltonian within a controlled prethermal window?
5. Floquet Phases and Discrete Time Crystals — Which eigenstate, response, rigidity, lifetime, and thermodynamic criteria define Floquet phases and discrete time crystals?
6. Quantum-Scar Models and Exceptional Eigenstates — How do quantum-scar models produce exceptional nonthermal eigenstates and revivals inside an otherwise thermal spectrum?
7. Hilbert-Space Fragmentation and Constrained Dynamics — How do kinetic constraints split Hilbert space into invariant sectors, and which observables distinguish fragmentation from symmetry sectors, scars, or localization?
8. Driven-Dissipative Condensate, Cavity, and Circuit Matter — How do coherent drive, loss, measurement, and interaction create steady states and transitions in condensate, cavity, circuit, and programmable matter?
9. Dynamical Phase Transitions and Loschmidt Diagnostics — What do nonanalytic Loschmidt-rate functions or dynamical order parameters establish, and how are they related—or not—to equilibrium phases?
10. Heating, Finite Size, and Open-System Evidence — How are heating, prethermal plateaus, recurrences, loss, bath memory, and measurement backaction separated in nonequilibrium quantum-matter evidence?

How the ideas fit together

A high-frequency drive can produce an effective Hamiltonian over a long prethermal interval while still causing heating at later times. Micromotion and the observation protocol are part of that effective description Bukov, D’Alessio, and Polkovnikov 2015, §§ 2–3.

Constraints, scars, and fragmentation can preserve memory by different mechanisms, while baths introduce loss and noise. Similar revivals or slow relaxation therefore do not identify one mechanism. Compare the signal’s time window with heating, recurrence, and environmental times before interpreting temporal order or a stationary phase.

The comparison below separates the assumptions and tests needed for these different conclusions.

Regime comparison

Regime	Protocol and controlling object	Primary observable	Expected lifetime or size trend	Main alternative	Strongest bounded claim
Dephasing	Sudden quench; post-quench energy gaps	Local correlator or imbalance	Stationary fluctuations shrink with effective dimension; recurrences move with size	Coarse time resolution	Observable equilibrates over a declared window
Kibble–Zurek production	Critical ramp; $\hat t$ and $\hat\xi$	Defect density or correlation length	Power law in ramp time, then finite-size saturation	Post-ramp coarsening	Freeze-out scaling in an independently fixed critical window
Nonthermal fixed point	Far-from-equilibrium quench; scaling function	$n(k,t)$ collapse and transported invariant	Scaling range widens before infrared or ultraviolet cutoff	Transient redistribution or imaging convolution	Finite-window self-similar dynamics
Floquet engineering	Periodic drive; $H_{\mathrm{eff}}$ plus micromotion	Held-out stroboscopic and intraperiod observables	Error decreases with expansion order until resonances or noise dominate	Calibrated single-particle dressing only	Engineered dynamics within a prethermal window
Discrete time crystal	Periodic drive; quasienergy pairing or prethermal symmetry breaking	Rigid subharmonic correlations	Lifetime grows with size or drive frequency according to mechanism	Beat note, synchronization, or recurrence	Robust finite-window temporal order
Many-body scars	Special quench; exceptional spectral tower	State-selective fidelity and local revival	Tower reconstruction and perturbation trend persist with size	Accidental commensurability	Scar-enhanced revival dynamics in a specified model
Hilbert-space fragmentation	Kinetic constraint; Krylov connectivity graph	Sector participation and memory	Exact sectors remain disconnected; weak breaking gives coupling-dependent leakage	Disorder localization or slow energetics	Fragment-resolved dynamics or a prefragmented window
Driven-dissipative criticality	Coherent drive and loss; Liouvillian	Output correlations and relaxation gap	Gap closes only in a declared large-system or large-occupation limit	Finite-system switching	Metastability or open critical scaling, as tested
Loschmidt transition	Global quench; return-rate Fisher zeros	Return probability and critical-time scaling	Rounding sharpens with size	Smooth local crossover	Dynamical return-rate criticality
Bath- or measurement-limited behavior	Drive or quench plus environment; channel or memory kernel	Unconditional and conditional records	Rate follows bath coupling, probe cadence, or memory time	Intrinsic heating	Environment-bounded nonequilibrium response

Where to go next

Continue to Quantum-Matter Probes, Inference, and Evidence to map a detector signal to the correlation function it can constrain.

References

• Bukov, Marin, Luca D’Alessio, and Anatoli Polkovnikov. “Universal High-Frequency Behavior of Periodically Driven Systems: From Dynamical Stabilization to Floquet Engineering.” Advances in Physics 64, 139–226 (2015). DOI.