Quenches

A quantum quench is a controlled change of a Hamiltonian or state-preparation parameter on a timescale short compared with selected intrinsic dynamics, followed by time-resolved observation under the resulting generator. “Sudden” is therefore relative to a declared energy window. A pulse that is abrupt for collective modes may still be adiabatic for excluded electronic bands, while an ultrafast optical pump may populate those bands and invalidate a low-energy closed-system model.

This page is the quantum-matter protocol bridge. It owns switch calibration, energy-injection and model-validity ledgers, platform-specific observables, and the boundary between nearly isolated quenches and pump–probe driving. Pump–Probe Spectroscopy owns absorbed-fluence, optical-depth, trARPES, coherent-phonon, and light-induced-state measurement practice. Quantum Quenches is the canonical home for final-basis expansions, exact observable evolution, entanglement growth, work statistics, and worked models. Loschmidt Echo and Dynamical Phase Transitions Preview owns Fisher zeros and thermodynamic return-rate singularities.

The Switch Ledger

A reproducible quench report should specify:

Item	Required information
Preparation	Initial Hamiltonian, state, temperature or energy window, and symmetry sector
Control	Which field, interaction, lattice depth, hopping, geometry, or boundary is changed
Waveform	Full ramp or pulse shape, duration $\tau_q$, overshoot, ringing, and timing jitter
Final generator	Hamiltonian after the switch and any residual time dependence
Energy injection	Mean final energy, width, and occupation of excluded states
Geometry	Global, local, boundary, homogeneous, or spatially patterned change
Probes	Local densities, correlations, spectra, currents, entanglement, or return measurements
Limits	Size, observation time, boundary return, coherence, and environmental rates

Write a finite ramp as

$$H(t) = H_i + s(t) \left( H_f-H_i \right), \qquad s(0)=0, \qquad s(\tau_q)=1.$$

The actual post-ramp state is

$$|\psi(\tau_q)\rangle = \mathcal T \exp\!\left[ - \frac{i}{\hbar} \int_0^{\tau_q}H(t)\,dt \right] |\psi_0\rangle.$$

Only in an ideal sudden approximation is the state treated as unchanged during the switch:

$$|\psi(0^+)\rangle \approx |\psi(0^-)\rangle.$$

Replacing a measured waveform by an instantaneous step is a model assumption, not a definition.

Sudden for Which Degrees of Freedom?

Suppose a low-energy model has a characteristic scale $\Delta_{\mathrm{IR}}$, while excluded states begin at $\Delta_{\mathrm{UV}}$ with $\Delta_{\mathrm{UV}}\gg\Delta_{\mathrm{IR}}$. A useful selective-quench window is

$$\frac{\hbar}{\Delta_{\mathrm{UV}}} \ll \tau_q \ll \frac{\hbar}{\Delta_{\mathrm{IR}}}.$$

The right inequality makes the switch sudden for the retained slow dynamics. The left makes it comparatively adiabatic for high-energy states that the effective model omits.

No such window exists if the two scales are not separated. In that case one must simulate the finite ramp in a larger Hilbert space or weaken the claim.

Additional controls include:

• Landau–Zener transitions at avoided crossings;
• bandwidth-limited edges and pulse spectral weight;
• spatial inhomogeneity across the sample;
• calibration drift between preparation and evolution;
• heating and loss during the ramp;
• micromotion if the “quench” is implemented through a periodic drive.

Energy and Work after the Quench

After an ideal switch, energy is conserved with respect to $H_f$, not $H_i$. The mean and variance are

$$\overline E_f = \langle\psi_0|H_f|\psi_0\rangle, \qquad \left(\Delta E_f\right)^2 = \langle H_f^2\rangle_0 - \overline E_f^{\,2}.$$

For a system of $N$ local degrees of freedom, define an excess-energy density relative to the final ground state:

$$\varepsilon_{\mathrm{ex}} = \frac{ \overline E_f-E_{0,f} }{N}.$$

A global quench can inject an extensive energy, while a strictly local quench changes the total energy by order unity. The distinction controls late-time temperature, overlap catastrophe, and whether an intensive return-rate function is meaningful.

Work in an isolated quench is not a Hermitian operator evaluated once. Under a two-projective-measurement protocol it is a random variable,

$$W = E_m^{f}-E_n^{i},$$

with probabilities determined by initial energy outcomes and overlaps between initial and final eigenstates. Interferometric measurements can access the corresponding characteristic function, but a pump fluence is not automatically the quantum work distribution.

Return Amplitudes and Terminology

For a pure initial state evolving under $H_f$, the Loschmidt amplitude and return probability are

$$\mathcal G(t) = \langle\psi_0| e^{-iH_ft/\hbar} |\psi_0\rangle, \qquad \mathcal L(t) = \left| \mathcal G(t) \right|^2.$$

Literature sometimes calls $\mathcal L$ a Loschmidt echo. In precision work, reserve echo for a reversal or perturbation protocol comparing forward and backward evolution, and call $\mathcal L$ a return probability unless the protocol actually reverses dynamics.

Quantity	Operational meaning
Return probability	Project the evolved state onto the initial state
Fidelity	Compare two declared states; conventions may square the overlap or not
Echo	Compare evolution under forward and imperfectly reversed generators
Ramsey or ancilla signal	Interferometrically reconstruct a complex amplitude
DQPT rate function	Take a thermodynamic intensive logarithm and audit finite-size sharpening

A zero or cusp in a few-site return curve is not a thermodynamic dynamical phase transition. The canonical DQPT page owns the required size sequence, Fisher-zero structure, and order of limits.

Relaxation after Quenches

A quench defines the preparation; it does not predetermine the outcome. The same protocol can yield:

• coherent oscillations and revivals;
• dephasing toward a stationary local value;
• ordinary thermalization;
• generalized thermalization in an integrable system;
• a prethermal plateau followed by drift;
• many-body-localized memory;
• bath-dominated relaxation.

The post-quench observable

$$\langle O(t)\rangle = \langle\psi_0| e^{iH_ft/\hbar} O e^{-iH_ft/\hbar} |\psi_0\rangle$$

must be interpreted with conserved quantities and finite-size timescales. If

$$[H_f,\rho_0] = 0,$$

then every Schrödinger-picture expectation value is stationary even if $H_i\ne H_f$. Changing a Hamiltonian parameter is not sufficient; the prepared state must have coherence or population imbalance in the final eigenbasis.

Quantum Thermalization supplies the positive ensemble test after relaxation. A smooth trace alone establishes neither thermalization nor irreversibility.

Correlation Fronts and Locality

For local operators $A_x$ and $B_y$ separated by distance $r$, short-range lattice Hamiltonians satisfy a Lieb–Robinson-type bound

$$\left\| \left[ A_x(t),B_y \right] \right\| \le C \exp\!\left[ - \mu \left( r-v_{\mathrm{LR}}t \right) \right].$$

This defines an upper causal envelope, not a unique measured velocity. An experiment may extract:

• an onset velocity from the first resolvable signal;
• a peak velocity from the ridge of a correlation packet;
• a group velocity from a quasiparticle dispersion;
• a butterfly velocity from operator growth;
• an entanglement velocity from entropy production.

These velocities can differ. A measured front lying below $v_{\mathrm{LR}}$ is consistent with locality; it does not “measure the Lieb–Robinson velocity.”

Long-range interactions require modified bounds and can produce curved or leaky fronts. Trapped-ion power-law couplings therefore should not be analyzed with a nearest-neighbor light cone without qualification.

Cold Atoms, Ions, and Qubit Arrays

Platform	Typical quench	Strong observables	Main systematic
Optical-lattice atoms	Lattice-depth, interaction, tilt, or dimensional crossover	Site densities, parity correlations, momentum distribution, Rényi entropy	Trap, finite ramp, loss, imaging fidelity
One-dimensional gases	Split, join, interaction, or confinement change	Correlation functions and momentum distributions	Tube averaging and near-integrability
Trapped-ion spins	Transverse field or programmed coupling change	Site-resolved spin correlations and propagation	Long-range interactions and inhomogeneous coupling
Superconducting qubits	Gate-programmed Hamiltonian or local detuning switch	Tomography, return probability, entanglement spectrum	Trotter error, decoherence, calibration drift

Cold-atom quenches can approximate closed-system unitary evolution over a broad window. Their principal advantage is not perfect isolation but the ability to measure loss and dephasing independently and compare them with the intrinsic timescale.

The collapse-and-revival experiment in an optical lattice is a useful warning against equating collapse with irreversible relaxation. Discrete interaction phases can rephase, revealing coherent finite-system dynamics.

The measured correlation front after a Bose–Hubbard quench provides a spatial diagnostic stronger than a one-point decay: it tests propagation, velocity, and quasiparticle structure together.

Pump–Probe Quenches in Solids

An ultrafast pump changes occupations, screening, lattice coordinates, exchange couplings, or order-parameter landscapes. The probe samples a delayed response. Calling this a quench is useful only after declaring which effective Hamiltonian changes and which degrees of freedom remain explicit.

Model hierarchy

A credible solid-state interpretation separates:

1. the electromagnetic pulse and its spectral content;
2. electronic interband and intraband excitation;
3. electron–electron redistribution;
4. coherent phonons and structural motion;
5. electron–phonon energy transfer;
6. diffusion and coupling to the substrate;
7. the probe matrix element and penetration depth.

There may be no interval in which a time-independent isolated $H_f$ is adequate. A time-dependent or open-system description is then the correct model, even if the word “quench” remains convenient.

What a transient signal establishes

Signal	Direct support	Additional claim requiring controls
Gap-like spectral feature	Changed spectral weight in the probe window	A thermodynamic ordered phase
Coherent oscillation	A driven collective or lattice mode	The identity and equilibrium character of the mode
Fast order-parameter suppression	Loss of the measured order-sensitive response	Homogeneous melting throughout the sample
Josephson-plasma-like response	Transient coherent interlayer electrodynamics	Full equilibrium superconductivity
Long-lived metastable spectrum	A nonthermal state over the measured window	A new phase with a defined order parameter

Pump fluence, absorbed energy density, penetration-depth mismatch, and heating controls are part of the Hamiltonian inference. Transient resemblance to an equilibrium spectrum does not prove that the same equilibrium state has been created.

Evidence Workflow

1. Publish or reconstruct the switch waveform.
2. Identify the infrared and ultraviolet scales that define suddenness.
3. Simulate the finite ramp and compare it with the ideal step.
4. Measure injected energy, its spatial profile, and excluded-state population.
5. Declare global, local, boundary, or patterned geometry.
6. Choose probes that separately test populations, correlations, propagation, and coherence.
7. Track boundary-return, recurrence, loss, heating, and dephasing times.
8. Compare several switch durations and amplitudes.
9. Use the correct canonical page for the outcome: thermalization, prethermalization, localization, or DQPT.
10. State whether the effective evolution is closed, driven, open, or a controlled crossover among them.

Common Mistakes

• Calling a finite ramp instantaneous without a scale comparison.
• Requiring suddenness relative to every microscopic gap.
• Ignoring population transferred outside the effective model.
• Treating a pulse as a permanent Hamiltonian quench.
• Confusing return probability with a time-reversal echo.
• Calling a few-body zero a dynamical phase transition.
• Equating correlation-front, group, butterfly, and entanglement velocities.
• Treating a Lieb–Robinson bound as an equality.
• Inferring irreversibility from collapse without searching for revival.
• Calling every stationary post-quench value thermal.
• Ignoring pump/probe penetration-depth mismatch in solids.
• Translating a transient spectral resemblance directly into an equilibrium phase label.

Exercises

1. A selective suddenness window

A low-energy mode has $\Delta_{\mathrm{IR}}=20\ \mathrm{meV}$ and excluded bands begin at $\Delta_{\mathrm{UV}}=5.0\ \mathrm{eV}$. Using $\hbar=0.658\ \mathrm{eV\,fs}$, test whether a $5\ \mathrm{fs}$ switch lies in the selective-quench window.

Solution

The two inverse-energy times are

$$\frac{\hbar}{\Delta_{\mathrm{UV}}} = \frac{0.658}{5.0}\ \mathrm{fs} \approx 0.132\ \mathrm{fs},$$

and

$$\frac{\hbar}{\Delta_{\mathrm{IR}}} = \frac{0.658}{0.020}\ \mathrm{fs} \approx 32.9\ \mathrm{fs}.$$

Thus

$$0.132\ \mathrm{fs} \ll 5\ \mathrm{fs} \ll 32.9\ \mathrm{fs}.$$

The switch is sudden for the low-energy mode while comparatively adiabatic for the excluded bands, subject to matrix elements and intermediate resonances.

2. Local versus global injection

A local quench changes one bond by $\delta J$, while a global quench changes every one of $N$ bonds by the same amount. How do their injected energies scale generically?

Solution

The perturbation norm of the local quench is order $\delta J$, so its total injected energy is generically order unity as $N\to\infty$. Its energy density therefore vanishes like $N^{-1}$.

The global perturbation is a sum of $N$ local changes. At fixed quench amplitude it can inject order $N$ energy, leaving a finite excess-energy density. This is why global quenches can set a finite effective temperature while local quenches often create propagating disturbances on an otherwise unchanged background.

3. A quench with no dynamics

Show why a state satisfying $[H_f,\rho_0]=0$ remains stationary after the switch.

Solution

The evolved state is

$$\rho(t) = e^{-iH_ft/\hbar} \rho_0 e^{iH_ft/\hbar}.$$

If $\rho_0$ commutes with $H_f$, it also commutes with every function of $H_f$, so

$$\rho(t) = \rho_0.$$

All expectation values are stationary. The Hamiltonian may have changed, but the prepared state contains no coherence or population redistribution that the final generator can evolve.

4. Return or echo?

An experiment evolves under $H_f$ for time $t$ and projects onto the initial state. A second experiment evolves under $H_f$ and then attempts reversal with $-H_f+\delta V$. Name the two measured quantities.

Solution

The first measures the return probability

$$\mathcal L(t) = \left| \langle\psi_0| e^{-iH_ft/\hbar} |\psi_0\rangle \right|^2.$$

The second is a Loschmidt echo or fidelity-under-reversal protocol because it compares forward evolution with an imperfect inverse. Calling both “echo” without describing the sequence hides physically different sensitivities.

5. Which front velocity?

A correlation map shows first detectable weight at $r/t=3J a/\hbar$ and its maximum at $r/t=2J a/\hbar$. A band calculation gives a maximum quasiparticle group velocity $2.4J a/\hbar$. What should be reported?

Solution

Report an onset velocity near $3J a/\hbar$ and a peak or ridge velocity near $2J a/\hbar$, together with threshold and fitting uncertainty. The calculated maximum group velocity is $2.4J a/\hbar$.

The three numbers need not coincide because the detectable onset depends on tails and resolution, while the peak is weighted by occupations and matrix elements. None should automatically be called the Lieb–Robinson velocity, which is an upper-bound parameter.

6. Audit a pump–probe phase claim

A mid-infrared pump produces a transient gap-like optical feature and a sharp reflectivity edge for $3\ \mathrm{ps}$. List eight checks needed before claiming an equilibrium superconducting phase.

Solution

Required checks include:

1. absorbed fluence and electronic/lattice heating;
2. pump and probe penetration-depth mismatch;
3. homogeneous volume fraction;
4. spectral-weight transfer and conductivity sum rules;
5. phase-sensitive or inductive response, not only a gap-like feature;
6. competing explanations from coherent phonons or transient screening;
7. time-resolved structural and competing-order probes;
8. pulse-shape, polarization, and frequency controls;
9. relaxation into electrons, phonons, substrate, and environment;
10. comparison with the full equilibrium response, including dissipation;
11. reproducibility over fluence and initial temperature;
12. a model connecting the driven distribution to the inferred order parameter.

The direct result is a transient optical response compatible with selected superconducting signatures. Equivalence to a homogeneous equilibrium phase is a stronger claim.

Connections

• Quantum Quenches is the canonical home for final-basis weights, exact dynamics, entanglement growth, work statistics, and worked quench models.
• Loschmidt Echo and Dynamical Phase Transitions Preview owns return-rate singularities, Fisher zeros, finite-size scaling, and DQPT status.
• Quantum Thermalization owns the platform-facing test of post-quench local thermal values.
• Relaxation and Thermalization owns dephasing, equilibration bounds, and constrained ensembles.
• Prethermalization Preview owns long-lived intermediate plateaus and escape-time scaling.
• Time-Dependent Correlations fixes two-time, stationarity, spectral, and correlation-front conventions.
• Entanglement Entropy in Many-Body Systems owns spatial entanglement measures and scaling.
• Bose–Hubbard Model supplies a canonical interaction-quench and correlation-front platform.
• Open Quantum Systems owns reduced dynamics when loss, dephasing, or external relaxation cannot be neglected.

References

1. 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. Reviews quench protocols, methods, and nonequilibrium regimes.
2. P. Calabrese and J. Cardy, “Time Dependence of Correlation Functions Following a Quantum Quench,” Physical Review Letters 96, 136801 (2006), doi:10.1103/PhysRevLett.96.136801. Develops the quasiparticle and light-cone picture in one-dimensional critical systems.
3. E. H. Lieb and D. W. Robinson, “The Finite Group Velocity of Quantum Spin Systems,” Communications in Mathematical Physics 28, 251–257 (1972), doi:10.1007/BF01645779. Establishes the foundational locality bound.
4. S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb–Robinson Bounds and the Generation of Correlations and Topological Quantum Order,” Physical Review Letters 97, 050401 (2006), doi:10.1103/PhysRevLett.97.050401. Connects locality bounds to correlation and entanglement growth.
5. M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, “Collapse and Revival of the Matter Wave Field of a Bose–Einstein Condensate,” Nature 419, 51–54 (2002), doi:10.1038/nature00968. Demonstrates coherent interaction-driven collapse and revival after a lattice-depth jump.
6. T. Kinoshita, T. Wenger, and D. S. Weiss, “A Quantum Newton’s Cradle,” Nature 440, 900–903 (2006), doi:10.1038/nature04693. Shows long-lived post-quench dynamics near one-dimensional integrability.
7. M. Cheneau et al., “Light-Cone-Like Spreading of Correlations in a Quantum Many-Body System,” Nature 481, 484–487 (2012), doi:10.1038/nature10748. Measures a propagating correlation front after a Bose–Hubbard quench.
8. S. Trotzky et al., “Probing the Relaxation towards Equilibrium in an Isolated Strongly Correlated One-Dimensional Bose Gas,” Nature Physics 8, 325–330 (2012), doi:10.1038/nphys2232. Combines controlled quenches with time-dependent many-body calculations.
9. P. Richerme et al., “Non-Local Propagation of Correlations in Quantum Systems with Long-Range Interactions,” Nature 511, 198–201 (2014), doi:10.1038/nature13450. Tests modified propagation in trapped-ion systems.
10. P. Jurcevic et al., “Quasiparticle Engineering and Entanglement Propagation in a Quantum Many-Body System,” Nature 511, 202–205 (2014), doi:10.1038/nature13461. Resolves tunable correlation and entanglement propagation in an ion chain.
11. A. Silva, “Statistics of the Work Done on a Quantum Critical System by Quenching a Control Parameter,” Physical Review Letters 101, 120603 (2008), doi:10.1103/PhysRevLett.101.120603. Connects quench work statistics and return amplitudes.
12. M. Heyl, A. Polkovnikov, and S. Kehrein, “Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model,” Physical Review Letters 110, 135704 (2013), doi:10.1103/PhysRevLett.110.135704. Introduces DQPT return-rate singularities.
13. M. Heyl, “Dynamical Quantum Phase Transitions: A Review,” Reports on Progress in Physics 81, 054001 (2018), doi:10.1088/1361-6633/aaaf9a. Reviews definitions, solvable models, experiments, and limitations.
14. D. Fausti et al., “Light-Induced Superconductivity in a Stripe-Ordered Cuprate,” Science 331, 189–191 (2011), doi:10.1126/science.1197294. Reports transient interlayer electrodynamic signatures after mid-infrared excitation.
15. C. Giannetti et al., “Ultrafast Optical Spectroscopy of Strongly Correlated Materials and High-Temperature Superconductors: A Non-Equilibrium Approach,” Advances in Physics 65, 58–238 (2016), doi:10.1080/00018732.2016.1194044. Reviews pump–probe modeling, coupled relaxation channels, and the interpretation of transient states.
16. F. Schmitt et al., “Transient Electronic Structure and Melting of a Charge Density Wave in $\mathrm{TbTe}_3$,” Science 321, 1649–1652 (2008), doi:10.1126/science.1160778. Uses time-resolved photoemission to track a driven ordered state.

Further Reading

• P. Calabrese and J. Cardy, “Quantum Quenches in Extended Systems,” Journal of Statistical Mechanics: Theory and Experiment 2007, P06008 (2007), doi:10.1088/1742-5468/2007/06/P06008. A foundational field-theory treatment.
• M. A. Cazalilla and M. Rigol, “Focus on Dynamics and Thermalization in Isolated Quantum Many-Body Systems,” New Journal of Physics 12, 055006 (2010), doi:10.1088/1367-2630/12/5/055006. An overview of experimental and theoretical directions.