Quench Spectroscopy Overview

• Quench spectroscopy is a nonequilibrium method that abruptly changes system parameters to reveal excitation energies through oscillatory observables.
• It employs Fourier transforms of spatial and temporal correlation functions or nonlinear current signals to decode post-quench excitation structures.
• Practical implementations span lattice models, superconducting THz experiments, and impurity setups, each using tailored observables like Loschmidt echoes and spectral functions.

Quench spectroscopy denotes a class of nonequilibrium spectroscopic protocols in which a system is prepared in an initial state, subjected to a sudden quench at $t=0$, and then analyzed through the real-time evolution of observables whose oscillatory structure encodes the excitation content of the post-quench problem. Across the literature, the quench may be a global parameter change, a local perturbation, a tunneling switch, a projective measurement, or a broadband THz pulse, while the readout may be equal-time correlation functions, local observables, nonlinear currents, optical absorption edges, Loschmidt echoes, or time-dependent spectral functions (Villa et al., 2020, Puviani et al., 2021, Vasseur et al., 2014).

1. Scope and variants of the concept

The term is used for several closely related protocols rather than a single fixed experimental design. In lattice many-body systems, quench spectroscopy usually means preparing the ground state of an initial Hamiltonian $H(\lambda_i)$, abruptly changing a parameter to $\lambda_f$, and reconstructing excitation energies from the spatiotemporal Fourier transform of correlation functions or one-point observables (Villa et al., 2020). In superconducting THz experiments, “quench-drive spectroscopy” uses a short, few-cycle quench pulse and a delayed multicycle drive pulse, and the nonlinear current is analyzed in both real time and quench–drive delay (Puviani et al., 2021). In topological and impurity settings, the quench can be a sudden local tunneling switch or a single-photon absorption event, with the spectral information encoded in the Loschmidt echo or in edge singularities of optical absorption (Vasseur et al., 2014, Latta et al., 2011).

A distinct methodological extension is “measurement quench,” in which the Hamiltonian is left unchanged and the quench is produced solely by measurement-induced wavefunction collapse; the ensuing unitary dynamics under the original Hamiltonian then serves spectroscopic purposes (Bayat et al., 2018). Another extension is non-local quench spectroscopy, where non-local spin-string correlators are monitored because they coincide with fermionic correlators under the Jordan–Wigner mapping; this is required when the elementary excitations are non-local in the microscopic variables (Bocini et al., 2024).

Setting	Quench	Readout
Lattice many-body models	Global or local parameter change	Space-time Fourier transform of observables
THz superconducting spectroscopy	Few-cycle quench plus delayed multicycle drive	$j(t,\tau)$, $j(\omega,\tau)$, $j(\omega,\Omega)$
Majorana and impurity problems	Sudden local tunneling or optical absorption	$\mathcal{L}(t)$, threshold absorption, $A(\omega,t)$

A common misconception is that quench spectroscopy is simply pump–probe spectroscopy under another name. The superconducting literature explicitly distinguishes quench-drive spectroscopy from conventional pump–probe THz and from single-pulse third-harmonic generation: the quench is broadband and short, the drive is narrowband and multicycle, and the delay is itself used as a spectral knob (Puviani et al., 2021). Another misconception is that only local probes are relevant. In one-dimensional spin chains, local probes can access only continua, whereas non-local correlators are needed to recover sharp fermionic quasiparticle lines (Bocini et al., 2024).

2. Formal structure and signal extraction

In its most standard form, the protocol begins from the ground state of an initial Hamiltonian $H_i$, followed by unitary evolution under $H_f$,

$H(\lambda_i)$0

Expanding the initial state in the eigenbasis of $H(\lambda_i)$1 yields, for an observable $H(\lambda_i)$2,

$H(\lambda_i)$3

so peaks in the Fourier transform occur at frequencies $H(\lambda_i)$4 (Yu et al., 10 Jul 2025). For weak quenches dominated by transitions from the ground state, the strongest features typically occur at $H(\lambda_i)$5 (Yu et al., 10 Jul 2025).

For spatially resolved settings, the central object is often a quench spectral function obtained by Fourier transforming in both space and time. In the local-quench formalism,

$H(\lambda_i)$6

which makes explicit that the peaks arise from energy and momentum differences between post-quench eigenstates populated by the quench and connected by the chosen observable (Villa et al., 2020). In this sense, quench spectroscopy is selection-rule dependent: the measured operator determines which branches are visible.

Several concrete readout formulas recur across implementations. In the dipolar XY Rydberg simulator, linear spin-wave theory gives

$H(\lambda_i)$7

so fitting early-time oscillations of the time-dependent structure factor directly yields $H(\lambda_i)$8 (Chen et al., 2023). In superconducting quench-drive spectroscopy, the measured current depends on both real time $H(\lambda_i)$9 and delay $\lambda_f$0, and the two-dimensional spectrum is

$\lambda_f$1

with $\lambda_f$2 associated with harmonics of the drive and $\lambda_f$3 the delay frequency conjugate to $\lambda_f$4 (Puviani et al., 2021). In boundary and topological versions, the relevant quantity can instead be the Loschmidt echo,

$\lambda_f$5

whose long-time asymptotics diagnose the post-quench boundary condition and low-energy fixed point (Vasseur et al., 2014).

These formulations share a structural feature emphasized in the broader quench-dynamics literature: the quench itself acts as the pump, while the time-resolved observable is the spectroscopic probe. This viewpoint naturally connects oscillation frequencies, light-cone propagation, dephasing, and prethermal plateaus to the excitation structure and to the stability or breakdown of generalized Gibbs descriptions (Mitra, 2017).

3. Lattice many-body implementations and quantum simulators

Global and local quenches in one-dimensional lattice models provided an early systematic formulation of quench spectroscopy as a space-time Fourier analysis of nonequilibrium observables. In translationally invariant models, local quenches are sufficient to reconstruct momentum-resolved spectra from one-point observables, and the resulting quench spectral function can recover single-quasiparticle dispersions as well as energy-difference branches. For cosine-like bands $\lambda_f$6, the strongest conserving-quasiparticle features follow $\lambda_f$7 (Villa et al., 2020).

A direct experimental realization was demonstrated in a two-dimensional dipolar XY Rydberg simulator. There the system is initialized in coherent spin states aligned along $\lambda_f$8, the quench is the sudden onset of dipolar XY exchange dynamics, and the equal-time $\lambda_f$9 correlations are Fourier transformed to obtain a time-dependent structure factor (Chen et al., 2023). The ferromagnetic case exhibits elementary excitations behaving as linear spin waves, long-lived oscillations, super-ballistic correlation fronts $j(t,\tau)$0, and a nonanalytic small-$j(t,\tau)$1 dispersion $j(t,\tau)$2. The antiferromagnetic case instead displays $j(t,\tau)$3, a clear light cone with $j(t,\tau)$4, and strong damping attributed in the Supplemental Material to one-to-three magnon decay channels allowed by quartic Holstein–Primakoff terms (Chen et al., 2023).

Quench spectroscopy has also been used as a phase diagnostic in disordered bosonic systems. In the one-dimensional disordered Bose–Hubbard model, the momentum-resolved quench spectral function distinguishes the Mott insulator, Bose glass, and superfluid by the presence or absence of low-frequency weight and of a sound-like linear mode. Operationally, the Mott insulator is characterized by $j(t,\tau)$5, whereas the Bose glass and superfluid have $j(t,\tau)$6; the superfluid alone retains a well-defined linear lower edge $j(t,\tau)$7 at small $j(t,\tau)$8 (Villa et al., 2021). Spatially resolved spectroscopy via local density fluctuations supplies a second diagnostic through the typical gapped-region size $j(t,\tau)$9, with $j(\omega,\tau)$0 in the Mott phase, $j(\omega,\tau)$1 in the superfluid, and $j(\omega,\tau)$2 in the Bose glass (Villa et al., 2021).

The trapped-boson implementation shows how confinement modifies otherwise homogeneous quench spectra. In one-dimensional bosons in an optical lattice with harmonic trapping, the Mott-insulating spectrum retains a gap near $j(\omega,\tau)$3 but acquires a broadened band with width $j(\omega,\tau)$4 in DMRG and $j(\omega,\tau)$5 experimentally, together with a suppression beyond $j(\omega,\tau)$6 (Yu et al., 10 Jul 2025). The superfluid response remains gapless near $j(\omega,\tau)$7, and the inverse quench with larger amplitude yields the clearest trapped Mott spectrum (Yu et al., 10 Jul 2025).

One-dimensional spin chains introduce a sharper distinction between local and non-local probes. In Jordan–Wigner-mappable models, local spin observables typically excite multiparticle continua, whereas the non-local fermionic quench spectral function

$j(\omega,\tau)$8

yields sharp peaks at twice the fermionic quasiparticle energy (Bocini et al., 2024). This is the central result of non-local quench spectroscopy of fermionic excitations in 1D quantum spin chains. The XX-chain case gives peaks at $j(\omega,\tau)$9, while local probes instead display broad continua with edges characteristic of two-particle processes (Bocini et al., 2024).

The same dynamical logic extends to modes that are not isolated poles in linear response. In the gapless phase of the $j(\omega,\Omega)$0D XXZ spin chain, nonequilibrium quench spectroscopy identifies a symmetry-preserving underdamped amplitude mode on top of a Luttinger-liquid ground state, visible in the oscillations of the $j(\omega,\Omega)$1-symmetric observable $j(\omega,\Omega)$2. At $j(\omega,\Omega)$3, the dominant oscillation has period $j(\omega,\Omega)$4, corresponding to $j(\omega,\Omega)$5, and the amplitude scales with the quench strength $j(\omega,\Omega)$6 (Ha et al., 28 Jan 2026). The Bethe-ansatz analysis traces the mode to specific families of string excitations, especially two 2-strings at $j(\omega,\Omega)$7 (Ha et al., 28 Jan 2026).

A conceptually different but experimentally economical route is the measurement quench. Here a local projective measurement changes the state while leaving the Hamiltonian unchanged; spectroscopy then proceeds from the post-measurement time trace. In Ising chains this yields oscillations governed by matrix elements $j(\omega,\Omega)$8, while in a Kondo spin-chain emulation it provides nonequilibrium scaling collapse at fixed $j(\omega,\Omega)$9 and an exponential extraction of the Kondo cloud through $\mathcal{L}(t)$0, with $\mathcal{L}(t)$1 and $\mathcal{L}(t)$2 (Bayat et al., 2018).

4. Superconducting and quench-drive spectroscopy

In superconductors, quench spectroscopy takes a particularly developed form in THz nonlinear optics. The single-band BCS formulation of quench-drive spectroscopy combines a short, broadband quench $\mathcal{L}(t)$3 with a delayed, multicycle narrowband drive $\mathcal{L}(t)$4, and records the nonlinear current $\mathcal{L}(t)$5 under

$\mathcal{L}(t)$6

The dynamics are computed in the Anderson-pseudospin framework,

$\mathcal{L}(t)$7

with the electromagnetic field entering through minimal coupling and the dominant third-order response summarized by

$\mathcal{L}(t)$8

in inversion-symmetric $\mathcal{L}(t)$9-wave superconductors (Puviani et al., 2021).

The distinctive object is the two-dimensional spectrum $A(\omega,t)$0, obtained by Fourier transforming with respect to both real time and quench–drive delay. Pure third-harmonic generation produces vertical features at $A(\omega,t)$1 and $A(\omega,t)$2, whereas quench–drive mixing yields diagonal lines $A(\omega,t)$3 with $A(\omega,t)$4 (Puviani et al., 2021). Along these diagonals, the signal directly tracks the nonlinear susceptibility $A(\omega,t)$5, allowing extraction of resonances near $A(\omega,t)$6 without scanning the drive frequency. The third harmonic shows $A(\omega,t)$7-dependent modulation, visible as sidebands at $A(\omega,t)$8, $A(\omega,t)$9, and $H_i$0 in $H_i$1, and polarization control can suppress or preserve specific pathways (Puviani et al., 2021).

The cuprate version adapts the same logic to a clean $H_i$2-wave BCS superconductor with $H_i$3. In that setting, the Higgs mode is broadened by nodal quasiparticles, but the delay-dependent third-harmonic signal still carries a peak in the modulation spectrum at $H_i$4, together with sidebands near $H_i$5 (Puviani et al., 2022). The theory also models a phase-fluctuating pseudogap state by attaching random momentum-dependent phases to the Cooper pairs; moderate incoherence strongly suppresses THG and removes the sharp $H_i$6 signature, while stronger pulses can transiently increase $H_i$7 and partially restore phase coherence (Puviani et al., 2022).

A symmetry-resolved generalization makes the role of irreducible representations explicit. In the phase-fluctuating $H_i$8-wave state above $H_i$9, the equilibrium order parameter vanishes because global phase coherence is lost, yet a short quench can transiently align phases of pre-existing pairs. The resulting third-order current factorizes into a susceptibility and Raman-like vertices,

$H_f$0

which can be decomposed into $H_f$1, $H_f$2, and $H_f$3 channels by polarization geometry (Puviani, 2024). A central result is that fully incoherent pairs with an equilibrium vanishing order parameter can still produce a third-harmonic signal after a quench. In the resulting two-dimensional spectrum $H_f$4, the $H_f$5 vertical features are drive-only, diagonal lines encode quench–drive mixing, $H_f$6 dominates THG, and $H_f$7 controls quench-induced gap enhancement (Puviani, 2024).

These superconducting variants make especially clear that quench spectroscopy is not merely the time-domain counterpart of linear response. The delay axis in quench-drive spectroscopy is itself a conjugate spectral variable, and in cuprates it has been proposed as a route to detect induced or increased phase coherence in the pseudogap regime through the reappearance of a delay-locked Higgs modulation peak and revived THG (Puviani et al., 2022).

5. Boundary, impurity, and topological realizations

A different branch of quench spectroscopy uses local boundary quenches to probe topological and impurity excitations. In the Majorana protocol, a metallic lead is suddenly coupled to the end of a topological superconducting wire. The post-quench low-energy Hamiltonian is a chiral fermion coupled to an end Majorana zero mode, and the central observable is the Loschmidt echo $H_f$8 (Vasseur et al., 2014). For $H_f$9,

$H(\lambda_i)$00

with the exponent fixed by a boundary-condition change from normal reflection to Andreev reflection and by a boundary condition changing operator of dimension $H(\lambda_i)$01 (Vasseur et al., 2014). This exponent remains robust for interacting Luttinger-liquid leads with $H(\lambda_i)$02, for spinful and multichannel leads in the weak-tunneling regime, and even in the presence of an additional bound level when the Majorana coupling is nonzero (Vasseur et al., 2014). In a trivial gapped phase, by contrast, the exponent is non-universal and small, $H(\lambda_i)$03, so the quench distinguishes a Majorana end mode from a trivial subgap state in the same device geometry (Vasseur et al., 2014).

Optical quench spectroscopy of Kondo correlations realizes a related idea in semiconductor quantum dots tunnel-coupled to a fermionic reservoir. Absorption of a single photon on the $H(\lambda_i)$04 transition suddenly changes the local Hamiltonian by creating a doubly occupied dot and effectively turning off the spin-exchange interaction with the reservoir (Latta et al., 2011). The absorption rate near threshold obeys

$H(\lambda_i)$05

where $H(\lambda_i)$06 is set by the change of scattering phase shifts across the quench. In the Kondo-singlet to local-singlet quench, the dominant value is $H(\lambda_i)$07 at $H(\lambda_i)$08, while at $H(\lambda_i)$09 the extracted exponents are $H(\lambda_i)$10 for the blue transition and $H(\lambda_i)$11 for the red transition (Latta et al., 2011). The associated edge singularity is an optical manifestation of Anderson orthogonality catastrophe, and the scaling collapse of $H(\lambda_i)$12 versus $H(\lambda_i)$13 demonstrates Kondo universality in an optical spectrum (Latta et al., 2011).

Time-dependent impurity spectral functions supply another readout layer. For the Anderson impurity model after a sudden quench, the time-dependent numerical renormalization group yields retarded, advanced, lesser, and greater Green functions for several time-reference choices, and defines

$H(\lambda_i)$14

with $H(\lambda_i)$15 directly determining time-resolved photoemission for a Gaussian probe pulse (Nghiem et al., 2019). In the symmetric-to-symmetric Kondo quench considered there, the high-energy satellite peaks rearrange on the charge timescale $H(\lambda_i)$16, whereas the Kondo resonance develops on the much longer spin timescale $H(\lambda_i)$17 (Nghiem et al., 2019). The different choices of time reference shift how the transient buildup appears, but the long-time spectral function tends to the final equilibrium one, while the occupied density of states shows imperfect thermalization characteristic of single-quench TDNRG (Nghiem et al., 2019).

These impurity and boundary cases clarify that quench spectroscopy need not rely on spatial Fourier transforms. The same general logic can be implemented through threshold power laws, occupied density of states, or overlap dynamics, provided the sudden local change prepares a coherent superposition that is diagnostic of the post-quench fixed point.

6. Extensions, limitations, and outlook

The scope of quench spectroscopy has expanded beyond isolated Hermitian systems. In a resonant Bose gas quenched to strong interactions near a Feshbach resonance, post-quench dynamics shows many-body Rabi-like oscillations between the condensate and finite-momentum quasiparticle pairs, followed by a pre-thermalized nonequilibrium steady state with a broad stationary momentum distribution, a $H(\lambda_i)$18-type tail through Tan’s contact, and characteristic signatures in the structure function and RF spectroscopy (Yin et al., 2016). In this setting, quench spectroscopy is less about resolving a single dispersion branch than about diagnosing interaction strength, prethermalization times $H(\lambda_i)$19, and short-range correlations through their temporal fingerprints (Yin et al., 2016).

The method has also been generalized explicitly to dissipative and non-Hermitian lattice models. For the open Bose–Hubbard chain confined in the superfluid phase, a sudden global quench of dissipations and repulsive interactions produces equal-time correlators whose quench spectral function still reconstructs the isolated post-quench Bogoliubov dispersion, supplemented by a $H(\lambda_i)$20 background from the loss quench (Despres, 2024). For the non-Hermitian transverse-field Ising chain in the paramagnetic phase, the quench spectral function reveals peaks at $H(\lambda_i)$21, so the spectroscopy tracks twice the real part of the complex single-particle energy, while the imaginary part governs the exponential envelope (Despres, 2024). This suggests that the notion of quench spectroscopy is broader than unitary Hamiltonian evolution, although the interpretation of linewidths and envelopes becomes model dependent.

Several limitations recur across implementations. Finite observation windows impose a nominal resolution $H(\lambda_i)$22, finite size broadens or splits peaks, and boundary reflections limit the useful time window in real-space reconstructions (Yu et al., 10 Jul 2025, Bocini et al., 2024). In superconducting quench-drive spectroscopy, moderate fields, short or narrowband pulses, and slow relaxation underpin the quasiequilibrium interpretation, while strong-field, strongly nonequilibrium regimes may require Keldysh nonequilibrium Green’s functions (Puviani et al., 2021). In trapped bosonic systems, harmonic confinement broadens the Mott band and mixes momentum components (Yu et al., 10 Jul 2025). In Majorana boundary quenches, the quench must be effectively instantaneous relative to $H(\lambda_i)$23, the superconducting gap must remain clean, and finite Majorana splitting $H(\lambda_i)$24 restricts the available power-law window (Vasseur et al., 2014). In superconducting pseudogap settings, phase incoherence can strongly suppress THG and remove the sharp delay-frequency peak (Puviani et al., 2022).

Another persistent issue is operator selectivity. Local probes can be sufficient for bosonic quasiparticles or for one-point local quench spectroscopy in translation-invariant models, but they can completely miss non-local quasiparticles, as shown by the 1D fermionic spin-chain case (Villa et al., 2020, Bocini et al., 2024). Conversely, non-local observables can be experimentally costly because they require parity strings or site-resolved readout. This tension between selectivity and accessibility is intrinsic to the method rather than incidental.

The recent literature points toward several concrete extensions already articulated in the underlying works: multi-band and unconventional superconductors with Leggett or Bardasis–Schrieffer modes, different power-law exponents and frustrated geometries in quantum simulators, higher dimensions, dissipative and non-Hermitian settings, and full pump-pump-probe or more general multidimensional coherent protocols (Puviani et al., 2021, Chen et al., 2023, Despres, 2024). A plausible implication is that quench spectroscopy is evolving from a niche post-quench diagnostic into a general framework for extracting excitation structure from controlled nonequilibrium dynamics, with the precise observable and transform chosen to match the conserved quantities, symmetry sector, and experimental platform.