Spatiotemporal Quenches & Critical Cooling

Preparing a critical ground state is difficult because the gap protecting adiabatic evolution closes as the target is approached. A spatially uniform ramp can create excitations everywhere at once. Agarwal, Bhatt, and Sondhi proposed a different strategy: remove the gap along a moving front so that excitations are distributed unevenly in space and direction. The resulting cold region is the useful output. The protocol's success depends on causal structure and dispersion, not on eliminating every excitation in the entire sample.

Figure to read

Space-time energy density during a moving-front quench in a long-range Ising chain. Figure 1 in the source paper. Bernier, Simon; Agarwal, Kartiek. Spatiotemporal Quenches in Long-Range Hamiltonians, Physical Review B 108, 024310 (2023). Paper. CC BY 4.0. Original manuscript graphic; no alterations.

Locate the cold region and the narrow regions carrying excess energy. The color scale reports energy relative to the critical spectral bandwidth. This example uses interaction exponent alpha = 6; it is not representative of every long-range regime.

The conventional benchmark

Near a continuous quantum transition, long relaxation times make a finite-rate sweep nonadiabatic. Adiabatic perturbation theory and Kibble-Zurek reasoning relate the resulting excitations to the closing gap and ramp rate. Those ideas establish why simply moving slowly can become expensive. They also emphasize that the relevant error depends on what is measured: excitation density, excess energy, correlation length, and global fidelity need not scale identically. 1

Spatial inhomogeneity supplies an additional control parameter. Earlier work on inhomogeneous quantum transitions studied how an ordered region can influence neighboring regions and suppress defect formation when a transition front moves sufficiently slowly. That mechanism is distinct from the superluminal-front construction: here the front outruns the emergent low-energy propagation speed and exploits the resulting directional distribution of excitations. Both belong in the comparison, but they should not be merged into a single causal argument. 2

How the moving-front idea works

Begin with a gapped state that is easier to prepare than the desired critical state. Let a parameter controlling the gap change locally as a function of position and time. In an approximately relativistic low-energy theory, a suitable spacetime transformation relates the moving-front problem to a simpler quench. Excitations propagating in different directions experience different Doppler shifts. Near the relevant velocity limit, much of the excess energy is concentrated away from an extended cold region. 3

The word “superluminal” refers to the effective velocity $c$ of the model's low-energy excitations. A pattern of externally applied local controls can move faster than that emergent velocity without sending information faster than light in vacuum. The distinction is experimentally important: the challenge is spatial and temporal control resolution, not a violation of relativistic causality. A front must also be smooth enough to avoid excessive ultraviolet excitations outside the effective theory.

Evidence and extensions

The original work combines solvable field-theory examples with a numerical illustration in a spin chain. Its attractive preparation-time scaling applies under the stated model and quality assumptions. It should not be advertised as a universal speedup for arbitrary Hamiltonians or as a guarantee of high global fidelity for the full sample. The paper case study makes the measurement region and discarded hot region explicit. 3

Bernier and Agarwal's long-range Ising-chain study asks what happens when dispersion departs from the simple relativistic picture. The cooling minimum can weaken or disappear in regimes where the relevant excitations no longer share the needed propagation structure. Their later two-dimensional work tests an interacting transverse-field Ising system and finds useful cooling near the emergent velocity for accessible sizes. Together these studies turn a kinematic idea into a set of model-dependent numerical tests. 4 5

What should be compared next?

Counterdiabatic and optimal-control methods offer complementary routes. A fair comparison fixes the available controls, maximum amplitudes, total runtime, initial state, and the region in which quality is assessed. A method requiring long-range many-body operators should not be compared with a local front solely through its formal error scaling. Conversely, a front's spatial-addressing cost is part of its resource budget. 6

The open questions concern robustness to finite initial temperature, imperfect front trajectories, and interactions that spoil the effective continuum description. Energy profiles and correlations should be measured together; a cold-looking local observable can conceal residual long-wavelength errors. Follow geometry and causality for hyperbolic quenches and the connection to chiral prethermalization. The central design lesson is to specify where the excitations go, not merely how many are generated.

References

1. De Grandi, C.; Polkovnikov, A.. Adiabatic perturbation theory: from Landau-Zener problem to quenching through a quantum critical point. "Quantum Quenching, Annealing and Computation", Eds. A. Das, A. Chandra and B. K. Chakrabarti, Lect. Notes in Phys., vol. 802 (Springer, Heidelberg 2010). Open manuscript.
2. Dziarmaga, Jacek; Rams, Marek M.. Dynamics of an inhomogeneous quantum phase transition. New J. Phys. 12, 055007 (2010). Open manuscript.
3. Agarwal, Kartiek; Bhatt, R. N.; Sondhi, S. L.. Fast Preparation of Critical Ground States Using Superluminal Fronts. Physical Review Letters 120, 210604 (2018). Open manuscript.
4. Bernier, Simon; Agarwal, Kartiek. Spatiotemporal Quenches in Long-Range Hamiltonians. Physical Review B 108, 024310 (2023). Open manuscript.
5. Bernier, Simon; Agarwal, Kartiek. Spatiotemporal quenches for efficient critical ground state preparation in the two-dimensional transverse field Ising model. Physical Review B 111, 054311 (2025). Open manuscript.
6. Guéry-Odelin, D.; Ruschhaupt, A.; Kiely, A.; Torrontegui, E.; Martínez-Garaot, S.; Muga, J. G.. Shortcuts to adiabaticity: concepts, methods, and applications. Rev. Mod. Phys. 91, 045001 (2019). Open manuscript.

Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.