Double-Quench Protocol in Quantum Dynamics

• Double-quench protocol is a nonequilibrium driving scheme characterized by two successive sudden Hamiltonian changes separated by a controllable waiting time, enabling access to dynamical quantum phase transitions.
• It uses a sequence of three Hamiltonians to manipulate steady-state quantum correlations, with implementations in models like the XY-chain, SSH, and Kitaev chains, and experimental realizations in trapped ions and ultracold atoms.
• The protocol optimizes entanglement measures and reveals persistent nonanalytic signatures in observables such as the Loschmidt echo, offering practical ways to amplify quantum correlations and investigate dynamical transitions.

A double-quench protocol is a nonequilibrium driving scheme in which a quantum system undergoes two successive sudden parameter changes separated by a controllable waiting time. In the formulations considered for the spin-$\tfrac12$ transverse-field XY chain and for one-dimensional two-band systems, the protocol is used to manipulate post-quench steady states, amplify quantum correlations, and engineer dynamical quantum phase transitions (DQPTs) with behavior that is not accessible under a single quench alone (Kheiri et al., 2022, Hou et al., 2022).

1. Protocol definition and temporal structure

The protocol is specified by three Hamiltonians, conventionally denoted $H_0$, $H_1$, and $H_2$. For the XY-chain setting, these correspond to three parameter pairs $(h_0,\gamma_0)$, $(h_1,\gamma_1)$, and $(h_2,\gamma_2)$, while in the two-band DQPT setting they correspond to three Bloch Hamiltonians before and after the two quenches. The temporal structure is common to both formulations: the system is prepared with $H_0$, quenched at $t=0$ to $H_1$, evolved for a finite interval, and then quenched again to $H_0$0, after which the subsequent dynamics are monitored (Kheiri et al., 2022, Hou et al., 2022).

Time interval	Hamiltonian	Evolution
$H_0$1	$H_0$2	Preparation in the initial state
$H_0$3 or $H_0$4	$H_0$5	First post-quench evolution
$H_0$6 or $H_0$7	$H_0$8	Second post-quench evolution

In the XY-chain protocol, the chain is prepared in the ground state $H_0$9. For $H_1$0, the state evolves as

$H_1$1

At $H_1$2, the second quench $H_1$3 is applied, and for $H_1$4,

$H_1$5

By choosing $H_1$6 in phase $H_1$7, $H_1$8 in phase $H_1$9, and $H_2$0 in phase $H_2$1, one can explore all inter-phase passages (Kheiri et al., 2022).

In the two-band DQPT framework, the time-dependent Hamiltonian is

$H_2$2

For pure states, the Loschmidt amplitude is $H_2$3; for thermal mixed states, the finite-temperature Loschmidt amplitude is

$H_2$4

with rate function

$H_2$5

This dual usage already indicates that “double quench” names a protocol rather than a single observable: the same temporal pattern supports both correlation-based and Loschmidt-based diagnostics (Hou et al., 2022).

2. XY-chain realization in the transverse-field anisotropic model

For the one-dimensional nearest-neighbor spin-$H_2$6 XY chain in a transverse field, the Hamiltonian is

$H_2$7

with periodic boundary conditions and units $H_2$8. Its zero-temperature phase diagram has a ferromagnetic (FM) phase for $H_2$9, a paramagnetic (PM) phase for $(h_0,\gamma_0)$0, and critical line $(h_0,\gamma_0)$1 (Kheiri et al., 2022).

Jordan–Wigner and Fourier transformation map the model to a free-fermion form,

$(h_0,\gamma_0)$2

with

$(h_0,\gamma_0)$3

A Bogoliubov rotation by angle $(h_0,\gamma_0)$4 diagonalizes each $(h_0,\gamma_0)$5-sector: $(h_0,\gamma_0)$6

The quench dynamics are encoded in two-point correlators such as

$(h_0,\gamma_0)$7

For $(h_0,\gamma_0)$8, the correlators depend on the change $(h_0,\gamma_0)$9; for $(h_1,\gamma_1)$0, the expressions involve both $(h_1,\gamma_1)$1 and $(h_1,\gamma_1)$2, together with phases $(h_1,\gamma_1)$3. The appearance of $(h_1,\gamma_1)$4 is the origin of the protocol’s explicit waiting-time dependence (Kheiri et al., 2022).

The steady state after the second quench is defined through a long-time average of the two-site reduced density matrix,

$(h_1,\gamma_1)$5

Because oscillatory terms of the form $(h_1,\gamma_1)$6 average to zero, $(h_1,\gamma_1)$7 becomes block-diagonal in the post-quench eigenbasis. Equivalently, in each $(h_1,\gamma_1)$8-sector the off-diagonal Bogoliubov coherence decays, leaving only occupation probabilities proportional to $(h_1,\gamma_1)$9 (Kheiri et al., 2022).

3. Steady-state quantum correlations and critical nonanalyticity

The principal observables studied in the XY-chain implementation are nearest-neighbor concurrence and quantum discord. The reduced density matrix for two neighboring spins has the form

$(h_2,\gamma_2)$0

with $(h_2,\gamma_2)$1, $(h_2,\gamma_2)$2, and occupation-based matrix elements such as $(h_2,\gamma_2)$3. The concurrence is

$(h_2,\gamma_2)$4

while quantum discord is defined by

$(h_2,\gamma_2)$5

where $(h_2,\gamma_2)$6 and the classical part $(h_2,\gamma_2)$7 is obtained by optimizing over projective measurements on one spin (Kheiri et al., 2022).

A central result is that the steady-state correlators $(h_2,\gamma_2)$8 and $(h_2,\gamma_2)$9 retain a residual dependence on the accumulated phase $H_0$0. As a consequence, both $H_0$1 and $H_0$2 oscillate as functions of $H_0$3, and scanning $H_0$4 identifies optimal times at which the steady-state concurrence or discord is maximized. The intermediate quench point $H_0$5 is crucial: quenching first into the FM region and then into the PM region, or vice versa, can amplify the final steady-state entanglement well above both the single-quench outcome and the ground-state value at $H_0$6 (Kheiri et al., 2022).

The same framework also diagnoses nonequilibrium criticality. When the final quench parameter $H_0$7 crosses the equilibrium critical line, one observes a cusp, described as a nonanalytic kink, in $H_0$8 or $H_0$9 exactly at $t=0$0. This is inherited from the singular behavior of the Bogoliubov angle $t=0$1 at the gap-closing mode $t=0$2. Equivalently, the long-time average of the Loschmidt echo rate has a cusp there, and the same nonanalyticity is carried into the steady correlations through the expansion of $t=0$3 in the post-quench energy eigenbasis (Kheiri et al., 2022).

A recurrent misconception is that long-time quench physics must be weaker than equilibrium ground-state correlations. The XY-chain results explicitly show the opposite possibility: steady-state quantum correlations can be strikingly greater than the equilibrium ones in the ground state under appropriate single- and double-quench passages. Another misconception is that only conventional observables detect nonequilibrium transitions; here the nonanalyticity is visible directly in steady-state concurrence and discord (Kheiri et al., 2022).

4. Double quench in one-dimensional two-band systems and metamorphic DQPT

In a broader two-band setting, the instantaneous Hamiltonian may be written as

$t=0$4

where $t=0$5 is the band gap and $t=0$6 is a unit Bloch vector. Because the post-quench Hamiltonians are quadratic and block-diagonal in $t=0$7, the time-evolution operators in each momentum sector take the form

$t=0$8

$t=0$9

with $H_1$0 and $H_1$1 (Hou et al., 2022).

Within this framework, the double-quench protocol supports a distinct class of DQPTs termed metamorphic DQPTs. Ordinary DQPTs are associated with zeros of the Loschmidt amplitude at discrete critical times, producing nonanalytic kinks in the rate function. By contrast, a metamorphic DQPT occurs when there exists a critical momentum $H_1$2 and waiting time $H_1$3 such that

$H_1$4

together with the duration fine-tuning condition

$H_1$5

Under these conditions,

$H_1$6

so the rate function remains singular for every $H_1$7. In the formulation of the source work, the final state continually has no overlap with the initial state (Hou et al., 2022).

This phenomenon occurs at zero as well as finite temperatures. The mixed-state formalism makes the temperature dependence explicit through $H_1$8, while the geometric conditions for metamorphic DQPT depend on Bloch-vector relations that are independent of $H_1$9. The work also notes that if $H_0$00 is not tuned exactly to $H_0$01, a small offset $H_0$02 yields a characteristic logarithmic divergence,

$H_0$03

which provides an additional signature (Hou et al., 2022).

5. Canonical model realizations: SSH and Kitaev chains

The Su–Schrieffer–Heeger (SSH) model and the Kitaev chain provide explicit realizations of the two-band double-quench construction. In the SSH case,

$H_0$04

with

$H_0$05

Using hopping pairs $H_0$06, $H_0$07, and $H_0$08, the condition $H_0$09 gives

$H_0$10

which is admissible whenever

$H_0$11

This is precisely the condition that the post-first-quench SSH chain and the post-second-quench chain lie in opposite topological phases. Choosing $H_0$12 enforces $H_0$13, and the special waiting time is

$H_0$14

For the numerical example $H_0$15, $H_0$16, $H_0$17, $H_0$18, and $H_0$19, one finds $H_0$20 and, for $H_0$21, $H_0$22; ordinary DQPT kinks occur at $H_0$23, while at $H_0$24 a single metamorphic DQPT appears and the rate function remains divergent thereafter (Hou et al., 2022).

For the one-dimensional Kitaev $H_0$25-wave superconductor,

$H_0$26

with $H_0$27 and $H_0$28. The three Hamiltonians are labeled by $H_0$29, $H_0$30, and $H_0$31. The critical momentum is determined by

$H_0$32

which yields

$H_0$33

subject to $H_0$34. Again taking $H_0$35, the waiting time is

$H_0$36

For $H_0$37, $H_0$38, $H_0$39, and $H_0$40, two solutions $H_0$41 and $H_0$42 exist, hence two candidate waiting times $H_0$43 and $H_0$44; choosing $H_0$45 yields a metamorphic DQPT at $H_0$46 and permanent orthogonality thereafter (Hou et al., 2022).

These examples clarify an important distinction. In the SSH and Kitaev settings, the waiting time is tuned to force a Loschmidt zero that persists for all later times. In the XY-chain setting, the waiting time is scanned to optimize steady-state entanglement or discord. The protocol is structurally the same in both cases, but the target observable and the physical signature are different.

6. Experimental platforms and operational scope

The XY-chain version of the double-quench protocol is described as feasible in trapped-ion and ultracold-atom experiments. Trapped-ion chains can realize the transverse-field XY Hamiltonian by driving two Raman beams that induce spin–spin couplings $H_0$47 together with an additional carrier implementing the transverse field $H_0$48. The quoted scales are typical couplings $H_0$49 kHz, fields $H_0$50–$H_0$51 kHz, and experiment times up to a few ms. Sudden quenches are produced by rapidly changing the detuning of one Raman beam, while state-dependent fluorescence and full two-spin tomography allow reconstruction of $H_0$52 (Kheiri et al., 2022).

In ultracold atoms, two hyperfine states $H_0$53 and $H_0$54 in an optical lattice can be mapped to an effective spin chain via second-order superexchange. The anisotropy $H_0$55 is tuned by slightly tilting the lattice or modulating the tunneling rates, and the transverse field $H_0$56 is implemented by a real magnetic field or a differential light shift. The stated scales are $H_0$57–$H_0$58 and hold times $H_0$59 up to hundreds of ms. Single-site imaging and noise-correlation methods measure $H_0$60. In both platforms, the double quench consists of two fast changes of the control lasers or fields separated by a programmable delay $H_0$61; scanning $H_0$62 identifies times of maximal steady-state entanglement, and varying the second quench amplitude maps the nonequilibrium critical cusp at $H_0$63 (Kheiri et al., 2022).

For metamorphic DQPTs, the experimental prescription is correspondingly direct: prepare the initial thermal state $H_0$64, perform the first quench $H_0$65, evolve for time $H_0$66, apply the second quench $H_0$67, and measure the Loschmidt echo $H_0$68 through interferometric schemes or projective return-probability measurements. Scanning $H_0$69 seeks the special value satisfying $H_0$70; at $H_0$71, the return probability drops to zero at all $H_0$72, and plotting $H_0$73 shows an abrupt divergence at $H_0$74 and divergences thereafter (Hou et al., 2022).

In practical terms, one implementation strategy in the XY-chain context is to choose the initial point deep in one phase, the intermediate point in the opposite phase, the final point across the critical line, and then tune $H_0$75 so as to maximize $H_0$76. A plausible implication is that the double-quench protocol should be understood less as a single-purpose quench sequence than as a controllable nonequilibrium architecture: by changing the intermediate Hamiltonian and the waiting time, it can either amplify steady-state quantum correlations or enforce persistent orthogonality in Loschmidt dynamics, depending on the model and observable under consideration.