--- title: Double-Quench Protocol in Quantum Dynamics url: https://www.emergentmind.com/topics/double-quench-protocol type: topic --- # Double-Quench Protocol in Quantum Dynamics 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 [2202.06341] [2202.10532]. ## 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 \(H_2\), after which the subsequent dynamics are monitored [2202.06341] [2202.10532]. | Time interval | Hamiltonian | Evolution | |---|---|---| | \(t<0\) | \(H_0\) | Preparation in the initial state | | \(0\le t\le T\) or \(0\le t<\tau\) | \(H_1\) | First post-quench evolution | | \(t\ge T\) or \(t\ge\tau\) | \(H_2\) | Second post-quench evolution | In the XY-chain protocol, the chain is prepared in the ground state \(\ket{\Psi(0)}=\ket{\mathrm{GS}(h_0,\gamma_0)}\). For \(0T\), \[ \ket{\Psi(t)}=U_2(t-T)\ket{\Psi(T)}, \qquad U_2(\tau)=\exp[-\,i\,H_2 \tau]. \] By choosing \((h_0,\gamma_0)\) in phase \(A\), \((h_1,\gamma_1)\) in phase \(B\), and \((h_2,\gamma_2)\) in phase \(C\), one can explore all inter-phase passages [2202.06341]. In the two-band DQPT framework, the time-dependent Hamiltonian is \[ H(t)= \begin{cases} H_0,& t<0,\\ H_1,& 0\le t<\tau,\\ H_2,& t\ge \tau. \end{cases} \] For pure states, the Loschmidt amplitude is \(G(t)=\bra{\psi(0)}e^{-\,iHt}\ket{\psi(0)}\); for thermal mixed states, the finite-temperature Loschmidt amplitude is \[ G_\rho(t)=\Tr[\rho(0)e^{-\,iHt}], \] with rate function \[ g(t)=-\frac1N\ln |G_\rho(t)|^2. \] 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 [2202.10532]. ## 2. XY-chain realization in the transverse-field anisotropic model For the one-dimensional nearest-neighbor spin-\(\tfrac12\) XY chain in a transverse field, the Hamiltonian is \[ H(h,\gamma)= -\sum_{j=1}^N\Bigl[\tfrac{1+\gamma}{2}\sigma^x_j\sigma^x_{j+1} +\tfrac{1-\gamma}{2}\sigma^y_j\sigma^y_{j+1}\Bigr] -h\sum_{j=1}^N\sigma^z_j, \] with periodic boundary conditions and units \(J=1\). Its zero-temperature phase diagram has a ferromagnetic (FM) phase for \(|h|<1\), a paramagnetic (PM) phase for \(|h|>1\), and critical line \(h_c=1\) [2202.06341]. Jordan–Wigner and Fourier transformation map the model to a free-fermion form, \[ H=\sum_{k>0}\Bigl[{\cal A}_k(a_k^\dagger a_k+a_{-k}^\dagger a_{-k}) +i{\cal B}_k(a_k^\dagger a_{-k}^\dagger+a_{-k}a_k)\Bigr], \] with \[ {\cal A}_k= -[\cos k+h], \qquad {\cal B}_k=\gamma \sin k. \] A Bogoliubov rotation by angle \(\theta_k\) diagonalizes each \(k\)-sector: \[ H=\sum_k \varepsilon_k\Bigl(\alpha_k^\dagger\alpha_k-\tfrac12\Bigr), \qquad \varepsilon_k=\sqrt{{\cal A}_k^2+{\cal B}_k^2}, \qquad \tan 2\theta_k=-\,{\cal B}_k/{\cal A}_k. \] The quench dynamics are encoded in two-point correlators such as \[ R_{j,j+1}(t)=\langle a_j^\dagger a_{j+1}\rangle,\qquad S_{j,j+1}(t)=\langle a_j^\dagger a_{j+1}^\dagger\rangle. \] For \(0T\), the expressions involve both \(\Phi_k^1\) and \(\Phi_k^2=\theta_k^2-\theta_k^1\), together with phases \(e^{\pm i\varepsilon_k^1T}\). The appearance of \(\varepsilon_k^1 T\) is the origin of the protocol’s explicit waiting-time dependence [2202.06341]. The steady state after the second quench is defined through a long-time average of the two-site reduced density matrix, \[ \rho_{j,j+1}^{(\mathrm{ss})} =\lim_{\tau\to\infty}\frac1\tau\int_T^{T+\tau}dt\,\rho_{j,j+1}(t). \] Because oscillatory terms of the form \(e^{\pm i(E_n-E_{n'})t}\) average to zero, \(\rho^{(\mathrm{ss})}\) becomes block-diagonal in the post-quench eigenbasis. Equivalently, in each \(k\)-sector the off-diagonal Bogoliubov coherence decays, leaving only occupation probabilities proportional to \(|g_n|^2\) [2202.06341]. ## 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 \[ \rho= \begin{pmatrix} X^+ &0&0&-f^*\\ 0&Y^+&Z^*&0\\ 0&Z&Y^-&0\\ f&0&0&X^- \end{pmatrix}, \] with \(Z=\langle a_i^\dagger a_{i+1}\rangle\), \(f=\langle a_i^\dagger a_{i+1}^\dagger\rangle\), and occupation-based matrix elements such as \(X^+=\langle n_i n_{i+1}\rangle\). The concurrence is \[ C(\rho)=\max\{0,\;2|Z|-\sqrt{X^+X^-},\;2|f|-\sqrt{Y^+Y^-}\}, \] while quantum discord is defined by \[ \mathrm{QD}(\rho)=\mathcal I(\rho)-\mathcal C(\rho), \] where \(\mathcal I(\rho)=S(\rho_i)+S(\rho_{i+1})-S(\rho_{i,i+1})\) and the classical part \(\mathcal C(\rho)\) is obtained by optimizing over projective measurements on one spin [2202.06341]. A central result is that the steady-state correlators \(R^{(\mathrm{ss})}\) and \(S^{(\mathrm{ss})}\) retain a residual dependence on the accumulated phase \(\varepsilon_k^1 T\). As a consequence, both \(C_{\rm ss}\) and \(\mathrm{QD}_{\rm ss}\) oscillate as functions of \(T\), and scanning \(T\) identifies optimal times at which the steady-state concurrence or discord is maximized. The intermediate quench point \((h_1,\gamma_1)\) 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 \((h_2,\gamma_2)\) [2202.06341]. The same framework also diagnoses nonequilibrium criticality. When the final quench parameter \((h_2,\gamma_2)\) crosses the equilibrium critical line, one observes a cusp, described as a nonanalytic kink, in \(C_{\rm ss}(h_2)\) or \(\mathrm{QD}_{\rm ss}(h_2)\) exactly at \(h_2=1\). This is inherited from the singular behavior of the Bogoliubov angle \(\theta_k\) at the gap-closing mode \(k=\pi\). 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 \(\rho_{\rm ss}\) in the post-quench energy eigenbasis [2202.06341]. 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 [2202.06341]. ## 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 \[ H_k=E_k\,\openone_{2\times2}+\tfrac12\,\Delta_k\,\hat n_k\cdot \vec\sigma, \] where \(\Delta_k\) is the band gap and \(\hat n_k\) is a unit Bloch vector. Because the post-quench Hamiltonians are quadratic and block-diagonal in \(k\), the time-evolution operators in each momentum sector take the form \[ U_{1k}(t)=e^{-\,iE_{1k}t}\bigl[\cos(\omega_{1k}t)\openone-i\sin(\omega_{1k}t)\hat n_{1k}\cdot\vec\sigma\bigr], \] \[ U_{2k}(t-\tau)=e^{-\,iE_{2k}(t-\tau)}\bigl[\cos[\omega_{2k}(t-\tau)]\openone-i\sin[\omega_{2k}(t-\tau)]\hat n_{2k}\cdot\vec\sigma\bigr], \] with \(\omega_{1k}=\tfrac{\Delta_{1k}}2\) and \(\omega_{2k}=\tfrac{\Delta_{2k}}2\) [2202.10532]. 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 \(k_c\) and waiting time \(\tau^*\) such that \[ \hat n_{1k_c}\perp \hat n_{2k_c}, \qquad \hat n_{0k_c}=\hat n_{2k_c}, \] together with the duration fine-tuning condition \[ \omega_{1k_c}\tau^*=\Bigl(n+\tfrac12\Bigr)\pi,\qquad n\in\mathbb Z. \] Under these conditions, \[ G_\rho(t\ge \tau^*)=0, \] so the rate function remains singular for every \(t>\tau^*\). In the formulation of the source work, the final state continually has no overlap with the initial state [2202.10532]. This phenomenon occurs at zero as well as finite temperatures. The mixed-state formalism makes the temperature dependence explicit through \(G_\rho(t)=\Tr[\rho(0)e^{-\,iHt}]\), while the geometric conditions for metamorphic DQPT depend on Bloch-vector relations that are independent of \(\beta\). The work also notes that if \(\tau\) is not tuned exactly to \(\tau^*\), a small offset \(\epsilon=\tau-\tau^*\) yields a characteristic logarithmic divergence, \[ g(t\ge \tau)\supset -\ln|\epsilon|, \] which provides an additional signature [2202.10532]. ## 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, \[ H_k=\tfrac12\,\Delta_k \begin{pmatrix} 0 & f_k^*\\ f_k & 0 \end{pmatrix}, \qquad \Delta_k=2\sqrt{J_1^2+J_2^2+2J_1J_2\cos k}, \] with \[ \hat n_k=\frac{1}{\Delta_k}(-J_1-J_2\cos k,\,-J_2\sin k,\,0). \] Using hopping pairs \((J_{01},J_{02})\), \((J_{11},J_{12})\), and \((J_{21},J_{22})\), the condition \(\hat n_{1k}\cdot \hat n_{2k}=0\) gives \[ \cos k_c=-\frac{J_{11}J_{21}+J_{12}J_{22}}{J_{11}J_{22}+J_{12}J_{21}}, \] which is admissible whenever \[ (J_{11}-J_{12})(J_{21}-J_{22})\le 0. \] This is precisely the condition that the post-first-quench SSH chain and the post-second-quench chain lie in opposite topological phases. Choosing \(H_2=H_0\) enforces \(\hat n_{2k}=\hat n_{0k}\), and the special waiting time is \[ \tau^*=\frac{(n+\frac12)\pi}{\omega_{1k_c}}. \] For the numerical example \(J_{11}=0.4\), \(J_{12}=0.8\), \(J_{21}=J_{01}=1.0\), \(J_{22}=J_{02}=0.8\), and \(T=3J_{01}\), one finds \(k_c\approx \arccos(-0.8)\) and, for \(n=1\), \(\tau^*\approx 11.04/J_{01}\); ordinary DQPT kinks occur at \(t\tau^*\), and plotting \(g(t)\) shows an abrupt divergence at \(t=\tau^*\) and divergences thereafter [2202.10532]. 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 \(T\) so as to maximize \(C_{\rm ss}\). 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. Source: https://www.emergentmind.com/topics/double-quench-protocol