Theory of metastability in discrete-time open quantum dynamics

Metastability in open system dynamics describes the phenomena of initial relaxation to long-lived metastable states before decaying to the asymptotic stable states. It has been found in continuous-time stochastic dynamics of both classical and quantum systems. However, many cases of open quantum system dynamics are intrinsically discrete, and the evolution within each discrete time interval is described by an arbitrary quantum channel, which often cannot be generated by continuous-time master equations. Here we develop a general theory of metastability in discrete-time open quantum dynamics, described by sequential repetitive quantum channels. We apply the general metastability theory to a typical class of quantum channels on a target system, induced by an ancilla qubit with a pure-dephasing coupling to the target system and under Ramsey sequences. Interesting metastable behaviors are predicted and numerically demonstrated by decomposing the average dynamics of sequential quantum channels into stochastic trajectories. We also present examples of applications in quantum state and dynamics engineering of a target quantum system with an ancilla qubit.

Figure 1

(a) Schematic of sequential quantum channels. For an initial state $|\rho \rangle \rangle$ of a target system, its final state becomes ${\widehat{\mathrm{\Phi }}}^m|\rho \rangle \rangle$ after applying the quantum channel for $m$ times. (b) The eigenvectors of a quantum channel can be divided into three categories: the fixed points with eigenvalue $\lambda =1$, rotating points with $\lambda =e^{i\phi }(\phi \ne 0)$, and decaying points with $\left|\lambda \right|<1$. The decaying points with $\left|\lambda \right|\approx 1$ are also called metastable points. Note that the complex eigenvalues of a quantum channel always come in conjugate pairs. The area shaded in yellow labels a metastable region and quantum metastability emerges when the gap (represented by an arrow) between the smallest eigenvalue in this region and the next smaller one outside the region is relatively large. (c) Schematic of the quantum circuit for sequential RIMs, where $R_{\varphi }\left(\theta \right)=e^{-i(cos\varphi {\sigma }_q^x+sin\varphi {\sigma }_q^y)\theta /2}$ is the ancilla rotation operator and $U_{\alpha }=e^{-i[{(-1)}^{\alpha }B+\gamma C]t}$ is a unitary operator of the target system conditioned on the ancilla state ${|\alpha \rangle }_q$ ($\alpha =0,1$).

Figure 2

Metastability in sequential RIMs of an ancilla qubit coupled to a single target qubit. (a)–(d) Measurement statistics of sequential RIMs from Monte Carlo simulations (with $\gamma =0.05$), where we present four stages of evolutions of measurement polarization statistics. (a) When $m$ is relatively small, the width of peaks is large and the two peaks have overlaps. (b) For a larger $m$, quantum metastability emerges and there appears two well-distinguished distribution peaks corresponding to two EMSs. (c) The two peaks gradually disappear as $m$ increases beyond the metastable region. (d) Finally, there appears a single peak corresponding to the maximally mixed state $\mathbb{I}/2$ when $m\gg {\mu }^{\prime }$. (e) $\langle {\sigma }_z\rangle$ of the target qubit for three classes of trajectories in Monte Carlo simulations with different $\gamma$: the target qubit is either polarized to nearly $|0\rangle$ (upper lines) or $|1\rangle$ (lower lines), or depolarized to $\mathbb{I}/2$ (black line in the middle). The metastable region of the case that $\gamma =0.05$ is marked with a gradient yellow shade. The boundaries of the region (${\mu }^{″}=1/|ln|{\lambda }_3\left|\right|$ and ${\mu }^{\prime }=1/|ln|{\lambda }_2\left|\right|$) are calculated by numerically diagonalizing the channel [also see (f), the gap emerges between ${\lambda }_2$ and ${\lambda }_3]$ and are labeled with maroon (dark gray) and green (light gray) dotted dash vertical lines. Inset: Proportion of samples in polarized regions to total samples (also for $\gamma =0.05$). (f) Spectrum of the channel on the target system as a function of $\gamma$, where the spectra corresponding to $\gamma$ in (e) are labeled. All the simulations contains ${10}^4$ samples with $\mathrm{\Delta }\varphi =\pi /2$.

Figure 3

Metastability in sequential RIMs of an ancilla qubit coupled to two target spins with zero magnetic field. (a)–(d) Measurement statistics of sequential RIMs from Monte Carlo simulation for two target qubits, in which we present four stages of the system evolution. (a) When the time of measurements $m$ is small, the different peaks are not fully distinguishable due to the large distribution width. (b) For larger $m$, the four peaks corresponding to four EMSs can be well distinguished. (c) The peaks corresponding to the two EMSs ($|22\rangle \rangle$ and $|33\rangle \rangle$) disappear, while those relevant to $|11\rangle \rangle$ and $|44\rangle \rangle$ still remain. (d) All the peaks are merged into a single one corresponding to the only fixed point. (e) The trajectory of evolution of fidelity. Upper lines are $F_1$ and $F_4$, decreasing more slowly than the lower lines showing $F_2$ and $F_3$, which indicates faster vanishing of EMSs relevant to $|22\rangle \rangle$ and $|33\rangle \rangle$. Inset: The proportion of samples corresponding to each trajectory and the green (uppermost) dot represents the proportion of samples except for four trajectories. (f) Spectra of the quantum channel. In all Monte Carlo simulations, we use ${10}^4$ samples and the parameters are $\mathrm{\Delta }\varphi =\pi /2$, $|{\mathbf{A}}_1|=0.585\mathrm{kHz}$, $|{\mathbf{A}}_2|=0.890\mathrm{kHz}$, $|D_{12}|=11.6\mathrm{Hz}$, and $\left|\right|C\left|\right|/\left|\right|B\left|\right|=0.0316$.

Figure 4

Monte Carlo simulations for sequential RIMs and two target spins with a weak magnetic field. The evolution of measurement statistics can be seen from (a) to (d), which shows that the metastability theory still applies to the system with a Zeeman term. All the Monte Carlo simulations contain ${10}^4$ samples, and the parameters are $\mathrm{\Delta }\varphi =\pi /2$, $|{\mathbf{A}}_1|=0.138\mathrm{MHz}$, $|{\mathbf{A}}_2|=0.517\mathrm{MHz}$, $|D_{12}|=60.5\mathrm{Hz}$, ${\omega }_L=13.5$ KHz, and $\left|\right|C\left|\right|/\left|\right|B\left|\right|=0.0412$.

Figure 5

Schematic of an $N$-pulse CPMG sequence with $N$ $\pi -$flips at $t=\tau ,3\tau ,...,(2N-1)\tau$.

Figure 6

Monte Carlo simulations for sequential DD sequences and two target spins. As $m$ grows, the peaks emerge and become apparent (a), (b) since they are corresponding to the four EMSs, then they (c) vanish gradually as $m$ approaching the boundary of metastable region, which (d) finally collapse to the maximally mixed state. All the Monte Carlo simulations contain ${10}^4$ samples, and the parameters are $\mathrm{\Delta }\varphi =\pi /2$, $|{\mathbf{A}}_1|=5\mathrm{kHz}$, $|{\mathbf{A}}_2|=6\mathrm{kHz}$, ${\omega }_L=135\mathrm{kHz}$, ${\mathrm{\Delta }}_{\omega }={10}^{-3}{\omega }_L$, $N=32$.

Figure 7

Monte Carlo simulations for sequential RIMs and a single target qubit with (a) dephasing noise and (b) relaxation noise. The noise rates are shown in the figure, and the other parameters are the same as those in Fig. 2 of the main text.