Universal quantum control over non-Hermitian continuous-variable systems

TL;DR

This paper develops a universal quantum control framework for non-Hermitian continuous-variable systems by exploiting time-dependent ancillary frames and a gauge potential, avoiding spectrum-based strategies. A key result is that upper triangularization of the rotated Hamiltonian in the stationary ancillary frame enables two Heisenberg-picture passages that drive exact, norm-controlled evolutions in the full Hilbert space. When applied to a cavity-magnonic system, the method enables perfect transfer of arbitrary states between modes and can realize unidirectional perfect absorption, independent of PT-symmetry or exceptional points. The approach provides a general, deterministic tool for coherent control of non-Hermitian bosonic systems with broad potential applications in quantum information processing and photonic/magnonic technologies.

Abstract

Current studies of non-Hermitian continuous-variable systems heavily revolved around the singularities in the eigen-spectrum by mimicking their discrete-variable counterparts. The growing discussions over the nonunitary features in time evolution, however, are limited in scalability and controllability. We here develop a general theory to control an arbitrary number of bosonic modes under the time-dependent non-Hermitian Hamiltonian. Far beyond the subspace of few excitations, our control theory operates in the full Hilbert space within the Heisenberg framework and exploits the gauge potential underlying the instantaneous frames rather than the eigen-spectrum. In particular, the instantaneous frames are defined by time-dependent ancillary operators as linear combinations of laboratory-frame operators; while the associated gauge potential arises from the unitary transformation connecting the time-dependent and stationary ancillary frames. We find that the upper triangularization condition of non-Hermitian Hamiltonian's coefficient matrix in the stationary ancillary frame gives rise to nonadiabatic passages of two time-dependent ancillary operators, also leading to exact solutions of the time-dependent Schrödinger equation. At the end of these passages, the probability conservation of the system wavefunction can be automatically restored without artificial normalization. Our theory is exemplified with the perfect and nonreciprocal state transfers in a cavity magnonic system. The former holds for arbitrary initial states and is irrelevant to both parity-time symmetry of the coefficient matrix and exceptional points of eigen-spectrum; and the latter is consistent with the coherent perfect absorbtion. We essentially constructs the universal quantum control (UQC) theory for the non-Hermitian continuous-variable systems, promising a powerful and reliable approach for their coherent control.

Figures (5)

• Figure 1: Sketch of an open cavity magnonic system under control. More than the coherent exchange coupling with strength $J$ and phase $\varphi$ between the cavity mode $a$ and the magnon mode $b$, the cooperative coupling of the two-mode system to the environment gives rise to the gain or loss rate $\gamma_a$ of the cavity mode, the loss rate $\gamma_b$ of the magnon mode, and the dissipation coupling $i\Gamma e^{i\Theta}$ between them. $\Theta$ can be $0$ or $\pi$ in experiments Wang2019Nonreciprocity.
• Figure 2: Fidelity dynamics during the Fock-state transfer $|\psi(0)\rangle=|5\rangle_a|0\rangle_b\rightarrow|\psi(\tau)\rangle=|0\rangle_a|5\rangle_b$ under the $\mathcal{PT}$-symmetric Hamiltonian, using the passage $\mu_1^\dagger(t)$ in the cavity-magnonic system for (a) avoiding EPs and (b) crossing EPs. The associated dynamics of the real and imaginary parts of the energies $E_\pm$ in Eq. (\ref{['eigenenergy']}) is plotted in (c) and (d) for avoiding and crossing EPs, respectively. With $\theta(t)$ in Eq. (\ref{['paratheta']}), the coherent coupling strength $J(t)$ and the detuning $\Delta(t)$ are constrained by Eq. (\ref{['Hconstr']}). $\varphi_a=\pi$, $\varphi=\pi/2$, $\Gamma=0$, and $\gamma_a=\gamma_b$, where $\gamma_a$ satisfies Eq. (\ref{['condvanifi']}) with $\lambda=\pi$ in (a) and (c), and $\lambda=4\pi$ in (b) and (d). Then $f_i(\tau)-f_i(0)=0$ for both avoiding and crossing EPs.
• Figure 3: Fidelity dynamics during the Fock-state transfer $|\psi(0)\rangle=|5\rangle_a|0\rangle_b\rightarrow|\psi(\tau)\rangle=|0\rangle_a|5\rangle_b$ under a $\mathcal{PT}$-symmetric-broken Hamiltonian, using the passage $\mu_1^\dagger(t)$ in the cavity-magnonic system for (a) avoiding EPs and (b) crossing EPs. The associated dynamics of the real and imaginary parts of the energies $E_\pm$ in Eq. (\ref{['eigenenergy']}) is shown in (c) and (d) for avoiding and crossing EPs, respectively. $\varphi_a=0$. Both $\Gamma$ and $\gamma_a$ are given by Eq. (\ref{['condvanifi']}) with $\lambda=0.5$ in (a) and (c), and $\lambda=1.2$ in (b) and (d). The other parameters are the same as Fig. \ref{['ConverFockPT']}. And $f_i(\tau)-f_i(0)=0$ for both avoiding and crossing EPs.
• Figure 4: Fidelity dynamics during the perfect transfer of (a) the binomial code state $|\phi_b,0\rangle\rightarrow|0,\phi_b\rangle$ with $|\phi_b\rangle=(\sqrt{3}|2\rangle+|6\rangle)/2$Michael2016NewClass, (b) the coherent state $|\alpha,0\rangle\rightarrow|0,\alpha\rangle$ with $\alpha=5$, (c) the cat state $|\rm cat,0\rangle\rightarrow|0,\rm cat\rangle$, where $|\rm cat\rangle=(|\alpha\rangle+|-\alpha\rangle)/\sqrt{2}$ with $\alpha=5$, and (d) the thermal state $\rho_{\rm th}\otimes|0\rangle\langle0|\rightarrow|0\rangle\langle0|\otimes\rho_{\rm th}$, where $\rho_{\rm th}=\sum_np_n|n\rangle\langle n|$ with $p_n=(\bar{n}^n)/(1+\bar{n})^{n+1}$ and $\bar{n}=5$. The parameters are the same as Figs. \ref{['ConverFock']}(b) and (d).
• Figure 5: Fidelity dynamics about the individual states $|5\rangle_a|0\rangle_b$ and $|0\rangle_a|5\rangle_b$ with various $\lambda$, the coupling strength or the dissipation rate of the system, under a $\mathcal{PT}$-symmetric-broken Hamiltonian. Inset: Numerical results for the logarithm of the nonreciprocity $\log_{10}[S_{21}(\tau)/S_{12}(\tau)]$ as a function of $\lambda$. The other parameters are the same as Fig. \ref{['ConverFock']}. The initial state of system is $|\psi(0)\rangle=|0\rangle_a|5\rangle_b$.