From Liouville equation to universal quantum control: A study of generating ultra highly squeezed states

Abstract

Within a unified framework, we reveal that the seemingly disparate control approaches for classical and quantum continuous-variable systems are interconnected via differential manifolds of the ancillary representations. For classical systems, the ancillary representation is defined by the time-dependent ancillary canonical variables resulting from a symplectic transformation over the original canonical variables. Under the conditions of the Hamilton-Jacobi equation, the ancillary canonical variables act as dynamical invariants to guide the system nonadiabatically through the entire phase space. The second quantization of the Liouville equation for the canonical variables leads to the Heisenberg equation for the relevant ancillary operators, which is found to be a sufficient condition to yield nonadiabatic passages towards arbitrary target states in both Hermitian and non-Hermitian systems and constrained exact solutions of the time-dependent Schroedinger equation. Using the non-Hermitian Hamiltonian rigorously derived from the Lindblad master equation, our theory is exemplified by the generation of single-mode squeezed states with a squeezing level of 29.3 dB and double-mode squeezed states with 20.5 dB, respectively.

Figures (2)

• Figure 1: Dynamics of (a) the fidelity with respect to the target single-mode squeezed state $|\psi_{\rm tar}(\tau)\rangle$ and (b) the squeezing level $\mathcal{S}(t)$. The Hilbert-space is truncated at $N=1400$. The driving intensity $\Omega(t)$ and the eigenfrequency $\omega(t)$ are set by Eq. (\ref{['CondiOne']}), with $\varphi=\pi/2$, $\alpha(t)=0$, and $\theta(t)=\pi t/(2T)$ with a time constant $T\sim0.5\mu$s. The gain or loss rate $\gamma$ is defined in Eq. (\ref{['imagphase']}) with $\tau_1=6\tau/7$. $\lambda=2$ for $\tau/T=3/2$ and $\lambda=0.2$ for $\tau/T=9/4$. $\gamma>0$ when $t\in[0,\tau_1]$ and $\gamma<0$ when $t\in[\tau_1,\tau]$. $|\gamma T|\sim0.5$ and $\Omega(t)\sim10-10^3\gamma$. All these parameters are experimentally practical in circuit-QED systems Frattini2024ObservationHajr2024HighBeaulieu2025Observation.
• Figure 2: Dynamics of (a) the fidelity with respect to the target two-mode squeezed vacuum state $|\psi_{\rm tar}(\tau)\rangle$ and (b) the relevant squeezing level $\mathcal{S}(t)$. The Hilbert-space of the two-mode system is truncated at $N=100\otimes100$. The squeezing coupling strength $g(t)$, and the eigenfrequencies $\omega_1(t)$ and $\omega_2(t)$ are chosen according to Eq. (\ref{['Condi']}), where the gain or loss rates $\gamma_1$ and $\gamma_2$ are constrained by Eq. (\ref{['condifi']}) with $\lambda=0.02$. $|\gamma_1T|\sim|\gamma_2T|\sim0.5$ and $g(t)\sim10\gamma_1$. The other parameters are set the same as Fig. \ref{['onesqueez']}.