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.

• The paper establishes a unified framework that maps classical Liouville dynamics to quantum Heisenberg evolution using ancillary variables.
• It demonstrates strong numerical evidence achieving squeezing levels of 29.3 dB for single-mode and 20.5 dB for two-mode systems, surpassing previous benchmarks.
• The scheme offers operational protocols for robust control in non-Hermitian settings, enabling high-fidelity state engineering for quantum information applications.

Unified Quantum Control Framework Based on the Liouville Equation

Interconnection of Classical and Quantum Control Protocols

The paper "From Liouville equation to universal quantum control: A study of generating ultra highly squeezed states" (2604.02635) establishes a comprehensive operational framework that unifies control strategies for both classical and quantum continuous-variable systems. The central theoretical advance is the interpretation of dynamical invariants—arising from the Hamilton-Jacobi equation (HJE) in classical mechanics—as ancillary variables, whose evolution is governed by the Liouville equation in phase space. This constructs a bridge wherein the symplectic transformation of canonical variables in classical systems is mirrored by a symplectic transformation of bosonic operators under second quantization in quantum systems, leading to an equivalent Heisenberg equation.

For classical systems, ancillary variables manipulated by symplectic transformations and generator functions attain time-dependent invariance, guiding nonadiabatic transport within the phase space. Imposing the HJE ensures that the transformed Hamiltonian vanishes, so these variables become constant along system trajectories. The explicit Liouville equation $\partial J_k/\partial t = -i \{H, J_k\}$ (for $J_k$ ancillary variables) condenses the nonadiabatic dynamics into a compact form suitable for both analytical and numerical solutions.

Second Quantization and Universal Quantum Control

Upon second quantization, the classical Liouville formalism is mapped to the quantum Heisenberg equation for time-dependent ancillary operators, which serves as a sufficient condition for nonadiabatic passage to arbitrary target states in both Hermitian and non-Hermitian settings. The symplectic transformations—parametrized through time-dependent squeezing strength and phase—preserve canonical commutation relations, allowing for the novel construction of control protocols that remain valid in open quantum systems described by the Lindblad master equation.

The framework proves that solving the Heisenberg equation for ancillary operators yields exact, constrained solutions to the time-dependent Schrödinger equation. This is predicated on the dynamical decoupling of these modes, enforced by the vanishing of off-diagonal elements in the transformed Hamiltonian. For non-Hermitian systems, the gauge-invariant global phase is generally complex, but through a two-stage gain/loss control, the imaginary component can be nullified at the target moment, circumventing artificial state normalization and ensuring probability conservation.

Ultra High Squeezing Generation: Strong Numerical Evidence

The principal application is the generation of single-mode and two-mode squeezed states with squeezing levels significantly exceeding prevailing theoretical and experimental benchmarks. The protocol employs time-dependent non-Hermitian Hamiltonians derived rigorously from the Lindblad equation, with symplectic control over the driving intensity, eigenfrequency, and gain/loss rate. For single-mode systems, the squeezing level reaches 29.3 dB; for two-mode systems, the protocol achieves 20.5 dB, compared to previous theoretical limits of approximately 15 dB and experimental achievements below 15 dB for single-mode and 10 dB for two-mode [Vahlbruch2016Detection, Cai2025Quantum, Sutherland2021Universal, Eberle2013Stable].

The fidelity to target squeezed states approaches unity for single-mode and exceeds 0.95 for two-mode at modest evolution times, confirmed by numerical integrations with large Hilbert-space truncations (up to $N=1400$ for single-mode). All parameters used are experimentally feasible, particularly within circuit-QED platforms.

Practical and Theoretical Implications

Practically, achieving squeezing beyond 20 dB opens new domains for quantum information protocols reliant on high squeezing thresholds, such as high-fidelity quantum teleportation, GKP encoding, and FTQC based on cluster states. The formalism offers operational recipes for implementing nonadiabatic control protocols that can circumvent limitations imposed by dissipation, demanding only time-dependent manipulation of ancillary variables and accessible gain/loss modulation.

Theoretically, the unified structure provides a versatile toolkit for exploring shortcuts-to-adiabaticity, scale-invariant classical transport, and dissipative stabilization in both closed and open systems. The mapping of the Liouville equation to the Heisenberg equation under symplectic transformations is extensible to nonlinear control scenarios, demonstrated with multiphoton parametric down-conversion.

Speculation on Future Developments

The established framework could directly inform the design of scalable quantum platforms where ultra-high squeezing is a deterministic resource, significantly enhancing error correction, metrology, and entanglement generation capabilities. Its applicability to non-Hermitian settings indicates potential for robust state engineering in environments with engineered dissipation or gain, a scenario increasingly relevant in hybrid quantum systems and photonic networks.

Generalizing the ancillary invariants to nonlinear and multimode networks can further unlock exotic control possibilities, including realization of non-Gaussian resources and universal holonomic quantum computation via geometric phase encoding [Michael2017Geometry, Zhang2023Geometric]. The formal structure invites extension to quantum-classical hybrid systems and systems governed by pseudo-Hermitian Hamiltonians, intensifying interest in universal control paradigms for open quantum systems.

Conclusion

The paper provides a unified geometrical and operational basis for quantum control protocols, encapsulating both classical dynamics and quantum evolution under time-dependent control fields. By leveraging dynamical invariance and symplectic transformation principles, the authors demonstrate powerful schemes for preparing ultra-highly squeezed states. Strong numerical results, theoretical rigor, and the universality of the approach signal substantive implications for quantum information processing and state engineering. The framework's extensibility to nonlinear systems and non-Hermitian environments foregrounds its relevance to future developments in quantum technologies and control theory.