Mode-selective excitation in parametrically driven coupled quantum oscillators

Abstract: A parametrically driven classical harmonic oscillator exhibits resonant instability when driven at twice its natural frequency, with the lowest energy configuration remaining unaffected by the drive. In contrast, the ground state of the quantum mechanical counterpart shows a non-trivial response to such a drive due to the spatial delocalization of the wavefunction. The standard realization of PR involves modulating the natural frequency of the oscillator. Here we study a different drive protocol in which the coupling between two such quantum harmonic oscillators is modulated parametrically. We show that the drive frequency can in principle be tuned to selectively excite any desired normal mode, while leaving the other close to its ground state. Only states with even quantum numbers in each normal mode are populated. Within the parametric resonance window the excitations follow a power-law decay with respect to occupation number, in contrast to the exponential decay observed off-resonance. We also briefly discuss how this framework can be extended to a system of N coupled oscillators

• The paper introduces a method for targeted excitation in coupled quantum oscillators using parametric modulation of the coupling, revealing even-parity selection rules.
• It contrasts degenerate and nondegenerate cases, showing power-law decay in resonant modes and exponential decay in nonresonant modes.
• The study offers a scalable framework for controlling energy absorption and excitation distribution, essential for advanced quantum state engineering.

Mode-Selective Excitation in Parametrically Driven Coupled Quantum Oscillators

Introduction

This paper (2605.26227) investigates the dynamics of quantum harmonic oscillators under parametric modulation of their coupling, contrasting this protocol with conventional parametric driving where individual oscillator frequencies are time-modulated. The study is rooted in the theory of quantum parametric resonance (QPR), which classically leads to instability only for excited states and quantum-mechanically generates a non-trivial response even from the ground state. The motivation is derived from numerous applications in quantum technologies (e.g., superconducting circuits, optomechanics, ion traps), where targeted mode excitation is essential.

Hamiltonian and Normal Mode Analysis

The system under consideration is a two-dimensional isotropic quantum harmonic oscillator with time-dependent coupling $\kappa(t)$ between degrees of freedom. The analysis begins with the time-dependent Schrödinger equation for the coupled system:

$$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$

Upon transformation to the normal mode basis $(Q_+, Q_-)$ (in-phase and out-of-phase combinations), the Hamiltonian decouples into a sum of two independent oscillators with time-dependent natural frequencies:

$\Omega_\pm^2(\tau) = 1 \pm \beta(\tau)$

where $\beta(\tau)$ encodes the parametric modulation.

Ground State Dynamics and Selection Rules

The ground state evolves under quadratic time-dependent Hamiltonians, thus retaining its Gaussian form but with modified parameters. The explicit ansatz allows analytical and numerical tracking of the probability amplitudes for occupation of excited states in the normal-mode basis.

A strict selection rule emerges: only states with even quantum numbers in each normal mode are populated. This is caused by parity preservation under parametric driving; the wavefunction remains even under $Q_\pm \rightarrow -Q_\pm$, preventing transitions to odd eigenstates (i.e., $p_{n_\pm}$ vanishes for odd $n_\pm$).

Numerical Results and Excitation Distribution

The mode excitation behavior is classified into two regimes based on the static coupling parameter $\beta_0$:

• Degenerate Case ($\beta_0=0$): Each mode shares the same resonance condition and exhibits symmetric excitation. Resonant driving ($$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$0) distributes occupation probability broadly across several energy levels. The excitation probability per state follows a power-law decay in occupation number ($$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$1), indicating non-negligible population in higher states. Off-resonance, occupation decays exponentially with $$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$2 ($$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$3).
• Nondegenerate Case ($$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$4): Degeneracy lifting splits resonance windows for each mode. With appropriate detuning, one mode enters resonance and is selectively excited (power-law excitation spectrum), while the other remains near its ground state exhibiting only exponential decay in occupation probability. Hence, the drive frequency enables mode-selective excitation, leaving nonresonant modes effectively unperturbed.

The crossover from exponential to power-law decay constitutes a robust diagnostic of quantum parametric resonance.

Energy Pumping and Resonance Windows

Energy absorption profiles show twin sharp peaks at mode-specific resonance frequencies $$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$5 and $$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$6. Numerical evaluation of the Hamiltonian expectation confirms that maximized energy pumping occurs only when the drive frequency matches the resonance condition for a specific mode, with exponential growth in absorbed energy with the number of drive cycles.

Generalization to N-Coupled Oscillators

The theoretical framework extends to $$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$7 oscillators with quadratic coupling, provided the static coupling matrix is symmetric. Diagonalization yields $$H = -\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2) + \frac{1}{2}m\omega_0^2(x^2 + y^2) + \kappa(t) xy$$8 normal modes, each with independent parametric modulation. The Gaussian ansatz holds due to quadraticity, and the selection rule forbidding odd quantum numbers in each mode remains valid. One can selectively excite individual modes in sufficiently separated frequency regimes.

Implications and Future Directions

The implications are most prominent for systems demanding targeted excitation, such as cavity arrays, superconducting circuits, or quantum-dot ensembles. The ability to populate only even-parity states in a controlled mode-selective fashion is potentially beneficial for state engineering, entanglement generation, or precision metrology, as in parametric amplifiers or cat-qubit architectures.

Future research directions include:

• Dissipation and Environmental Effects: Realistic quantum systems exhibit relaxation and dephasing, which may alter or erase mode-selectivity and power-law excitation scaling. Detailed analysis can reveal persistent structures or the onset of dissipative phase transitions.
• Drive Protocol Optimization: The effect of finite ramp and decay times for the parametric drive (non-instantaneous switching) on resonance windows and excitation selectivity merits systematic investigation.
• Experimental Realization and Characterization: Mapping theory to experiment in platforms like optomechanical arrays, superconducting cavities, or trapped ions will validate predictions and potentially inspire novel quantum control protocols.

Conclusion

This study establishes a theoretically rigorous foundation for mode-selective excitation in coupled quantum oscillators via parametric modulation of coupling. The derived selection rules, excitation statistics, and energy absorption characteristics, as well as the generalizability to many-mode systems, furnish a comprehensive toolkit for quantum-state control. The resonance window analysis and occupation number scaling provide practical diagnostics for operation in quantum parametric resonance regimes. The approach opens avenues for targeted state engineering in advanced quantum hardware and informs further exploration of dissipative dynamics and control optimization.