Dissipation across the ultrastrong-coupling regime of nanomechanical quantum Rabi systems

Technical Summary: Dissipation across the ultrastrong-coupling regime of nanomechanical quantum Rabi systems

Problem Statement
Nanomechanical resonators, particularly suspended carbon nanotubes coupled to double quantum dots, offer a promising platform for mechanical qubits by introducing significant anharmonicity through ultrastrong electromechanical coupling (USC). While the closed system is well-described by the Quantum Rabi Model (QRM), a consistent and practical description of dissipation remains a significant challenge across the broad parameter space explored in current experiments.

Conventional approaches face critical limitations:

1. Local Lindblad Equations: These derive dissipation by neglecting interactions between subsystems. In the USC regime, this leads to unphysical heating artifacts.
2. Full Secular Approximation (FSA): While avoiding artificial heating, FSA imposes strict conditions on the separation of transition frequencies (requiring $\Gamma \ll \chi$, where $\Gamma$ is the decay rate and $\chi$ is the anharmonicity). This restricts its validity to regimes of large anharmonicity. However, current experiments routinely operate in regimes of reduced anharmonicity, strong driving, or increased dissipation where FSA breaks down.

There is a lack of a unified theoretical framework capable of describing dissipative dynamics from weakly anharmonic operating regimes to the strongly anharmonic mechanical-qubit regime, particularly when the secular approximation is invalid.

Methodology
The authors employ a Born-Markov framework for the open Quantum Rabi Model, utilizing the Slowly Varying Bath Spectrum (SVS) approximation. This approach yields a Lindblad master equation that remains valid beyond the constraints of the full secular approximation.

Key methodological steps include:

• System Modeling: The system is modeled as a suspended carbon nanotube (mechanical oscillator) coupled to a double quantum dot (two-level system, TLS). The Hamiltonian is the standard QRM.
• Analytical Approximations:
  • In the lower USC regime ($g \ll \omega_m$), the system is mapped via fourth-order perturbation theory to an anharmonic Kerr oscillator.
  • In the deep-strong coupling regime ($g \ge \omega_m$), a Born-Oppenheimer approximation is used, revealing an effective double-well potential.
• Derivation of the SVS Lindblad Equation: The authors derive a master equation where the collapse operators are constructed from the eigenstates of the full interacting Hamiltonian. The rates are parametrized by a minimal set of environmental parameters (decay rate $\gamma$, temperature $T$, and pure dephasing rate $\gamma_\phi$).
• Validation: The SVS approach is shown to recover the local Lindblad equation in the Jaynes-Cummings limit and the FSA in the limit of sufficiently small decay rates relative to anharmonicity.
• Numerical Implementation: The master equation is solved numerically to calculate experimentally accessible observables, including the two-phonon correlation function $g^{(2)}(0)$, the thermal displacement spectrum $S_{xx}(\omega)$, and the mean-square displacement under driving.

Key Contributions and Results

1. Unified Description of Dissipation: The SVS Lindblad equation provides a continuous description across the parameter space, bridging the gap between weakly and strongly anharmonic regimes. It remains valid where the secular approximation fails (i.e., when $\Gamma \sim \chi$ or $\Gamma > \chi$).

2. Phonon Blockade and Correlation Functions:

   • The authors analyze the two-phonon correlation function $g^{(2)}(0)$ to characterize the transition from phonon blockade (anti-bunching) to coherent behavior.
   • They find that the onset of phonon blockade coincides with the parameter regime where the full secular approximation loses validity.
   • The SVS approach quantitatively describes the continuous evolution of phonon blockade with coupling strength and dissipation, a regime inaccessible to conventional methods.

3. Thermal Spectrum and Coherence Decay:

   • At finite temperatures, the conventional secular approximation systematically overestimates the decay rate of correlations and the linewidth of the mechanical spectrum $S_{xx}(\omega)$.
   • The SVS approach reveals that transfer of coherence between different eigenstates counteracts the temperature-induced increase in the decay rate.
   • A temperature-dependent shift in the peak of the correlation function is identified, which vanishes in both the harmonic limit and the large-anharmonicity limit.

4. Driven Systems and Robustness:

   • Under finite driving, the SVS dissipator remains applicable for drive amplitudes substantially larger than those allowed by the FSA (which requires the drive to be smaller than the anharmonicity $\chi$). The SVS approach only requires the drive to be smaller than the transition frequencies ($\omega_m$).
   • In regimes of strong driving and dissipation, the observable mean-square displacement $\langle x^2 \rangle$ mimics the response of a classical Duffing oscillator. However, the underlying steady state retains Wigner negativity, confirming the system remains non-classical despite the classical appearance of the observable.

5. Effect of Detuning:

   • The study investigates the effect of detuning the double quantum dot, which reduces the induced anharmonicity $\chi$ and introduces a new decoherence channel.
   • As detuning increases, $\chi$ can become arbitrarily small, rendering the FSA invalid. The SVS approach successfully describes the system across the entire detuning range, predicting experimentally relevant spectra even when anharmonicity is negligible.

Significance
The paper establishes a unified theoretical framework for interpreting and optimizing nanotube-based mechanical qubits across the full range of conditions encountered in present-day experiments, not merely in the idealized regimes where existing treatments were designed. By providing a consistent description of dissipation that avoids unphysical heating and remains valid when the secular approximation breaks down, the SVS Lindblad equation enables quantitative predictions for observables in the ultrastrong and deep-strong coupling regimes. This is crucial for understanding the transition between quantum and classical behaviors in hybrid electromechanical systems and for accurately modeling devices operating with reduced anharmonicity or under strong driving.