Simultaneous cooling of degenerate mechanical modes in unresolved sideband regime via optical and mechanical nonlinearities

1. Problem Statement

The paper addresses two primary challenges in the simultaneous ground-state cooling of multiple degenerate mechanical modes (modes with identical or nearly identical frequencies) within optomechanical systems:

• The Dark-Mode Effect: When two or more degenerate mechanical oscillators couple to a single common optical cavity mode, a "dark mode" is formed. This hybrid mode is decoupled from the optical cavity, preventing the extraction of thermal excitations through the standard cooling channel. Consequently, the dark mode cannot be cooled to the ground state.
• The Resolved Sideband Limitation: Standard sideband cooling requires the optical cavity decay rate ($\kappa$) to be smaller than the mechanical frequency ($\omega_m$), known as the resolved sideband regime ($\kappa < \omega_m$). This condition necessitates high-finesse cavities and limits the size of mechanical resonators. Cooling in the unresolved sideband regime ($\kappa > \omega_m$) is generally difficult or impossible for degenerate modes using standard linear techniques.

2. Methodology

The authors propose a theoretical scheme utilizing a hybrid approach involving both mechanical and optical nonlinearities to overcome these obstacles.

• System Model: The system consists of one optical cavity containing a second-order nonlinear medium ($\chi^{(2)}$) and $N$ mechanical resonators. The cavity supports a fundamental mode ($a_1$) and a second-order mode ($a_2$).
• Breaking the Dark Mode (Mechanical Nonlinearity):
  • The authors introduce Duffing nonlinearities (mechanical nonlinearities) for each mechanical oscillator.
  • They demonstrate that if the Duffing nonlinearity amplitudes ($\Lambda_j$) of the degenerate modes are not identical (i.e., $\Lambda_i \neq \Lambda_j$), the symmetry required to form a perfect dark mode is broken.
  • This asymmetry creates an interaction between the previously decoupled dark mode and the bright mode (which couples to the cavity), allowing thermal energy to be extracted.
• Enabling Unresolved Sideband Cooling (Optical Nonlinearity):
  • To operate in the unresolved sideband regime ($\kappa \gg \omega_m$), the authors introduce a second-order nonlinear medium inside the cavity.
  • This medium couples the fundamental mode ($a_1$) and the second-harmonic mode ($a_2$).
  • The optical nonlinearity ($\chi$) effectively modifies the cooling dynamics, suppressing Stokes heating processes and enabling ground-state cooling even when the cavity decay rate is much larger than the mechanical frequency.
• Theoretical Framework:
  • The system is modeled using a Hamiltonian including free energy, driving fields, optomechanical coupling, Duffing terms, and the $\chi^{(2)}$ interaction.
  • The dynamics are analyzed using Quantum Langevin Equations and linearized around steady-state amplitudes.
  • Stability and cooling performance are evaluated using the covariance matrix method and the Routh-Hurwitz criterion.

3. Key Contributions

• Simultaneous Ground-State Cooling in Unresolved Regime: The paper demonstrates, for the first time (in this context), that simultaneous ground-state cooling of multiple degenerate mechanical modes is achievable in the unresolved sideband regime ($\kappa \gg \omega_m$).
• Novel Dark-Mode Breaking Mechanism: Unlike previous schemes that used auxiliary cavities or phase-dependent interactions, this work proposes breaking the dark mode solely through asymmetric mechanical nonlinearities (Duffing terms).
• Synergy of Nonlinearities: The work establishes a critical synergy where mechanical nonlinearities break the dark mode, while optical nonlinearities enable the cooling process to function efficiently despite the large cavity decay rate.

4. Key Results

The authors present numerical simulations for systems with 2, 3, and 4 degenerate mechanical oscillators:

• Two Degenerate Modes:
  • Without nonlinearities, the mean phonon number ($n$) remains high ($>1$) due to the dark mode effect.
  • Introducing different Duffing amplitudes ($\Lambda_1 \neq \Lambda_2$) breaks the dark mode, reducing $n$.
  • In the unresolved regime ($\kappa = 20\omega_1$), standard cooling fails ($n \gg 1$). However, with the addition of optical nonlinearity ($|\chi| = 10\omega_1$) and asymmetric Duffing terms, both modes are cooled to the ground state ($n < 1$).
• Three and Four Degenerate Modes:
  • The scheme scales effectively. For $N=3$ and $N=4$, the mean phonon numbers of all modes drop below 1 simultaneously.
  • The simulations show that if the nonlinearities of any pair of modes are too close (e.g., $\Lambda_i \approx \Lambda_j$), a dark mode re-emerges between that specific pair, causing a spike in their phonon numbers. Therefore, distinct nonlinearity amplitudes are crucial for all modes.
• Role of Optical Nonlinearity:
  • In the highly unresolved regime, the optical nonlinearity suppresses Stokes heating, which is the dominant heating mechanism.
  • The authors note a trade-off: in the resolved regime, optical nonlinearity can sometimes be detrimental due to photon generation heating, but in the unresolved regime, the suppression of Stokes heating dominates, making it highly beneficial.

5. Significance

• Experimental Feasibility: The scheme relaxes the stringent requirements for high-finesse cavities (resolved sideband condition), allowing for the cooling of larger mechanical resonators which are often easier to fabricate but harder to cool.
• Scalability: The method provides a pathway to cool complex multimode mechanical systems (arrays of oscillators) simultaneously, which is essential for quantum information processing and quantum sensing applications involving macroscopic objects.
• Fundamental Physics: It offers a new mechanism to control dark modes using intrinsic mechanical nonlinearities rather than external auxiliary systems, simplifying the experimental architecture.

In conclusion, the paper proposes a robust theoretical framework that overcomes the dual limitations of dark-mode effects and the unresolved sideband regime, paving the way for the experimental realization of simultaneous ground-state cooling in complex optomechanical systems.