Bimodal phase transition in a periodically modulated $Λ$-type three-level system

Technical Summary: Bimodal Phase Transition in a Periodically Modulated $\Lambda$-type Three-Level System

Problem Statement
Quantum phase transitions (QPTs) in cavity-QED systems are traditionally studied in the thermodynamic limit ($N \to \infty$) or within finite-component systems. While critical phenomena have been observed in two-level systems (2LS) and recently in three-level systems (3LS), experimental realization often requires coupling strengths that exceed the validity of the Rotating Wave Approximation (RWA). Specifically, achieving superradiant phases in a static $\Lambda$-type three-level system (3LS) coupled to a double-mode cavity typically demands coupling constants ($g$) that are orders of magnitude larger than the cavity frequencies ($\Omega$), pushing the system into a non-perturbative regime where standard analytical models (like the Jaynes-Cummings model) break down. The authors address the challenge of accessing exotic bimodal superradiant phases in a finite-component 3LS while remaining within the critical coupling limits of the static Hamiltonian.

Methodology
The authors propose a theoretical framework involving a $\Lambda$-type three-level atom embedded in a double-mode cavity, described by an extended three-level Jaynes-Cummings (3L-JC) Hamiltonian. To overcome the limitations of static coupling strengths, they introduce a periodic modulation (driving field) applied specifically to the $|3\rangle_{at} \leftrightarrow |2\rangle_{at}$ transition.

The methodology proceeds through the following steps:

1. Hamiltonian Formulation: The system is initially modeled with a static Hamiltonian ($H_{JC}$) and a time-dependent driving term ($H_D(t) = A_D \cos(\omega_D t)(\sigma_{33} - \sigma_{22})$).
2. Unitary Transformations: The authors perform a series of unitary transformations to move the system into a rotating frame. This involves:
   • Transforming to the frame of the external drive to handle the time-dependence.
   • Applying the Jacobi-Anger identity to expand the interaction terms, generating sidebands involving Bessel functions ($J_n(\theta)$).
   • Applying a second unitary transformation to define effective frequencies ($\tilde{\omega}_i, \tilde{\Omega}_i$) and effective coupling strengths ($g_{r1}, g_{r2}$).
3. Approximations and Effective Hamiltonian: By imposing conditions where the driving frequency $\omega_D$ is much larger than the detunings and coupling strengths, fast-oscillating terms are averaged out. The counter-rotating terms are neglected under the condition that their ratio to the detuning is negligible ($\ll 1$). This yields an effective 3L-JC Hamiltonian ($\tilde{H}_{3JC}$) where the system parameters (atomic transition frequencies, cavity frequencies, and coupling strengths) are functions of the driving parameters (amplitude $A_D$ and frequency $\omega_D$).
4. Validation: The validity of these approximations is verified using the Loschmidt echo (time-evolved fidelity). The authors compare the ground state wavefunctions of the exact time-evolved Hamiltonian against those of the approximate effective Hamiltonian, demonstrating high overlap ($>0.99$) across relevant parameter ranges.

Key Contributions and Results

• Derivation of Effective Parameters: The study demonstrates that periodic modulation allows for the tuning of effective coupling strengths ($g_{r1}, g_{r2}$) relative to effective cavity frequencies. Specifically, the effective couplings are scaled by Bessel functions of the modulation index $\theta = A_D/\omega_D$.
• Accessing Superradiant Phases at Low Static Coupling: The primary result is that by tuning the modulation parameters, the system can access bimodal superradiant phases (characterized by non-zero photon occupation in one or both cavity modes) even when the static coupling constants ($g_1, g_2$) are two orders of magnitude smaller than the critical coupling values required for the undriven system.
• Phase Diagram Characterization: The authors map out the ground state phase diagram of the effective Hamiltonian. They identify distinct phases:
  • Normal Phase: $|0, 0\rangle$ (zero photons in both modes).
  • $y_1$-type Phase: Excitation in mode-1 only ($|n_{ph}, 0\rangle$).
  • $y_2$-type Phase: Excitation in mode-2 only ($|0, m_{ph}\rangle$).
  • Mixed Phase: Simultaneous excitation in both modes ($|n_{ph}, m_{ph}\rangle$).
• Bimodal Dynamics: The results show that varying the driving amplitude and detuning allows for transitions between these phases. Notably, the system can exhibit higher-order superradiant states (e.g., $|0, 5\rangle$ or $|5, 0\rangle$) in parameter regimes where the static Hamiltonian would remain in the normal phase.
• Experimental Feasibility: The paper suggests that such a system could be realized using a single InAs/GaAs quantum dot under a perpendicular magnetic field, utilizing orthogonally polarized cavity modes to map the electronic spin and trion states to the $\Lambda$-type levels.

Significance and Claims
The authors claim that their work provides a mechanism to manipulate quantum phases in a finite-component three-level system without violating the RWA or requiring unphysical coupling strengths. By utilizing external periodic modulation, the system effectively "renormalizes" its parameters, allowing the exploration of exotic bimodal dressed-state phases within a lower, experimentally accessible parameter range.

The paper positions this work as an advancement over previous studies on two-level systems, noting that the presence of two interacting cavity modes in their model creates a significantly richer and more complex phase structure with a larger domain for higher-order phases. The findings are intended to facilitate further exploration of phase transitions in finite-sized quantum optical models beyond the standard two-level approximation, with potential relevance to qutrits, quantum sensing, and quantum batteries. The authors remain modest regarding immediate applications, framing the work as a theoretical investigation into the manipulation of quantum phases in an extended Jaynes-Cummings regime.