Modeling Quantum Optomechanical STIRAP

Abstract

Quantum optomechanical STIRAP (Stimulated Raman Adiabatic Passage) is investigated for a system of two mechanical modes coupled to an optical mode. We show analytically that in a system without loss, fractional STIRAP can generate a mechanical Bell state from a single phonon Fock state of one of the mechanical modes with the other mechanical mode in the vacuum state, and a product state from a coherent state. Relative phases between Fock basis components in the final state of STIRAP are determined by the phonon-number parity of the initial state. Furthermore, the system is numerically studied to determine the effects of dissipation, and it is concluded that high-fidelity entanglement can be achieved via fractional STIRAP using state-of-the-art cryogenic cooling and mechanical devices. Finally, an interferometric protocol using time-reversed fractional STIRAP is proposed to quantify entanglement between two mechanical modes.

Figures (15)

• Figure 1: Time evolution of STIRAP of the superposition state $\frac{1}{\sqrt{2}}(\ket{0}_1+\ket{1}_1)$ at 10mK. The solid curves track the single phonon probability amplitudes for the first mechanical resonator (red) and second mechanical resonator (blue). The dotted Gaussian profiles represent the counterintuitive pulse sequence, where the Stokes pulse (red) leads the pump pulse (blue). Inset shows Hinton plots of the truncated reduced density matrices $\rho_{12}$ for the initial (left) and final (right) states. The black dashed line shows the entanglement evolution quantified by the negativity $\mathcal{N}(\rho_{12})$, which reaches a peak value of $\mathcal{N} \approx 0.25$ during maximal pulse overlap at 0ms. The state transfer at 2.0ms achieves a fidelity of 0.98 when measured against the state $\frac{1}{\sqrt{2}}(\ket{0}_2-\ket{1}_2)$.
• Figure 2: Time evolution of STIRAP of the state $\rho^i_1$ at 1K. Solid: Single-phonon probability amplitude, $n_1$ (red) and $n_2$ (blue) for the first and second resonators, respectively. Dotted: Gaussian profiles, indicating the STIRAP Stokes (red) and pump (blue) pulse sequence. Inset Hinton plots are included for -0.5ms (left) and 0.5ms (right), representing the density matrix at the start and end of STIRAP, with the color mapping (blue for positive, red for negative) indicating the phase of the off-diagonal elements. Black Dashed: Negativity $\mathcal{N}(\rho_{12})$, reaching a peak during the pulse overlap at 0ms. The final state transfer fidelity is 0.82 when measured against the state $\rho^f_2$.
• Figure 3: Time-evolution of the single phonon probability amplitude of a single phonon Fock state during fractional STIRAP and, reverse fractional STIRAP at 10mK. At time 2 ms the pulses are off and the system is close to the Bell state $\ket{\Psi^-}$. Red Solid: Mechanical mode 1. Blue Solid: Mechanical mode 2. Red Dotted: Stokes pulse. Blue Dotted: Pump pulse. Inset Hinton plot displays the real part of the density matrix $\rho_{12}$ at 2.0ms, with red and blue regions indicating positive and negative values that characterize the coherence of the evolved state. Black Dashed: Negativity, $\mathcal{N}(\rho_{12})$. The negativity reaches a stable plateau of $\mathcal{N} \approx 0.48$ between 0ms and 4ms. The fidelity of this process against a single phonon Fock state is 0.971.
• Figure 4: Wigner quasi-probability distribution $W(x,p)$ for the density matrix $\rho_{12}$ at 2.0ms in Figure \ref{['fig:fSTIRAP_comparison']}. Here, $x$ and $p$ represent the dimensionless phase space coordinates corresponding to the mechanical modes' position and momentum relative to the zero-point fluctuations. Red and blue regions indicate positive and negative values, respectively, with the negative components characterizing the non-classical nature of the state, which has fidelity of 0.98 when compared to $\ket{\psi^-}$ Bell state.
• Figure 5: Time evolution of the single phonon population of mode 1 (red solid) and mode 2 (blue solid) during fSTIRAP at 1K of the states $\rho^\mathrm i_1$ and $\rho^\mathrm b_2$. Dotted red and blue lines show the fSTIRAP pulse shapes, with pulse widths of $\sigma_1 = \sigma_2 = 0.15ms$. Black Dashed: Negativity $\mathcal{N}(\rho_{12})$ with a maximum at $\sim$0.6ms of $\mathcal{N}(\rho_{12})\approx 0.25$ followed by a steady decay. The state transfer has a fidelity of 0.77, compared to the state $\ket{\Psi^-}$ at 1ms.