Bath-free squeezed phonon lasing via intrinsic ion-phonon coupling

Abstract

We present a theoretical model for realizing squeezed lasing in a trapped-ion system without relying on engineered baths or tailored dissipative reservoirs. Our approach leverages the intrinsic ion-phonon interactions, where two trapped ions, each interacting with a shared vibrational mode, are driven on both red- and blue-sideband transitions. This enables the creation of a squeezed state of motion through the dynamic coupling between the ions' internal states and the phonon mode. Unlike traditional methods that require bath engineering, our model demonstrates that squeezed lasing can be achieved through a direct manipulation of ion-phonon interactions, with no external reservoirs required. We explore the steady-state behavior of the system, analyzing the onset of lasing, gain-loss balance, and the role of the squeezing parameter in shaping the phonon field's statistical properties. Furthermore, we show how external coherent drives can stabilize phase coherence and achieve controlled quadrature squeezing, offering a simple yet effective method for achieving squeezed lasing in quantum mechanical systems. Our findings provide new insights into the realization of squeezed states in phonon-based systems, with potential applications in quantum metrology and information processing.

Figures (5)

• Figure 1: Schematic of the bath-free squeezed phonon laser configuration. Two ions are driven asymmetrically on their red- and blue-sideband transitions. One ion acts as a heating agent to provide gain, while the other acts as a cooling agent to introduce loss for the shared motional mode. This setup generalizes the standard phonon laser architecture to the squeezed domain by leveraging intrinsic ion-phonon interactions without the need for external reservoir engineering.
• Figure 2: Steady-state Wigner functions of the phonon mode. (a) Below-threshold regime at $g_{1}=0.01\gamma_{1},$ exhibiting a near-symmetric profile characteristic of a squeezed vacuum state. (b) Above-threshold regime at $g_{1}=0.2\gamma_{1},$ showing a displaced, elliptically deformed structure consistent with a random-phase squeezed coherent state. Simulation parameters are fixed at $g_{2}=0.15\gamma_{1}$ and $\gamma_{2}=2.5\gamma_{1}.$
• Figure 3: Signatures of the squeezed-lasing transition. (a) Steady-state ratio of gain to loss, $\left\langle \mathcal{G}\right\rangle /\left\langle \mathcal{K}\right\rangle ,$ as a function of the effective heating coupling $g_{1},$ with the threshold identified at $g_{1,\text{th}}\approx0.12\gamma_{1}.$ (b) Second-order coherence $g^{(2)}\left(0\right)$ (left axis) and mean phonon number $\left\langle n\right\rangle$ (right axis) of the phonon mode as functions of $g_{1}.$ (c) Emission linewidth $\Gamma$ verse $g_{1},$ demonstrating linewidth narrowing above the threshold. (d) Normalized emission spectrum $S\left(\omega\right)$ for various $g_{1}/\gamma_{1},$ where frequency is defined relative to the squeezed mode frequency $\omega_{s}.$
• Figure 4: Dependence of phonon statistics on the squeezing parameter $r$ in the above-threshold regime. (a) Second-order coherence function $g^{(2)}(0)$ and (b) mean phonon number $\left\langle n\right\rangle$ as functions of $r.$ Solid curves represent numerical results from the master equation, while dashed curves indicate analytical predictions based on the mapping to the squeezed frame. All results are obtained well above the lasing threshold.
• Figure 5: Phase-symmetry breaking and quadrature squeezing. (a) Steady-state Wigner function in the presence of a weak phase-stabilizing field. Dashed orange lines indicate the 0.1 contour of the Wigner distribution, compared to the black dotted contours of a coherent state with equal amplitude. (b) Quadrature variance $\Delta X_{\phi_{1}}^{2}$ as a function of the drive phase $\phi_{1},$ with the dashed line marking the vacuum-level fluctuations. Quadrature squeezing $\left(\Delta X_{\phi_{1}}^{2}<1\right)$ is observed near $\phi_{1}=0.5\pi$ and $1.5\pi.$ Shared parameters: $g_{1}=0.2\gamma_{1},g_{2}=0.15\gamma_{1},\gamma_{2}=2.5\gamma_{1},\theta=0;$ panel (a) uses $\phi_{1}=\pi/2.$