Phonon Laser in the Quantum Regime

We demonstrate a trapped-ion system with two competing dissipation channels, implemented independently on two ion species cotrapped in a Paul trap. By controlling coherent spin-oscillator couplings and optical pumping rates we explore the phase diagram of this system, which exhibits a regime analogous to that of a (phonon) laser but operates close to the quantum ground state with an average phonon number of $\overline{n}<10$. We demonstrate phase locking of the oscillator to an additional resonant drive, and also observe the phase diffusion of the resulting state under dissipation by reconstructing the quantum state from a measurement of the characteristic function.

Figure 1

(a) Schematic of the two dissipative channels. The red (blue) sideband plus engineered decay is applied to calcium (beryllium) and thus realizes cooling (heating) of the shared motional mode. (b) Sketch of heating and cooling process including the engineered decay in calcium and beryllium.

Figure 2

Simulated phase diagram plotted as a function of control parameters, with the average phonon number used to characterize the steady state. The values are generated using numerical simulations including up to 100 motional states, solving for the steady state. ${\gamma }_{\text{c}}$ is the internal state damping rate in the cooling ion while $1/{\kappa }_{\mathrm{c}}={\gamma }_{\mathrm{c}}/g_{\mathrm{c}}^2$, with the sideband coupling strength $g_{\text{c}}$. An effective beryllium decay $1/{\gamma }_{\mathrm{h}}=15.5\mathrm{\mu }\mathrm{s}$ and $1/{\kappa }_{\mathrm{h}}=0.18\mathrm{ms}$ are fixed. Red and light blue dots correspond to the data taken and shown in Fig. 3. White lines indicate the expected phase transitions at ${\gamma }_{\mathrm{c}}={\gamma }_{\mathrm{h}}$ and ${\kappa }_{\mathrm{c}}={\kappa }_{\mathrm{h}}$. Values of $\overline{n}>80$ are grayed out, they correspond to heating or high $\overline{n}$ lasing values.

Figure 3

Phonon distributions for different $1/{\kappa }_{\text{c}}$. (a)–(c) taken at $1/{\gamma }_{\mathrm{c}}=2.3\mathrm{\mu }\mathrm{s}$, show an increase in the average phonon number with an increase of $1/{\kappa }_{\text{c}}$. The measured phonon distributions (blue bars) agree well with a Poisson distribution of same $\overline{n}$ (gray). (d) Taken in the heating region at $1/{\kappa }_{\mathrm{c}}=0.28\mathrm{ms}$ and $1/{\gamma }_{\mathrm{c}}=19.9\mathrm{\mu }\mathrm{s}$ as indicated by the black triangle. Here, the reconstructed phonon distribution neither follows a thermal (green) nor a coherent (gray) distribution of the same $\overline{n}$. (e) Average phonon number as a function of $1/{\kappa }_{\text{c}}$, mean-field theory (dashed line), simulation (solid line), and experiment (dots with error bars) obtained from the phonon distributions. (f) Calcium carrier oscillations (driving $|0{⟩}_{\mathrm{c}}|n⟩\leftrightarrow |1{⟩}_{\mathrm{c}}|n⟩$) taken in the heating region corresponding to (d). The decrease in calcium carrier frequency for increasing lasing times shows the heating nature.

Figure 4

(a)–(d) Simulation (orange line) and experimental data (blue dots) of real and imaginary part of the characteristic function along the 90 degree axis in the unlocked (a),(b) and locked (c),(d) case. Symmetry breaking in the imaginary part (b) to (d) is observed. (e),(f) Simulated Wigner function (2D) and marginal probability (side plots, orange line) along two axes for the (e) unlocked and (f) phase locked phonon laser including experimental data (blue dots).

Figure 5

Phase diffusion. (a) Marginal probabilities along 90 degree axis for different lasing lengths. Starting from a coherent state with a well-defined phase, the lasing process introduces phase diffusion over time (simulation in orange, experiment in blue). (b),(c) Simulated Wigner functions (2D) with marginal probabilities for a dissipation duration of 0.5 ms and 1.5 ms.