Many-Body Effects in Dark-State Laser Cooling

TL;DR

This work introduces a unified many-body framework for two-photon dark-state cooling in Λ-system ions, enabling analytical insight across weak and strong spin-phonon coupling and extending to large ion crystals. By adiabatically eliminating the excited state, it derives an effective two-level model with dark and bright states coupled to motion, yielding Lorentzian spin-absorption spectra and clear formulas for cooling rates and final phonon occupations. It reveals a collective cooling speed-up in the strong-coupling regime via phonon exchange and a number-independent cooling rate in the weak-coupling regime, with the crossover η_z shifting with ion number. The results are validated against exact numerics and offer practical guidelines for optimizing cooling in large ion arrays, with implications for scalable quantum information processing and precision metrology.

Abstract

We develop a unified many-body theory of two-photon dark-state laser cooling, the workhorse for preparing trapped ions close to their motional quantum ground state. For ions with a $Λ$ level structure, driven by Raman lasers, we identify an ion-number-dependent crossover between weak and strong coupling where both the cooling rate and final temperature are simultaneously optimized. We obtain simple analytic results in both extremes: In the weak coupling limit, we show a Lorentzian spin-absorption spectrum determines the cooling rate and final occupation of the motional state, which are both independent of the number of ions. We also highlight the benefit of including an additional spin dependent force in this case. In the strong coupling regime, our theory reveals the role of collective dynamics arising from phonon exchange between dark and bright states, allowing us to explain the enhancement of the cooling rate with increasing ion number. Our analytic results agree closely with exact numerical simulations and provide experimentally accessible guidelines for optimizing cooling in large ion crystals, a key step toward scalable, high-fidelity trapped-ion quantum technologies.

Figures (10)

• Figure 1: Schematics of the cooling scheme. (a): A generic three level $\{\ket{g},\ket{e},\ket{r}\}$ atomic system is employed for cooling and it is driven by two Raman lasers (RL). (b) Such a level scheme can be realized in either 1D or 2D trapped ion crystals, where the RL couple (with rate $g_{\rm R}$) the internal states to the external collective motion of the ion crystal. Optical dipole-force lasers (ODFL) can add additional coupling (with rate $g_{\rm O}$) of the internal states $\{\ket{g},\ket{e}\}$ to the collective motion. (c) Effective two-level (dark $\ket{-}$ and bright $\ket{+}$ spin states) after adiabatically eliminating the state $\ket{r}$. The effective two-level spin system is coupled to the mechanical mode of the ions with rate $g_{\rm R}$ and $g_{\rm O}$. The mode is further coupled to the intrinsic thermal bath at temperature $T$.
• Figure 2: Effective single-ion cooling dynamics. (a): The dark steady state is populated far from resonance $\omega_{s}-\omega_{m}\neq 0$. (b) Cooling cycle starts for the resonant case $\omega_{s}-\omega_{m}=0$ in the weak spin-phonon coupling limit, where the spin escapes from dark steady state by absorbing a phonon from the mode, followed by a fast reinitialization to the same dark state. The dominant processes are shown with solid arrowhead lines, while weak processes are shown with dashed arrowhead lines. The red cross mark represents the absence of such a process. (c) Cooling cycle in the strong spin-phonon coupling regime. Both Rabi flopping (shown as blue circles) and subsequent cooling steps take place (see text).
• Figure 3: Cooling spectra and excited state scattering rate. (a) Spin in a dark steady state is characterized by its Lorentzian absorption spectrum which determines the cooling rate and final occupation number of the motional state (see text). (b) Steady-state (SS) scattering rate $(\gamma_{g}+\gamma_{e})\varrho_{rr}$ of the optically excited state $\ket{r}$: It features an underlying coherent population trapping profile for $\Omega_{g}=\Omega_{e}$ (see text).
• Figure 4: Cooling results in weak spin-phonon coupling regime. See text for parameters used. (a-d) Steady state occupation number $n_{f}$ as a function of single-photon Rabi drive strength $\Omega_{g}$ for different cases (see insets). Solid lines (green) represent the analytical solutions as determined by the effective two-level spin absorption spectrum, Eq. \ref{['eq:Lorentzian']}. The dashed lines (orange) are numerically exact simulations of the master equation for the full three-level system using Eq. \ref{['eq:MasterEq3L']}. (e-f) Time resolved dynamics for the mean phonon number at spin-phonon resonance condition. Solid (green) and dashed (red) lines are effective two-level analytical solutions obtained from the rate Eq. \ref{['eq:eq_rate']} for the cases shown (see inset). Open markers, diamond and circle, are the corresponding numerical simulation of the full three-level master equation. In the simulations, we truncated the Fock state space with a cutoff $n_{cut}=30$ for a thermal state with mean occupation $n_{\rm th}=4.6$. In all figures, the horizontal dotted-dashed lines (blue) represent the minimum possible mean phonon number for corresponding cases.
• Figure 5: Effective many-ion cooling dynamics analogous to Fig. \ref{['Fig:Fig3']}. (a) Sketch of the dynamics under the Hamiltonian in Eq. \ref{['eq:taviscummingsa']} for given $n_{\rm ex}$ (red box). The Hamiltonian exchanges spin and motion excitations but cannot change the total number of excitations. The grey side bars indicate which states are accessible to weak and strong coupling regimes, respectively. (b) Weak coupling. Dynamics is restricted to $N_+\approx0$; states that are only populated in high-order perturbation theory are greyed-out. Each line indicates a separate ion, which can be in the dark state (black) or the bright state (white). The green arrows then indicate independent spontaneous emission of each ion, reducing the number of excitations. The purple dashed line indicates off-resonant coupling in Eq. \ref{['eq:dicke']}. (c) Strong coupling, analogous to panel (b). Here, all states are accessible to the dynamics. The decay rate $2N_+\gamma_{1,\rm sc}$ is proportional to the number of bright ions, as indicated by one arrow per bright state. These combine to an effective decay rate of $\gamma_{N,\mathrm{sc}}^{(n_\mathrm{ex})}$ from $n_{\rm ex}$ to $n_{\rm ex} - 1$.