Laser cooling by sawtooth-wave adiabatic passage

We provide a theoretical analysis for a recently demonstrated cooling method. Two-level particles undergo successive adiabatic transfers upon interaction with counterpropagating laser beams that are repeatedly swept over the transition frequency. We show that particles with narrow linewidth transitions can be cooled to near the recoil limit. This cooling mechanism has a reduced reliance on spontaneous emission compared to Doppler cooling, and hence shows promise for application to systems lacking closed cycling transitions, such as molecules.

Figure 1

(a) Spatial setup of the standing wave (two counterpropagating lasers with frequency ${\omega }_L\left(t\right)$) and particle (circle) with velocity $v$ in the laboratory frame. (b) Energy level diagram of the bare internal states, separated by energy $\mathrm{\Delta }E=\hslash {\omega }_{\text{a}}$.

Figure 2

(Top) The laser frequency ${\omega }_L\left(t\right)$ as a function of time. The approximate resonance frequencies for a particle with velocity $v$ are labeled. The sweep range is chosen to be large enough such that both beams will become resonant with the particle at some time during the sweep. (Bottom) The ideal excited-state fraction $P_e$ in the adiabatic regime. The particle remains in the excited state for a time interval ${\tau }_{\text{e}}$.

Figure 3

The excited state fraction $P_e$ of a particle prepared in the state $|g,10\hslash k\rangle$ over one sweep. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=200,T_s=22,$ and ${\mathrm{\Omega }}_0=5$.

Figure 4

Eigenvalue energy versus detuning for the four coupled states given in Eq. (15). The inset shows the splitting of an $n=1$ Doppleron resonance, which is small in the high-velocity regime. The red arrows identify the physical path being considered. Dashed lines show the evolution of the uncoupled states. Values, in units of ${\omega }_r$, are ${\mathrm{\Omega }}_0=2,\alpha ={\omega }_r=1,T_s=50$. $p=4\hslash k.$

Figure 5

Root-mean-square momentum $p_{\text{rms}}$ (steplike curve; magenta online) for a particle starting in the state $|g,10\hslash k\rangle$ over five sweeps. The curve exhibiting rising and falling pulses shows the excited state fraction $P_e$ (cyan online). The particle experiences a significant reduction in $p_{\text{rms}}$ and is left in a superposition of the internal states. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=120,T_s=1000,$ and ${\mathrm{\Omega }}_0=1$.

Figure 6

$\mathrm{\Delta }{p}_{\text{rms}}$ vs $p_i$ [(a) and (c)] and $P_e$ vs $p_i$ [(b) and (d)] for a particle that started in the state $|g,p_i\rangle$ at the beginning of a sweep. The various momentum regions in Eq. (18) are labeled in Fig. 6 and are presented in all subplots by vertical, dashed lines. The horizontal, dashed line corresponds to $\mathrm{\Delta }{p}_{\text{rms}}=0$. The adiabaticity parameter $\kappa$ lies in the diabatic regime for plots (a) and (b) ($\kappa =0.5,{\mathrm{\Omega }}_0=9.5{\omega }_r$) and in the adiabatic regime for plots (c) and (d) ($\kappa =4,{\mathrm{\Omega }}_0=26.8{\omega }_r$). Values common to all plots, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=360$, $T_s=2$.

Figure 7

$\mathrm{\Delta }{p}_{\text{rms}}$ vs $p_i$ [(a) and (c)] and $P_e$ vs $p_i$ [(b) and (d)] for a particle that started in state $|e,p_i\rangle$ at the beginning of a sweep. The various momentum regions in Eq. (18) are labeled in Fig. 7 and are presented in all subplots by vertical, dashed lines. The adiabaticity parameter $\kappa$ lies in the diabatic regime for plots (a) and (b) ($\kappa =0.5,{\mathrm{\Omega }}_0=9.5{\omega }_r$) and in the adiabatic regime for plots (c) and (d) ($\kappa =4,{\mathrm{\Omega }}_0=26.8{\omega }_r$). Values common to all plots, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=360$, $T_s=2$.

Figure 8

Various quantities for a particle that is subject to a single sweep with $\gamma \ne 0$ as a function of its initial momentum $p_i$. (a) $\mathrm{\Delta }{p}_{\text{avg}}$ (average impulse, in units of $\hslash k$). (b) $P_e^{\text{ss}}$ (steady-state population). (c) $\xi$ (number of scattered photons). The vertical, dashed lines correspond to the four conditions $\left|kv\right|<{\mathrm{\Delta }}_s/2$ (where $\mathrm{\Delta }{p}_{\text{rms}}$ falls to zero) and $\left|kv\right|<{\mathrm{\Delta }}_s/4$ (the capture range). Each point was averaged over 1000 trajectories. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=1800$, $T_s=1.0$, ${\mathrm{\Omega }}_0=60,\gamma =1$.

Figure 9

$\mathrm{\Delta }{p}^{\pm }$ vs $p_i$. The sharp, linear feature in $\mathrm{\Delta }{p}^{+}$ is due to Bragg oscillations, which mix the particle between the $|\pm p_i\rangle$ states before it reaches resonance with the lasers. Each point is averaged over 1000 trajectories. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=1800$, $T_s=1.0$, ${\mathrm{\Omega }}_0=60,\gamma =1$.

Figure 10

Frequency profile for the simplified “sweep-wait” cooling scheme.

Figure 11

Cooling of a particle that begins in the state $|g,10\hslash k\rangle$ to the recoil limit after five sweep-wait cycles. (Top inset) A snapshot of the momentum distribution $P\left(p\right)$ halfway through the seventh sweep. (Bottom inset) A closer look at the cooling trajectory once the system reached equilibration. The recoil limit $2T_r$ is included as a horizontal, dashed line. This curve is the average of 100 trajectories. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=100,T_s=60$, and ${\mathrm{\Omega }}_0=2$.

Figure 12

Stationary values of $\langle p^2/2m\rangle$ as a function of ${\mathrm{\Omega }}_0$ in the sweep-wait sequence. The vertical, solid line labels the value of ${\mathrm{\Omega }}_0$ that divides the diabatic and adiabatic regimes. The recoil limit is represented as a horizontal, dashed line. Deep within the adiabatic regime, the temperature scales linearly with ${\mathrm{\Omega }}_0$. Values, in units of ${\omega }_r$, are ${\mathrm{\Delta }}_s=100,T_s=60.$ Each point is averaged over 500 trajectories.

Figure 13

$\langle {\widehat{p}}^2\rangle /2m$ vs the average number of scattered photons for SWAP cooling and Doppler cooling. For both methods, ${\omega }_r=\gamma =1$. The SWAP parameters, in units of ${\omega }_r$, are ${\mathrm{\Omega }}_0=28,{\mathrm{\Delta }}_s=391,T_s=1$, and Doppler parameters are $\mathrm{\Omega }=40,\delta =-40$. The SWAP cooling data is over 38 sweeps. Each curve is averaged over 1000 trajectories. We extract an average cooling efficiency for SWAP cooling of up to $5\hslash k$ per scattered photon.

Figure 14

(a) A first-order Doppleron resonance, characterized by the condition ${\beta }_f={\beta }_i-3$. (b) A Bragg resonance between the states $|g,\pm 2\hslash k\rangle$.