Molecular heat pump for rotational states

In this work, we investigate the theory for three different unidirectional population-transfer schemes in trapped multilevel systems which can be utilized to cool molecular ions. The approach we use exploits the laser-induced coupling between the internal and motional degrees of freedom so that the internal state of a molecule can be mapped onto the motion of that molecule in an external trapping potential. By sympathetically cooling the translational motion back into its ground state, the mapping process can be employed as part of a cooling scheme for molecular rotational levels. This step is achieved through a common mode involving a laser-cooled atom trapped alongside the molecule. For the coherent mapping, we will focus on adiabatic passage techniques which may be expected to provide robust and efficient population transfers. By applying far-detuned chirped adiabatic rapid passage pulses, we are able to achieve an efficiency of better than $98\%$ for realistic parameters and including spontaneous emission. Even though our main focus is on cooling molecular states, the analysis of the different adiabatic methods has general features which can be applied to atomic systems.

Figure 1

Overview of the molecular levels and the couplings which are included in the model. Near-resonant laser pulses are driving the transitions between the excited levels $|e,n\rangle$ and the rotational states $|J,n\rangle$. The excited state decays toward the states $|J\rangle$ at a rate ${\Gamma }_J$ and toward the uncoupled state $|u\rangle$ at a rate ${\Gamma }_u$. The laser pulses have time-dependent Rabi frequencies ${\Omega }_J^{s,p}$ and are detuned from the relevant transition frequency by ${\Delta }_J^{s,p}$.

Figure 2

Basic $\Lambda$ configuration used to investigate the transfer efficiency of various adiabatic passage schemes in Sec. 3. The pump and Stokes pulses, ${\Omega }_1^p(t)$ and ${\Omega }_0^s(t),$ respectively, will induce a two-photon Raman transition (which may be chirped) from state $|n=0,J=1\rangle$ to state $|n=1,J=0\rangle$. The best results are obtained if the initial population in $|n=0,J=0\rangle$ remains there. The off-resonant intermediate states $|e,0\rangle$ decays toward the two rotational states and toward the uncoupled state $|u\rangle$ at a rate ${\Gamma }_u$. The decay rate ${\Gamma }_J$ represents decay from the levels $e$ to the levels $n,J$. Other off-resonant states are included as shown.

Figure 3

(a) Population transfer for STIRAP with different delay times $\tau$ and pulse widths $T$. The calculations are for the $\Lambda$ system in Fig. 2. The Rabi frequency is ${\Omega }_0=\nu /20$. (b) Efficiency as a function of Rabi frequency for $T=4500/\nu$ and $\tau =3500/\nu$. In (a) and (b), the pump and Stokes pulse detunings are ${\Delta }_1^p={\Delta }_1^s=0$ and the other parameters are ${\Gamma }_J={\Gamma }_u=0.01\nu$, $\eta =0.1$, $P_{0,0}(-\infty )=0.3,$ and $P_{1,0}(-\infty )=0.7$.

Figure 4

Efficiency of the STIRAP process as a function of detuning ${\Delta }_1^p$ ($={\Delta }_1^s$), with decay, ${\Gamma }_J={\Gamma }_u=0.01\nu$ (solid line), and without decay, ${\Gamma }_J={\Gamma }_u=0$ (dashed line). The other parameters are as in Fig. 3 with the best values taken for $T=4500/\nu$, $\tau =3500/\nu$, and ${\Omega }_0=\nu /20$.

Figure 5

(a) Efficiency for SCRAP for different Rabi frequencies ${\Omega }_0$ and delay times $\tau$ for the $\Lambda$ system in Fig. 2. The pulse width is $T=800/\nu$ and the pump and Stokes pulse detunings are ${\Delta }_1^p={\Delta }_1^s=100\nu$. (b) Efficiency for a delay $\tau =320/\nu$. Other parameters are as in Fig. 3a.

Figure 6

(a) Efficiency of CARP for different Rabi frequencies ${\Omega }_0$ and chirp rates $\alpha$ for the $\Lambda$ system in Fig. 2. The delay is $\tau =0$, the pulse width is $T=800/\nu$, and the Stokes pulse detuning is given by ${\Delta }_1^s={\Delta }_1^p-\alpha t$. (b) Efficiency as a function of Rabi frequency ${\Omega }_0$ for a small chirp rate $\alpha =8\times {10}^{-6}{\nu }^2$. Other parameters are as in Fig. 3a with pump pulse detuning ${\Delta }_1^p=100\nu$.

Figure 7

Population transfer for the $\Lambda$ configuration of Fig. 2, for different decay rates $\Gamma ={\Gamma }_J={\Gamma }_u$ and the three methods we consider. The parameters are optimized at ${\Gamma }_J={\Gamma }_u=0.01\nu$ for each of the three methods and are listed separately in the following. For STIRAP (solid line): ${\Omega }_0=\nu /20$, $T=4500/\nu$, $\tau =3500/\nu$, and ${\Delta }_1^p={\Delta }_1^s=0$. For SCRAP (dashed line): ${\Omega }_0=7.5\nu$, $T=800/\nu$, $\tau =320/\nu$, and ${\Delta }_1^p={\Delta }_1^s=100\nu$. For CARP (dotted line): ${\Omega }_0=5\nu$, $T=800/\nu$, $\tau =0$, ${\Delta }_1^p=100\nu$, and ${\Delta }_1^s={\Delta }_1^p-\alpha t$ with $\alpha =4.69\times {10}^{-5}{\nu }^2$. Other parameters which are fixed for all three cases are $J_{\max }=1$, $n_{\max }=2$, $\eta =0.1$, $P_{0,0}(-\infty )=0.3$, and $P_{1,0}(-\infty )=0.7$.

Figure 8

(a) Initial thermal distribution for a system with $J_{\max }=5$ and $\beta B=0.15$, and (b) the final population distribution after the completion of five sequential chirped two-photon Raman transitions. The detuning for the pump pulse is ${\Delta }_J^p=100\nu$, the Rabi frequency is ${\Omega }_{0}T=5\nu$, and all the decay rates are equal to ${\Gamma }_J={\Gamma }_u=0.01\nu$. Other parameters are $\eta =0.1$, $\tau =0$, $\mathrm{\tau ̃}=4800/\nu$, $\alpha =16\times {10}^{-5}{\nu }^2$, $T=800/\nu ,$ and $n_{\max }=6$.

Figure 9

The (small) population loss as a function of time for the two-photon chirped Raman passage in a system with $J_{\max }=5$, see Fig. 8. The detuning for the pump pulse is equal to ${\Delta }_1^p=100\nu$ and the Rabi frequency is ${\Omega }_0=5\nu$. Other parameters are as in Fig. 8.