Cavity-controlled formation of ultracold molecules

When driving the $| i\rangle \leftrightarrow | e\rangle$ transition for a single atom pair with the PA laser, at most one electronic or photonic excitation can be brought into the system at a time. We can then model the system effectively by five quantum states denoted by $| m,n\rangle$, where m denotes the atomic/molecular quantum state and n the cavity photon number, see figure 2. Furthermore, we assume tight confinement in the Lamb-Dicke regime (${\rm{\hslash }}{\omega }_{\mathrm{trap}}\gg {E}_{\mathrm{recoil}}$), and therefore, we need not consider the external motion of the particles. The total energy of the atom pair, including the inter-particle interaction, is on the order of the ${\rm{\hslash }}{\omega }_{\mathrm{trap}}$ which is much smaller than the linewidth Γ of $| e\rangle$.

The couplings, detunings and decay rates of the five states are also shown in figure 2. The PA laser field (frequency ω_L) couples the asymptotic two-atom-state $| i,0\rangle$ and the excited molecular state $| e,0\rangle$ with Rabi frequency Ω and detuning Δ₁ = ω_ge − ω_L − ω_gi. The $| e,0\rangle$-state is coupled coherently at rate $2g$ to the state $| g,1\rangle$, which can decay to $| g,0\rangle$ at rate 2κ. The two-photon detuning Δ₂ = ω_c − ω_L − ω_gi is the frequency mismatch between emitted photon and cavity frequency ${\omega }_{c}$. The $| e,0\rangle$-state can also decay spontaneously at rates Γ_g and Γ_h to the states $| g,0\rangle$ and $| h,0\rangle$, respectively (Γ = Γ_g + Γ_h). Here, $| g,0\rangle$ represents the desired final state, while $| h,0\rangle$ represents all other possible states.

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The Hamiltonian of the coherently-coupled sub-system $| i,0\rangle ,| e,0\rangle ,| g,1\rangle$ in a frame rotating with the laser frequency reads in the rotating-wave approximation

$$\begin{eqnarray*}&&\hat{H}={\rm{\hslash }}{{\rm{\Delta }}}_{1}| e,0\rangle \langle e,0| +{\rm{\hslash }}{{\rm{\Delta }}}_{2}| g,1\rangle \langle g,1| +{\rm{\hslash }}\displaystyle \frac{{\rm{\Omega }}}{2}| e,0\rangle \langle i,0| +{\rm{\hslash }}g| g,1\rangle \langle e,0| +{\rm{h}}.{\rm{c}}.\end{eqnarray*}$$

To take into account the molecular and cavity decay processes, we solve the corresponding master equation in Lindblad form (see appendix). The system is initialized in $| i,0\rangle$ at t = 0. The goal is to transfer the population from this state as fast, efficiently and coherently as possible to the state $| g,0\rangle$ of the ground state molecule. We therefore define the efficiency η of the scheme by the probability to produce a ground-state molecule $| g,0\rangle$ via cavity decay,

$$\begin{eqnarray*}&&\eta =2\kappa {\int }_{t=0}^{\infty }{\rho }_{g1g1}(t){\rm{d}}t,\end{eqnarray*}$$

where ${\rho }_{g1g1}(t)$ is the population of the $| g,1\rangle$ state.

We can derive an analytical expression for η in the weak-driving limit, (${\rm{\Omega }}\ll {\rm{\Gamma }},{g}^{2}/\kappa$), which, as we will later see, exhibits optimal achievable efficiency for time-independent control parameters. In the weak-driving limit, the coherent couplings of the laser Ω and the cavity g slowly keep populating the spontanously decaying states $| e,0\rangle$ and $| g,1\rangle$. As a consequence, a quasi-steady superposition state $| D\rangle$ forms, which slowly decays in an exponential manner. As derived in the appendix,

$$\begin{eqnarray}&&| D\rangle \approx | i,0\rangle +\displaystyle \frac{{\rm{\Omega }}}{2{g}^{2}+{\rm{\Gamma }}(\kappa -{\rm{i}}{{\rm{\Delta }}}_{2})}[({\rm{i}}\kappa +{{\rm{\Delta }}}_{2})| e,0\rangle -g| g,1\rangle ],\end{eqnarray} \tag{ 1 }$$

when ignoring the slow exponential decay and assuming resonant drive, Δ₁ = 0. The decay takes place through the small populations in $| e,0\rangle$ and $| g,1\rangle$ which decay via Γ and κ, respectively. This yields the transfer efficiency,

$$\begin{eqnarray}&&{\eta }_{\mathrm{wd}}=\displaystyle \frac{2\kappa {\rho }_{g1g1}}{2\kappa {\rho }_{g1g1}+{\rm{\Gamma }}{\rho }_{e0e0}}=\displaystyle \frac{2C}{2C+1+{({{\rm{\Delta }}}_{2}/\kappa )}^{2}},\end{eqnarray} \tag{ 2 }$$

where ρ_jj are components of the desity matrix ${\rho }_{\mathrm{wd}}=| D\rangle \langle D|$. The efficiency η_wd is maximal on two-photon resonance, Δ₂ = 0, which we consider from now on. One can also interpret η_wd as the ratio of cavity-induced decay rate ${{\rm{\Gamma }}}_{\kappa }=2{g}^{2}/\kappa$ of $| e,0\rangle$ (via green arrows in figure 2) and the total decay rate Γ_κ + Γ of $| e,0\rangle$ (for Δ₂ = 0). To obtain significant cavity-stimulated PA, i.e. a large fraction in the $| g,0\rangle$ state, we therefore aim at $C\gg 1$. For example, a moderate cooperativity $C\gtrsim 5$ will already result in an efficiency of η_wd > 0.9 for producing the chosen molecular quantum state.

As already mentioned the weak-driving limit gives optimal results in terms of transfer efficiency. This can be made plausible as follows. For vanishing Δ₂ and κ, the states $| i,0\rangle ,| e,0\rangle ,| g,1\rangle$ form a Λ-system which exhibits a dark state $| D\rangle \propto 2g| i,0\rangle +{\rm{\Omega }}| g,1\rangle$. This state is long-lived with no spontaneous decay via Γ, because $| e,0\rangle$ is not populated. If we now allow for a small but finite $\kappa \ll {g}^{2}/{\rm{\Gamma }}$ to populate the desired final state $| g,0\rangle$, the dark state $| D\rangle$ will become a little bit lossy due to population of $| e,0\rangle$ (∝κ², see equation (1)) and subsequent decay via Γ. These unwanted losses, however, are much smaller than the wanted decay flux from $| g,1\rangle \to | g,0\rangle$ which is proportional to κ. In addition, in the weak driving limit, ${\rm{\Omega }}\to 0$, the dark state $| D\rangle$ mainly consists of state $| i,0\rangle$. Since this is precisely the initially prepared state the weak driving limit is optimal for efficient transfer from $| i,0\rangle$ to the molcular state $| g,0\rangle$.

Compared to a single atom, a molecule often exhibits a reduced dipole matrix element of the electronic transition and thus a reduced coupling strength g and cooperativity C. The reduction is to first approximation determined by the Franck–Condon-factor f_FC = Γ_g/Γ for the specific ro-vibrational transition in a molecule,

$$\begin{eqnarray*}&&g={g}_{\max }\sqrt{{f}_{\mathrm{FC}}}\quad \mathrm{and}\quad C={C}_{\max }{f}_{\mathrm{FC}},\end{eqnarray*}$$

where the coupling strength and cooperativity for a closed two-level system (${f}_{\mathrm{FC}}=1$) are given by

$$\begin{eqnarray}&&{g}_{\max }={d}_{\mathrm{el}}\sqrt{\displaystyle \frac{{\omega }_{{ge}}}{2{\rm{\hslash }}{\epsilon }_{0}V}}\quad \mathrm{and}\quad {C}_{\max }=\displaystyle \frac{{g}_{\max }^{2}}{\kappa {\rm{\Gamma }}},\end{eqnarray} \tag{ 3 }$$

respectively. Here, V is the volume of the cavity mode and d_el the dipole moment of the electronic molecular $| g\rangle \leftrightarrow | e\rangle$ transition with frequency ω_ge (for a CQED review, we refer the reader to e.g. [7]). In a diatomic molecule, d_el depends in general on the internuclear distance R, and the decay rate Γ is about 2 times larger as compared to an atomic excited state (due to a Dicke superradiance effect [15]). In a rubidium dimer (Rb₂), there are strong transitions between ro-vibrational states $v^{\prime}$ of the shallow well of the (1)³Π_g potential (which asymptotically correlates to the atomic states 5S_1/2 + 5P_3/2) and the states $v^{\prime\prime}$ of the a³Σ_u potential (5S_1/2 + 5S_1/2). In particular, the transition from ${1}_{g},v^{\prime} =8$ to $v^{\prime\prime} =0$ at a wavelength of 744 nm has the largest f_FC,max = 0.37, see [16]. Apart from the transition dipole moment and Franck–Condon factor, the coupling strength g depends also on the mode volume as $g\propto 1/\sqrt{V}$, thus V should be minimized. For the fundamental TEM₀₀ mode of a Fabry–Perot resonator, $V=\pi {w}_{0}^{2}L/4$, where L is the cavity length and w₀ the mode waist. Similar as in [17], we consider the dipole trap beams to enter from the side without being clipped by the mirror substrates, which puts a lower limit on L. The mode waist w₀ is typically limited by the numerical aperture of the in-/outcoupling optics. According to equation (3), in order to maximise C, κ should be minimized. However, as we will discuss in the next section a small κ might lead to unacceptably long transfer times. Furthermore, the transmission through the mirror coatings should dominate over unavoidable absorption and scattering losses in the coatings. In table 1, we give an example for realistic CQED parameters for the above mentioned molecular transition in Rb₂. For those parameters, an efficiency of η_wd > 0.9 could be achieved for vibrational transitions with ${f}_{\mathrm{FC}}\gtrsim 0.05$.

Table 1.  Example of a set of realistic CQED parameters. The parameters g_max, Γ, C_max are derived for the transitions between vibrational states of the (1)³Π_g and a³Σ_u potentials in Rb₂. Here, d_el ≈ 3 × 10⁻²⁹ C m, see [6, 18].

Parameter	Symbol	Value
Cavity length	L	280 μm
Cavity mode waist	w0	4.8 μm
Cavity finesse	${ \mathcal F }$	5 × 104
Coupling strength for fFC = 1	gmax	2π × 80 MHz
Cavity field decay rate	κ	2π × 5.4 MHz
Excited state decay rate	Γ	2π × 12 MHz
Cooperativity for fFC = 1	Cmax	100
Max. Franck–Condon factor	fFC,max	0.37 [16]

Ideally, the transfer from the atom pair $| i,0\rangle$ to the molecular state $| g,0\rangle$ should be efficient and fast. In the weak driving limit the transfer efficiency is nearly optimal but can be very slow as the transfer time is ∝κ⁻¹. In fact, it can be shown that the transfer rate, which is the exponential decay rate of $| D\rangle$, is given by

$$\begin{eqnarray}&&{R}_{\mathrm{wd}}=\displaystyle \frac{{{\rm{\Omega }}}^{2}}{{\rm{\Gamma }}(2C+1)}\mathop{=}\limits_{C\gg 1}\kappa \displaystyle \frac{{{\rm{\Omega }}}^{2}}{2{g}^{2}},\end{eqnarray} \tag{ 4 }$$

if Δ₁ = 0. Clearly, too slow a transfer can be problematic for many reasons, e.g. when the transfer time is on the order of the particle lifetime in the trap.

We therefore now consider the limit of strong driving where a short π-pulse (short compared to all timescales determined by rates) quasi instantaneously transfers the population from $| i,0\rangle$ to $| e,0\rangle$ at t = 0. From there, it partially decays into $| g,0\rangle$ via the cavity mode but it also partially decays via Γ into $| h,0\rangle$. The efficiency η_π of the short-π-pulse scheme is then reduced as compared to η_wd of the weak-driving limit,

$$\begin{eqnarray}&&{\eta }_{\pi }=\displaystyle \frac{2\kappa }{2\kappa +{\rm{\Gamma }}}{\eta }_{\mathrm{wd}},\end{eqnarray} \tag{ 5 }$$

where we set ω_c = ω_ge. Besides the decay, the dynamics exhibit a damped oscillation of the population between $| e,0\rangle$ and $| g,1\rangle$. The oscillation can be understood as a beat of the two eigenstates

$$\begin{eqnarray*}&&| {B}_{\pm }\rangle \approx \displaystyle \frac{1}{\sqrt{2}}(| e,0\rangle \pm | g,1\rangle )\end{eqnarray*}$$

of the coupled system, which are populated at t = 0 by the π-pulse as an equal superposition state. In order for η_π to reach the weak-driving efficiency η_wd, we need $\kappa \gg {\rm{\Gamma }}$, according to equation (5). However, for a large η_wd we need $\kappa \ll {g}^{2}/{\rm{\Gamma }}$. Both conditions can be simultanously met only when $g\gg {\rm{\Gamma }}$ ¹ . According to our parameters in table 1 and even when using a relatively large Franck–Condon factor of f_FC > 0.3, this condition is not easily reached. Assuming a more typical f_FC = 0.1, we obtain η_π ≈ 0.77 compared to η_wd ≈ 0.95. Therefore, in general, the short π-pulse does not seem to be ideal for the transfer.

Therefore, the question arises, how quickly the optical pumping can be done without substantial loss of efficiency compared to the weak-driving limit? To answer this question, we investigate how the transfer efficiency to the ground state $| g,0\rangle$ can be optimized when employing a resonant square pulse of duration t_p and Rabi frequency Ω² .

For this, we solve the time-dependent master equation of the five-level system numerically. In figure 3, some examples of the time-dependent populations are plotted. For Ω = κ/2, we almost reach the weak-driving efficiencies ${\eta }_{\mathrm{wd}}=\tfrac{20}{21}$ (${\eta }_{\mathrm{wd}}=\tfrac{2}{3}$) for C = 10 (C = 1), respectively. Using Γ = 2π × 12 MHz, one reaches 95% of the respective asymptotic value within t ≈ 3 μs (t ≈ 0.5 μs). For a higher Rabi frequency (here, Ω = 3κ), the transfer works faster but the efficiency is already significantly reduced.

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In an optimization, we determine the highest Rabi frequency Ω and shortest pulse duration t_p for which the inefficiency 1 − η of this pulsed scheme does not significantly increase compared to the weak-driving limit. In particular, we allow for an inefficiency

$$\begin{eqnarray*}&&1-\eta =(1+\varepsilon )(1-{\eta }_{\mathrm{wd}}),\end{eqnarray*}$$

where ε is the tolerance, which can be arbitrarily chosen. According to equation (2) η_wd depends only on the cooperativity C = g²/(κΓ). However, because Γ is fixed and g_max = 2π × 80 MHz is also an estimated upper limit (see table 1), η_wd essentially depends on κ and the Franck–Condon factors, since $g={g}_{\max }\sqrt{{f}_{\mathrm{FC}}}$. In figure 4(a), we plot 1 − η for four different values of f_FC as a function of κ, where we set ε = 0.1. Figure 4(a) can be used to determine κ for a desired η and a given f_FC. Using this κ we can then read off the minimum pulse duration t_p and the maximal Rabi frequency Ω from figure 4(b). It is noteworthy that in the range κ/2π ≲ 10 MHz, Ω ≈ κ, almost independent of f_FC and that t_p ∝ f_FC/κ². (For larger values of κ, Ω diverges and t_p vanishes, see also appendix 'Condition for short π-pulses'.)

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For clarity, we finally discuss an example: if an efficiency of η = 0.97 is desired on a transition with f_FC = 0.1, we require κ = 2π × 3 MHz, see figure 4(a). From figure 4(b) it follows that the pulse length of the PA laser should be at least t_p = 4 × 10⁻⁵ s and its Rabi frequency should be at most Ω = 2π × 3 MHz.

Using the measured data in [16] we can estimate what Rabi frequencies Ω for PA can be realistically reached in an experiment. For this, we take into account that a tightly trapped atom pair in a lattice site has a density of about ∼10¹⁵ cm⁻³. To reach the Rabi frequency of Ω = 2π × 3 MHz of our example, a laser power of about 700 mW would be needed when the atom pair is located in a laser focus with 10 μm waist.

If only a smaller Rabi frequency is experimentally available, e.g. due to a smaller laser power, still the same targeted efficiency of η = 0.97 can be achieved by simply increasing the laser pulse length which should roughly scale as t_p ∝ 1/ Ω², see equation (4).

In order to take into account the incoherent decay processes of the excited molecular state and the cavity photon, we use a master equation in Lindblad form,

$$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}\hat{\rho }}{{\rm{d}}t} & = & -\displaystyle \frac{{\rm{i}}}{{\rm{\hslash }}}[\hat{H},\hat{\rho }]+2\kappa { \mathcal D }[\hat{a},\hat{\rho }]+{{\rm{\Gamma }}}_{g}{ \mathcal D }[| g\rangle \langle e| ,\hat{\rho }]+{{\rm{\Gamma }}}_{h}{ \mathcal D }[| h\rangle \langle e| ,\hat{\rho }],\\ & & \mathrm{where}\ { \mathcal D }[\hat{b},\hat{\rho }]=\hat{b}\hat{\rho }{\hat{b}}^{\dagger }-\displaystyle \frac{1}{2}{\hat{b}}^{\dagger }\hat{b}\hat{\rho }-\displaystyle \frac{1}{2}\hat{\rho }{\hat{b}}^{\dagger }\hat{b}.\end{array}\end{eqnarray*}$$

Here, $\hat{\rho }$ denotes the density operator and $\hat{a}$ is the annihilation operator of the cavity field.

In the limit of weak driving (${\rm{\Omega }}\ll {\rm{\Gamma }},{g}^{2}/\kappa$), we can derive an analytical expression for the efficiency η_wd, equation (2). It is given by the ratio of the decay rates from $| g,1\rangle$ and $| e,0\rangle$ in 'quasi steady-state',

$$\begin{eqnarray*}&&{\eta }_{\mathrm{wd}}=\displaystyle \frac{2\kappa {\rho }_{g1g1}^{\mathrm{ss}}}{2\kappa {\rho }_{g1g1}^{\mathrm{ss}}+{\rm{\Gamma }}{\rho }_{e0e0}^{\mathrm{ss}}}={\left[1+\displaystyle \frac{{\rm{\Gamma }}}{2\kappa }\displaystyle \frac{{\rho }_{e0e0}^{\mathrm{ss}}}{{\rho }_{g1g1}^{\mathrm{ss}}}\right]}^{-1}.\end{eqnarray*}$$

Here, ${\rho }_{e0e0}^{\mathrm{ss}}$ and ${\rho }_{g1g1}^{\mathrm{ss}}$ are components of the steady-state density matrix ρ^ss which we calculate as follows. We formally close the system in figure 2 by introducing an artificial 'repump' rate ζ from the states $| g,0\rangle$ and $| h,0\rangle$ back to state $| i,0\rangle$, which can be, in principle, arbitrarily slow ($\zeta \to 0$). It turns out, however, that the population ratio ${\rho }_{e0e0}^{\mathrm{ss}}/{\rho }_{g1g1}^{\mathrm{ss}}$ is independent of ζ, turning the system effectively into a three-level system, since the populations of the levels $| g,0\rangle$ and $| h,0\rangle$ are not relevant. The solution for the density matrix ${\rho }^{\mathrm{ss}}=| {D}^{\mathrm{ss}}\rangle \langle {D}^{\mathrm{ss}}|$ in the weak-driving regime is, to first order in Ω,

$$\begin{eqnarray*}&&| {D}^{\mathrm{ss}}\rangle \approx | {\rm{i}},0\rangle +{\rm{\Omega }}\displaystyle \frac{({\rm{i}}\kappa +{{\rm{\Delta }}}_{2})| e,0\rangle -g| g,1\rangle }{2{g}^{2}+({\rm{\Gamma }}-2{\rm{i}}{{\rm{\Delta }}}_{1})(\kappa -{\rm{i}}{{\rm{\Delta }}}_{2})},\end{eqnarray*}$$

If the system is open, i.e. ζ = 0, this state slowly decays exponentially, which can be described by

$$\begin{eqnarray*}&&| D\rangle =\exp \left\{-\displaystyle \frac{{R}_{\mathrm{wd}}}{2}t\right\}| {D}_{\mathrm{ss}}\rangle ,\end{eqnarray*}$$

where R_wd is given, for Δ₁ = Δ₂ = 0, by equation (4).

For a chosen ε, short π-pulses (as discussed before) are efficient enough if

$$\begin{eqnarray*}&&(1-{\eta }_{\pi })\leqslant (1+\varepsilon )(1-{\eta }_{\mathrm{wd}}).\end{eqnarray*}$$

Using equations (2) and (5), this is equivalent to the following condition

$$\begin{eqnarray*}&&\displaystyle \frac{\kappa }{{\rm{\Gamma }}}\geqslant \sqrt{\displaystyle \frac{{g}^{2}}{\varepsilon {{\rm{\Gamma }}}^{2}}+\displaystyle \frac{1}{16}}-\displaystyle \frac{1}{4},\end{eqnarray*}$$

which can be approximated by $\kappa \gtrsim g/\sqrt{\varepsilon }$ for $g\gg {\rm{\Gamma }}\sqrt{\varepsilon }$.

The estimated number of pairs N ≈ 10³ in the optical cavity for typical experimental conditions corresponds to a thermal cloud of approx. 10⁶ atoms with diameter 2σ ≈ 40 μm in a three-dimensional cubic optical lattice with laser wavelength λ_latt ≈ 1 μm, which overlaps with the fundamental cavity mode with waist w₀ ≈ 5 μm.