Photo-induced two-body loss of ultracold molecules

TL;DR

This study reveals that laser-induced electronic excitation causes two-body loss in ultracold NaK molecules, challenging previous assumptions about three-body loss limitations and suggesting ways to suppress these losses.

Contribution

It demonstrates that laser excitation, not sticky collisions, is responsible for two-body loss, providing a new understanding of loss mechanisms in ultracold molecules.

Findings

Electronic excitation causes observed two-body loss.

Longer laser wavelengths can reduce losses.

Sticking times are shorter than previously thought.

Abstract

The lifetime of nonreactive ultracold bialkali gases was conjectured to be limited by sticky collisions amplifying three-body loss. We show that the sticking times were previously overestimated and do not support this hypothesis. We find that electronic excitation of NaK+NaK collision complexes by the trapping laser leads to the experimentally observed two-body loss. We calculate the excitation rate with a quasiclassical, statistical model employing ab initio potentials and transition dipole moments. Using longer laser wavelengths or repulsive box potentials may suppress the losses.

Figures

Equations

$${\tau }_{\mathrm{stick}}=2\pi \hslash \rho .$$

$$\rho =\frac{g_{\mathbf{N}Jp}8{\pi }^{3+\frac{D}{2}}{\hslash }^3{C}_{\mathbf{Nm}}(2J+1)}{\Gamma (\frac{D}{2})}\int G(\mathbf{q})[E-V(\mathbf{q}){]}^{\frac{D}{2}-1}d\mathbf{q}.$$

$$W_{\mathrm{i}\to \mathrm{f}}(\mathbf{q})=-\int d\omega \frac{b_{\mathrm{i}\to \mathrm{f}}(\mathbf{q},\omega )n}{c}\frac{dI}{d\omega },$$

$$b_{\mathrm{i}\to \mathrm{f}}(\mathbf{q},\omega )=\frac{\pi }{3{ϵ}_0{\hslash }^2}{\mu }_{\mathrm{i}\to \mathrm{f}}(\mathbf{q}{)}^{2}g(\mathbf{q},\omega ),$$

$$\begin{gathered} ⟨b_{\mathrm{i}\to \mathrm{f}}(\mathbf{q},\omega ){⟩}_{\mathbf{q}}=\frac{g_{\mathbf{N}Jp}8{\pi }^{3+\frac{D}{2}}{\hslash }^3{C}_{\mathbf{Nm}}(2J+1)}{3{ϵ}_0{\hslash }^2\Gamma (\frac{D}{2})\rho } \\ \int d\mathbf{q}G(\mathbf{q})[E_{\mathrm{tot}}-V(\mathbf{y}){]}^{\frac{D}{2}-1}{\mu }_{\mathrm{i}\to \mathrm{f}}^2(\mathbf{q})\delta \left[\frac{E_{\mathrm{gap}}(\mathbf{q})}{\hslash }-\omega \right]. \end{gathered}$$

$$\frac{d{n}_c}{dt}=-n_c\sum _{\mathrm{f}}{\Gamma }_{\mathrm{i}\to \mathrm{f}}=-n_c\sum _{\mathrm{f}}\frac{I_{\mathrm{tot}}⟨b_{\mathrm{i}\to \mathrm{f}}({\omega }_{\mathrm{laser}})⟩}{c},$$

Full text

Ultracold dipolar gases have exciting applications across physics and chemistry Carr et al. (2009); Krems et al. (2009). They can be used for high-precision measurements that challenge the Standard Model of particle physics Baron et al. (2014); Andreev et al. (2018), to model or simulate quantum many-body physics Micheli et al. (2006); Büchler et al. (2007); Cooper and Shlyapnikov (2009), and to study and control chemical reactions Krems (2008); Ospelkaus et al. (2010a). There are also promising schemes for quantum computation using ultracold dipolar gases DeMille (2002); Yelin et al. (2006); Ni et al. (2018).

The first ultracold dipolar molecules in their absolute rovibronic and hyperfine ground state were realized by Ospelkaus et al. Ospelkaus et al. (2010b). The KRb molecules used were reactive Ni et al. (2008); Ospelkaus et al. (2010a), which limits the lifetime of these ultracold gases. To avoid losses due to chemical reactions, several groups realized nonreactive ultracold dipolar gases of the bosonic 87Rb133Cs Takekoshi et al. (2014); Molony et al. (2014), 23Na87Rb Guo et al. (2016) and the fermionic 23Na40K Park et al. (2015); Seeßelberg et al. (2018a). However, losses were still observed at about the same rate that would be expected of reactive molecules Ye et al. (2018); Guo et al. (2018); Gregory et al. (2019). In all of these experiments the lifetime of the gas in the crossed optical dipole trap was limited to a few seconds Park et al. (2015) or less Takekoshi et al. (2014); Molony et al. (2014); Guo et al. (2016). Preventing this loss is of crucial importance for realizing higher molecular densities required for loading optical lattices and to improve the coherence time of ultracold molecules Park et al. (2017).

The loss mechanism in these nonreactive gases is not yet understood, but there are strong indications that the loss is caused by ultracold molecular collisions, which have been studied extensively in the literature Quéméner and Julienne (2012). Mayle et al. Mayle et al. (2012, 2013) proposed that the loss mechanism may be due to “sticky collisions”, the formation of long-lived collision complexes. What actually happens to these complexes that causes loss of molecules from the trap is unknown and the subject of this paper. We show that laser excitation of these complexes can explain the losses in the experiments.

Mayle et al. Mayle et al. (2012, 2013) propose a procedure to calculate the sticking time of a collision complex by calculating the density of states. Rice-Ramsperger-Kassel-Marcus (RRKM) theory Levine (2005) relates the sticking time $\tau_{\mathrm{stick}}$ of the collision to the density of states (DOS, $\rho$)

[TABLE]

Mayle et al. obtain the RRKM sticking time using multichannel quantum defect theory (MQDT) [Mayle et al. (2012, 2013), which treats the long range fully quantum mechanically but uses a simplified short range, parameterized by the DOS. However, in the accompanying paper Christianen et al. (2019a) we show there was an error in the DOS calculation, leading to sticking times of two to three orders of magnitude too large. A quasiclassical equation to accurately calculate the DOS for arbitrary potentials is found:

[TABLE]

Here, $E$ is the total energy in the system, $J$ is the total angular momentum quantum number, $p$ is the parity and $V(\bm{q})$ is the potential energy as a function of the Jacobi coordinates of the complex $\bm{q}=(R,r_{1},r_{2},\theta_{1},\theta_{2},\phi)$: the intermolecular distance $R$, the bond lengths $r_{1}$, $r_{2}$, the polar angles, $\theta_{1}$, $\theta_{2}$, and the dihedral angle, $\phi$ (see Ref. Christianen et al. (2019b)). The complex has $D=6$ internal degrees of freedom. The constant $C_{\bm{Nm}}$, parity factor $g_{\bm{N}Jp}$, and geometry factor $G(\bm{q})$ are defined in the accompanying paper Christianen et al. (2019a). The factor $g_{\bm{N}Jp}=1/2$ for collisions of heteronuclear diatoms.

We use Eq. (2) and the recently constructed, accurate potential energy surface (PES) Christianen et al. (2019b) to calculate the DOS for the NaK+NaK system. These calculations result in a DOS of $0.37~{}\mu$K*-1* (for $J=1$) and a sticking time of about 18 $\mu$s. We showed that this lifetime is not long enough for complex-diatom collisions to explain the experimental losses Christianen et al. (2019a).

In this paper, we show that the excitation of collision complexes by the trapping laser can explain the losses observed experimentally. In typical experiments, the diatoms are confined using a crossed optical dipole trap with lasers far red detuned (1064 nm Takekoshi et al. (2014); Ye et al. (2018); Park et al. (2015, 2017) or 1550 nm Molony et al. (2014); Seeßelberg et al. (2018a, b)) from the molecular $X^{1}\Sigma^{+}\rightarrow A^{1}\Sigma^{+}$ transition. However, the electronic excitation energies of the complex differ from those of the individual molecules and depend on the nuclear geometry of the complex. This means that even though the laser is red detuned for the diatoms, the laser may electronically excite the collision complex.

Figure 1 shows the ground state and low-lying singlet excited states of NaK+NaK as a function of the intermolecular distance for a planar configuration of $C_{2h}$ symmetry. Colors correspond to the different irreducible representations of the electronic states. The potentials were calculated ab initio using internally contracted multireference configuration interaction with Molpro Werner et al. , as described in more detail in the Supplement Christianen et al. (2019c). The lowest two excited states correlate asymptotically to $X^{1}\Sigma^{+}\rightarrow A^{1}\Sigma^{+}$ excitations of one of the NaK molecules. The third excited state of the complex correlates to both molecules excited to $a^{3}\Sigma^{+}$. Above that, we find excited states correlating to single $X^{1}\Sigma^{+}\rightarrow B^{1}\Pi$ excitations, as well as simultaneous excitation of one molecule to $a^{3}\Sigma^{+}$ and the other to $b{{}^{3}\Pi}$. These highest two thresholds are nearly degenerate and their order depends on the monomer bond lengths. Higher excited states correlating to Na$(S\rightarrow P)$ transitions and more highly excited triplet states exist, but do not contribute to the absorption. We see that the excited-state potentials are more strongly bound than the ground-state potential, e.g., the first excited state potential has a well depth in the order of 12 000 cm*-1* compared to the well depth of 4534 cm*-1* of the ground-state potential Christianen et al. (2019b). The dashed line shows the ground-state potential shifted up by 1064 nm. This curve crosses four excited-state potentials in the region where the ground-state potential is attractive, indicating that these excited states can be reached. Furthermore, strong transition dipole moments (TDMs) to these excited states exists as the NaK $A^{1}\Sigma^{+}$ and $B^{1}\Pi$ excited states correlate to parallel and perpendicular dipole-allowed K$(S\rightarrow P)$ transitions.

This establishes that the energies of the excited states are low enough for trapping-laser-induced electronic transitions of collision complexes to occur. Next, we calculate the rates of these transitions to determine whether these transitions occur within the lifetime of a complex. This is not possible in the current MQDT framework, since this model does not explicitly treat the short range. Therefore, we develop a statistical model to calculate the laser excitation rates using ab initio short-range potentials.

The rate equation for laser excitation of the complex for a frozen geometry, $\bm{q}$, from discrete state $\mathrm{i}$ to $\mathrm{f}$ is given byHilborn (1982)

[TABLE]

where $n$ is the particle density and $dI/d\omega$ is the spectral irradiance of the laser. The coefficient $b_{\mathrm{i\rightarrow f}}$ is given by Hilborn (1982)

[TABLE]

where $\mu_{\mathrm{i\rightarrow f}}(\bm{q})$ is the TDM and $g(\bm{q},\omega)$ is the lineshape of the transition. We set $n=n_{c}$, the number of collision complexes. We assume the collisions to be ergodic and use the same quasiclassical phase-space model used to calculate the sticking time. The nuclear motion is treated classically and electronic transitions can only occur when the ground-state electronic energy plus the photon energy matches the excited-state electronic energy, i.e., where the excited potentials cross the blue dashed line in Fig. 1.

We calculate the expectation value of $b_{\mathrm{i\rightarrow f}}({\bm{q}},\omega)$ over the accessible phase space as a function of $\omega$. We assume the linewidth of the transition is small compared to the variation of the energy gap between the ground state and the excited state with geometry, $E_{\mathrm{gap}}(\bm{q})$, and replace $g(\bm{q},\omega)$ by a delta function,

[TABLE]

The linewidth of the laser is also very small with respect to the variation of the excitation energy as a function of the geometry, such that $dI/d\omega$ can be replaced by ${I_{\mathrm{tot}}\delta(\omega_{\mathrm{laser}}-\omega)}$. We then sum over the electronic final states (f) to obtain

[TABLE]

where $\Gamma_{\mathrm{i\rightarrow f}}$ is the excitation rate from state i to f. We define $\Gamma_{\mathrm{laser}}=\sum_{\mathrm{f}}\Gamma_{\mathrm{i\rightarrow f}}$ and $\tau_{\mathrm{laser}}=\Gamma_{\mathrm{laser}}^{-1}$.

Before evaluating Eq. (6), we first qualitatively explore the properties of the TDM between the ground state and the low-lying excited states by again considering the one-dimensional cut for the $C_{2h}$ configuration. Due to symmetry, only $A_{u}$ and $B_{u}$ states have nonzero TDMs with the ground state. We show the TDMs of the two lower-lying $B_{u}$ states in Fig. 2. At long range, the $1B_{u}$ state corresponds to $(|AX\rangle-|XA\rangle)/\sqrt{2}$, the anti-symmetric combination of $X^{1}\Sigma^{+}$ to $A^{1}\Sigma^{+}$ excitations in either molecule. The $A$ excited state correlates to K$(S\rightarrow P)$ transitions. The molecular TDMs add constructively such that the $1B_{u}$ linestrength approaches twice the linestrength of the $S\rightarrow P$ transition of the K atom, $\mu^{2}\sim 17$ $(ea_{0})^{2}$. Nandy et al. (2012). Asymptotically, the $2B_{u}$ state corresponds to $a^{3}\Sigma^{+}+b^{3}\Pi$, which has zero TDM with the ground state. For $R<20$ $a_{0}$, these states mix such that the $1B_{u}$ TDM decreases and the $2B_{u}$ TDM increases, but the sum remains nearly constant. At yet shorter distances, $R<10$ $a_{0}$, short-range effects decrease all TDMs. The more highly excited states can be reached energetically only at very short range, where the TDMs drop substantially, such that the lowest three excited states dominate the excitation rate. In what follows, we include only these three excited states.

The evaluation of Eq. (5) requires global PESs and TDM surfaces, rather than the one-dimensional cuts discussed above. For the ground state, we use the GP9 PES from Ref. Christianen et al. (2019b). New PESs for the lowest three excited states are constructed to describe the gaps between the electronic energy levels. We also construct TDM surfaces for these excited states, and fit all surfaces using our machine-learning fitting method Christianen et al. (2019b). The details are described in the Supplementary Information Christianen et al. (2019c).

For geometries with lower symmetry, the excited states exhibit many avoided crossings and conical intersections that complicate the electronic structure calculations. Conical intersections are expected to occur, e.g., from the many crossings observed in Fig. 2, and because the energetic ordering of electronic states switches between arrangements; the $a^{3}\Sigma^{+}+a^{3}\Sigma^{+}$ state is the third excited state in the NaK+NaK arrangement, but the second excited state for Na${}_{2}+$K2. Intersections occur at intermediate geometries. Near conical intersections, the TDM and potential energy vary rapidly with nuclear geometry, and the individual surfaces are difficult to fit accurately. However, we are interested in the laser-excitation rate, which is computed as a phase-space average summed over electronically excited states, and is less sensitive to the quality of the fits for individual electronic states.

The global TDM surfaces constructed here exhibit features similar to those observed at the $C_{2h}$ symmetric configuration. At long range, the excited states have large TDMs due to their K$(S\rightarrow P)$ character. For lower-symmetry configurations, the lowest three excited states mix strongly and the linestrength distributes over the excited states, such that the individual squared TDMs vary with geometry, but the total remains approximately constant. At yet shorter range, the total TDM decreases.

\floatsetup

[figure]style=plain,subcapbesideposition=top

Figure 3 shows the laser excitation rate as a function of the frequency at a trap depth of 10$\mu$K. The colored lines indicate the laser transition rates to the individual excited states, the black line indicates the total. The grey shaded area indicates the error margin, which is estimated as described in the Supplement Christianen et al. (2019c). The vertical dotted lines indicate wavelengths of 10 $\mu$m, 1550 nm, and 1064 nm. At 1064 nm, the loss rate is $\Gamma_{\mathrm{laser}}=3.3$ $\mu$s*-1*, which corresponds to a lifetime for laser excitation of $\tau_{\mathrm{laser}}=0.30$ $\mu$s. At 1550 nm, the loss rate is reduced to 0.91 $\mu$s*-1*, corresponding to a lifetime of 1.1 $\mu$s. At either wavelength, the lifetime for laser excitation is much smaller than the sticking time, $\tau_{\mathrm{laser}}\ll\tau_{\mathrm{stick}}=18$ $\mu$s. Hence, essentially all complexes formed undergo laser excitation before they dissociate, such that complex formation manifests as effective two-body loss, in agreement with experimental observations in Refs. Takekoshi et al. (2014); Molony et al. (2014); Park et al. (2015); Ye et al. (2018); Gregory et al. (2019). Switching to a 10 $\mu$m laser wavelength would reduce the excitation rate by orders of magnitude to around 0.2 ms*-1*, which is much slower than the complex dissociation rate.

Figure 4 shows the laser-excitation lifetime, $\tau_{\mathrm{laser}}$, of the complexes (in blue, left-hand axis) and the half-life of the NaK molecules in the trap $\tau_{\mathrm{NaK}}$ (in red, right-hand axis), as a function of the laser intensity. The solid, dashed, and dash-dotted lines show results for wavelengths of 1064 nm, 1550 nm, and 10 $\mu$m, respectively. The lifetime was calculated using an initial trap molecule density of $0.4\cdot 10^{11}$cm*-3*, temperature of 500 nK Park et al. (2015), and a diatom-diatom $p$-wave collision rate of $3\cdot 10^{-11}$ cm3 s*-1* based on multichannel quantum defect theory Mayle et al. (2013). We include both the laser excitation loss mechanism and sticking-amplified three-body loss with a rate of $1.1\cdot 10^{-10}$ cm*-3s-1*, as described in Ref. Christianen et al. (2019a). The estimated lifetime of the complex with respect to complex-molecule collisions $\tau_{3b}=1/[n(0)k_{3b}]$, where $n(0)$ is the initial molecular density and $k_{3b}$ the molecule-complex collision rate. At 1064 nm and 1550 nm, in the typical range of experimental intensities, we find laser excitation leads to effective two-body loss, limited by the formation of collision complexes, and is insensitive to small changes in wavelength and intensity. For a 10 $\mu$m wavelength, however, the lifetime of the molecules becomes orders of magnitude larger and strongly intensity-dependent. Changing the trapping laser wavelength to 10 $\mu$m emerges as a straightforward way to strongly reduce the losses. However, even at this wavelength the laser excitation of collision complexes is dominant over sticking-amplified three-body loss.

Possibilities for reducing trapping-laser-induced loss of bialkali molecules are the following:

1. Box potentials. Rather than switching to very long wavelengths, a promising avenue may be to use shorter wavelengths to realize repulsive potential walls of blue-detuned trapping light Davidson et al. (1995), such that molecules are trapped between them in the dark. Such uniform box potentials have been realized for ultracold atoms, for both Bose-Einstein condensates Gaunt et al. (2013) and Fermi gases Mukherjee et al. (2017).

2. Preventing molecular collisions. If molecular collisions are suppressed, fewer collision complexes are formed that can be excited by the lasers. Several ways to prevent molecular collisions have been proposed, such as using optical lattices to confine the molecules de Miranda et al. (2011), or inducing repulsive interactions between colliding molecules using static González-Martínez et al. (2017), or using microwave fields Karman and Hutson (2018); Lassablière and Quéméner (2018).

3. The trapping-laser excitation loss mechanism would be suppressed completely if the molecules were confined by non-optical traps, such as a magnetic trap. This would require preparation of the molecules in a low-field seeking Zeeman state, and may require nonzero electron spin.

We note that any additional effects not taken into account here—such as an increased sticking time due to external fields or hyperfine coupling, or excitations to higher excited states—can only increase the loss. Such effects could increase trapping-laser-induced loss at lower intensities, while they do not affect the high-intensity limit where complex formation is the rate-limiting step, see Fig. 4.

To summarize, we find that experimentally observed losses of nonreactive ultracold molecules cannot be attributed to sticky collisions, but they can be explained by electronic excitation of collision complexes by the trapping laser. The loss can be reduced by using significantly lower intensities and longer wavelengths. The loss can be prevented by using a magnetic instead of optical trap, repulsive box potentials, or by preventing molecular collisions. The theory is illustrated for NaK+NaK collisions. Other bialkalis, such as NaRb and RbCs, have comparable electronic structure and longer sticking times, such that we expect trapping-laser-induced loss to also be the major cause of losses for these systems.

I Acknowledgements

We thank Simon Cornish, Alan Jamison, Edvardas Narevicius, Kang-Kuen Ni, Sebastian Will, and Zoe Yan for useful discussions. M.Z. acknowledges support from NSF, AFOSR, and the Gordon and Betty Moore Foundation through grant GBMF5279. T.K. is supported by NWO Rubicon grant 019.172EN.007 and an NSF grant to ITAMP.