Improved magneto–optical trapping of a diatomic molecule

For the common type-I MOT, an $F\to F^{\prime} =F+1$ cycling transition is employed, where F is the total angular momentum quantum number and the prime indicates the excited state. This level structure ensures that there are no dark Zeeman sublevels in the ground state. Typically for type-I MOTs, the magnitude of the ground-state magnetic g-factor, g, is less than that of the excited-state g-factor, g'. In this case the transition shifted closest to resonance with the red-detuned MOT light is dictated by the sign of the magnetic g-factor in the excited state, g'. In this common case the correct arrangement of circular polarizations is intuitive and clear. For the nominal case considered here ($z\gt 0$, local magnetic field ${{B}_{z}}\gt 0$, laser beam red-detuned and propagating towards $-z$), when g' is negative a trapping laser should be circularly polarized to drive ${{\sigma }^{+}}$ transitions; see figure 2(c). If g' is positive the circular polarization should be reversed to drive ${{\sigma }^{-}}$ transitions.

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The rarely-used type-II MOT operates on an $F\to F^{\prime} =F$ or $F\to F^{\prime} =F-1$ cycling transition where certain 'dark' ground-state sub levels are not coupled to the excited state by the confining laser light. Without a mechanism to 'remix' dark states into bright states, such as Larmor precession or optical transitions driven by orthogonal or anti-trapping MOT beams, the scattering rate will go to zero. Furthermore, the scattering rate from the MOT beam counter-propagating to a particle's displacement from the trap center must be greater than the scattering rate from the co-propagating MOT beam to ensure a confining force. In contrast to the case of type-I MOTs, there has been no widely accepted understanding of the mechanism behind the confining force in a type-II MOT, making the correct choice of circular polarizations in these systems difficult. Despite this lack of understanding, type-II MOTs have been reported in several atomic systems [17–20], and these results inspired our group's recent work demonstrating the magneto–optical trapping of a diatomic molecule for the first time [15]. The rotational degree of freedom in molecules requires the cycling transition of a molecular MOT to correspond to a type-II system [21].

In [15] the confining transitions for the original SrF MOT were selected using the sign of the difference in the magnetic moment between the ground and excited states of the cycling transition. In particular, for each ground-state hyperfine manifold (and for the nominal magnetic field gradient), we considered a deviation from the trap center along a given laser axis. The polarization of the laser counter-propagating to this deviation was chosen to excite the transition Zeeman shifted closest to resonance with the red-detuned light. For the ${{X}^{2}}{{\Sigma }^{+}}\to {{A}^{2}}{{\Pi }_{1/2}}$ cycling transition, the ground-state g-factor is much larger in magnitude than the excited-state g-factor, i.e. $|g|\gg |g^{\prime} |$. Nominally, g = 2 for a $^{2}\Sigma$ state and $g^{\prime} =0$ for a $^{2}{{\Pi }_{1/2}}$ state [22], though due to spin-orbit mixing effects both values are known to differ from the nominal. We proceeded assuming that $g^{\prime} \approx 0$, resulting in the circular polarizations being dictated by the sign of the g-factor for each hyperfine ground state; see figure 2(f). Although this was effective at producing a MOT of SrF, similar to atomic type-II MOTs, the source of the small but non-zero restoring force observed was not understood.

To highlight why the source of the restoring force was unclear in type-II systems, consider a lambda system in 1D with a restoring MOT beam driving (for example) ${{\sigma }^{-}}$ transitions from one ground-state sublevel and an anti-restoring MOT beam driving ${{\sigma }^{+}}$ transitions from the other ground-state sublevel. Assuming that the excited state decays into each ground state with equal probability, and that both beams have the same frequency and intensity, then when both ground states are degenerate (figure 1(a)) the scattering rate is the same for each transition and there is no restoring force. Now consider the case where a magnetic field shifts the sublevel addressed by the restoring beam closer to resonance and the sublevel addressed by the anti-restoring beam further from resonance (figure 1(b)). In this case the excitation rate is smaller from the sublevel further from resonance, so more population accumulates there than in the sublevel closer to resonance. In this picture it can be shown [23] that the reduced excitation rate from the sublevel further from resonance, addressed by the anti-restoring beam, is cancelled exactly by the population imbalance between the two sublevels. Hence the scattering rates from the restoring and anti-restoring beams remain equal, and there is no net restoring force for all velocities and any applied magnetic field. We note in passing that the gradient force can be 'rectified' and provide confinement in this system for sufficiently high laser intensity and a different polarization configuration [24].

Numerical simulations recently performed by M Tarbutt led him to propose a number of general polarization rules to be followed when magneto–optically trapping [16]. Crucially, this work proposes that the restoring force present in a MOT relies on the excited-state g-factor, g', being non-zero. Herein lies a potential explanation of the mechanism behind the confining force in a type-II MOT. The Zeeman splitting in the excited state is essential in order to break the symmetry between confining and anti-confining transitions, allowing a net restoring force to be applied provided that the correct circular polarization is chosen. The larger the magnitude of g', the more this symmetry is broken and the greater the confining force applied. (Surprisingly, this symmetry argument also holds for type-I MOTs where $F\lt F^{\prime}$: if $g^{\prime} =0$ there is no restoring force. However we know of no physical example of such a case since type-I MOTs typically operate on an $nS\to nP$ transition where $|g|\lt |g^{\prime} |$.)

When considering which circular polarization is required to magneto–optically trap, a key consideration is determining which ground-state m_F sublevel is populated most often. The $X\to A$ transition in SrF is an instructive example as it includes all three dipole-allowed changes in angular momenta, namely $F\to F^{\prime} =F-1$, $F\to F^{\prime} =F$, and $F\to F^{\prime} =F+1$ (figures 2(a)–(d)). Note that for this transition in SrF, g' is negative; if g' were positive the confining polarizations would be reversed. In addition, since the ground-state g-factor does not lead to a restoring force, we ignore ground-state Zeeman shifts in the ensuing discussion.

The simplest case, $F=0\to F^{\prime} =1$ (figure 2(c)), is that of a type-I MOT with no dark states. Here it is clear that the confining MOT beam should be polarized to drive ${{\sigma }^{+}}$ transitions.

For $F=1\to F^{\prime} =1$ (figures 2(b) and (d)) ${{\sigma }^{\mp }}$ transitions out of ${{m}_{F}}=\pm 1$ form a pure lambda system and (as discussed above) no restoring force can be applied. However, transitions out of ${{m}_{F}}=0$ can lead to a restoring force provided that the restoring MOT beam is polarized to drive ${{\sigma }^{+}}$ transitions into the $m_{F}^{\prime }=\;+1$ excited state, which is shifted closer to resonance; from here molecules decay into ${{m}_{F}}=\;0,+1$ with equal probability. Although ${{m}_{F}}=\;+1$ is dark to the restoring beam, orthogonal MOT beams can drive transitions out of this sublevel and potentially repopulate ${{m}_{F}}=0$, from which the restoring beam can apply a force again. In this scheme the majority of photons scattered are from orthogonal MOT beams; transitions from ${{m}_{F}}=0$ are occasionally driven by the restoring beam and rarely by the anti-restoring beam. Additionally, when ${{m}_{F}}=-1$ is occasionally populated, transitions can be driven by the restoring MOT beam but not by the anti-restoring beam.

For the final case, $F=2\to F^{\prime} =1$ (figure 2(a)), again the transition closest to resonance is to the $m_{F}^{\prime }=\;+1$ sublevel, which decays and populates ${{m}_{F}}=\;+2$ with $60\;\%$ probability, ${{m}_{F}}=\;+1$ with $30\;\%$ probability, and ${{m}_{F}}=0$ with $10\;\%$ probability [25]. These branching ratios and the resonance condition highlight that the restoring MOT beam should be polarized to drive ${{\sigma }^{-}}$ transitions. The biased branching ratios result in a restoring force in this configuration despite the rarely-visited ${{m}_{F}}=-2,-1$ sublevels being dark to the restoring MOT beam and transitions from ${{m}_{F}}=0$ being driven more often by the anti-restoring than the restoring MOT beam.

Following the polarization rules developed in [16] and outlined above, a revised trapping scheme for SrF was proposed in [16] (figure 2(e)) that differs from that used for the original MOT demonstration in [15] (figure 2(f)). Simulations of this new scheme in [16] predicted both tighter confinement and a larger capture volume than those present in the original setup.

Several features of the MOT produced with the new scheme are strikingly different from those using the original scheme. For example, MOT LIF images show that the trapped cloud is spatially smaller than for the original trapping scheme [15] (figure 3). A two-dimensional Gaussian fit to the LIF intensity profile gives radial and axial rms widths of ${{\rho }_{{\rm rms}}}=1.9(1)$ mm and ${{z}_{{\rm rms}}}=2.1(1)$ mm respectively, which may be compared to values ${{\rho }_{{\rm rms}}}=4.1(1)$ mm and ${{z}_{{\rm rms}}}=2.6(1)$ mm previously. This indicates that the confining and/or damping forces within the trap have changed.

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The spontaneous scattering rate, ${{R}_{{\rm sc}}}$, is measured using the same method presented in [15]. In short, the ${{\mathcal{L}}_{21}}$ laser is extinguished at t = 58.6 ms and the LIF decay constant, ${{\tau }_{\nu =2}}$, is measured as molecules are optically pumped into the now dark ${{X}^{2}}{{\Sigma }^{+}}(\nu =2)$ state. We measure ${{\tau }_{\nu =2}}=0.74_{-0.27}^{+0.34}$ ms and, using the vibrational branching fraction ${{b}_{02}}\approx 0.004$, find that ${{R}_{{\rm sc}}}\approx 3.8_{-1.2}^{+2.2}\times {{10}^{6}}$ s⁻¹. This is not significantly different from the scattering rate in the original configuration. The detection system is the same as used in [15], with a measured efficiency of $\sim 0.8\;\%$; the measured value of $1.1\times {{10}^{6}}$ detected photoelectron counts imaged at t = 100 ms, with camera exposure $\Delta {{t}_{{\rm exp} }}=60$ ms, corresponds to a trapped population at the end of the slowing period (i.e. at t = 35 ms) of ${{N}_{{\rm MOT}}}\approx 500$ molecules, roughly $1.7\times$ greater than in [15]. The peak density is ${{n}_{{\rm MOT}}}\approx 4000$ cm⁻³, roughly $7\times$ greater than that measured in [15].

To directly probe the confining and damping forces within the trap, the MOT response to a rapid displacement of the trap center is measured. During the loading phase, a magnetic bias field offsets the trap center $\approx 5$ mm radially, along the axis of the molecular beam. The bias field is then rapidly switched off to release the trapped molecules into the unbiased potential. To avoid significant LIF from the slowed molecular beam, the displaced MOT is released at t = 59 ms. The molecular cloud's position is measured as a function of time using short camera exposures, $\Delta {{t}_{{\rm exp} }}=2$ ms, and compared to data taken using the original trapping scheme [15] as shown in figure 4.

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The cloud moves with damped harmonic motion about the trap center with radial frequency ${{\omega }_{\rho }}=2\pi \times 76(5)$ Hz and damping coefficient $\alpha /{{m}_{{\rm SrF}}}=310(30)$ s⁻¹, where ${{m}_{{\rm SrF}}}$ is the mass. The new trapping scheme used in this work produces a potential with trap frequencies over $4\times$ greater than those measured in our original trap, where ${{\omega }_{\rho }}=2\pi \times 17.2(6)$ Hz [15], indicating that the trap spring constants (${{\kappa }_{\rho ,z}}={{m}_{{\rm SrF}}}\omega _{\rho ,z}^{2}$) are 20 × greater. In addition to tighter confinement, the measured damping coefficient is increased by a factor of 2.

Once the trap frequency and cloud size are known, the equipartition theorem can be used to calculate the radial MOT temperature, ${{T}_{\rho }}$. We find ${{T}_{\rho }}={{m}_{{\rm SrF}}}\omega _{\rho }^{2}\rho _{{\rm rms}}^{2}/{{k}_{B}}=11(2)$ mK, where k_B is the Boltzmann constant. Assuming ${{\omega }_{z}}=\sqrt{2}{{\omega }_{\rho }}$, which applies to atomic MOTs in quadrupole magnetic fields [25], the calculated axial trap frequency ${{\omega }_{z}}=2\pi \times 107(7)$ Hz and measured MOT axial width, ${{z}_{{\rm rms}}}$, correspond to an axial temperature ${{T}_{z}}=26(3)$ mK. The MOT temperature extracted using this method is ${{T}_{{\rm MOT}}}={{(T_{\rho }^{2}{{T}_{z}})}^{1/3}}=14(5)$ mK.

An independent measurement of the MOT temperature is made using the release-and-recapture method. In this method trapped molecules are released from the trap for a variable time of flight $\Delta {{t}_{{\rm TOF}}}$ before the trap is switched back on; this recaptures a fraction of the initial molecules that depends on their mean velocity and therefore temperature [29]. To avoid LIF from the slowed molecular beam, the MOT is released (by extinguishing the ${{\mathcal{L}}_{21}}$ laser) at a fixed time of $t={{t}_{{\rm rel}}}=100$ ms, and the time of flight $\Delta {{t}_{{\rm TOF}}}$ is varied from 0 to 30 ms. After each release, the ${{\mathcal{L}}_{21}}$ laser is turned on at $t={{t}_{{\rm rel}}}+\Delta {{t}_{{\rm TOF}}}$ to recapture the remaining molecules, and imaging begins at $t={{t}_{{\rm rel}}}+\Delta {{t}_{{\rm TOF}}}+3$ ms using $\Delta {{t}_{{\rm exp} }}=60$ ms (figure 6).

To extract the MOT temperature, the measured recaptured fraction as a function of $\Delta {{t}_{{\rm TOF}}}$ is compared to a Monte–Carlo simulation. The model assumes a spherical trap volume with radius ${{r}_{{\rm cap}}}$; molecule velocities are drawn from an isotropic Boltzmann distribution, and the effects of gravity are included. Molecules inside the trap volume are assumed to be recaptured with $100\;\%$ efficiency while those outside are assumed to be lost. MOT LIF images are used to infer the initial spatial distribution of the MOT prior to release, assuming that the MOT is radially symmetric. While the uncertainty in ${{r}_{{\rm cap}}}$ limits the absolute accuracy of extracted temperatures, it does not influence comparative measurements between data sets to detect changes in temperature, provided that the MOT parameters during recapture are fixed. We fix ${{r}_{{\rm cap}}}={{d}_{\lambda }}/2$, where ${{d}_{\lambda }}=23$ mm is the MOT beam diameter, to obtain an upper limit on the isotropic temperature, ${{T}_{{\rm iso}}}$. We find ${{T}_{{\rm iso}}}\lt 12.5\pm 1.5$ mK, in good agreement with the MOT temperature derived from the MOT oscillation measurement.

The MOT temperature is increased by $\sim 5\times$ compared to the SrF MOT formed with the original trapping scheme and described in [15], where ${{T}_{{\rm MOT}}}=2.3(4)$ mK. The damping coefficient measured for the new configuration in this work indicates an increased cooling rate, but evidently this increased cooling is overwhelmed by an increase in the heating rate. (Alternatively, the heating rate may be similar to that in [15], while the increased trap depth allows the MOT to retain hotter molecules that would have quickly escaped from the shallow trap in the previous work.) Increased heating could conceivably be due to the decreased MOT size and hence increased mean laser intensity experienced by a trapped molecule. This change is not reflected in the measured value of ${{R}_{{\rm sc}}}$, so that any increased heating cannot be explained by an increase in spontaneous emission random-walk heating, but there may be additional heating mechanisms that are sensitive to laser intensity or to the particular mix of polarizations and frequencies applied [30]. Unfortunately, no estimate is given in [16] of the temperature or damping force for the new configuration. However, we note that, in [16], the calculated damping coefficient for the original MOT configuration exceeded the measured value in [15] by over an order of magnitude. Thus, it appears that both the heating and cooling rates in these type-II MOTs remain poorly understood.

The MOT lifetime, ${{\tau }_{{\rm MOT}}}$, is determined by measuring MOT LIF in the trapping region as a function of time. Here we measure ${{\tau }_{{\rm MOT}}}=136(2)$ ms. This corresponds to 12 trap periods and is over $2\times$ longer than the trap lifetime measured for the original trapping scheme, where ${{\tau }_{{\rm MOT}}}=56(4)$ ms (figure 5). With the original trapping scheme, ${{\tau }_{{\rm MOT}}}$ was observed to be strongly dependent on the MOT beam diameter (${{d}_{\lambda }}$), and we concluded that the dominant loss mechanism was 'boil-off' due to the low ratio of trap depth to MOT temperature [15]. With the new trapping scheme, we measure ${{\tau }_{{\rm MOT}}}$ to be largely independent of ${{d}_{\lambda }}$, indicating that the trap depth to MOT temperature ratio has increased and is no longer a limiting factor in the trap lifetime.

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An upper limit on the trap depth, $U_{{\rm MOT}}^{{\rm max} }=\frac{1}{2}{{\kappa }_{\rho }}{{({{d}_{\lambda }}/2)}^{2}}$, is estimated assuming that ${{\kappa }_{\rho }}$ is constant to the edges of the MOT beam. This gives $U_{{\rm MOT}}^{{\rm max} }/{{k}_{B}}=190(30)$ mK $\approx 15\;{{T}_{{\rm MOT}}}$. The ratio of trap depth to MOT temperature is thus increased by $\sim 4\times$ compared to [15] but remains in stark contrast to atomic MOTs where ${{U}_{{\rm MOT}}}/{{k}_{B}}\approx {{10}^{3}}{{T}_{{\rm MOT}}}$.

Additional measurements were made to determine whether the observed lifetime was limited primarily by scattering from background gas in our chamber. To do this, the MOT lifetime was measured as a function of background pressure by varying the pumping speed in the trapping region while the He flow rate from the beam source was fixed at 5 standard cubic centimeters per minute (sccm). By extrapolating a linear fit to the total loss rate versus pressure data down to our typical operating pressure of ≈2 × 10⁻⁹ torr, we estimate that the loss rate due to collisions with background He, ${{\Gamma }_{{\rm He}}}=0.9(4)$ s⁻¹, is only a small fraction of the current total loss rate $1/{{\tau }_{{\rm MOT}}}=7.4(1)$ s⁻¹. Hence reducing the background pressure will not significantly improve the MOT lifetime. We note in passing that the MOT lifetime is also not sensitive to ballistic He from the cryogenic buffer gas source: it changes negligibly for increased flow rates up to 20 sccm.

In the absence of the $\mathcal{L}_{00}^{N=3}$ rotational repumping laser, the MOT lifetime was observed to be ${{\tau }_{{\rm MOT}}}=39(1)$ ms, shorter than the lifetime measured using the original trapping scheme [15]. Given the increased trap depth to MOT temperature ratio, and measured independence of ${{\tau }_{{\rm MOT}}}$ on ${{d}_{\lambda }}$, this result was surprising. While searching for the dominant loss mechanism from the new trap, a $60\;\%$ decrease in the ${{\mathcal{L}}_{00}}$ laser intensity was found to increase the MOT lifetime by $\approx 50\;\%$, indicating that the dominant loss mechanism was related to scattering photons. Adding the $\mathcal{L}_{00}^{N=3}$ laser (while at full ${{\mathcal{L}}_{00}}$ intensity) to repump population from the dark ${{X}^{2}}{{\Sigma }^{+}}(v=0,N=3)$ state increased the MOT lifetime by $\sim 4\times$, to its current value.

To understand the rate of pumping into the N=3 state, we considered the effect of off-resonant pumping into this state by the trapping lasers. The ${{A}^{2}}{{\Pi }_{1/2}}(v^{\prime} =0,J=3/2)$ excited state, which decays into the ${{X}^{2}}{{\Sigma }^{+}}(v=0,N=3)$ ground state with $30\;\%$ probability, lies 17 GHz above the excited state of our cycling transition [31]. The measured value of ${{R}_{{\rm sc}}}$ can be used to calculate the rate of off-resonant excitation to the ${{A}^{2}}{{\Pi }_{1/2}}(v^{\prime} =0,J=3/2)$ state, ${{R}_{{\rm sc}}}(\Delta )$. We find ${{R}_{{\rm sc}}}(\Delta =17\;{\rm GHz})=1$ s⁻¹, corresponding to ${{\tau }_{{\rm MOT}}}\approx 3$ s. Clearly this effect cannot explain the observed loss rate. We have also considered hyperfine mixing in the ground and excited states of the cycling transition which can populate the ${{X}^{2}}{{\Sigma }^{+}}(v=0,N=3)$ state. Simple estimates of these effects give branching ratios $\lesssim {{10}^{-7}}$ which are insufficient to explain the loss rate observed.

A plausible explanation for the pumping mechanism is that our trapping laser is contaminated by broadband amplified spontaneous emission (ASE), with non-negligible spectral density at the frequency of the ${{X}^{2}}{{\Sigma }^{+}}(v=0,N=1)\to {{A}^{2}}{{\Pi }_{1/2}}(v^{\prime} =0,J=3/2)$ transition. The type of semiconductor tapered amplifier used to boost the ${{\mathcal{L}}_{00}}$ laser power is known to be susceptible to emitting ASE. Although preliminary measurements using the original trapping scheme showed no evidence of significant loss into N=3, this loss mechanism may have contributed towards limiting the final MOT lifetime reported in our previous work [15]. However, limits on the loss into N=3 from preliminary data indicate that the loss rate was less then than it is now. This is not inconsistent with the proposed explanation, since the level of ASE from a tapered amplifier can change due to aging of the gain medium and/or minor changes in alignment and hence could be larger now than before. While this loss mechanism is not intrinsic to our physical system, it may occur in other experiments that use similar laser technology, so we highlight the issue with the hope of helping others avoid the same problem in the future.

Table 1 compares the MOT parameters we have determined in this work using the new configuration suggested by Tarbutt [16] and in our original observation of a molecular MOT [15].

Table 1.  Comparison of SrF MOT parameters determined using the new and original [15] trapping schemes. * Loss into N = 3 may have contributed a non-negligible fraction of the total loss rate that determined the original MOT lifetime.

Parameter	This work	Original work	Unit
Number of molecules, ${{N}_{{\rm MOT}}}$	500	300
Density, ${{n}_{{\rm MOT}}}$	4000	600	${\rm c}{{{\rm m}}^{-3}}$
Temperature, ${{T}_{{\rm MOT}}}$	13(1)	2.3(4)	mK
Radial trap frequency, ${{\omega }_{\rho }}$	$2\pi \times 76(5)$	$2\pi \times 17.2(6)$	Hz
Damping coefficient, $\alpha /{{m}_{{\rm SrF}}}$	310(30)	140(10)	${{{\rm s}}^{-1}}$
Optimal detuning, Δ	$-2\pi \times 10$	$-2\pi \times 8$	MHz
Optimal gradient, ${\rm d}B/{\rm d}z$	9	15	${\rm G}\;{\rm c}{{{\rm m}}^{-1}}$
Gradients with observed MOT	4–40	4–30	${\rm G}\;{\rm c}{{{\rm m}}^{-1}}$
$U_{{\rm MOT}}^{{\rm max} }/{{k}_{B}}T$	15	4
Lifetime, ${{\tau }_{{\rm MOT}}}$	136(2)	$56{{(4)}^{*}}$	ms
Scattering rate, ${{R}_{{\rm sc}}}$ (×106)	$3.8_{-1.2}^{+2.2}$	$4.3_{-2.2}^{+4.1}$	${{{\rm s}}^{-1}}$
Radial width, ${{\rho }_{{\rm rms}}}$	1.9(1)	4.1(1)	mm
Axial width, ${{z}_{{\rm rms}}}$	2.1(1)	2.6(1)	mm
MOT beam full diameter, ${{d}_{\lambda }}$	23	23	mm
MOT beam $1/{{e}^{2}}$ diameter	14	14	mm