Magnetic trapping and coherent control of laser-cooled molecules

We demonstrate coherent microwave control of the rotational, hyperfine and Zeeman states of ultracold CaF molecules, and the magnetic trapping of these molecules in a single, selectable quantum state. We trap about $5\times 10^{3}$ molecules for 2 s at a temperature of 65(11) $\mu$K and a density of $1.2 \times 10^{5}$ cm$^{-3}$. We measure the state-specific loss rate due to collisions with background helium.

tional transition. A pulse of CaF emitted at time t = 0 from a cryogenic buffer gas source [33] is slowed down by frequency-chirped counter-propagating laser light [34]. The slowest molecules are captured in a MOT [25] where the main laser drives the A 2 Π 1/2 (v = 0, J = 1/2) ← X 2 Σ + (v = 0, N = 1) transition, with an intensity of I and a detuning of δ. The linewidth of this transition is Γ = 2π × 8.3 MHz. Three additional lasers repump population that leaks to the v = 1, 2 and 3 vibrational levels of the X state. The MOT and magnetic trap share the same in-vacuum coils [25], which produce an axial field gradient B . Using I = I max = 400 mW cm −2 , δ = −0.75Γ and B = 30 G cm −1 , we routinely capture 2 × 10 4 molecules with a peak density of n = 6 × 10 5 cm −3 and a temperature of T = 11 mK. Figure 1(a) shows the energy levels most relevant to the present work, which we label |N, F, M F . The ground rotational state, N = 0, is split into two hyperfine components, F = 0 and 1, while the first excited rotational state, N = 1, is split by spin-rotation and hyperfine interactions into four components with F ∈ {1, 0, 1, 2}. To address these, the MOT laser is tuned near F = 0, the sideband of a 48 MHz acousto-optic modulator (AOM) addresses the upper F = 1 level and the sidebands of an electro-optic modulator (EOM) address the F = 2 and lower F = 1 levels. The light from the AOM and EOM have opposite circular polarizations, implementing a dual-frequency MOT [35]. Since our previous work, we have changed the EOM frequency from 74.5 MHz to 70.5 MHz. This has increased the density by a factor of 4, mainly by increasing the MOT spring constant. Figure 1(b) illustrates the new control steps we implement and presents their timings. Each sequence begins with a fluorescence image taken at t = 40 ms, used to determine the number of molecules, N mol , in each MOT. We first compress the MOT by increasing B linearly between t = 40 and 50 ms, and holding the higher B until t = 55 ms. Figure 2 shows n as a function of B in the compressed MOT (cMOT). Increasing B to 113 G cm −1 , increases n to 3.4 × 10 6 cm −3 , a factor of 5.3 greater than in the standard MOT. If N mol and T are conserved in the compression, we expect n ∝ (B ) 3/2 , resulting in a factor of 7.3 increase in density. We find that N mol is conserved, but that T increases, which explains the smaller observed factor. For all subsequent data, we use B = 69 G cm −1 in the cMOT, giving n ≈ 2 × 10 6 cm −3 .  Following the cMOT, we lower the temperature using a procedure similar to the one described previously [23], and illustrated in Fig. 1(b). Between t = 55 and 59 ms, I is ramped down to 0.1I max , where it is held until the MOT coils and laser are switched off at t = 63 ms. At transitions can all be driven. Bias coils cancel the background magnetic field and apply a constant, uniform field of ∼ 60 mG, sufficient to resolve the Zeeman sub-levels, but small enough not to disrupt the molasses cooling. Figure 3(a) shows the depletion of the N = 1 population as a function of the microwave angular frequency, ω, as it is scanned through the magnetically-insensitive |1, 0, 0 → |0, 1, 0 transition at ω 0 . The microwave pulse has a duration of τ µ1 = 140 µs, and the Rabi frequency is Ω = π/τ µ1 . Molecules transferred to N = 0 are decoupled from the MOT light. Thus, we measure the number of molecules remaining in N = 1 by turning the MOT back on and imaging the fluorescence after a time τ hold , typically 30 ms. This number, divided by the initial number in the MOT, is the fraction recaptured. The line in Fig. 3(a) is a fit to the data using the model y 0 + Af (Ω, ω − ω 0 , τ µ1 ), where y 0 is the fraction re-captured without the microwave pulse, A is an amplitude, and f is the usual Rabi lineshape for a two level system. We fix τ µ1 and Ω, leaving y 0 , A and ω 0 as free parameters. The fit is a good one, and gives y 0 = 0.57, consistent with the MOT lifetime, and A = −0.32. The microwave transfer efficiency is MW = A/(y 0 OP ) = 94 %. We infer that, in the relevant polarization, the microwave intensity at the molecules is 64 nW cm −2 . Figure 3(b) shows similar data for the magnetically-sensitive transition |1, 0, 0 → |0, 1, 1 . We drive a π-pulse with a shorter duration of τ µ1 = 40 µs in order to reduce the effects of magnetic field inhomogeneities and fluctuations. Their effects are still visible in the data, producing a slight broadening relative to the model, a poorer fit in the wings, and a lower efficiency of MW = 87 %. The inferred intensity in the relevant polarization is 780 nW cm −2 .
Figure 3(c) shows Rabi oscillations on the magnetically-sensitive |1, 0, 0 → |0, 1, 1 transition. We measure the percentage recaptured versus τ µ1 , with the microwave frequency on resonance and the microwave power held constant. To model these data we found it necessary to include two imperfections. The first relates to the microwave synthesizer, which we discovered has a transient frequency drift when switched. This frequency change is well modeled by ω(t ) = ω ∞ − ∆ω e −t /τ where ω ∞ /(2π) is the frequency at long times, t is the time since the start of the pulse, ∆ω/(2π) ≈ 7 kHz is the total frequency change and τ ≈ 105 µs is the timescale. This has no observable effect on the lineshapes in Fig. 3(a,b), but causes a slight frequency shift in the line centre, a noticeable chirp in the frequency of the Rabi oscillations and a slight reduction in their contrast. The second imperfection is due to gradients of intensity and polarization produced by the standing wave component of the microwave field, and is the main reason for the gradual reduction in the contrast of the Rabi oscillations with increasing τ µ1 . To model these effects, we first solve the two-level optical Bloch equations with the measured drift in frequency included. This gives a function y 0 + Ag(Ω, ω ∞ − ω 0 , τ µ1 ). We average this over a Gaussian distribution of Rabi frequencies with a width of ∆Ω. The solid line in Fig. 3(c) is a fit to this model, with y 0 , A, ω ∞ − ω 0 , Ω and ∆Ω as free parameters. We find ∆Ω/Ω = 0.16, which is reasonable since the distance from node to antinode of the standing wave component is 3.5 mm, comparable to the size of the molecule cloud.
With most molecules now transferred to N = 0, we push those remaining in N = 1 out of the trap region by turning on the slowing light for 1 ms. This leaves a pure sample of molecules in a single state. We then either turn on the magnetic trap or apply a second microwave pulse, of duration τ µ2 , to transfer back to a selected sublevel of N = 1. Figure 3(d) shows Rabi oscillations on the |0, 1, 1 → |1, 2, 2 transition as τ µ2 is varied. The percentage recaptured is zero when τ µ2 = 0, showing that we indeed have a pure sample. The line is a fit using the same model described above and is seen to fit well. A πpulse takes 100 µs, implying a microwave intensity in the appropriate polarization of 42 nW cm −2 . The efficiency of this second π-pulse is 75 %.
With these procedures, we can trap molecules in any of the weak-field seeking states shown in Fig. 1(a). Here, we demonstrate trapping in the |0, 1, 1 and |1, 1, 2 states. We turn the magnetic trap on with B = 30 G cm −1 after either the first or second microwave pulse, then detect the number remaining after a time τ mag by turning the MOT light on and imaging the fluorescence. When trapping molecules in N = 0, which do not fluoresce in the MOT light, we transfer back to N = 1 using the |0, 1, 1 → |1, 2, 2 transition prior to detection. Conveniently, this transition is magneticallyinsensitive so can be driven while the molecules are magnetically trapped. Indeed, we observe Rabi oscillations on this transition, similar to those shown in Fig. 3(d), even when the molecules are trapped. The number of molecules in the trap fits well to a single-exponential decay, N mol (τ mag ) = N mol (0) exp(−R loss τ mag ), which we attribute mainly to collisions with helium gas from the buffer-gas source. Figure 4 shows the loss rate, R loss , as a function of the helium flow rate, for molecules in each of the two states, showing a linear dependence in both cases. The gradients are 2.03(8) and 2.42(16) s −1 sccm −1 for the |0, 1, 1 and |1, 1, 2 states respectively, differing by 2.2 σ. Extrapolating to zero flow, the loss rates for the two states are 0.30(3) and 0.17(6) s −1 , differing by 1.9 σ. These rates are close to the loss rate due to vibrational excitation by room temperature blackbody radiation [36], which is 0.22 s −1 for all the states shown in Fig. 1(a).
To investigate whether the molecules are heated in the magnetic trap, we have measured their temperature prior to trapping and for various τ mag . Before turning on the trap, the radial, axial, and geometric mean temperatures are rate in the trap is consistent with zero and has an upper limit of 37 µK s −1 . We load about 5 × 10 3 molecules into the trap, and the cloud has radial and axial rms radii of σ ρ = 1.37(1) mm and σ z = 1.44(2) mm. The density is 1.2 × 10 5 cm −3 , and could be increased by increasing B . The phase-space density is 2.6 × 10 −12 .
In summary, we have compressed our MOT to increase the density of molecules, demonstrated coherent control of their rotational, hyperfine and magnetic states, and transferred them to a conservative trap. The entire sequence takes only 75 ms. We have measured the trap loss rate for two selected states, which is a prototype measurement for future experiments studying state-selective elastic and inelastic collisions between co-trapped ultracold atoms and molecules [37]. Our demonstrations of long lifetimes and low heating rates in a magnetic trap are important for reaching lower temperatures by sympathetic cooling [38,39]. The magnetically trapped molecules can now easily be transported to experiments more conveniently located away from the MOT region [40]. The quantum state control we demonstrate is required for measurements with ultracold molecules that test fundamental physics [41][42][43][44]. With sufficient control, quantum information can be stored within hyperfine states and the dipole-dipole interaction turned on when needed for information processing [6]. Controlled microwaveinduced dipoles are crucial for simulating spin Hamiltonians [7,10,45,46], studying topological superfluids [47] and enhancing evaporative cooling [48]. For a realistic spacing of 0.5 µm, the dipole-dipole interaction energy is a few kHz, comparable to the resolution we achieve. Thus, our work demonstrates many of the key capabilities needed for the applications of laser-cooled molecules.
We are grateful to J. Dyne, G. Marinaro and V. Gerulis for technical assistance. The research has received fund-