Magneto-Optical Trap for Polar Molecules
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec A Magneto-Optical Trap for Polar Molecules
Benjamin K. Stuhl, ∗ Brian C. Sawyer, Dajun Wang, and Jun Ye
JILA, National Institute of Standards and Technology and the University of ColoradoDepartment of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA (Dated: December 12, 2008)We propose a method for laser cooling and trapping a substantial class of polar molecules, and in particulartitanium (II) oxide (TiO). This method uses pulsed electric fields to nonadiabatically remix the ground-statemagnetic sublevels of the molecule, allowing us to build a magneto-optical trap (MOT) based on a quasi-cycling J ′ = J ′′ − m K and trapping it with a radiation-pumping-limited lifetimeon the order of 80 ms.
PACS numbers: 37.10.Pq, 37.10.Mn, 37.10.Vz
The field of ultracold polar molecules has recently madegreat strides. Coherent optical transfer of magneto-associatedmolecules can now produce ultracold molecular gases in theX S ( v =
0) ground state with densities of 10 cm − andtranslational temperatures of 350 nK [1]. Incoherent photo-association techniques can reach the X S ( v =
0) ground stateat 100 m K [2]. With these temperatures and the reasonablylarge electric dipole moments available from heteronuclearbialkali molecules (e. g. 0.76 D for X S ( v =
0) KRb [3]),progress towards quantum simulations of condensed mattersystems [4, 5] and quantum computation [6, 7] should berapid. In fields such as ultracold chemistry [8], access tomolecular species beyond the bialkali family is of great in-terest. Arbitrary species can be cooled to the kelvin regimethrough buffer-gas cooling [9, 10], while Stark deceleration[11, 12] reaches the tens of millikelvin level for selected lightmolecules. Unfortunately, there is no demonstrated tech-nique to further compress and cool the lukewarm molecularclouds resulting from the latter two techniques. Even cavity-mediated schemes for molecular laser cooling [13, 14, 15, 16],while in the abstract highly attractive methods for coolinga broad, chemically interesting set of molecules, have sofar been unable to cool these lukewarm samples, due to theschemes’ low scattering rates [15], small cavity mode vol-umes [16], and requirement of multiparticle collective effects[13, 14].Direct, free-space laser cooling and trapping would be theideal method for producing ultracold molecules, just as it isfor atoms. Unfortunately, atoms are in general much easier tolaser cool than molecules, due to the latter’s glaring lack of cy-cling transitions. Laser cooling generally requires electronictransitions, as vibrational and rotational transitions have im-practically long excited state lifetimes unless a cavity is used[16]. Unfortunately, these “electronic” transitions are neverpurely electronic. Rather, they are rovibronic, and decay intovarious rotational, vibrational, or hyperfine excited states, aswell as the original ground state [17].The branching ratios of these rovibronic decays, however,are governed by the molecular structure and the dipole selec-tion rules. This implies that a clever choice of molecule cangreatly reduce the number of possible decays. Decays into
FIG. 1: (color online, not to scale). (a) The electronic level struc-ture of TiO and the transitions of interest for laser cooling. TheX D ground state is split by the spin-orbit interaction into the threeX D − sublevels, of which the X D level is the lowest. Each sub-level contains a vibrational ladder, while each vibrational level con-tains a ladder of rotationally excited states (not shown). Ti Ohas zero nuclear spin and thus there is no hyperfine structure. Theground-state L doublet (not shown) is much less than the naturallinewidth of the E P ← X D transition. The solid arrow denotes the v ′ = ← v ′′ = v ′ = ← v ′′ = v ′ = ← v ′′ = q [19] next to each decay. (b) Therotational and L − doublet structure of the E P electronic excitedstate. The states are interleaved, as the rotational splitting is smallerthan the L − doublet splitting; a and b denote the parity states. Boththe cooling and repump lasers address the J ′ = , a state. excited hyperfine states are impossible in molecules with zeronuclear spin, as these molecules have no hyperfine structure.The Franck-Condon ratios for decay back to the ground vibra-tional level can be quite good (99+% for selected molecules[18]). However, the only constraint on decays to rotationallyexcited levels is that all decays satisfy the total angular mo-mentum selection rule D J = , ± J ′′ is greaterthan the excited state angular momentum J ′ , two of these threedecays are forbidden. In this case, the molecule must fol-low the angular momentum cycle J ′′ → J ′ = J ′′ − → J ′′ , andso only one laser is required per relevant vibrational level —making laser cooling of these molecules truly practical.Thus, by combining these various transition closure crite-ria, we can identify a class of molecules that are exceptionallygood candidates for laser cooling: they have no net nuclearspin, good Franck-Condon overlaps, and their ground or low-est metastable state has a higher angular momentum than thefirst accessible electronically excited state. For non-singletmolecules, the excited electronic level must also not be a S state, as the lack of spin-orbit splitting in S states means thatthe excited state can decay across the spin-orbit ladder.We have identified a number of molecules that satisfy allof the above requirements. TiO and TiS are both satisfactoryin their absolute ground states. Metastable FeC, ZrO, HfO,ThO, SeO, and the like are promising [20]. We expect thatsome other, as-yet uncharacterized metal oxides, sulfides, andcarbides should also have the necessary electronic structure. Ifone is willing to accept some hyperfine structure as the priceof chemical diversity, some metal hydrides and metal halidesmay provide additional suitable candidates.Of these candidates, we chose to focus on TiO, due to itsviability in its absolute ground state and the breadth of spec-troscopy and theory available in the literature [19, 21, 22, 23,24, 25, 26, 27, 28]. A simplified level structure of TiO isshown in Fig. 1. The lowest ground state of TiO is the X D ,with spin-orbit constant A ( X D ) = .
61 cm − and rotationalconstant B ( X D ) = .
534 cm − [28]. The lowest excited state isthe E P with A ( E P ) = .
82 cm − and B ( E P ) = .
515 cm − [23]. (While the d S + and a D level are energetically belowthe E P level, the inter-system branching ratio is expected tobe very small.) As Fig. 1 shows, the Franck-Condon fac-tors [19, 21] for the E P − X D band are quite favorable,yielding a population leak of 3 × − scatter − (or a meanof ∼ I sat ), and Franck-Condon factors for the cooling and repumplines are summarized in Table I. Note that these transitions areall accessible with diode lasers. The saturation intensities areextremely low, as the natural linewidth g of the E P − X D transition is on the order of 2 p × J ′ = J ′′ + J ′ > J ′′ means that the atom can alwaysscatter from the correct beam, as shown in Fig. 2(a). Thestandard MOT will therefore not work for molecules usingthe aforementioned J ′ = J ′′ − TABLE I: The wavelengths, Franck-Condon factors, and saturationintensities of the cooling and repump transitions of TiO. n ′′ l , n ′′ [ nm ] Franck-Condonfactor q n ′′ a estimated I sat [ m W / cm ] b c c a from [19], except for the second value of q b estimated for a two-level system with g = p × / q v ′′ c calculated using the diatomic molecular constants of [22] a magnetic field can break the degeneracy of the ground-state magnetic sublevels and thus provide beam selectivity, the | m J ′′ | = J ′′ stretched states are effectively dark states, as theycan only interact with one of the laser beams, not both [31].What is needed, then, is a way to continually remix theground-state sublevels so that all the molecules spend somefraction of their time in bright states. Fortunately, polarmolecules provide just the handle needed to accomplish this:the effective magnetic (B) and electric (E) moments of a polarmolecule depend in different ways on m J . Thus, applying asudden (i.e. nonadiabatic) electric field orthogonal (or at leastnonparallel) to the local magnetic field reprojects the total an-gular momentum against a new axis, randomizing m J (andthe L − doublet state) by coupling the two L − doublet mani-folds together. At high remix rates and high laser saturations,the molecules’ time is equally divided across the 2 ( J ′′ + ) ground and 2 J ′ + L doublet), butthey can only decay while they are in an excited state. Thus,while the molecules are effectively always bright, the maxi-mum photon scattering rate is only J ′ + ( J ′′ + )+( J ′ + ) g . Suchremixing of the ground-state magnetic sublevels allows thebuilding of a new kind of trap, the electrostatically remixedmagneto-optical trap (ER-MOT). The ER-MOT operation isshown in Fig. 2(b). Note that, as the local direction of thequadrupole B-field spans all of 4 p steradians over the MOTvolume, a single E-field pulse will be parallel to the local B-field in some region and therefore ineffective at remixing the m J ′ s there. This hole can, however, easily be closed by ap-plying a second E-field pulse, nonparallel to the first. A basicER-MOT design is shown in Fig. 2(c).To build an ER-MOT with TiO, there is a minor technicalcomplication. To leading order, the molecular magnetic mo-ment can be written as m = m B m J ( g L L + g S S ) W J ( J + ) Since g L ≈ g S ≈
2, the magnetic moment of the X D ( L = S = −
1) state is small, likely on the order of am B , or thefine-structure constant times the Bohr magneton. In contrast,while W = P state, the large L − doublet splittingindicates strong mixing with higher electronic excited states,and so by analogy with the B P optical Zeeman measure-ments of [32], we estimate the magnetic moment to be ∼ D . This, combined with the narrownessof the E P ↔ X D transition, implies that the dynamics of a FIG. 2: (color online). (a) The level structure of a traditional MOT.The local magnetic field strength and orientation combined withthe Doppler shift enhance the scattering from the laser beam thatprovides the damping and restoring forces and suppresses scatter-ing from the counter-propagating beam. Since J ′ > J ′′ , the groundstate(s) are always able to scatter from every beam. (b) The levelstructure of the ER-MOT. The local magnetic field still governswhich laser is preferentially scattered, but angular momentum con-servation forbids some ground states from interacting with the pre-ferred beam. To overcome this, the ground-state magnetic sublevelpopulations are remixed by pulsed electric fields, as represented bythe dashed lines. (c) A sample ER-MOT design. A pair of elec-tromagnet coils are aligned in anti-Helmholtz fashion to produce aquadrupole field. Six beams of the cooling laser are converged onthe center with their polarizations oriented as usual for a MOT, but aset of four open-mesh grids are added. The grids are pulsed in pairs(e. g., first the X-axis pair and then the Y-axis pair) to produce thedipole electric fields needed to remix the magnetic sublevels. Thecenter is also illuminated by the repump lasers (not shown). TiO ER-MOT will have more in common with narrow-linealkaline-earth MOTs [33] than normal alkali metal MOTs.Given these predicted magnetic moments, the magnetic gradi-ent in a TiO ER-MOT must be .
100 G / cm. This gradient canbe easily achieved with water-cooled electromagnets [12] orrare-earth permanent magnets [34]. In contrast, the large ( ≈ L − doublet spacing [26] mean that electricfields of only 1 V / cm will give Stark shifts of about 50 g —far more than the Zeeman shift within the ER-MOT and thussufficient to reproject m J . These small fields can easily beswitched with rise times on the order of 10 ns (a frequency of2800 g and 56 times the Larmor frequency due to the electricfield), and thus nonadiabaticity is assured.A final concern regarding the viability of the TiO ER-MOTis that either the electrostatic or magnetic fields might some-how cause population loss by mixing in rotationally excited J ′ > J ′′ > ∼ − rotational splitting [Fig. 1(b)]. Neither the Zeeman or Stark FIG. 3: (color online). Number (upper, black) and temperature(lower, red) time-of-flight plots for the loading of a molecular packetinto a simulated TiO ER-MOT. The initial spike on the number plot isthe molecular packet flying through the ER-MOT volume; the broadhump is the actual captured molecules. The decay of the moleculenumber is due to radiation pumping of the captured population intoexcited v ′′ ≥ ± shifts within the ER-MOT are anticipated to be larger than ∼
100 MHz, and so the perturbative probability to leave thedesired J ′ = (cid:0)
100 MHz30 GHz (cid:1) ≈ − . This is muchsmaller than the 3 × − scatter − loss rate from decays to v ′′ ≥ g = p ×
32 kHz, a magnetic dipole moment of am B , and anelectric dipole moment of 3 D. We used a 1 / e laser waistdiameter of 6 cm. Our code treated photon scattering andmolecule kinematics semiclassically and approximated theelectrostatic remixing as a sudden reprojection of the diag-onalized Zeeman Hamiltonian wavefunction against a newStark + Zeeman Hamiltonian. In addition, the simulationsused a set of 60 additional red-detuned frequency components(4 . g spacing, 7 . − . ± . / scentered in orientation around ˆx, with an opening half-angleof 0.3 rad. The magnetic field gradient was 51 G / cm and theelectric field was pulsed to 4 V / cm for 100 ns at a rate of50 kHz. The cooling laser was detuned by 3 . g to the red,with a saturation parameter of s cool = . s repump = . m K. The ER-MOTlifetime was limited to about 80 ms by radiation pumping intovibrationally-excited dark states. The estimated capture ve-locity was 5 . / s, which when taking the rotational distri- FIG. 4: Fractional capture vs electrostatic remix rate after 160 ms ofsimulation time for the TiO ER-MOT. Error bars represent statisticsover multiple simulation runs. The curve is only a guide to the eye. bution into account allows the capture of about 0.02% of a4.2 K thermal distribution. If one assumes a 4.2 K sourceflux of 10 s − [30], the calculated capture velocity, lifetime,and ER-MOT radius predict an ER-MOT number of 10 anda density of 10 cm − .In addition, we studied the importance of the electrostaticremixing to the ER-MOT operation. Figure 4 plots themolecule number after 160 ms against the electrostatic remixfrequency G remix . The plot clearly shows the importance ofthe electrostatic remixing. At G remix ≪ g p , no molecules arecaptured, since the molecules are optically pumped out of thebright state much faster than they are remixed back into it.For G remix . g p the capture efficiency rises with increasing G remix , and then the efficiency saturates around G remix = g p ,as the molecules become evenly divided among the variousground sublevels. As additional validation checks on the sim-ulation code, we verified that turning off the repump lasersdoes indeed inhibit the formation of an ER-MOT by pumpingthe entire population into vibrationally excited states. We alsoverified that we could reproduce the experimental Yb MOTof [29] by modifying the code to simulate a J ′ = J ′′ + J ′ = J ′′ − ∗ Electronic address: [email protected][1] K.-K. Ni et al. , Science , 231 (2008).[2] J. M. Sage et al. , Phys. Rev. Lett. , 203001 (2005).[3] S. Kotochigova et al. , Phys. Rev. A , 022501 (2003).[4] K. Gòral et al. , Phys. Rev. Lett. , 170406 (2002).[5] A. Micheli et al. , Nature Physics , 341 (2006).[6] D. DeMille, Phys. Rev. Lett. , 067901 (2002).[7] C. Lee and E. A. Ostrovskaya, Phys. Rev. A , 062321 (2005).[8] E. R. Hudson et al. , Phys. Rev. A , 063404 (2006). R. V.Krems, Phys. Chem. Chem. Phys. , 4079 (2008).[9] J. D. Weinstein et al. , Nature , 148 (1998).[10] W. C. Campbell et al. , Phys. Rev. Lett. , 213001 (2007).[11] H. L. Bethlem and G. Meijer, Int. Rev. Phys. Chem. , 73(2003).[12] B. C. Sawyer et al. , Phys. Rev. Lett. , 253002 (2007).[13] B. L. Lev et al. , Phys. Rev. A , 023402 (2008).[14] P. Domokos and H. Ritsch, Phys. Rev. Lett. , 253003 (2002).[15] G. Morigi et al. , Phys. Rev. Lett. , 073001 (2007).[16] A. André et al. , Nature Physics , 636 (2006).[17] J. T. Bahns et al. , J. Chem. Phys. , 9689 (1996).[18] M. D. D. Rosa, Eur. Phys. J. D , 395 (2004).[19] I. M. Hedgecock et al. , Astron. Astrophys. , 667 (1995).[20] D. DeMille, private communication (2008).[21] N. V. Dobrodey, Astron. Astrophys. , 642 (2001).[22] T. Gustavsson et al. , J. Mol. Spectrosc. , 56 (1991).[23] K. Kobayashi et al. , J. Mol. Spectrosc. , 133 (2002).[24] S. R. Langhoff, Astrophys. J. , 1007 (1997).[25] C. Lundevall, J. Mol. Spectrosc. , 93 (1998).[26] K. Namiki et al. , J. Mol. Spectrosc. , 176 (1998).[27] T. C. Steimle and W. Virgo, Chem. Phys. Lett. , 30 (2003).[28] C. Amiot et al. , J. Chem. Phys. , 4375 (1995).[29] T. Kuwamoto et al. , Phys. Rev. A. , R745 (1999).[30] D. Egorov et al. , Phys. Rev. A , 043401 (2002).[31] H. J. Metcalf and P. van der Straten, Laser Cooling and Trap-ping (Springer, New York, 2001).[32] W. Virgo and T. C. Steimle, Astrophys. J. , 567 (2005).[33] T. Mukaiyama et al. , Phys. Rev. Lett. , 113002 (2003).[34] B. C. Sawyer et al. , Phys. Rev. Lett.101