Dark-state sideband cooling in an atomic ensemble
DDark-state sideband cooling in an atomic ensemble
Chang Huang, Shijie Chai, and Shau-Yu Lan ∗ Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore (Dated: January 13, 2021)We utilize the dark-state in a Λ-type three-level system to cool an ensemble of Rb atoms in anoptical lattice [Morigi et al ., Phys. Rev. Lett. , 4458 (2000)]. The common suppression of thecarrier transition of atoms with different vibrational frequencies allows them to reach a sub-recoiltemperature of 100 nK after being released from the optical lattice. A nearly zero vibrational quan-tum number is determined from the time-of-flight measurements and adiabatic expansion process.The features of sideband cooling are examined in various parameter spaces. Our results show thatdark-state sideband cooling is a simple and compelling method for preparing a large ensemble ofatoms into their vibrational ground state of a harmonic potential and can be generalized to differentspecies of atoms and molecules for studying ultra-cold physics that demands recoil temperature andbelow. Cooling of atoms is indispensable in many quantumscience experiments [3]. To achieve an extremely lowtemperature of atoms, various stages of cooling mecha-nisms have to be employed. Doppler and sub-Dopplercooling through lasers are conventionally used to removethe kinetic energy of atoms from room temperature downto the few hundreds or tens of micro-kelvin range. Toreach tens of nano-kelvin or an even lower tempera-ture, evaporative cooling [4] and delta-kick cooling [5] arethe workhorses whose cooling efficiencies, however, arelargely affected by the initial temperature. To bridge thegap, free-space Raman cooling [6], Raman sideband cool-ing [7, 8], or degenerate Raman sideband cooling [9, 10]can be deployed to prepare atoms in the few hundreds ofnano-kelvin range. Nevertheless, these methods requiretuning the laser frequencies or external magnetic fields tomatch the vibrational levels to minimize the heating pro-cess. For a sizable atomic ensemble in a harmonic poten-tial with a distribution of vibrational frequencies over dif-ferent lattice sites, simultaneously attaining a high cool-ing efficiency and short cooling time is challenging.Alternatively, dark-state sideband cooling offers awider cooling bandwidth by tailoring the cooling laserabsorption spectrum through quantum interference. Thedark-state has been utilized in cooling atoms to sub-recoiltemperatures in terms of velocity selective coherent pop-ulation trapping [11]. However, it is restricted to atomsthat have F = 1 and F (cid:48) = 1 energy levels. Combinedwith polarization gradient cooling (PGC) to form graymolasses cooling, the dark-state has been used to coolthe temperature of atoms down to the few micro-kelvinrange. For trapped atoms, the dark-state has been ap-plied in cooling a chain of trapped ions with differentvibrational frequencies to their vibrational ground statewithin 1 ms [1, 2, 12–18]. For an ensemble of neutralatoms in an optical lattice [19–21], cooling to the groundstate as well as sub-recoil temperature using the dark-state has not been demonstrated. Here, we realize dark- ∗ [email protected] state cooling of an atomic ensemble in an optical latticeto the vibrational ground state. With the aid of adia-batic cooling, we have observed a sub-recoil temperatureof atoms at 100 nK, or 0.27 T r , after releasing them fromthe optical lattice, where T r is the recoil temperature. (a) (b) … F =2 n =0 n =1 n =2 … … F =3 n =0 n =1 n =2 … F ′=3 Δ MOT PGC
TOF lattice control probe3.53.8 t (ms) probe lattice control FIG. 1. Experimental details. (a) Configuration of the opticallattice and cooling beams. All the laser beams are aligned inthe horizonal plane. The white circles indicate the polariza-tion of the laser beams perpendicular to the x - y plane. Thedouble arrows represent the polarization of the laser beamslying on the x - y plane. The lattice potential is in units of u . The x and y coordinates are in units of k − . (b) Rele-vant energy levels. The control beam modifies the absorptionspectrum of the probe beam such that the cooling transi-tions ( n to n (cid:48) < n ) dominate the heating transitions ( n to n (cid:48) ≥ n ) when the frequencies of the two beams are adjustedto the two-photon resonance condition. (c) Timing sequencefor dark-state sideband cooling (not to scale). The conventional sideband cooling depends upon thecompetition between the absorption of the cooling beamfrom the vibrational level n to n (cid:48) < n (the cooling tran-sitions or red sidebands) and n to n (cid:48) ≥ n (blue sidebands a r X i v : . [ phy s i c s . a t o m - ph ] J a n and carrier transitions), where n is the vibrational quan-tum number. This is usually done by tuning the coolingbeam frequency resonant on the n to n − n to n (cid:48) = n carrier transi-tions, which limits the cooling duration and efficiency.In dark-state cooling, two cooling beams, which wedenote as the control and probe beams here, couple twonarrow linewidth ground states to an excited state sepa-rately. The control beam tailors the absorption spectrumof the probe beam such that the n to n (cid:48) ≥ n transi-tions are greatly suppressed, while the transitions from n to n (cid:48) < n remain allowed. In the dressed-state pic-ture, the three-level system and two cooling lasers formthe condition of coherent population trapping (CPT)[22].The eigenstates of CPT consist of one dark-state and twobright-states. The dark-state corresponds to the carriertransitions, and the energy shift of one of the bright statesfrom the dark-state can be adjusted to match the redsideband transitions which are shifted from the carriertransitions by [2] δ = 12 ( (cid:113) ∆ + Ω + Ω − | ∆ | ) , (1)where ∆ is the single-photon detuning of the control andprobe beams from the excited state, Ω p is the probe beamRabi frequency, and Ω c is the control beam Rabi fre-quency. When δ is tuned to the vibrational frequency f of the harmonic potential by the Rabi frequencies ofthe control and probe beams, the cooling process takesplace. For a large ensemble in an optical lattice withdifferent trapping frequencies, the mode profiles of thecooling beams can, in principle, be tailored to match δ with f , while the carrier transitions are strongly sup-pressed for all f . The cooling cycle is completed whenatoms decay back to the ground states in the Lamb-Dickeregime, which preserves the vibrational quantum num-ber. In dark-state cooling, both ground states are in-volved in the cooling process, while in conventional Ra-man sideband cooling, only one ground state is used forthe cooling.In our experiment, the optical lattice is formed by threeco-planar free-running laser beams separated by 120 ◦ ,and the polarization of the lattice beams is perpendicularto the lattice plane, as shown in Fig. 1(a). The latticepotential can be written as U = − u (cid:20)
32 + cos( √ kx ) + cos (cid:32) √ kx − ky (cid:33) + cos (cid:32) √ kx + 3 ky (cid:33) (cid:21) , (2)where u is the single beam potential, k is the wavenum-ber of the light, and x and y are the spatial coordinates. OD E O M f r e q u e n c y ( M H z ) ( a ) n t o n ' > n n t o n ' < n OD E O M f r e q u e n c y ( M H z ) ( b ) n t o n ' = n OD F r e q u e n c y ( M H z )
FIG. 2. (a) Measurements of the probe beam absorption asa function of the probe beam frequency through tuning theEOM frequency. The absorption is characterized by the op-tical depth (OD). The control beam is fixed at +20 MHzdetuned from the F (cid:48) =3 state. The inset shows the Zeemanspectrum of the F =2 to the F =3 states after minimizing theambient magnetic field. The absorption peaks from differentZeeman states collapse into a single peak. The curve is a Voigtfunction fitted to the experimental data with a full width athalf maximum (FWHM) of 25 kHz. (b) Measurements of theprobe beam absorption spectrum as a function of the probebeam EOM frequency from 2835 to 2842 MHz in (a) with finerresolution. The dashed line indicates the suppressed carriertransitions. To the right and left sides of the dashed lineare red sideband and blue sideband transitions, respectively.Each data point is an average of five experimental runs, andthe bars represent the standard errors. When the trap depth is large compared to the recoil en-ergy, the potential can be approximated as a harmonicpotential around x =0 and y =0 as U ≈ − u [ − k ( x + y )] , (3)with oscillation frequency f = π (cid:113) uk m , where m is themass of the particle. Each lattice beam has a 1/ e waistof 3.5 mm with 90-mW power. The frequency of the lat-tice beams is 12 GHz red-detuned from the D line, andthe corresponding peak vibrational frequency is f = 70kHz [23]. The control and probe beams are generatedfrom the same extended cavity diode laser, whose fre-quency is locked to the crossover peak between the F = 3to F (cid:48) = 2 and F = 3 to F (cid:48) = 3 transitions. The controlbeam passes through a double-pass acoustic-optical mod-ulator (AOM) for switching and frequency-shifting of ap-proximately +200 MHz. The probe beam passes throughanother double-pass AOM at approximately +400 MHz,followed by an electro-optical modulator (EOM). Thepositive first sideband of the EOM transmits through apassive optical cavity for the experiment. The frequencyof the EOM governs the detuning of the probe beam.The probe beam overlaps with one of the latticebeams and has a 1/ e waist of 2 mm, and the con-trol beam with a 1/ e waist of 3 mm is aligned at15 ◦ from the probe beam and retro-reflected to bal-ance the radiation pressure. The polarizations of theprobe and control beams are linear and perpendicular toeach other. The Lamb-Dicke parameter along the y -axis η y =( k c cos 45 ◦ + k p cos 60 ◦ ) (cid:113) ¯ h πmf is 0.23, where k c and k p are the wavenumbers of the control and probe beamsand cos 45 ◦ and cos 60 ◦ account for the angles betweenthe cooling beams and the cooling axis y .An ensemble of cold Rb atoms at 20 µ K is preparedby switching on the magneto-optical trap (MOT) for 1s succeeded by PGC on the D line. The three-level Λ-type system consists of hyperfine ground states F = 2and F = 3, and the excited state F (cid:48) = 3 of the D line, as shown in Fig. 1(b). The sequence of the dark-state sideband cooling is shown in Fig. 1(c). The opticallattice is turned on 550 µ s before the control beam. Thecontrol beam is on for 150 µ s to ensure all the atoms arepopulated in the F = 2 state. After a few millisecondsof the cooling sequence, the intensity of the probe beamis gradually reduced to allow atoms to accumulate in the F = 2 state. The lattice beam intensity is then rampedoff adiabatically in a few hundred microseconds.To reduce the broadening of the hyperfine groundstates from the Zeeman splitting, we apply three-axiscompensating magnetic fields to minimize the ambientmagnetic field. The residual magnetic field is character-ized by microwave spectroscopy. We apply a microwavepulse of 0.5 ms to transfer atoms from the F = 2 to F = 3 states and detect the population on the F = 3state through absorption detection. The inset in Fig.2(a) shows the population of atoms in the F = 3 state asa function of microwave frequency. The full width at halfmaximum of the Voigt fitting function implies a residualmagnetic field of 10 mG over 1 mm of the atomic cloud.The Fano-type absorption spectrum is measured for il-lustration by scanning the probe beam frequency throughthe EOM, as shown in Fig. 2(a). The control beam isset at +20 MHz from the resonance at 0.66 mW, and theprobe beam is 0.34 mW. The relatively narrow transmis-sion peak of the asymmetric spectrum is magnified in ( b ) Fluorescence (arb. units) y ( m m ) W i t h c o o l i n g 7 m s 2 4 m s ( a )
Fluorescence (arb. units) y ( m m ) W i t h o u t c o o l i n g
7 m s
Temperature ( m K) E O M f r e q u e n c y ( M H z ) ( c )
Temperature ( m K) C o n t r o l b e a m p o w e r ( m W ) ( d )
FIG. 3. (a) Time-of-flight images projected on the y -axis at 7and 24 ms after releasing the atoms from the optical lattice.The images are from the data point with EOM frequency of2836.522 MHz in (c). (b)Time-of-flight images projected onthe y -axis at the same times as in (a) without any sidebandcooling. (c) Temperature measurements with varying probebeam detuning through tuning the EOM frequency. The con-trol beam is fixed at +20 MHz detuned from the F (cid:48) =3 state.(d) Temperature measurements with varying control beampower. The probe beam power is 0.15 mW. Each data pointin (c) and (d) is an average of 20 experimental runs, and thebars represent the standard errors. The systematic error fromthe magnification of the imaging method is not included. Fig. 2(b). The dip corresponds to the dark-state wherethe carrier transitions are minimized while the small peakallows red sideband transitions over a range of lattice fre-quencies which is determined by the width of the peak.The temperature of the atomic ensemble is measuredby the time-of-flight (TOF) method, where the two-dimensional fluorescence images of the atomic cloud arerecorded 7 and 24 ms after the atoms are released fromthe optical lattice. We use the MOT beams to irradiatethe atoms, and the fluorescence is collected by a cameraalong the z -axis. The systematic error from the uncer-tainty of the magnification of the imaging system on thetemperature measurements is estimated to be about 20%(10% estimated error of the imaging system). For eachtwo-dimensional image of the atomic cloud, we sum upthe fluorescence photons on the camera pixels along the x -axis to obtain the distribution of the atomic cloud onthe y -axis, as shown in Figs. 3(a) and 3(b). We ob-serve two distributions corresponding to different tem-peratures. The hotter one is mainly due to the impre-cision of our cooling beams’ alignment with the atomiccloud. We fit the experimental data with two Gaussianfunctions, and the data presented throughout the rest ofthe article are the results of the colder atomic cloud. Thefitted Gaussian widths are used to determine the temper-ature of the atomic cloud. The position shift of the 24-msimage from the 7-ms image is due to the angle betweenthe line-of-sight of the camera and gravity. For compar-ison, we plot the distributions of the atoms after PGCwithout any sideband cooling in Fig. 3(b).Figure 3(c) shows the temperature of the atoms alongthe y -axis as a function of the two-photon detuning withthe single-photon detuning of the control beam set at+20 MHz. The power of the control beam is 0.33 mW,which shifts the narrow absorption peak by δ = 140 kHz,approximately two times the peak vibrational frequencyof the optical lattice. The best cooling result is observedat two-photon resonance, which agrees with the dip ofthe absorption profile shown in Fig. 2(b).While the dark state sideband cooling of atoms to thevibrational ground state has been demonstrated in vari-ous systems, the temperature T of the atoms that can beachieved in combination with other cooling methods hasnever been explored. Here, we include adiabatic coolingafter sideband cooling by switching off the optical latticebeam gradually. To extract only the dark-state sidebandcooling performance, the mean vibrational number n af-ter cooling is calculated by an equation that describes theexpansion of the atoms from tight-binding bands into thefree-particle band [24]12 k B T = E R ( Qk ) f B + f − f B ) , (4)where k B is the Boltzmann constant, E R is the recoil en-ergy, Q = 3 k/ y -axis,and f B = n n is the Boltzmann factor. For T = 100(20)nK, the mean vibrational number n is 0.07(4), while n = 6 before dark-state sideband cooling. We note that n is an average over different lattice sites with differ-ent trapping frequencies. Using Eq.(5) of Ref. [1], thecooling-beams peak Rabi frequencies, the lattice vibra-tional frequency, and the cooling-beams single-photondetuning, we calculate the theoretical mean vibrationalnumber n th =0.01. As the cooling is expected to be moreefficient at the center than at the side of the atomic en-semble, our experimental result agrees reasonably wellwith the theory.We vary the control beam power to shift the absorptionpeak and observe that the temperature decreases withsmaller δ , as shown in Fig. 3(d). This implies that thedark-state sideband cooling is also favorable for atoms inthe potential with small vibrational frequency. When δ decreases, the absorption probability of atoms with small f increases, which results in more efficient cooling. Al-though the absorption probability decreases for atomswith large f , it is still relatively large compared to theheating transition. As a result, the cooling occurs for awide range of f .We also measure the temperature as a function of thesingle-photon detuning with fixed control beam power,as shown in Fig. 4(a). The result follows the trend ofEq. (1), where the increase of the single-photon detun- Temperature ( m K) S i n g l e p h o t o n d e t u n i n g ( M H z ) ( a )
Temperature ( m K) C o o l i n g d u r a t i o n ( m s ) ( b )
FIG. 4. Temperature measurements as a function of (a)single-photon detuning and (b) cooling duration. Each datapoint is an average of 20 experimental runs, and the bars rep-resent the standard errors. The systematic error from themagnification of the imaging method is not included. ing reduces the frequency shift. Figure 4(b) presents thetemperature as a function of cooling time. The increaseof the temperature after 3 ms is mainly due to the single-photon scattering from the optical lattice beams at a rateof 10 s − . As the third dimension of the temperature ofthe atoms is not cooled by the dark-state sideband cool-ing, heating due to single-photon scattering from the lat-tice beams in the third dimension pushes the atoms awayfrom the center of the two-dimensional lattice, which de-creases the cooling efficiency.With an optical depth of1 using isotropic polarization of the probe beam and asize ensemble along the probe beam direction of 2 mm,the atom density is N =4 × cm − . The correspond-ing phase-space density at a temperature of 100 nK is N λ = 8 × − , where λ dB is the thermal de Brogliewavelength.In summary, we demonstrated cooling of atoms tothe vibrational ground state in an optical lattice andsub-recoil temperature with the aid of adiabatic cool-ing within a few milliseconds. Our results are in goodqualitative agreement with the theoretical predictions forthe explored parameter ranges. The lowest temperatureachieved, in units of the recoil temperature, is a factor of4 lower than the previous result [10]. In addition, the ex-perimental implementation is relatively simple comparedto other sideband cooling schemes. This method can beapplied to atoms trapped in two- or three-dimensionaloptical lattices whose trapping frequencies vary acrossdifferent lattice sites due to the Gaussian distributionof the laser beams. 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