Suppressed spontaneous emission for coherent momentum transfer
Xueping Long, Scarlett S. Yu, Andrew M. Jayich, Wesley C. Campbell
SSuppressed spontaneous emission for coherent momentum transfer
Xueping Long, ∗ Scarlett S. Yu, Andrew M. Jayich, and Wesley C. Campbell University of California Los Angeles University of California Santa Barbara (Dated: July 23, 2019)Strong optical forces with minimal spontaneous emission are desired for molecular decelerationand atom interferometry applications. We report experimental benchmarking of such a stimulatedoptical force driven by ultrafast laser pulses. We apply this technique to accelerate atoms, demon-strating up to an average of 19 (cid:126) k momentum transfers per spontaneous emission event. Thisrepresents more than an order of magnitude improvement in suppression of spontaneous emissioncompared to radiative scattering forces. For molecular beam slowing, this technique is capable ofdelivering a many-fold increase in the achievable time-averaged force to significantly reduce boththe slowing distance and detrimental losses to dark vibrational states. The directed, narrow-band light emitted by lasers hasbeen used to great effect to manipulate the motion ofgas-phase atoms, leading to a diverse set of applica-tions [1–4]. In contrast to atoms, the rich internal struc-tures of polar molecules and their readily available long-range and anisotropic dipolar interactions make ultra-cold molecules uniquely promising candidates for preci-sion measurements [5–9], quantum information process-ing [10–15] and quantum chemistry [16, 17]. However, formolecules, spontaneous emission populates excited vibra-tional states, which has largely precluded the adaptationof atomic laser cooling techniques for molecules.Recently, the workhorse of ultracold atomic physics,the magneto-optical trap (MOT), has been success-fully demonstrated with some carefully chosen diatomicmolecules [18–23]. Despite this substantial step forward,the largest number of molecules that have been trappedin a MOT ( ≈ [22]) is still orders of magnitude lessthan a typical atomic MOT, limited by the small fractionof molecules that can be slowed from a molecular beam tothe MOT capture speed [24]. Further, extension of thistechnique to molecules with higher vibrational branchingprobability (such as polyatomics) will likely require newmethods for beam deceleration.While the most commonly used laser decelerationmethods employ spontaneous radiation pressure, thetime-averaged force is limited to a low value by the needto wait for spontaneous decay after each (cid:126) k of momen-tum transfer. For molecules, slowing via spontaneousscattering has been limited to a handful of specially-chosen diatomic species [25–30] with extremely low vibra-tional branching probabilities [31]. Moreover, multiplemolecular transitions must be driven that connect variousground states to the same excited state, which further re-duces the time-averaged force that can be achieved [32].As a result, radiative deceleration of molecular beamsleads to long slowing lengths and low trap capture ef-ficiencies associated with molecule loss from transversevelocity spread and spontaneously populated excited vi-brational states.For atom interferometry [33, 34] (including fast entan- gling gates with trapped ions [35–37]), coherent forces areneeded to manipulate phase space separation. In thesecases, even a single spontaneously emitted photon cancarry “which way” information that will decohere thesuperposition, entirely precluding the use of spontaneousradiation pressure for these applications. Further, strongforces are desired to effect large separation in a short in-teraction time, and coherent, spin-dependent momentumkicks [38] are particularly attractive [39–41].To address these needs, various optical forces that uti-lize stimulated emission are being pursued. For stimu-lated forces, a reasonable figure of merit for evaluatingthe gain in requisite cycle closure of stimulated over spon-taneous scattering is the average momentum transferred(in units of the photon momentum, (cid:126) k ) per spontaneousemission event, which we denote by the symbol Υ. Forspontaneous scattering, Υ = 1. For most stimulated scat-tering schemes, the stimulated processes can be drivenquickly compared to the spontaneous emission lifetime,and the stimulated force can therefore be stronger thanthe spontaneous scattering force by a factor of approxi-mately Υ.In this Letter, we demonstrate and benchmark an op-tical force derived entirely from stimulated scattering ofmode-locked (ML) laser pulses [42, 43], shown in Fig. 1.Early work on this technique showed order-of-unity forcegains over spontaneous scattering [44–46]. Here, by usinga pre-cooled sample of atoms to benchmark and optimizethe force, we show that its performance can be substan-tially improved. We are able to achieve an average ofΥ = (19 +6 − ) momentum transfers of (cid:126) k per spontaneousemission event. This potentially extends optical deceler-ation to molecules with state leakage probabilities an or-der of magnitude worse than currently used species, suchas complex polyatomics [47] and molecules well-suited toprecision measurement [9, 48–50].The stimulated force we demonstrate here is generatedby the fast repetition of a cycle in which a time-ordered,counterpropagating pair of picosecond laser pulses (“ π -pulses”) illuminate the sample. As illustrated in Fig. 1(see also [42–46]), a ground-state molecule from a molec- a r X i v : . [ phy s i c s . a t o m - ph ] J u l FIG. 1. Pulse sequence of the stimulated force. As describedbelow, the delay time τ between pump and dump pulses ischosen to be much smaller than the spontaneous emissionlifetime ( τ (cid:28) /γ ). ular beam is first excited by absorbing a photon fromthe “pump pulse” that is counter-propagating with re-spect to the molecular beam, thereby losing momentum (cid:126) k . The molecule is then immediately illuminated bya co-propagating “dump pulse,” which deterministicallydrives the molecule back to its original ground state viastimulated emission and removes another (cid:126) k of momen-tum from the molecule. The direction of the force is setby the order in which the pulses arrive, which introducesthe necessary asymmetry to establish a preferred direc-tion. This cycle can be repeated rapidly to create anapproximately continuous deceleration force that can bemuch stronger than spontaneous scattering. The broadspectrum coverage of the ultrafast laser pulses allows forsimultaneous deceleration of molecules from a wide rangeof velocities, and further augmentation of this schemewith adiabatic rapid passage and single-photon coolinghas been studied theoretically [43].We demonstrate and benchmark this force on a MOTof 10 pre-cooled (120 ± µ K Rb atoms using the S / → P / transition. As illustrated in Fig. 2, theML laser pulses are generated from a Ti:sapphire laseremitting 30 ps pulses at 780 nm at a repetition pe-riod of 12 . γ/ × s − , where the comb tooth visibilityscales as V ∝ sech( T rep γ/
2) [51]), a Pockels cell is usedfor pulse picking (power extinction ratio = 7 × − ) , in-creasing the pulse-to-pulse separation time from 12 . T rep = 250 ns. The 1 / e intensity diameter of theML laser beams at the position of atomic cloud is w = Time (ns)
Frequency (MHz) / γ = 26 ns 12.5 ns080 MHz γ /2 π = 6 MHz4 MHz FIG. 2. Time domain (upper) and frequency domain (lower)illustration of single-beam processes in this work. The MLlaser generates 30 ps pulses at 12 . > .
99% decay probability betweenpulses. The excited state probability for an atom excited bythe first pulse is represented by the yellow area in the timedomain figure. The corresponding atomic spectrum is shownin the frequency domain figure. (0 . ± .
03) mm, but non-gaussian variations are alsopresent, as discussed below.The MOT light and magnetic fields are turned off be-fore the ML laser pulse trains are introduced. The dumpbeam path is made ≈
10 cm longer than the pump beampath to set a τ = (310 ±
60) ps intra-pulse-pair delay.This delay distance ensures no temporal overlap betweenpump and dump pulses, and can be reduced to nearlythe pulse duration if it becomes a limiting factor in ap-plications.To calibrate and match the effective average pulse flu-ence from both beams, the atoms are illuminated withsingle pulses as the pulse energy is scanned, shown inFig. 3. Fluorescence from spontaneous decays collectedperpendicular to the ML beam propagation directionshows clear, coherent Rabi flops, an observation that ismade possible despite the short lifetime of this transi-tion (1 /γ = 26 ns) by the ultrafast excitation. The pe-riod of the Rabi flops allows us to infer the pulse area,shown as the top axis in Fig. 3. We model the ex-cited state probability ( P ( θ o )) as a function of pulse area θ ≡ (cid:82) d t Ω( t ) ∝ √ pulse energy as coherent evolution av-eraged over a normal distribution of pulse areas with av-erage value θ o and standard deviation σ θ , P ( θ o ) = 12 (cid:16) − e − σ θ / cos ( θ o ) (cid:17) , (1)shown as a dashed, purple curve in Fig. 3.When the sequential, counter-propagating pump-then-dump pulses illuminate the atoms, the relative phase be-tween them is spatially dependent on a length scale of afraction of an optical wavelength. Since the atom cloudis large compared to λ , we model the ensemble-averagedinteraction as devoid of intra-pulse-pair coherence. Theexcited state probability after the pump-dump sequenceis given by P (2)ex ( θ o ) = 12 (cid:16) − e − σ θ / cos ( θ o ) (cid:17) −
12 e − γτ × (cid:18) − e − σ θ / cos ( θ o ) + 12 e − σ θ cos (2 θ o ) (cid:19) , (2)where we assume that the two pulses have the same pulsearea and τ is the time delay between the pulses. Eq. 2 iscombined with the probability of spontaneous emissionbetween the pump and the dump pulse to give the ex-pected number of spontaneous emissions per pulse pair, (cid:104) N γ (cid:105) = 1 −
34 e − γτ + (1 − e − γτ )(2 ¯ P − −
14 e − γτ (2 ¯ P − , (3)where ¯ P ≡ P ( θ o = π ) is the average single-pulse popula-tion transfer fidelity at the π -pulse condition from Eq. 1.Here we assume that any residual excited state popula-tion decays before the next pulse pair, the probability ofwhich was made greater than 0 . π -pulse condition is determined by finding themaximum of the single pulse (average fluorescence ofpump-only and dump-only pulses) and local minimum ofthe pump-then-dump fluorescence signals, respectively,which coincide at the same pulse area. The measured π -pulse energy agrees with the theoretical prediction for atransform-limited, 30 ps pulse to within 20%, confirmingthat the laser pulses in this work are nearly transformlimited. Using Eq. 1 and Eq. 3, the ratio between thedetected single-pulse and pump-then-dump fluorescencesignals at the π -pulse condition does not require a cali-bration of the fluorescence collection efficiency, and pro-vides a stand-alone measurement of σ θ /θ o = (0 . ± . ≈ . σ θ is known, and Fig. 3 shows that the average π -pulse population transfer fidelity is ≈
98% for eachbeam. As shown in Fig. 3, this model overshoots the
Pulse energy [ nJ ] P r o b a b ili t y Pump PulseDump PulseBoth Pulses
Inferred pulse area [ ]
FIG. 3. Coherent Rabi flops on an optical frequency elec-tric dipole transition. The oscillation period allows identi-fication and matching of the π -pulse pulse energy for bothbeams. Fluorescence is collected from the atom cloud trans-verse to the acceleration direction for single pulses from thepump beam (red), dump beam (blue), or a pump-then-dumpsequence (black). The π -pulse condition is satisfied slightlyabove 1 nJ, and the probability of spontaneous emission (thevertical axis) is calibrated from the measurements using Eqs. 1and 3, which are shown as dashed curves. data for intermediate pulse areas ( θ o (cid:54) = nπ ), which webelieve is caused by the finite optical depth of the sampleleading to preferential emission in the forward direction[52]. Since the actual force is implemented at the π -pulsecondition, we are primarily concerned with the agreementbetween our model and the data at this location. As away to check the consistency of this model, we comparethe predicted vs. measured fluorescence at the 2 π -pulsecondition for a single beam; the measured excited statefraction is (5 . ± . ± (cid:104) ∆ p (cid:105) (cid:126) k = 18 e − γτ (cid:0) P −
1) + (2 ¯ P − (cid:1) . (4)The assumption of full spontaneous emission of any ex-cited population before the next pulse pair yields an ex-pression for the number of (cid:126) k photon momenta (alongthe pump beam propagation direction) transferred perspontaneous emission event,Υ = (cid:32) P e − γτ (cid:0) P −
1) + (2 ¯ P − (cid:1) − (cid:33) − . (5)Taking ¯ P = (0 . ± . ± FIG. 4. Effect of varying pulse energy on the arrival timesof the atoms at the TOF detection position. Each verticalcross-section is a TOF trace (see [53]). The dashed guidelines represent the theoretical arrival times if the indicatedmomentum had been transfered to the atoms. These diag-nostic data were taken before optimizing the force, and thearrival times of the fastest 10% atoms corresponds to Υ ≈ of the two beams in space or variations in ¯ P that mayappear as atoms are accelerated along the beam. SinceEq. 5 diverges as ¯ P → τ = 0), small additionalimperfections in pulse area may significantly decrease Υ.To obtain a better measurement of Υ, we benchmarkthe momentum transfer itself by applying 1000 repeatedpump-dump sequences (i.e. 2000 total pulses) to theatoms followed by TOF measurements that allow us todetermine the time-averaged force. A resonant, cw laserbeam centered 4 − π -pulse energy, the stimulated force be-comes more efficient, resulting in better acceleration andearlier arrival times. Further, as the population transferfidelity nears 1, the arrival time distribution of the atomsnarrows, as expected from the reduction in quantum pro-jection noise associated with the outcome of each pulsepair becoming more deterministic. Even so, the widthof the distribution at the π -pulse condition far exceedswhat would be expected for uncorrelated pulse area fluc-tuations (which would contribute an arrival time spreadof order 10 µ s, compared to the observed width of order100 µ s), and is instead caused by the systematic varia-tions in population transfer fidelity associated with thenonuniform transverse intensity profiles of the beams.To quantify the acceleration, TOF measurements areused to optimize the force and are then performed at 5different absorption beam locations along the trajectoryof the accelerated atoms to control for the initial position and its spread (see [53]). As discussed above, since thespatial profile of the pulses is not perfectly uniform, somelocations in space experience systematically higher pop-ulation transfer fidelity than others, and it is these thatrepresent the ensemble most interesting for consideringfuture applications to molecules. During optimizationof the beam positions and strengths, the arrival time ofthe fastest moving 10% of the atoms was minimized. Forthese atoms, we obtain a velocity of (11 . ± .
3) m / s, cor-responding to a total momentum transfer of (1820 ± (cid:126) k from 2000 pulses and a momentum transfer efficiency of(91 ± π -pulse popula-tion transfer fidelity, yielding ¯ P = (0 . ± . +6 − . The lowervalues of ¯ P and Υ measured from in situ accelerationmeasurements as compared to those inferred from few-pulse fluorescence experiments (e.g. Fig. 3) highlight theneed to perform measurements of this kind by measuringthe actual momentum transfer, which is sensitive to morepotential systematic effects than observations of internalstate dynamics. For instance, due to the finite extinc-tion ratio of the pulse picker, small comb tooth effectscan become important between pulse pairs during longpushing sequences. We find that the momentum trans-fer efficiency is insensitive to the comb tooth positionas long as the nearest-resonant tooth is not close to thethe atomic transition, a condition that is maintained forthese experiments with the passive stability of the laser(see [53]).Comparison of this measurement of Υ to other meth-ods in the literature is complicated by the fact that veryfew demonstrations of stimulated slowing techniques re-port the average gains in cycle closure that they are de-signed to provide (though a recent demonstration of thebichromatic force on polyatomic molecules [54] achievedΥ = (3 . ± .
7) [55]). Two other performance indicatorsare more common: the excited state fraction, which de-termines the ensemble-averaged radiative decay rate, andthe force gain factor, which is the ratio of the magnitudeof the stimulated force over the theoretical maximum ra-diative force for an ideal two-level system. The time-averaged excited state fraction induced by the bichro-matic force for a two-level system can be optimized to41%, though it could be improved further to 24% witha four-color force scheme [56]. The pulsed scheme inthis work can be viewed as a polychromatic limit ofthe bichromatic force, and the time-averaged pump-then-dump excited state fraction achieved here is (1 . ± . . ± . / s to a full stop in 22 cm, thereby suppressingmolecular losses due to transverse motion and increas-ing the molecular flux. 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