Observation of Quantum Interference and Coherent Control in a Photochemical Reaction

David B. Blasing,1 Jesús Perez-Ríos,2 Yangqian Yan,1 Sourav Dutta,1,3,†

Chuan-Hsun Li,4 Qi Zhou,1,5 and Yong P. Chen1,4,5,*1Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

2School of Materials Sciences and Technology, Universidad del Turabo, Gurabo, Puerto Rico 00778, USA3Department of Physics, Indian Institute of Science Education and Research, Bhopal 462066, India

4School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA5Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA

(Received 27 April 2018; published 16 August 2018)

Coherent control of reactants remains a long-standing challenge in quantum chemistry. In particular, wehave studied laser-induced molecular formation (photoassociation) in a Raman-dressed spin-orbit-coupled87Rb Bose-Einstein condensate, whose spin quantum state is a superposition of multiple bare spincomponents. In contrast to the notably different photoassociation-induced fractional atom losses observedfor the bare spin components of a statistical mixture, a superposition state with a comparable spincomposition displays the same fractional loss on every spin component. We interpret this as thesuperposition state itself undergoing photoassociation. For superposition states induced by a large Ramancoupling and zero Raman detuning, we observe a nearly complete suppression of the photoassociation rate.This suppression is consistent with a model based upon quantum destructive interference between twophotoassociation pathways for colliding atoms with different spin combinations. This model also explainsthe measured dependence of the photoassociation rate on the Raman detuning at a moderate Ramancoupling. Our work thus suggests that preparing atoms in quantum superpositions may representa powerful new technique to coherently control photochemical reactions.

DOI: 10.1103/PhysRevLett.121.073202

Quantum coherent control of atomic processes has been asignificant triumphof atomic,molecular, and optical physics.Extending such coherent control tomolecular processes is anactive and interesting research area. In particular, the study ofcoherent control of photochemical molecular processeshas focused on light-based control or control of the initialand final quantum states (for reviews see Refs. [1–4]).Theoretical studies have concerned both the manipulationof light parameters, such as the pulse trains, polarization,relative phases, etc., [5–12] and the initial or final quantumstates [13]. Experimentally, tailored light pulses have beenshown to control isomerization, photoassociation (PA), andphotodissociation [14–21]. However, there is much lesserexperimental study of influencing molecular processes bycoherently controlling the reactants. Such a difficulty canarise from incoherent population in many scattering statesdue to finite experimental temperatures or an incompleteunderstanding of the quantum molecular processes.In this work, we explore the following question: What

happens in a chemical reaction if the reactants are preparedin quantum superposition states? Here we report spin-dependent PA experiments using a 87Rb Bose-Einsteincondensate (BEC). Photoassociation [22] is a light-assistedchemical process where two atoms absorb a photon whilescattering, and bind into an excited molecule. Our Bose-Einstein condensates are at ultracold temperatures and

populate only a small number of scattering channels. Inour experiment, we have prepared atoms in spin-momen-tum quantum superposition states, and a pair of such atomscan couple to an excited molecular state simultaneouslythrough two atomic scattering channels. These two chan-nels of different spin combinations both contribute to thecoupling, but with opposite sign due to opposite Clebsch-Gordan (CG) coefficients. The relative amplitudes of thetwo contributions depend on the superposition state. Thespin portion of a representative scattering state is shown inFig. 1(a), along with the relevant molecular potentialenergy curves plotted against the internuclear separationR in units of Bohr radius a0 [23]. Our system exploitsthe intrinsic quantum nature of PA and our tunable super-position states of the reactant atoms allows us to observea nearly total suppression of the molecular formation,thus representing a significant step forward for coherentchemistry.Our experiment begins with a 87Rb BEC of ∼1.5 × 104

atoms in the f ¼ 1 hyperfine state, which is produced via alloptical evaporation in a cross-beamoptical dipole trap createdby a 1550 nm laser [24]. The trap has a characteristic trapfrequency ω¼ðωxωyωzÞ1=3∼2π×ð140×140×37Þ1=3Hz¼2π×ð90HzÞ. Tuning the magnetic field during the evapora-tion can lead to a BEC with baremf ¼ −1; 0;þ1 spin state,or a statistical mixture of all three.

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After preparing a bare BEC in spin state mf ¼ 0, we canload the BEC into a spin-momentum superposition byadiabatically applying a pair of counterpropagating Ramanbeams with wavelengths near 790.17 nm, see Fig. 1(b). TheRaman beams couple the mf states, as shown in Fig. 1(c),and “dress” the atoms into superpositions of the mf spinstates and mechanical momenta:

P3i¼1 Cijmf; pii ¼

C−1j−1;ℏðqþ 2krÞi þ C0j0;ℏqi þ Cþ1jþ1;ℏðq − 2krÞi,where p and ℏq are the mechanical momentum andquasimomentum along y, respectively, ℏkr is the singlephoton recoil momentum of the Raman lasers with wave-lengths near 790.17 nm, and ℏ is the reduced Planck’s

constant. The quasimomentum is the canonical momentumconjugate of the position along y. For more details of oursetup, see Refs. [25,26]. Our Raman coupling scheme islargely similar to previous works [27,28]. The Hamiltonianfor the Raman light-atom interaction is

H¼

0B@

ℏ22m ðqþ 2krÞ2 − δ ΩR

20

ΩR2

ℏ22mq

2 − ϵqΩR2

0 ΩR2

ℏ22m ðq− 2krÞ2 þ δ

1CA;

ð1Þ

where m is the 87Rb mass, δ is the Raman detuning,ΩR is the Raman coupling (calculated from the measuredfrequency of resonant Raman-Rabi oscillations), Er ¼ℏ2k2r=2m ¼ h × ð3.68 kHzÞ is the recoil energy, and ϵq ¼0.65Er is the quadratic Zeeman shift. Unless otherwisestated, ΩR and δ carry uncertainties of 10% and �0.5Er,respectively. The eigenstates of Eq. (1) are the “dressed”spin-momentum superposition states described above. Theq-dependent eigenenergies of the atoms form the dressedbands, with a few representative examples shown inFig. 1(d). Red, blue, and green colors reflect the proportionof mf ¼ −1, 0, and þ1 in the spin-momentum super-position state. The small dots represent the BECs adiabati-cally prepared at the band minimum. Such Raman couplinghas been used previously to induce synthetic partial waves[29], modify an s-wave Feshbach resonance [30], andcouple singlet and triplet scattering states [31].After preparing our BEC in a statistical mixture or a spin-

momentum superposition state, we apply a PA laser withwavelength 781.70 nm and typical intensity IPA of a fewW=cm2 for a time tPA of a few ms [32]. The frequency ofthe PA laser is tuned to the ð2Þ1g excited molecular statewith vibrational quantum number 152 [see again Fig. 1(a)],and corresponds to the PA line ϵ in the Fig. 1 of Ref. [33].After the PA pulse, the PA, Raman (in the case of thesuperposition state), and the dipole lasers are simultane-ously switched off to allow the BEC to undergo 15 ms oftime-of-flight (TOF) expansion. During the initial portionof this expansion, we use a Stern-Gerlach magnetic fieldgradient to separate atoms in the differentmf spin states. Atthe end of the TOF expansion, we apply absorption-basedimaging to extract the atom numbers in different spin-momentum projections. Then, this sequence is repeated atvarious PA frequencies to obtain a PA spectrum.To extract the PA rate constant, kPA, due to a PA process,

we follow a procedure similar to those in Refs. [34,35].The two-body rate equation describing the time-dependentBEC density of the atoms participating in PA, ρðt; rÞ, isdρðt;rÞ=dt¼−kPAρ2ðt;rÞ. In our experiment, Γstim ≪ Γspon

(where Γstim and Γspon are the stimulated and spontaneousemission rates, respectively), and kPA has a Lorenztiandependence with respect to the PA frequency [36].

FIG. 1. Energy level diagrams and experimental setup.(a) Relevant molecular potential energy curves. Depicted atthe right is a scattering state of a pair of atoms (the tricoloredspheres) whose spin part of their quantum state is a superpositionof different bare mf spin states. Beneath is the decomposition ofthe scattering state as that of various pairs of atoms (themonocolored spheres) with bare mf spin states. The super-position coefficients are denoted C−1, C0, and Cþ1; red, blue,and green represent mf ¼ −1, 0, and þ1, respectively. Thenonzero CG coefficients for the jF ¼ 0; mF ¼ 0i component ofthe various pairs of bare spins in the scattering state are shownnear the thin black arrows. (b) Laser geometry. Two Raman laserswith angular frequencies ωR þ ΔωR and ωR propagate along �yand have linear polarizations along x and z. The frequencydifference between the Raman beams ΔωR=2π is 3.5 MHz. A PAlaser propagates in the x-z plane (due to spatial constraints) andhas a linear polarization with components along all three axes.(c) Atomic energy diagram. A Zeeman bias magnetic field,jBBiasj ≈ 5 G is used to tune the Raman detuning δ. (d) Dressedband structures for Raman coupling ΩR ¼ 0, 1.1, 3.2, 8.0, and12Er, all with δ ¼ 0. Dots represent BECs adiabatically loaded tothe band minima at q ¼ 0. Panels (a) to (c) are not to scale.

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Solving for ρðt; rÞ and spatially integrating it over a Thomas-Fermi BEC density profile yields an expression for the atomnumber, NðηÞ, remaining after a PA pulse,

NðηÞ ¼ N0

15

2η−5=2

�η1=2 þ 1

3η3=2

− ð1þ ηÞ1=2tanh−1½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiη=ð1þ ηÞ

p��; ð2Þ

where η ¼ kPAρ0tPA is a dimensionless parameter indicatingthe strength of the PA pulse, ρ0 is the peak atomic density atthe center of the BEC, and N0 is the off-resonant atomnumber with no PA loss. We denote the resonant kPA for aBEC composed of mf ¼ 0 bare spin (or spin-momentumsuperposition) states as k0;0 (or ksup).First, as shown in Fig. 2,we compared the PAofBECs in a

spin statistical mixture to that of BECs in a spin-momentumsuperposition state, with a nearly identical atom number andspin composition in the two cases. For the spin-momentumsuperposition state, we used ΩR ¼ 8.0Er and δ ¼ 0Er [seealso Fig. 1(d)]. In panels (a) and (b), we show the opticaldensity (OD) images for PAboth on andoff resonance for the

spin statistical mixture (a) and the spin-momentum super-position state (b). For the spin statistical mixture, the PA-induced loss for the mf ¼ 0 component is notably largerthan that for themf ¼ �1 components. The lower reductionof the OD was due to the lower ρ0 for the mf ¼ �1

components and that each molecule formed by the PAprocess reduced the mf ¼ 0 atom number by two, but formf ¼ �1, only by one each. However, for PA on a spin-momentum superposition state, we observed comparablePA-induced loss among all three mf components. Inpanels (c) and (d), for each mf component and the total,we show the normalized atom number (N=N0) at various PAdetunings (ΔνPA) from the resonance for the statisticalmixture (c) and the spin-momentum superposition state(d). Each PA spectrum for every mf component or the totalwas fitted to Eq. (2) to extract the appropriate N0 andthen normalized. The N0 were ∼ð1.2; 7.0; 1.1Þ × 103 andð1.5; 6.9; 1.3Þ × 103 for the ðmf ¼ −1; 0;þ1Þ componentsof the statistical mixture and superposition state, respec-tively. For the spin statistical mixture, ð79� 2Þ% of themf ¼ 0 atoms were lost on resonance, but less than ∼25%were lost for the mf ¼ �1 components. For the dressedBECs, allmf components lost ð36� 2Þ%. All the data weretaken using PA pulses with identical parameters.To further explore this phenomenon, we prepared BECs

with atoms in several spin-momentum superposition (ormf ¼ 0 spin) states by using ΩR ¼ 0, 1.1, 3.2, and 12Er

with δ ¼ 0Er (or δ ∼ 100Er) and plotted the normalized PAspectra in Figs. 3(a)–3(d) using red squares (or blackcircles). With δ ∼ 100Er, the Raman beams did not dressthe atoms into spin-momentum superposition states, andthese BECs therefore remained in the mf ¼ 0 bare spinstate and displayed comparable loss ð∼40%Þ for all ΩR.However, the loss for BECs in spin-momentum super-positions decreases with increasing ΩR. At ΩR ¼ 12Er, noloss is apparent. For all the data in panels (a) to (d), we usedcomparable total off-resonant BEC atom numbers[N0 ∼ ð1.1� 0.1Þ × 104] and square PA pulses with tPA ¼3.2 ms and IPA ¼ 6.0� 0.7 W=cm2.We also fitted photoassociation spectra measured with

δ ¼ 0 (or δ ∼ 100Er) to Eq. (2) and extracted ksup (or k0;0),and then plotted ksup=k0;0 in Fig. 3(e). We used PA pulseswith IPA ¼ 6.1� 0.7 W=cm2 and tPA of 2–8 ms to inducea repeatable but unsaturated loss of (10–40)%. Alsoincluded are solid bands, which are predictions forksup=k0;0 derived as follows. The molecular hyperfine stateexcited by our chosen PA transition has total angularmomentumF ¼ 1 and nuclear spin I ¼ 1, and only couplesto a pair of colliding atoms (represented by subscripts a andb in the following) whose total angular momentumjF¼faþfb;mF¼mf;aþmf;bi¼j0;0i. Using the singleparticle basis, jfa;mf;aijfb;mf;bi, jF ¼ 0; mF ¼ 0i ¼ðj1;þ1ij1;−1i þ j1;−1ij1;þ1i − j1; 0ij1; 0iÞ= ffiffiffi

3p

. Thus,

(a) (b)

(c) (d)

FIG. 2. Photoassociation of BECswith atoms in a spin statisticalmixture (with Raman coupling ΩR ¼ 0Er) versus a spin-momen-tum superposition state with similar atom number and spincomposition (ΩR ¼ 8.0Er and Raman detuning δ ¼ 0.0Er). (a),(b): The average optical density (OD) images with PA off and onresonance for the spin statistical mixture (a) and the spin-momentum superposition state (b). (c),(d): The extracted normal-ized atom numbers of themf components and the total of BECs atvarious PAdetunings (ΔνPA) from resonance for the spin statisticalmixture (c) and spin-momentum superposition (d). The atomnumbers of everymf component or the total are normalized by thecorresponding fitted values of the off-resonant atom number N0

and the error bars are the standard error of the mean. Both the ODimages and data points are averages of 5 to 7 experimental runs.

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there are two allowed pathways for the PA transition (afteraccommodating the indistinguishability of bosons): one inwhich both atoms have mf ¼ 0 and another with atoms inmf ¼ �1. Both pathways contribute to this PA process withopposite signs due to the oppositeCGcoefficients (�1=

ffiffiffi3

p).

We note kPA ∝ jhψmoljd · Ejψ scatij2 with a spin independentproportionality factor [37], where jψ scati and jψmoli are thetotal wave functions for the scattering state and molecularstate, respectively, E is the electric field of the PA laser, and d

is the dipole operator. The spin portion of jψ scati for twoRaman dressed atoms (labeled by subscripts a and b) isPþ1

i¼−1Pþ1

j¼−1 CiCjjf ¼ 1; mf ¼ iia ⊗ jf ¼ 1; mf ¼ jib.Using the CG coefficients, the probability amplitude of thejF ¼ 0; mF ¼ 0i component of the scatteringwave functionis therefore ð2C−1Cþ1 − C2

0Þ=ffiffiffi3

p. We arrive at [38]

ksup=k0;0 ¼ jC20j2 þ 4jC−1Cþ1j2 − 4Re½C2

0C�−1C

�þ1�: ð3Þ

Describing the entire superposition BEC with one PArate constant ksup is supported by our observation in Fig. 3that all the bare spin components of the superposition statedisplay identical fractional losses. In the limit of ΩR → 0with δ¼0, ksup=k0;0→1 since C0→1 and C−1 ¼ Cþ1 → 0.However, interestingly, ksup=k0;0 → 0 for large ΩR because

C0 → 1=ffiffiffi2

pand C−1 ¼ Cþ1 → 1=2 [all the superposition

coefficients Ci can be calculated by diagonalizing Eq. (1)].In this case the third term of Eq. (3) cancels the first two,thus no molecular formation is predicted even with reso-nant PA light. This happens despite PA being allowed onboth channels for associating two mf ¼ 0 atoms andassociating two mf ¼ �1 atoms [see Fig. 1(a)]. Thiscomplete destructive interference comes from the oppositeCG coefficients (�1=

ffiffiffi3

p). In Fig. 3, and also later in Fig. 4,

the orange (blue) bands are theoretical predictions fromEq. (3) that show the range of theoretical predictionsresulting from our typical experimental uncertainties,including (excluding) the destructive interference effect,

FIG. 3. Photoassociation of atoms in spin-momentum super-positions at various values of the Raman couplingΩR. Panels (a)–(d): Normalized atom loss for BECs of atoms in spin-momentumsuperpositions (or spin state mf ¼ 0) with Raman detuningδ ¼ 0� 0.5Er (or δ > 100Er) are shown as red squares (orblack circles). The values of ΩR=Er were: 0 (a), 1.1 (b), 3.2 (c),and 12 (d). Error bars are the standard error of the mean of 6experimental runs. Panel (e): Normalized photoassociation rate,ksup=k0;0, for BECs at various ΩR with δ ¼ 0Er. The orange(blue) bands are theoretical predictions with (without) thedestructive interference term in Eq. (3). The band boundariesreflect one standard deviation of the predicted values given ourexperimental uncertainties.

FIG. 4. Normalized photoassociation rate, ksup=k0;0, for BECsat Raman detuning δ from −2.5 to þ2.5Er with Raman couplingΩR ¼ 5.4Er. The orange (blue) bands are theoretical predictionswith (without) the destructive interference term in Eq. (3). Theband boundaries reflect one standard deviation of the predictedvalues given our experimental uncertainties. Inset: Dressed bandstructures for ΩR ¼ 5.4Er and δ ¼ −2, 0, and 2Er. The dotsrepresent BECs prepared at the band minima.

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the last term of Eq. (3). The nearly total suppression ofksup=k0;0 at large ΩR with δ ¼ 0Er is consistent with theprediction of Eq. (3) with the destructive interference term.We also studied PA on BECs prepared with ΩR ¼ 5.4Er

and δ from −2.5 to þ2.5Er. Figure 4 shows ksup=k0;0 vs δ,measured using square PA pulses with tPA ¼ 5.5 ms andIPA ¼ 5.7� 0.2 W=cm2. The experimental error barsreflect the aggregate uncertainty of ksup=k0;0. The insetcontains calculated band structures for δ ¼ −2, 0 andþ2Er.Using δ beyond ∼∓2Er polarizes the dressed BEC intomajoritymf ¼ �1 and collisions between such atoms do notcontribute to the jF ¼ 0; mF ¼ 0i channel. This is consis-tent with our observed ksup=k0;0 → 0 with increasing jδj.This suppression is again consistent with Eq. (3) becauseC0

and one of C�1 vanish, predicting ksup=k0;0 → 0.Atoms in spin-momentum superpositions are novel

reactants, and thus PA of such atoms represents a newtype of photochemistry. We interpret observing the samefractional loss on all components of a spin-momentumsuperposition state as an indication that it is the super-position state itself that undergoes PA. Further, the differenttotal fractional loss between BECs of a spin statisticalmixture and a spin-momentum superposition state demon-strates a significant modification to the PA process due tothe coherent quantum superposition. Lastly, we interpretthe nearly full suppression of ksup=k0;0 as resulting fromdestructive interference between the two out-of-phasepathways (mf ¼ 0, mf ¼ 0 and the mf ¼ þ1; mf ¼ −1).Our scattering state simultaneously accesses these twopathways as it couples to the chosen excited molecularstate. Thus our observations suggest that scattering states ofatoms in quantum superpositions may offer a powerful newapproach to coherently control photochemical reactions.

We acknowledge helpful conversations with Su-Ju Wangand Abraham Olson. We acknowledge NSF GrantNo. PHY-1708134 and the Purdue University OVPRResearch Incentive Grant No. 206732 for partial supportof our research. D. B. B. also acknowledges a PurdueResearch Foundation Ph.D. Fellowship. Y. Y. and Q. Z.acknowledge startup funding from Purdue University.

*yongchen@purdue.edu†Present address: Tata Institute of Fundamental Research,Homi Bhabha Road, Colaba, Mumbai 400005, India.

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[38] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.121.073202 for a deri-vation of Eq. (3). In writing Eq. (3), we have ignored in theFranck-Condon factor the relativemomentum imparted fromthe Raman beams. For the purpose of our PA resonance, thisis justifiable since the length scale set by the wavelength ofthe Raman beams is ∼1 μm, but the length scale of the PAprocess is significantly shorter, less than 2 nm.

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