APS/123-QED

Observation of Quantum Interference and Coherent Control in a PhotochemicalReaction

David B. Blasing,1 Jesus Perez-Rıos,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, IN 47907

2School of Materials Sciences and Technology, Universidad del Turabo, Gurabo, Puerto Rico 007783Department of Physics, Indian Institute of Science Education and Research, Bhopal 462066, India

4School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 479075Purdue Quantum Center, Purdue University, West Lafayette, IN 47907

(Dated: August 10, 2018)

Coherent control of reactants remains a longstanding challenge in quantum chemistry. In par-ticular, we have studied laser-induced molecular formation (photoassociation) in a Raman-dressedspin-orbit-coupled 87Rb Bose-Einstein condensate, whose spin quantum state is a superposition ofmultiple bare spin components. In contrast to the notably different photoassociation-induced frac-tional atom losses observed for the bare spin components of a statistical mixture, a superpositionstate with a comparable spin composition displays the same fractional loss on every spin component.We interpret this as the superposition state itself undergoing photoassociation. For superpositionstates induced by a large Raman coupling and zero Raman detuning, we observe a nearly completesuppression of the photoassociation rate. This suppression is consistent with a model based uponquantum destructive interference between two photoassociation pathways for colliding atoms withdifferent spin combinations. This model also explains the measured dependence of the photoasso-ciation rate on the Raman detuning at a moderate Raman coupling. Our work thus suggests thatpreparing atoms in quantum superpositions may represent a powerful new technique to coherentlycontrol photochemical reactions.

Quantum coherent control of atomic processes hasbeen a significant triumph of atomic, molecular, and op-tical physics. Extending such coherent control to molecu-lar processes is an active and interesting research area. Inparticular, the study of coherent control of photochemicalmolecular processes has focused on light-based control orcontrol of the initial and final quantum states (for reviewssee Refs. [1–4]). Theoretical studies have concerned boththe manipulation of light parameters, such as the pulsetrains, polarization, relative phases, etc., [5–12] and theinitial or final quantum states [13]. Experimentally, tai-lored light pulses have been shown to control isomeriza-tion, photoassociation (PA), and photodissociation [14–21]. However, there is much lesser experimental study ofinfluencing molecular processes by coherently controllingthe reactants. Such a difficulty can arise from incoherentpopulation in many scattering states due to finite exper-imental temperatures or an incomplete understanding ofthe quantum molecular processes.

In this work, we explore the question: what hap-pens in a chemical reaction if the reactants are pre-pared in quantum superposition states? Here we re-port spin-dependent PA experiments using a 87Rb Bose-Einstein condensate (BEC). Photoassociation [22] is alight-assisted chemical process where two atoms absorb aphoton while scattering, and bind into an excited molec-ular state. Our Bose-Einstein condensates are at ultra-cold temperatures and populate only a small number ofscattering channels. In our experiment we have preparedatoms in spin-momentum quantum superposition states,and a pair of such atoms can couple to an excited molec-

ular state simultaneously through two atomic scatteringchannels. These two channels of different spin combina-tions both contribute to the coupling, but with oppositesign due to opposite Clebsch-Gordan (CG) coefficients.The relative amplitudes of the two contributions dependon the superposition state. The spin portion of a repre-sentative scattering state is shown in Fig. 1 (a), alongwith the relevant molecular potential energy curves plot-ted against the internuclear separation R in units of Bohrradius a0 [23]. Our system exploits the intrinsic quantumnature of PA and our tunable superposition states of thereactant atoms allows us to observe a nearly total sup-pression of the molecular formation, thus representing asignificant step forward for coherent chemistry.

Our experiment begins with a 87Rb BEC of ∼ 1.5×104

atoms in the f = 1 hyperfine state, which is produced viaall optical evaporation in a cross-beam optical dipole trapcreated by a 1550 nm laser. The trap has a characteristictrap frequency ω = (ωxωyωz)1/3 ∼ 2π × (140 × 140 ×37)1/3 Hz = 2π × (90 Hz) [24]. Tuning the magneticfield during the evaporation can lead to a BEC with baremf = −1, 0,+1 spin state, or a statistical mixture of allthree.

After preparing a bare BEC in spin state mf = 0,we can load the BEC into a spin-momentum super-position by adiabatically applying a pair of counter-propagating Raman beams with wavelengths near 790.17nm, see Fig. 1 (b). The Raman beams couple the mf

states, as shown in Fig. 1 (c), and “dress” the atomsinto superpositions of the mf spin states and mechan-

arX

iv:1

807.

1160

8v3

[qu

ant-

ph]

9 A

ug 2

018

2

ical momenta:3∑

i=1

Ci |mf , p〉i = C−1 |−1, ~(q + 2kr)〉 +

C0 |0, ~q〉+C+1 |+1, ~(q − 2kr)〉, where p and ~q are themechanical momentum and quasimomentum along y re-spectively, ~kr is the single photon recoil momentum ofthe Raman lasers with wavelengths near 790.17 nm, and~ is the reduced Planck’s constant. The quasimomen-tum is the canonical momentum conjugate of the posi-tion along y. For more details of our setup, see Refs.[25, 26]. Our Raman coupling scheme is largely simi-lar to previous works [27, 28]. The Hamiltonian for theRaman light-atom interaction is

H =

~2

2m (q + 2kr)2 − δ ΩR

2 0ΩR

2~2

2mq2 − εq ΩR

2

0 ΩR

2~2

2m (q − 2kr)2 + δ

,

(1)where m is the 87Rb mass, δ is the Raman detuning,ΩR is the Raman coupling (calculated from the mea-sured frequency of resonant Raman-Rabi oscillations),Er = ~2k2

r/2m = h × (3.68 kHz) is the recoil energy,and εq = 0.65Er is the quadratic Zeeman shift. Un-less otherwise stated, ΩR and δ carry uncertainties of10% and ± 0.5Er respectively. The eigenstates of Eq.1 are the “dressed” spin-momentum superposition statesdescribed above. The q-dependent eigen-energies of theatoms form the “dressed” bands, with a few representa-tive examples shown in Fig. 1 (d). Red, blue, and greencolors reflecting the proportion of mf = −1, 0, and +1 inthe spin-momentum superposition state. The small dotsrepresent the BECs adiabatically prepared at the bandminimum. Such Raman-coupling has been used previ-ously to induce synthetic partial waves [29], modify ans-wave Feshbach resonance [30], and couple singlet andtriplet scattering states [31].

After preparing our BEC in a statistical mixture ora spin-momentum superposition state, we apply a PAlaser with wavelength 781.70 nm and typical intensityIPA of a few W/cm2 for a time tPA of a few ms [32]. Thefrequency of the PA laser is tuned to the (2)1g excitedmolecular state with vibrational quantum number 152(see again Fig. 1 (a)), and corresponds to the PA lineε in the Fig. 1 of Ref. [33]. After the PA pulse, thePA, Raman (in the case of the superposition state), andthe dipole lasers are simultaneously switched off to allowthe BEC to undergo 15 ms of time-of-flight expansion(TOF). During the initial portion of this expansion, weuse a Stern-Gerlach magnetic field gradient to separateatoms in the different mf spin states. At the end of theTOF expansion, we apply absorption-based imaging toextract the atom numbers in different spin-momentumprojections. Then, this sequence is repeated at variousPA frequencies to obtain a PA spectrum.

To extract the PA rate constant, kPA, due to a PAprocess, we follow a procedure similar to those in Refs.

=

+

+

+

…0+1-1

f =1, mf mf

+1-1

-1 -1

0

>|F=0,mF=0

-1/

+1/

0+1-1

0

-1 -1

+1 -1+1/

+

2C-1C+1-C02 2∝

(b) (c)

q/kr

(a)

-1

0

δħ∆ωR

5P3/25P1/2

~~~

ħ∆ωR +1 δεq

~~

~~~

5P3/2

5P1/2

Energy/E

r

(d)

FIG. 1. Energy level diagrams and experimental setup. (a)Relevant molecular potential energy curves. Depicted at theright is a scattering state of a pair of atoms (the tri-coloredspheres) whose spin part of their quantum state is a superposi-tion of different bare mf spin states. Beneath is the decompo-sition of the scattering state as that of various pairs of atoms(the mono-colored spheres) with bare mf spin states. Thesuperposition coefficients are denoted C−1, C0, and C+1; red,blue, and green represent mf = -1, 0, and +1, respectively.The non-zero CG coefficients for the |F = 0,mF = 0〉 compo-nent of the various pairs of bare spins in the scattering stateare shown near the thin black arrows. (b) Laser geometry.Two Raman lasers with angular frequencies ωR + ∆ωR andωR propagate along ±y and have linear polarizations along xand z. The frequency difference between the Raman beams∆ωR/2π is 3.5 MHz. A PA laser propagates in the x-z plane(due to spatial constraints) and has a linear polarization withcomponents along all three axes. (c) Atomic energy diagram.

A Zeeman bias magnetic field, | ~BBias| ≈ 5 G is used to tunethe Raman detuning δ. (d) Dressed band structures for Ra-man coupling ΩR = 0, 1.1, 3.2, 8.0 and 12Er, all with δ = 0.Dots represent BECs adiabatically loaded to the band minimaat q = 0. Panels (a) to (c) are not to scale.

[34, 35]. The two body rate equation describing the time-dependent BEC density of the atoms participating in PA,ρ(t, ~r), is dρ(t, ~r)/dt = −kPAρ

2(t, ~r). In our experiment,Γstim << Γspon (where Γstim and Γspon are the stimu-lated and spontaneous emission rates, respectively), andkPA has a Lorenztian dependence with respect to the PAfrequency [36]. Solving for ρ(t, ~r) and spatially integrat-ing it over a Thomas-Fermi BEC density profile yields anexpression for the atom number, N(η), remaining after aPA pulse,

3

N/N

0

(c) (d)

Spin Statistical Mixture

Spin-Momentum Superposition

PA off-resonance

PA on-resonance

PA off-resonance

PA on-resonance

mf = -1 0 +1mf = -1 0 +1

OD

x

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OD

x

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OD

x

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OD

x

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

mf = -1 0 +1mf = -1 0 +1

+2kr

0kr

-2kr

+2kr

0kr

-2kr

ODOD

x

y

O.D

.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8

0.6

0.4

0.2

0.0

(a) (b)

FIG. 2. Photoassociation of BECs with atoms in a spin sta-tistical mixture (with Raman coupling ΩR = 0 Er) versus aspin-momentum superposition state with similar atom num-ber and spin composition (ΩR = 8.0 Er and Raman detuningδ = 0.0 Er). (a) and (b): the average optical density (OD)images with PA off and on resonance for the spin statisticalmixture (a) and the spin-momentum superposition state (b).(c) and (d): the extracted normalized atom numbers of themf components and the total of BECs at various PA detun-ings (∆νPA) from resonance for the spin statistical mixture(c) and spin-momentum superposition (d). The atom num-bers of every mf component or the total are normalized bythe corresponding fitted values of the off-resonant atom num-ber N0 and the error bars are the standard error of the mean.Both the OD images and data points are averages of 5 to 7experimental runs.

N(η) =N015

2η−5/2[η1/2 +

1

3η3/2−

(1 + η)1/2tanh−1(√η/(1 + η))],

(2)

where η = kPAρ0tPA is a dimensionless parameter in-dicating the strength of the PA pulse, ρ0 is the peakatomic density at the center of the BEC, and N0 is theoff-resonant atom number with no PA loss. We denotethe resonant kPA for a BEC composed of mf = 0 barespin (or spin-momentum superposition) states as k0,0 (orksup).

First, as shown in Fig. 2, we compared PA of BECsin a spin statistical mixture to that of BECs in a spin-momentum superposition state, with a nearly identicalatom number and spin composition in the two cases.For the spin-momentum superposition state, we usedΩR = 8.0Er and δ = 0Er (see also Fig. 1 (d)). In pan-els (a) and (b), we show the optical density (OD) imagesfor PA both on and off-resonance for the spin statisticalmixture (a) and the spin-momentum superposition state(b). For the spin statistical mixture, the PA-induced loss

for the mf = 0 component is notably larger than thatfor the mf = ±1 components. The lower reduction ofthe OD was due to the lower ρ0 for the mf = ± 1 com-ponents and that each molecule formed by the PA pro-cess reduced the mf = 0 atom number by two, but formf = ±1, only by one each. However, for PA on a spin-momentum superposition state, we observed comparablePA-induced loss among all three mf components. In pan-els (c) and (d), for each mf component and the total, weshow the normalized atom number (N/N0) at various PAdetunings (∆νPA) from the resonance for the statisticalmixture (c) and the spin-momentum superposition state(d). Each PA spectrum for every mf component or thetotal was fitted to Eq. 2 to extract the appropriate N0

and then normalized. The N0 were ∼ (1.2,7.0,1.1)×103

and (1.5,6.9,1.3)×103 for the (mf = −1, 0,+1) compo-nents of the statistical mixture and superposition staterespectively. For the spin statistical mixture, (79 ± 2)%of the mf = 0 atoms were lost on resonance, but lessthan ∼ 25% were lost for the mf = ±1 components. Forthe dressed BECs, all mf components lost (36 ± 2)%.All the data were taken using PA pulses with identicalparameters.

To further explore this phenomenon, we preparedBECs with atoms in several spin-momentum superpo-sition (or mf = 0 spin) states by using ΩR = 0, 1.1 , 3.2,and 12 Er with δ = 0Er (or δ ∼ 100Er) and plotted thenormalized PA spectra in Fig. 3 (a)-(d) using red squares(or black circles). With δ ∼ 100Er, the Raman beamsdid not dress the atoms into spin-momentum superpo-sition states, and these BECs therefore remained in themf = 0 bare spin state and displayed comparable loss(∼ 40%) for all ΩR. However, the loss for BECs in spin-momentum superpositions decreases with increasing ΩR.At ΩR = 12Er, no loss is apparent. For all the data inpanels (a) to (d), we used comparable total off-resonantBEC atom numbers (N0 ∼ (1.1± 0.1)× 104) and squarePA pulses with tPA = 3.2 ms and IPA = 6.0±0.7 W/cm2.

We also fitted photoassociation spectra measured withδ = 0 (or δ ∼ 100Er) to Eq. 2 and extractedksup (or k0,0), and then plotted ksup/k0,0 in Fig. 3(e). We used PA pulses with IPA = 6.1 ± 0.7 W/cm2

and tPA of 2 to 8 ms to induce a repeatable butunsaturated loss of (10 to 40)%. Also included aresolid bands, which are predictions for ksup/k0,0 derivedas follows. The molecular hyperfine state excited byour chosen PA transition has total angular momentumF = 1 and nuclear spin I = 1, and only couples toa pair of colliding atoms (represented by subscripts aand b in the following) whose total angular momentum|F = fa + fb,mF = mf,a +mf,b〉 = |0, 0〉. Using the sin-gle particle basis, |fa,mf,a〉 |fb,mf,b〉, |F = 0,mF = 0〉 =(|1,+1〉 |1,−1〉+ |1,−1〉 |1,+1〉 − |1, 0〉 |1, 0〉)/

√3. Thus,

there are two allowed pathways for the PA transition (af-ter accommodating the indistinguishability of bosons):one in which both atoms have mf = 0 and another with

4

FIG. 3. Photoassociation of atoms in spin-momentum su-perpositions at various values of the Raman coupling ΩR.Panels (a)-(d), normalized atom loss for BECs of atoms inspin-momentum superpositions (or spin state mf = 0) withRaman detuning δ = 0± 0.5Er (or δ > 100Er) are shown asred squares (or black circles). The values of ΩR/Er were: 0(a), 1.1 (b), 3.2 (c), and 12 (d). Error bars are the standarderror of the mean of 6 experimental runs. Panel (e), normal-ized photoassociation rate, ksup/k0,0, for BECs at various ΩR

with δ = 0 Er. The orange (blue) bands are theoretical pre-dictions with (without) the destructive interference term inEq. 3. The band boundaries reflect one standard deviationof the predicted values given our experimental uncertainties.

atoms in mf = ±1. Both pathways contribute to this PAprocess with opposite signs due to the opposite Clebsch-Gordan (CG) coefficients (±1/

√3). We note kPA ∝

| 〈ψmol| ~d · ~E |ψscat〉 |2 with a spin independent propor-tionality factor [37], where |ψscat〉 and |ψmol〉 are the totalwavefunctions for the scattering state and molecular staterespectively, ~E is the electric field of the PA laser, and~d is the dipole operator. The spin portion of |ψscat〉 fortwo Raman dressed atoms (labeled by subscripts a and

b) is+1∑

i=−1

+1∑j=−1

CiCj |f = 1,mf = i〉a ⊗ |f = 1,mf = j〉b.

Using the CG coefficients, the probability amplitude ofthe |F = 0,mF = 0〉 component of the scattering wave-function is therefore (2C−1C+1 − C2

0 )/√

3. We arrive at[38]

ksup/k0,0 = |C20 |2 + 4|C−1C+1|2 − 4Re[C2

0C∗−1C

∗+1]. (3)

Describing the entire superposition BEC with one PArate constant ksup is supported by our observation in Fig.3 that all the bare spin components of the superpositionstate display identical fractional losses. In the limit ofΩR → 0 with δ = 0, ksup/k0,0 → 1 since C0 → 1 andC−1 = C+1 → 0. However, interestingly, ksup/k0,0 → 0for large ΩR because C0 → 1/

√2 and C−1 = C+1 → 1/2

(all the superposition coefficients Ci can be calculated bydiagonalizing Eq. 1). In this case the third term of Eq. 3cancels the first two, thus no molecular formation is pre-dicted even with resonant PA light. This happens despitePA being allowed on both channels for associating twomf = 0 atoms and associating two mf = ±1 atoms (seeFig. 1 (a)). This complete destructive interference comesfrom the opposite CG coefficients (±1/

√3). In Fig. 3,

and also later in Fig. 4, the orange (blue) bands are the-oretical predictions from Eq. 3 that show the range oftheoretical predictions resulting from our typical exper-imental uncertainties, including (excluding) the destruc-tive interference effect, the last term of Eq. 3. The nearlytotal suppression of ksup/k0,0 at large ΩR with δ = 0Er

is consistent with the prediction of Eq. 3 with the de-structive interference term.

We also studied PA on BECs prepared with ΩR =5.4Er and δ from −2.5 to +2.5Er. Figure 4 showsksup/k0,0 vs δ, measured using square PA pulses withtPA = 5.5 ms and IPA = 5.7 ± 0.2 W/cm2. The ex-perimental error bars reflect the aggregate uncertainty ofksup/k0,0. The inset contains calculated band structuresfor δ = −2, 0 and +2Er. Increasing |δ| beyond ∼ ∓ 2Er

polarizes the dressed BEC into majority mf = ± 1 andcollisions between such atoms do not contribute to the|F = 0,mF = 0〉 channel. This is consistent with our ob-served ksup/k0,0 → 0 with increasing |δ|. This suppres-sion is again consistent with Eq. 3 because C0 and oneof C±1 vanish, predicting ksup/k0,0 → 0.

Atoms in spin-momentum superpositions are novel re-actants, and thus PA of such atoms represents a new typeof photochemistry. We interpret observing the same frac-tional loss on all components of a spin-momentum super-position state as an indication that it is the superpositionstate itself that undergoes PA. Further, the different totalfractional loss between BECs of a spin statistical mixtureand a spin-momentum superposition state demonstratesa significant modification to the PA process due to thecoherent quantum superposition. Lastly, we interpret the

5

-2 0 2-4

-3

-2

-1

0

/ = 0 Er

/ = -2 Er/ = +2 Er

+R = 5.4 Er

-2 0 2q / kr

Ener

gy /

E r

FIG. 4. Normalized photoassociation rate, ksup/k0,0, forBECs at Raman detuning δ from−2.5 to +2.5Er with Ramancoupling ΩR = 5.4 Er. The orange (blue) bands are theoret-ical predictions with (without) the destructive interferenceterm in Eq. 3. The band boundaries reflect one standarddeviation of the predicted values given our experimental un-certainties. Inset: dressed band structures for ΩR = 5.4Er

and δ = −2, 0, and 2Er. The dots represent BECs preparedat the band minima.

nearly full suppression of ksup/k0,0 as resulting from de-structive interference between the two out-of-phase path-ways (mf = 0,mf = 0 and the mf = +1,mf = −1). Ourscattering state simultaneously accesses these two path-ways as it couples to the chosen excited molecular state.Thus our observations suggest that scattering states ofatoms in quantum superpositions may offer a powerfulnew approach to coherently control photochemical reac-tions.

ACKNOWLEDGEMENTS

We acknowledge helpful conversations with Su-JuWang and Abraham Olson. We acknowledge NSF GrantPHY-1708134 and the Purdue University OVPR Re-search Incentive Grant 206732 for partial support of ourresearch. D.B.B. also acknowledges a Purdue ResearchFoundation Ph.D. Fellowship. Y.Y. and Q.Z. acknowl-edge startup funding from Purdue University.

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[37] See the Supplemental Information.[38] See the Supplemental Information. In writing Eq. 3, we

have ignored in the Franck-Condon factor the relativemomentum imparted from the Raman beams. For thepurpose of our PA resonance, this is justifiable since thelength scale set by the wavelength of the Raman beamsis ∼ 1µm, but the length scale of the PA process is sig-nificantly shorter, less than 2 nm.