Optics of a Gas of Coherently Spinning Molecules

Uri Steinitz, Yehiam Prior, and Ilya Sh. Averbukh*

Department of Chemical Physics, Weizmann Institute of Science, 234 Herzl Street, Rehovot 76100, Israel(Received 27 August 2013; published 10 January 2014)

We consider the optical properties of a gas of molecules that are brought to fast unidirectional spinningby a pulsed laser field. It is shown that a circularly polarized probe light passing through the medium invertsits polarization handedness and experiences a frequency shift controllable by the sense and the rate ofmolecular rotation. Our analysis is supported by two recent experiments on the laser-induced rotationalDoppler effect in molecular gases and provides a good qualitative and quantitative description of the ex-perimental observations.

DOI: 10.1103/PhysRevLett.112.013004 PACS numbers: 33.20.Fb, 42.25.Ja, 42.25.Lc

When a circularly polarized photon is scattered forwardfrom an anisotropic body, its polarization handedness maybe inverted, and the scattering is accompanied by anexchange of angular momentum ΔL ¼ 2ℏ between thephoton and the body. For a body rotating at a frequencyΩmuch smaller than the light frequency, the kinetic energyof rotation is modified by ΩΔL ¼ 2ℏΩ. This should becompensated by the photon energy change; therefore,the frequency of the scattered photon becomes shiftedby 2Ω. This phenomenon, called the rotational Dopplershift [1–5], is a macroscopic classical analog of the rota-tional Raman effect [6]. It has been observed in the pastby using mechanical rotation of optical elements [7–10]and electro-optic effects in a nonlinear crystal subject toa rotating microwave electric field [11].In recent years, there has been an ever-growing interest in

aligning gas molecules by ultrashort laser pulses (for recentreviews, see Refs. [12,13]; earlier developments aredescribed in Ref. [14]). The research resulted in numerousachievements ranging from differentiation ofmolecular spe-cies through enhancement of filamentation effects to highharmonic generation control and attosecond pulse genera-tion.Severalmethodshavebeensuggestedanddemonstratedfor converting the transient molecular alignment into a con-certed unidirectional molecular rotation, including the tech-niquesof“opticalcentrifuge” [15–17],“molecularpropeller”[18–20], and “chiral train” of laser pulses [21–23].Ultrafast optics of a gas of coherently spinning mole-

cules only recently became a subject of intense experimen-tal investigations [24,25], and the related theoretical studiesare still in their infancy. Our Letter presents the first (to thebest of our knowledge) theoretical analysis of the opticalproperties of a generic nonstationary anisotropic mediumin which both the magnitude of birefringence and the direc-tion of birefringence axis are time dependent. In particular,we investigate the propagation of light through a gas of uni-directionally rotating molecules, analyze polarization andspectral changes due to the energy and angular momentumexchange with the gas, and demonstrate that the light may

experience a THz-range rotational Doppler frequency shiftunder experimentally feasible conditions. Our analysis con-siders in a unified way the two recent experiments on theoptics of coherently spinning molecules [24,25] whoseresults are in accord with our theory. The Letter describesa theoretical framework in which current and future experi-ments regarding these fundamental issues in molecular andoptical physics can be analyzed and understood.Consider an ensemble of unidirectionally rotating linear

molecules prepared by one of the above techniques[15–23]. The simplest of them (“molecular propeller”[18–20]) uses a sequential excitation by two time delayedpump pulses with different linear polarizations. The firstpulse aligns the molecules, and the second one applies abiased torque to them, thus, causing the unidirectionalrotation. We denote the molecular polarizability anisotropyby Δα ¼ α∥–α⊥ (α∥, α⊥ are the polarizability values alongand perpendicular to the molecular axis, respectively).The pump pulses are assumed to be collinear and to propa-gate at the same speed, c=n, as in the undisturbedand unaligned medium. Here, c is the speed of light, n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Nαϵ–10

pis the refractive index, α ¼ ðα∥ þ 2α⊥Þ=3 is

the orientation-averaged molecular polarizability, N theconcentration of the molecules, and ϵ0 is the vacuum per-mittivity. For undepleted pump pulses (as is the case fornonresonant excitation in transparent gases), the rotationaldynamics of molecules at any given distance down thepropagation line depends only on the time elapsed sincethe pulsed pump acted at that same location. Followingthe excitation, the molecules continue their field-free rota-tion, each at a constant angular velocity.The electric field Ei of an incident probe pulse follows

Maxwell’s equations

∂zzEi–1

c2∂ttEi ¼

1

c2∂ttðχ↔EiÞ; (1)

where χ↔is the electric susceptibility tensor of the medium.

In the paraxial approximation, the only significant field

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components are those perpendicular to the propagation axisz. We express the susceptibility in the basis of circularpolarization (CP) states,

χ↔ðz; tÞ ¼ N

�α −

Δα2

�hcos2θ0i–

1

3

��I

þ NΔα2

�G ρ�

ρ G

�;

ρ ¼ hsin2θe2iϕi;G ¼ hcos2θ0 − cos2θi: (2)

Here, I is the unity matrix, and the averaging is made overall the molecules in a unit volume. We use the standardspherical angles θ and φ to denote the instantaneous orien-tation of the molecules (the polar angle θ is measured fromthe propagation direction z). The angle θ0 denotes θ at themoment of the probe arrival at the molecule’s location. Themolecular rotation frequency is assumed to be small com-pared with the optical carrier frequency ωi, thus, allowingthe calculation of the susceptibility by the coordinate trans-formation to the molecules’ rotating frame.In what follows, we consider a delayed probe pulse,

Ei¼ εiðz;tÞexpiωiðt–z=ViÞ, with a slowly varying envelopeεiðz; tÞ (of two CP components) propagating collinearlywith the pumps. We use the first time-independentterm of the polarizability in Eq. (2) to define the velocityvalue Vi ¼ cf1þ ϵ–10 N½α–Δαðhcos2θ0i–1=3Þ=2�g–1=2 thatis the common term for both CP components. We rearrangeEq. (1) by neglecting the second order derivatives and trans-forming to the probe’s time frame ðz0; τÞ≡ ðz; t–z=ViÞ:

∂z0 εi ¼ –iβi

�G ρ�

ρ G

�εi þ 2

βiωi

∂τ

��G ρ�

ρ G

�εi

�;

βi ¼μ0NΔαωiVi

4; (3)

where μ0 is the vacuum permeability. Here, ρ andG (and allother time-varying quantities) depend only on the time τelapsed since the pumps arrived at each location. This is jus-tified as long as the pump-probe speed difference is smallenough so that the probe “surfs” on the seemingly unchang-ing wake of the pump throughout the interaction length, l[i.e., when l ≪ Δtalc=ðNjΔαjϵ–10 Þ, where Δtal is the typicaltime scale of the molecular alignment dynamics].We use a basis transformation R to diagonalize the

matrix that appears in Eq. (3),

�G ρ�

ρ G

�¼�− ρ�

jρjρ�jρj

1 1

��G− jρj 0

0 Gþ jρj

��− ρ2jρj

12

ρ2jρj

12

�

≡RðτÞKðτÞR–1ðτÞ:

Here, again, R and K are functions of τ only. Physically,the transformation R defines the instantaneous birefrin-gence axes of the medium. We present the two componentsof the pulse in the K eigenvectors basis as ðεþ ε–ÞT ¼R–1εi. Equation (3) may be replaced by two scalarequations:

�∂z0–

2βiωi

ðG∓jρjÞ∂τ

�ε� ¼ –iβiðG∓jρjÞε�

þ 2βiωi

½∂τðG∓jρjÞ�ε�

þ iβiΦ:

ωiðG� jρjÞε∓; (4)

where Φ is the argument of ρ≡ jρj exp iΦ. Equations (4)[or Eqs. (3)] describe a rich variety of phenomena in a time-dependent birefringent medium. In particular, the secondterm on the rhs of Eq. (4) is responsible for amplificationor attenuation of the field amplitudes ε� in a nonstationaryanisotropic medium with “nonrotating” birefringence axes.The third term on the rhs of Eq. (4) describes a nonadiabaticcoupling between the ε� amplitudes in a medium withrotating birefringence, which also results in a change ofthe CP amplitudes (see, e.g., Ref. [26]). However, thecumulative change of amplitudes due to these terms is smallas long as the difference in propagation time of the two fieldcomponents through the medium is smaller than themolecular alignment time scale [l ≪ Δtalc=ðNjΔαjϵ–10 Þ],and as long as the frequency of the birefringence axis rota-tion is slower than the optical frequency, jΦ

:j ≪ ωi. Under

these conditions, which prevail in the recent experiments[24,25], we may neglect these terms.The remaining uncoupled equations have a simple sol-

ution, which we combine and obtain the total CP fieldat location z as a function of the incoming field envelopeεiðz ¼ 0Þ:

EiðzÞ≈eiωiτ exp ðiβiGzÞ

×�cos ðβijρjzÞIþ i sin ðβijρjzÞ

�0 e–iΦ

eiΦ 0

��εið0Þ:

(5)

This expression describes a Rabi-like oscillation betweenthe two CP components. We identify the z-dependent phasemodulation terms, especially the exponent containing G.The rotational Doppler shift effect is described by the expo-nents of Φ in the off-diagonal elements of the matrix in theright-hand side of Eq. (5). To validate our solution, we haveanalyzed polarization and spectral transformations of aprobe pulse for various medium-preparation scenarios(see below) and found a good agreement between theresults based on the analytical Eq. (5) and those obtained

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by solving Eq. (4) numerically (finite difference timedomain calculations).Next we discuss the effect of specific molecular excita-

tion scenarios on the probe pulse. We first examine the sim-plest model case in which the molecular orientation isrestricted to be perpendicular to the laser propagation direc-tion (z axis). We assume that when the maximum of theprobe pulse arrives, the molecules are perfectly alignedand have a normal distribution of angular velocity φ

:with

a nonzero average Ω and a standard deviation σ. It can beshown that in this case jρj ¼ exp ð–2τ2σ2Þ, Φ ¼ 2Ωτ, andG ¼ 0. For a left-CP input field εLðz ¼ 0Þ, Eq. (5) providesthe following expression for the generated right-CP field atthe exit of the medium:

E RðzÞ ≈ i sin ðzβie–2τ2σ2Þeiðωþ2ΩÞτεLð0Þ: (6)

The apparent þ2Ω frequency shift in the generated field isinverted when the input field has a right-CP handedness.Moreover, when the angular velocity deviation σ is zero(all the molecules rotate in unison and with the samespeed), Eq. (6) shows a simple Rabi-type oscillation andfrequency shift that is consistent with that of light propa-gating through a rotating wave plate [8,27]. Our simplifiedclassical model of a “molecular wave plate” corresponds toexperiment [28], in which highly efficient single-sidebandfrequency conversion was indeed observed via coherentdriving of a single rovibrational Raman transition inmolecular deuterium.We proceed now to a more sophisticated 3D model of

molecules brought into fast unidirectional rotation by a pairof delayed pump pulses according to the “molecular pro-peller” scheme [18–20]. To calculate the ρ and G time-dependent values, we used both classical Monte Carlosimulation [20] (for short pump-probe delays) and fullquantum mechanical calculation (when the delay becomescomparable to the molecular rotational revival time). In thelatter case, we analyzed the rotational dynamics of quantumwave packets and averaged over the thermal distribution ofthe initial rotational states (similar to Ref. [18]).Figure 1 depicts the calculated frequency content of the

output pulse of inverted circular polarization for differenttime delays between the CP probe and the last pump pulse(left panels), as compared to the recent experiment [24](right-hand side). This example corresponds to nitrogenmolecules of air at ambient conditions, and the delaysare chosen around the half-revival time of the nitrogen,Trev ¼ 8.3 ps [14]. The kick strength [29] of each of thepump pulses is P ¼ 5, close to the estimated experimentalvalue. Figure 1(a) shows a spectrogram for a control case inwhichonlya singlepumppulse is employed.Thevertical linemarks the central wavelength of the incident pulse. A con-siderable spectral broadening is observed in the half-revivalregion (both in simulations and in the experiment) becauseofthe phase modulation effect, but no visible spectral shift is

present. However, if unidirectional rotation is present (seethe middle and the lower panels of Fig. 1), the spectrogramsare shifted to the red or to the blue, depending on the relativehandedness of the probe field polarization and inducedmolecular rotation. In our simulation, the rotationalDoppler shift reaches the level of about 5 THz. For an800 nmprobe pulse, this translates to awavelength deviationof about ∼10 nm, close to the spectral width of the pulsesused in the experiments [24]. The predicted value of the shiftand its direction are in a good agreement with the reportedexperimental observations [see Figs. 1(b) and 1(c)].As an additional and independent test of our theory, we

consider another recent experiment [25], in which unidirec-tionally rotating O2 molecules were produced using the“optical centrifuge” scheme. In that experiment, moleculeswere optically spun at a continuously increasing rotationalfrequency, up to J ∼ 69ℏ, after which the centrifuge wasabruptly switched off (only odd values of J are allowedfor 16O2 molecule because of its nuclear spin statistics).The optical properties of the molecular medium wereprobed by a delayed CP probe pulse during the centrifugeoperation and after its termination.To model this situation, we analyzed the time depend-

ence of the functionsG and ρ during the acceleration period(when the molecules are trapped by the centrifuge field)both classically and quantum mechanically, with a goodagreement between the two approaches. The long-time

FIG. 1 (color online). Spectrum of the output field of invertedcircular polarization vs pump-probe delay for 14N2 moleculesbrought to unidirectional rotation by the two-pulse “molecularpropeller” scheme. Color depicts the magnitude of the signal (ar-bitrary scaling). The left column shows calculated spectrum; theright one presents experimental data from Ref. [24] (with permis-sion). The central wavelength of the probe is 792 nm, indicatedby a vertical line. Panel (a) displays results for a control single-pulse excitation for which no chiral effect exists. Panel (b) cor-responds to molecules having the same sense of rotation as theelectric vector of the input CP pulse. Panel (c) presents results forthe opposite sense of molecular rotation.

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evolution of these functions after the driving field was cutoff and acceleration had stopped was treated quantummechanically in order to account for quantum revivals[14] of the released rotational wave packets. Figure 2(a)shows our results for the spectrum of the oppositely circu-larly polarized signal at the output of the medium as a func-tion of the probe delay with respect to the start of thecentrifuge. The calculated signal shows good qualitativeand quantitative agreement with the results of the reportedexperimental observations [25] [see Fig. 2(b)]. At the accel-eration stage, it exhibits linearly growing frequency shiftcorresponding to twice the instantaneous rotation fre-quency of the centrifuge. In the field-free regime (afterthe end of the centrifuge at ∼60 ps), the excited rotationalwave packet produces a revival oscillatory pattern of thesignal strength, as shown in the insets of Fig. 2. The revivalperiod changes with time because of the centrifugal distor-tion effect in the molecular rotational spectrum. The exactposition of the revival peaks depends on the relative phasesaccumulated by different rotational states during the cen-trifugation and in the process of the release from the cen-trifuge [25]. We matched our results to the experiment bytuning the phases of different states of the final rotationalwave packet to fit the position of a single, arbitrarily chosen

peak of the signal experimentally recorded near ∼300 psdelay. As a result, a very good agreement between the cal-culated and measured revival signals has been achieved inthe whole postcentrifugation region [compare the insets ofFigs. 2(a) and 2(b)].Finally, we discuss an additional manifestation of the

chirality transfer from the molecular medium to a probelight, when the input probe is linearly polarized. The lattermay be regarded as a pair of circularly polarized pulses ofopposite handedness. When the probe encounters a gas ofunidirectionally rotating molecules, each of these CP com-ponents partially transforms into a “daughter” CP pulse ofopposite chirality. The two mirror CP daughter pulsesappear in phase; they have opposite rotational Doppler fre-quency shifts but equal amplitudes; thus, they effectivelycombine into a linearly polarized field. However, the polari-zation direction determined by the relative phase betweenthe two pulses rotates with time at twice the mean fre-quency of the molecular spinning, resulting in a wave ofautonomously rotating linear polarization (WARP) featuredin Fig. 3. This kind of polychromatic light has been createdin the past [30] using a mechanically rotating wave plate.The mechanisms considered in our Letter allow for gener-ating the WARPs with the polarization rotation frequencywhich is many orders of magnitude higher.In conclusion, we presented a theoretical analysis of the

optical properties of a gas of unidirectionally rotating mol-ecules and showed that such a coherent collective spinningsubstantially shifts the spectrum of a CP light pulse passingthrough the medium. The direction of the shift is deter-mined by the relative handedness of the molecular rotationand the CP pulse, and the shift value is controlled by therate of molecular rotation. Our treatment considers in a

FIG. 2 (color online). (a) Calculated spectrum of the outputfield of inverted circular polarization vs probe delay for 16O2

molecules spun by the “optical centrifuge” to J ∼ 69ℏ. (b) Thecorresponding experimental results (from Ref. [25], with permis-sion). The insets show the signal intensity (a.u.) after the end ofcentrifugation.

FIG. 3 (color online). The electric field of the WARP pulse(arbitrary field units, projections in a lighter shade, color repre-sents the instantaneous linear polarization angle) resulting fromthe propagation of a linearly polarized probe through a gas ofunidirectionally rotating nitrogen molecules. The simulationwas performed assuming a double-pulse “molecular propeller”excitation scheme at atmospheric conditions.

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unified way two recent experiments on the optics of gaseswith unidirectionally spinning molecules [24,25] and provi-des a good qualitative and quantitative description ofthe measurement results. We also discussed using themolecular-induced frequency shift to prepare a pulse whoselinear polarization continuously rotates at a THz-rangespeed. This “twisted polarization” mode may become aninteresting and useful addition to the gallery of femtosecondshaped pulses, especially in view of high and tunable rota-tion frequency and a relatively simple preparation method.

We appreciate useful discussions with Johannes Floß,Erez Gershnabel, Robert J. Gordon, Yuri Khodorkovsky,Omer Korech, Aleksey Korobenko, Alexander Milner,Valery Milner, and Moshe Shapiro. This work was partlysupported by the Israel Science Foundation (ISF Grant No.601/10) and the Deutsche Forschungsgemeinschaft (DFGGrant No. LE 2138/2-1). Y. P. acknowledges support asthe Sherman Professorial Chair. I. A. acknowledges supportas the Patricia Elman Bildner Professorial Chair and appre-ciates hospitality of the Peter Wall Institute for AdvancedStudies (UBC, Vancouver). This research was made pos-sible in part by the historic generosity of the HaroldPerlman Family.

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