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PHYSICAL REVIEW A OCTOBER 1997VOLUME 56, NUMBER 4

Molecular emission tomography of anharmonic vibrations

L. J. Waxer,1 I. A. Walmsley,1 and W. Vogel21The Institute of Optics, University of Rochester, Rochester, New York 14627

2Fachbereich Physik, Universita¨t Rostock, Universita¨tsplatz 3, D-18051 Rostock, Germany~Received 25 June 1997!

We describe a method for the determination of the quantum state of molecular vibrations in a regime wherethe anharmonicities are significant. The method is based on the direct analysis of the observed time-resolvedspectrum of resonance fluorescence of the molecular sample.@S1050-2947~97!50810-9#

PACS number~s!: 03.65.Bz, 42.50.Vk, 33.80.2b, 42.50.Dv

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Phase-space tomography provides a means for the msurement of the quantum state of massive particles movinharmonic potentials. Tomographic reconstruction is basedthe fact that the time-dependent position distribution sampover a single period of the motion contains complete infmation on the quantum state. This property of harmonic stems was first noted in the context of measurements ofradiation field@1#, and quantum state reconstruction via tmography was in fact first demonstrated for a quantum fiof massless particles by Smitheyet al. @2#, who used theinverse Radon transform to reconstruct the Wigner functof a radiation mode from the phase-sensitive field amplitudistributions measured by balanced homodyning. The unirelationship between the Wigner function and the denmatrix in a field-strength~or rotated quadrature! basis allowsthe latter to be obtained by a Fourier transform.

Phase-space tomography has also been applied to restructing the quantum state of molecular vibrations@3#. Inmolecular emission tomography~MET! the time- andfrequency-resolved fluorescence spectrum of the molecuthe quantity that yields information on the quantum st@4,5#. The spectrum can be related to the time-dependposition distributions ~the particular analog of phasedependent quadrature distributions of the field!, provided theanharmonicities and the difference of vibrational frequencin the two electronic states of the molecule are small. Unthese circumstances the inverse Radon transform also apto the reconstruction of the vibrational quantum statesmolecules. The unavoidable uncertainty inherent in themultaneous measurement of two classically incompatquantities means that reconstruction is limited to a smootversion of the Wigner function.

Apart from the well-known inverse Radon transformthere have been numerous efforts to simplify the reconsttion procedure, primarily for the radiation field. The mesured quadrature distributions are related by a twofold Frier transform ~for which simple and efficient alogrithmexist! to the density matrix in a field strength~rotatedquadrature! representation@6–8#. Reconstruction procedurehave also been developed for the density matrix in the nber representation@9#. In this case, sampling functions aused to map the measured data onto the density matrix@10#.The two-time correlation functions of non-stationary ligfields have been measured by this method@11#. Anothermethod, which reconstructs the quantum state by measu

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its number-state statistics after introducing prescribedplacements@12#, has recently been applied to the motionquantum state of a trapped atom@13#.

All of these approaches apply to the reconstructionquantum states of harmonic oscillators. However, maphysical systems are only approximated by motion in hmonic potentials. In the specific cases of molecular vibtional and Rydberg electronic wave packets, for exampeffects of anharmonicities play an important role in the dnamics. Since nonclassical states have been prepared inanharmonic systems@14–16#, it is necessary to developmethods for reconstructing quantum states that do not relythe unique properties of harmonic potentials. It has beshown quite generally that reconstruction of the density mtrix in a number-state representation~i.e., in the basis ofenergy eigenstates of the potential! from the complete time-dependent position distribution is possible even in the anhmonic case@17#. The difficulty in applying this method to thereconstruction of quantum states of molecules resides infact that it requires the desired time-dependent positiontributions to be measured with both precise temporal aprecise spatial resolution, and this isa priori impossible us-ing time-resolved spectra.

Other approaches to reconstructing the quantum statemolecular vibrations also require precise measurements.example, the method of Shapiro@18# is based on measurements of the nondispersed, time-dependent fluorescenctensity and the time-integrated, dispersed fluorescence strum sampled with both precise temporal and precise speresolution, respectively. Because this technique requiresinversion of both linear equations of high dimension amatrices that may be singular, it is unclear how the unavoable imprecision of laboratory measurements will affectquality of the reconstruction.

In this paper we propose a reconstruction method tallows us to reconstruct the density matrix of a system evoing in an anharmonic potential directly from measurabquantities. We apply it to the case of molecular vibratioand take the data to consist of a set of time-resolved, sptrally dispersed fluorescence measurements. Thus, aMET, our method makes use of the full information inherein the time- and frequency-resolved fluorescence signal.method is demonstrated for realistic anharmonic potentithe vibrational mode of the Na2 dimer.

R2491 © 1997 The American Physical Society

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RAPID COMMUNICATIONS

R2492 56L. J. WAXER, I. A. WALMSLEY, AND W. VOGEL

The time- and frequency-resolved molecular fluorescesignal measured in the laboratory can be written as a linfunctional of the density matrix. This expression providedirect relationship between the measured signal and thesity matrix elements we seek to reconstruct, and it will bestarting point for the reconstruction method. We takelight field emitted by the molecular sample to be represenby source-field operators, which are related to the multile~vibronic! transition operators of the given molecule accoing to @7#

Es~1 !~ t !5(

knf knA 1 2

kn ~ t2r /c! . ~1!

The operatorA 1 2kn transforms the Born-Oppenheimer sta

u2,n& to the stateu1,k& at transition frequencyv 2 1nk , where

1,2 denote the electronic states andk,n the vibrational statesin the corresponding electronic manifolds. The coefficief kn are of the formf kn}(v 2 1

nk )21^kun&2 where 1^kun&2 are

the Franck-Condon overlaps of the vibrational wave futions in the two electronic states, and the proportionality ftor is not needed for the following. The time-delayr /c de-scribes the retardation. Note that the free-field operatornot contribute to the measured spectrum, since the fieltaken to be initially in the vacuum, and the detected signaa normally ordered expectation.

The molecular fluorescence is resolved in both time afrequency. Thus the source field passes a spectral filter atime gate before detection. The negative frequency parthe source field operator at the detectorEHB

(2) is therefore@4#

EHB~2 !~ t !5E

2`

t

dt8H* ~ t2t8,V!B* ~ t8,T!Es~2 !~ t8! . ~2!

where H and B are the response functions of the spectfilter and the time gate, respectively. The spectral filtertaken to be a Lorentzian characterized by its settingquency V and bandwidthg. For the time-gate functionB(t,T) we use a Gaussian of durationG21 that is peaked at5T, such thatg!G.

A photodetector records the time integrated intensitythe filtered field. Considered as a function of both the settfrequency and the time, it represents a frequency- and tiresolved spectrum,

S~V,T!5E2`

1`

dt^EHB~2 !~ t !EHB

~1 !~ t !&. ~3!

Eventually we discard damping effects on the time scaleinterest, so that the time dependence of the transition optor A 1 2

nm is determined by its free evolution. The moleculein a state in which the vibrational and electronic degreesfreedom are entangled. Because we detect only spontanemission, the excited electronic state component is projeout, and the time-dependent spectrum reflects only the vitional dynamics in this state. Based on these assumptionsstraightforward to derive an explicit expression for the msured fluorescence spectrum. Combining Eqs.~1!–~3!, aftersome algebra we arrive at

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S~V,T!5K(n,m

rnm(k

f knf km* exp~2 innmT!

3g~V2v 2 1nk !g~V2v 2 1

mk ! , ~4!

where rnm is the vibrational density matrix in the exciteelectronic stateu2& and nnm is the difference of vibrationafrequencies in the excited electronic state. The blurring fution g is given byg(v)5exp@2v2/4G2# and the unimportantconstantK specifies the correct units of the spectrum.

Equation ~4! provides a direct relationship between thmeasured signalS(V,T) and the elements of the densimatrix. An important feature of this relationship is that fany given spectral filter setting frequencyV, the densitymatrix elementrnm contributes to the detected signal onlythe frequency componentnnm . Thus, one can identify theamplitude and phase of each of the density matrix elemeby taking a Fourier transform ofS(V,T) at one particularvalue of V and identifying the appropriate amplitude. Taccomplish this requires both an extremely long time seand extremely fine time resolution, so that the maximufrequency of oscillation associated with the mooff2diagonal density matrix element can be resolved. Indition, if there are degeneracies, which occur where beatquencies between different pairs of levels are identical, tone will not be able to distinguish the contributions to tsignal from the degenerate density matrix elements. Evethose cases in which there are degenerate frequencies anwhich long time series are not available, it is still possiblereconstruct the density matrix elements by making use ofadditional information provided by taking several time serat different frequency filter settingsV; this is the basis of ourreconstruction algorithm.

A small, easily invertible linear system can be developfrom Eq. ~4!. A time series is obtained by sampling thfunction S8(V,T)5S(V,T)G(T;t), where G(T;t) is atruncating function of durationt. The Fourier transform ofthis series with respect toT is

S8~V,n!5K(n,m

rnm(k

f knf km* g~V2v 2 1mk !

3g~V2v 2 1nk !G~n2nnm! . ~5!

For a particular value ofn, chosen typically to lie near amaximum ofG(n2nnm), there will be severalrnm that con-tribute to S8(V,n) because of degeneracies or becauset isnot long enough to allow the separation of individual befrequencies.

In either case, it is possible to resolve the contributioneach density matrix element to the time-series spectrumfrequencyn by forming a linear system consisting of a setdifferent time series, each associated with different valuethe frequency filter settingV. This system can then be inverted to find the particular contributing density matrix ements. We define a column vectorS with elementsSj suchthat

Sj5 S8~V j ,n! . ~6!

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RAPID COMMUNICATIONS

56 R2493MOLECULAR EMISSION TOMOGRAPHY OF ANHARMONIC . . .

This column vector is chosen to have as many elementN~corresponding to choosingN different values ofV! as thereare possible density matrix elements that contributeS8(V,n) at that particular value ofn. This number will de-pend on the length of the sampling window compared wthe minimum inverse energy-level spacing of the system,form of the sampling function, and the signal-to-noise raTypically it is a number on the order of 10. Let the set ofNdensity matrix elements contributing bernimi

. Then

S5FR , ~7!

where the matrixF has elements

Fi j 5K(k

f knif kmi* g~V j2v 2 1

nik !g~V j2v 2 1mik ! , ~8!

and the column vectorR has elements

Ri5rnimiG~n2nnimi

! ~ i 51, . . . ,N! . ~9!

Provided that the inverse ofF exists, then this system ieasily inverted to yield the density matrix elements in E~9!. It has yet to be established thatF is not singular. In mostsituations of relevance to experiment there are sets of paeters for which it is not, and an inverse may readily be fouOne requires in particular that either the Franck-Condontors for the pair of statesu2,ni&, u2,mi&→u1,k& be signifi-cantly different from those for statesu2,nj&, u2,mj&, or thatthe optical frequencies of the corresponding transitionsquite different. These criteria are usually met in even weaanharmonic systems.

There are always degeneracies at dc, however, sinceis the frequency at which all the diagonal matrix elemecontribute to the signal. To obtain these from a time serieimpossible. However, they are easily obtained from the tiintegrated spectrum in the manner described by Shapiro@18#.

As an example, we show a simulated reconstruction ofreduced density matrix for a molecular vibration in tA1Su

1 state of the sodium dimer, measured by its fluorcence to theX1Sg

1 state. To demonstrate the robustnessthe method, we apply it to a quite complicated mesoscoSchrodinger-cat state, simulated by superposing tcoherent-state wave packets that are separated by one-hvibrational period. The mean vibrational quantum numbereach wave packet was set atn59, and about eight statewere populated around this quantum number. The tempresolution was taken to be 20% of the classical vibratioperiod ~310 fs!, or G50.0167 fs21, and the total samplingtime was t523 ps, corresponding roughly to the onquarter fractional revival time. In this case we used up todifferent time scans~each corresponding to a particular valof V!. In order to insure that the matrixF was not singular,the values ofV were chosen to be equally spaced acrossfluorescence spectrum of the molecular wave packets~i.e.,between 630 nm and 810 nm, corresponding to the classturning points for this energy!. In this particular case thewave packet occupied few enough states that there werdegeneracies. The minimum sampling timet was chosenbased on the largest number of fluorescence frequenciescould be used without the matrixF becoming overly singu-

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FIG. 1. Modulusurnmu of the reconstructed density matrix of thmesoscopic Schro¨dinger-cat state.

FIG. 2. Relative errore i j , as defined in Eq.~10!, of the recon-struction of Fig. 1.

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RAPID COMMUNICATIONS

R2494 56L. J. WAXER, I. A. WALMSLEY, AND W. VOGEL

lar. Note that, in contrast to the reconstruction method@17#, it is not necessary to use a sampling time as long ashalf revival.

Figure 1 shows the modulus of the density matrix ements, in the basis of the vibrational eigenstates of the mecule, reconstructed from simulated time series data uthe linear system method. A relative error for the reconstrtion may be calculated in the following manner:

e i j 5ur i j act2r i j rcn

u/max@r i j act# . ~10!

Here,r i j actandr i j rcn

are the actual and reconstructed densmatrix elements, respectively. The size of the error is calated relative to the largest density matrix element,r99act

inthis case. Figure 2 shows that the errors are on the ordeonly 1%. The reconstruction can be accomplished even wnoisy data by taking more spectral samples and/or sampfor longer timest.

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In summary we propose a method for state reconstrucof the vibrational mode of an anharmonic oscillator fromtime-dependent spectrum of fluorescence. This methodquires a knowledge of the Franck-Condon factors betwthe initial and final electronic states of the molecule, but donot require a detailed knowledge of the potential. It takinto account the blurring effects due to limited time and frquency resolution, and allows one to reconstruct the denmatrix elements directly from a signal measured in the laratory. The reconstruction uses a small linear system, anquite robust and relatively insensitive to amplitude noisethe measured signal. Further, it is possible to reconstcomplicated quantum states exhibiting extreme formsquantum interference, whose Wigner functions are negaover significant regions of phase space.

This work was supported by the National Science Fodation and the Deutsche Forschungsgemeinschaft.

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