Molecular quantum beat spectroscopy

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15 December 1990

Full length article

Molecular quantum beat spectroscopy H. Bitto a n d J. R o b e r t H u b e r Physikalisch-Chemisches lnstitut der Universitgit Ziirich, Winterthurerstrasse 190, CH-805 7 Ziirich, Switzerland

Received 19 July 1990

The present account of high-resolution molecular quantum beat spectroscopyemphasizesthe versatility of this powerful Doppler-free method. Hlustrations are drawn from work on the six-atomicmolecule propynal carried out in the authors' laboratory. Examples of anisotropic Zeeman, Stark and hyperline quantum beats, of isotropic molecular quantum beats and of multi-level quantum beats are discussed with regard to intramolecular relaxation dynamics and the determination of structural parameters such as nuclear hyperfineconstants, electricdipole moments, spin-orbit matrix elementsand asymmetry-splittings. 1. Introduction

The commonly used frequency domain spectra of atoms and molecules are interpreted in terms of transitions between stationary states or eigenstates ( I k ) ) . Time domain spectroscopy, on the other hand; is a convenient method to characterize nonstationary states ( I V) ) which are described as linear combinations or cohereht superpositions of eigenstates as given by I ~u(t) ) = ~ ak e x p ( - i E k t / h ) I k ) .

( 1)

k

Hence, following excitation of a non-stationary state the contribution of each eigenstate, being subject to a periodic phase change, introduces oscillations into the time evolution which is manifested as an intensity modulation in the time-resolved emission decay. This phenomenon is called quantum beats. Since the modulation pattern is determined by the energies E k of the superposed eigenstates, or more specifically by their energy differences, the oscillatory fluorescence decay can be utilized for time domain spectroscopy. Though the basic concept is simple and coherent superposition is at the heart of quantum mechanics, it was not until the early sixties that theory and experimental techniques were established for time domain spectroscopy in the optical region. In the first experimental demonstration Alexandrov [ 1 ] and, independently, Dodd et al. [ 2 ] used short light pulses 184

generated by shuttered spectral lamps to produce atoms in coherent superposition states showing quantum beats. Since the quantum beat frequencies are much smaller than the optical transition frequencies, quantum beats are essentially not affected by the Doppler effect, It is this very property which makes quantum beats an excellent tool for high-resolution spectroscopy. The aim of early quantum beat experiments was mainly to demonstrate the phenomenon. Only the advent of tunable dye lasers, making available intense and narrow-band light pulses, intensified the interest in quantum beats and many studies on atoms and diatomics have since been published. For a review of these experiments the reader is referred to the articles by Alexandrov [ 3 ], Dodd and Series [ 4 ], and Haroche [ 5 ]. Here, we highlight the experiments by Gornik et al. [6] who carried out the first laser experiments detecting Zeeman quantum beats in ytterbium atoms and the experiments by Haroche and coworkers who observed hyperfine [ 7 ] and fine structure quantum beats in Cs atoms [8 ]. Wallenstein and coworkers [9 ] reported the first quantum beats in a diatomic molecule (I2), while Hese et al. [10] introduced an electric field to produce level splittings for Stark quantum beats in Ba atoms and diatomic molecules. With the introduction of supersonic beam techniques in molecular spectroscopy spectral conges-

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tion in polyatomic molecules could be reduced to such a degree that McDonald and coworkers were able to detect the first quantum beats in a polyatomic molecule in 1978 [I 1 ]. Soon after, several research groups applied this method mainly to explore intra-molecular state couplings and the dynamics of energy redistribution in molecules [ 1216 ]. In this context we refer in particular to the work of Zewail and coworkers [ 13 ] who used picosecond laser pulses to produce coherences between rotational or vibrational states and to the investigations of Lira et al. [ 14 ] and K o m m a n d e u r et al. [ 15 ] on heterocyclic molecules. The work in Zfirich was started about 10 years ago [ 17,18 ]. Inspired by the quantum beat experiments on atoms mentioned above and on N O / b y Brucat and Zare [ 19 ] we began to develop quantum beat spectroscopy for high resolution spectroscopy of polyatomic molecules with the goal to study not only the dynamics but especially the molecular structure in excited states [2029 ]. By high resolution we refer to the frequency domain which should not be confused with high temporal resolution obtained with pico- or femtosecond pulses. Under the latter conditions the beat frequencies exceed the Doppler width so that the corresponding energy differences can usually be measured by conventional spectroscopy. Using illustrations drawn from the work carried out in the authors' laboratory we give an account on high resolution quantum beat spectroscopy. Following a simple theoretical treatment of the fluorescence decay of coherently excited states in section 2, the experimental conditions for detecting quantum beats in molecules are outlined in section 3. In section 4 we discuss quantum oscillations which exhibit an isotropic radiation pattern - often called molecular quantum beats - and in section 5 we consider those with an anisotropic emission behavior, in particular, hyperfine quantum beats, Zeeman, and Stark quantum beats. Section 6 deals with the complex decay of a coherent multi-level system and its information content for determination of statistical properties of molecular energy levels. The article is closed by addressing further developments of time-resolved spectroscopy.

15 December 1990

2. T h e o r e t i c a l a s p e c t s

We consider the four-level system shown in fig. I which represents states of an atom or molecule. The state [g) is the ground or initial state, [a) and [b) closely spaced excited states and If) the final state. A short laser pulse of appropriate frequency and of a Fourier bandwidth larger than the energy splitting Eabwill coherently excite the two eigenstates [a ) and ]b) thus preparing the superposition state I q / ( t = 0 ) ) =Ca l a ) +cb I b ) •

(2)

I f the laser pulse is much shorter than the mean lifetime of the excited states and any precession frequency, the time evolution of the superposition state can be described by I ~ ( t ) ) =ca exp(-iO~at) ]a ) +eb exp(-io~bt) I b ) , (3) where wa=Ega/h and o~b=Egb/h are the optical transition frequencies between the ground state and the Ia ) and Ib ) states, respectively. Assuming a constant laser pulse intensity over the energy range Eab the coefficients ca and eb can be replaced by the transition dipole matrix element ( a I/~ Ig ) and ( b [/t Ig ) , respectively, and after introduction of the phenomenological decay constants Ya for l a ) and Yb for Ib ) we obtain

Ib)

A

~l\

AE=hw.b

If) Ig)

-

Fig. 1. Four level system. States la)and Ib)are coherently excited from a singleground state lg). The coherence is evidenced by an interference effect (quantum beat) when the emission decay to a common final state [f) is observed.

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I ~'(t) ) = (al zzlg) exp[ - (ioga +)'a/2)t] la) + ( b [ # l g) exp[ - (i09b +Tb/2)t] [ b ) .

(4)

The time development of this superposition state monitored via the emission intensity l#(t) is given by the expression

I~q(t) ~ I (fl #1 ~(t) ) 12 ,

(5)

which after inclusion of eq. (4) becomes

lfl(t) ~ I]lfa[2ag e x p [ -- (io) a d-~a/2 )t ] +/Zyb/tbgexp [ -- (itOb + 7b/2) t] [2 ,

(6)

Time (us)

where the matrix elements ( i I# Ik ) =/Z~k. Having taken the modulus, the terms may be arranged as follows

(b)

Ifl( t) ~ I lz~gl211Zfa l2 e x p ( - 7~t) + [/zbg121/~yb12exp ( -- 7bt)

+ 21 [.~ag].~bg~.~fa~.~fbIe x p [ (~a "Jff~b ) t / 2 ] cos (Og~bt + 0 ) ,

(7) where we have introduced the phase angle 0 to maintain the generality of the expression. The first two terms describe the independent (incoherent) decays of the eigenstates Ia ) and Ib ) , respectively, while the last term - the cross or coherence term - describes the oscillatory decay referred to as the quantum beat or interference effect. Eq. (7) reveals the physics of the decay process and shows that time-resolved measurements of the emission permit us to determine small energy differences from the beat frequency tOab~-Eab/h. Fig. 2 exhibits two superimposed quantum beats from two sets of two coherently excited levels in the polyatomic molecule propynal. The example illustrates the high energy resolution of this method which is limited only by the lifetime of the excited states. The two quantum beat frequencies are separated by a mere 200 kHz as revealed by the Fourier transform from the time domain into the frequency (energy) domain. Such spectra provide a convenient way to analyse frequencies and intensities of even complicated molecular quantum beat structures as illustrated with the multi-level quantum beats of fig. 3. Before discussing the different aspects and the versatility of the method we emphasize the fundamental quantum-mechanical nature of the quantum beat 186

10 20 Frequency (MHz)

Fig. 2. Quantum beats in the fluorescence decay of the 5~ band of S~ propynal. The time domain signal (a) is Fourier transformed and the real part of the complex Fourier transform is displayed (b).

phenomenon. In the language of quantum mechanics, the situation of exciting a superposition state I~) is equivalent to that of two states, Ia ) and Ib ) , sharing a photon. If you know the path of the photon, i.e. from which of the two states it is emitted, you loose the interference pattern and if you observe the interference you loose the information about the path. In other words, the operators associated with the path information and the interference phenomenon do not commute [ 30 ]. Quantum beats are the effects of a single particle (atom or molecule) and a single photon scattered by two indistinguishable channels. In order to observe quantum beats an experimentalist must ensure that both channels remain indistinguishable during the measurement process. In a normal quantum beat experiment an ensemble of particles is excited with a laser pulse short

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i

Finally, the method is fundamental, simple, direct, and sensitive.

°-~c~l. I

IIL,II

15 December 1990

(a)

[7.

3. Experimental aspects

0

, 1

, 2

i 3

i 4

T i m e (~s)

(b)

20

40 Frequency (MHz)

60

Fig. 3. Complex quantum beat decay. This fluorescence decay (a) is observed in the same vibronic band as the one shown in fig. 2. The Fourier transform (b) reveals a rather complex spectrum which is due to coherent excitation of at least 7 states.

enough to be considered a J-pulse. This pulse defines the time zero and the signals of the independently emitted photons are accumulated to form the measured oscillatory decay signal. Although a quantum electrodynamic treatment is required for a full understanding [5,31,32], this simple description already demonstrates the strength of the quantum beat phenomenon when applied to molecular spectroscopy. It provides (i) Doppler-free spectroscopy, since only small energy differences are measured, • (ii) high energy resolution that is limited only by the mean lifetime of the coherently excited states, and (iii) spectroscopy which is not subject to saturation effects [33] owing to the fact that the measurement of the beat phenomenon starts after the excitation process.

Although quantum beats of atoms and of molecules are based on the same physical principles the experimental conditions to detect this phenomenon in polyatomic molecules is more demanding than in atoms due to the increased number of degrees of freedom. The rotational and vibrational degrees of freedom dramatically increase the density of states and thus the n u m b e r of transitions. As a consequence of the optical selection rules, many of these states cannot be excited coherently so that a mixture of incoherent and coherent states is initially prepared Since incoherently excited states contribute only to the non-modulated fluorescence, the relative modulation depth of the quantum beats is reduced to such a degree that quantum beat measurements are usually not feasible. In particular, broad-band excitation provided by electron bombardment or sudden impact - often used in atomic quantum beat spectroscopy or in beam foil spectroscopy [ 34 ] - can therefore not be employed for molecules. The selectivity of the excitation process and hence the relative intensity of the beat amplitude is, however, greatly increased by cooling the molecules to low temperatures and by using narrow-band laser pulses. With supersonic jet expansion seeded in a noble gas the molecules are conveniently and effectively cooled to a rotational temperature of a few K. In combination with pulsed laser excitation, a pulsed valve is most appropriate for this purpose. I t produces short pulses ( ~ 2 0 0 ~ts) of high particle density ( l0 is per cm 3) and, operating at moderate repetition frequencies (e.g. 20 Hz) it requires only small pumps to maintain vacuum (~< 10 -4 mbar). Moreover, a pulsed jet offers the benefit of small sample consumption which is particularly important when expensive isotopic species are investigated. Fig. 4 shows the classical experimental arrangement, i.e. the molecular beam, the laser beam and the detector axis are mutually perpendicular. To avoid molecular collisions during the time evolution of the superposition state the laser crosses the molecular beam about 100 to 200 nozzle diameters ( ~ = 0 . 3 m m ) below 187

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V

U ~,f

.

Fig. 4. Schematic diagram of the apparatus. The vacuum charnbcr is surrounded by three pairs of Helmholtz coils. The pulsed molecular beam (from valve V) crosses the laser beam and the detection axis at right angles. Some quantum beat experiments require specific laser and detection polarization (eL and ev) selected with a half wave plate (2/2) o r a polarizer (P), respectively.

the orifice, where the mean distance between particles has become larger than 1000 A. In most cases meaningful spectroscopic measurements are only obtained when the Earth's magnetic field is carefully compensated in order to avoid undesired Zeeman splitting. This is achieved by introducing three paris of Helmholtz coils around the vacuum chamber [ 22 ]. Commercially available dye lasers deliver tunable radiation of 1-2 GHz bandwidth in pulses of ~ 5 ns which is suitable for selective excitation of single rotational states in small polyatomic molecules such as propynal ( H C - C - C H O ) . This bandwidth is adequate for most of the quantum beat experiments described in this article, although it does not provide the ultimately achievable resolution o f 88 MHz (fwhm) given by a 5 ns (fwhm) Fourier transformlimited (FTL) pulse with a gaussian profile. Such pulses are useful in the high resolution experiments described in sections 5.1 and 6 [24,29 ]. The detection system consists of a photomultiplier tube which is connected to a transient digitizer or a digital storage oscilloscope. For the detection of isotropic quantum beats the total undispersed emission is imaged onto the photomultiplier, while in the case 188

15 December 1990

of anisotropic quantum beats a polarizer is added to detect the appropriate polarization component of the emission. Whenever possible, the detection bandwidth given by the convolution of the response functions of the photomultiplier, the amplifier, and the transient recorder (including the trigger jitter) should exceed the Fourier transform bandwidth of the excitation pulse. Moreover, the sampling rate of the transient recorder is required to be at least twice the detection bandwidth since the highest frequency which can be measured (Nyquist frequency) is half the sampling rate. In this case the Fourier transform bandwidth limits the highest observable quantum beat frequencies. The resolution is limited by the natural linewidth of the excited state, but from an experimental point of view, it is limited by the actual recording time T of the fluorescence. The spacing of adjacent frequency points in the discrete Fourier transform Afis given by Af = 1/ T so that for observing a decay with e.g. T = 10 ~ts, the Afvalue is 1043 kHz. The data analysis is carried out with a fast Fourier transform routine. The Fourier transform of a real function, such as the fluorescence decay, is in general a complex function which can be displayed in different ways. We use the real part of the quantum beat signal since it provides narrower lines than power or amplitude spectra. Further technical details of our apparatus, which has been designed for quantum beat spectroscopy on the nanosecond a n d microsecond t i m e scales, are found in recent articles [17,22,24 ].

4. Isotropic quantum beats

4.1. Isotropic quantum beats in zero field A coherent excitation of states with the same angn!ar momentum gives rise to an isotropic radiation pattern of the modulated emission. We illustrate these isotropic quantum beats for the case of interference between mixed singlet-triplet states in molecules. An electronically excited state of a polyatomic molecule can be described as a mixture between a pure singlet state l i ) and a pure triplet state [1) which results in two mixed states

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Ik>=cikli>+Clkll>,

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Ik'>=qk'li>+Clk'll>.

(8)

The states li> and 11> are often referred to as BornOppenheimer (BO) states and ]k> and )k' > as eigenstates of the molecular hamiltonian HMOL= H~o + Hso. The operator Hso describes the spin-orbit interaction. The energy spacing of the two eigenstates expressed in terms of the energy difference of the BO states AEe and the spin-orbit matrix element v~t is [ 22 ] riO&k, = (AE/~ + 4 I r a 12) I / 2

•

(9)

where COkk, is the beat frequency. According to eqs. ( 2 ) - (7) the time-dependent emission intensity I~q(t) of the coherently excited states Ik > and [k' > is then given by

Ifl(t)=C{ICikl4exp(--ykt)+

15 December 1990

termine the upper limit for which a spin-orbit matrix element can be measured by this method. Assuming toF as limiting factor we find toe > took' >t 2v~t/h. Fig. 5 shows an example ofisotropic quantum beats obtained after excitation of propynal into its first electronically excited state S,. Before we discuss this result, some spectroscopic details about our test molecule propynal are in order. Propynal is a planar molecule (cf. fig. 3) which belongs to the symmetry group C, and which can be treated as a nearly symmetric top. The transition dipole moment/~ of the SI,-So absorption is directed perpendicular to the molecular plane (C-type transition). The transition

]Cik' ]4exp(--yk't) (a)

+ 2 ]Cik l2icik ' ]2 exp[ -- (Tk'+ yk, ) t / 2 ] COS( tokk' t) } ,

(10) where ~'k and 7k' are the decay constants of Ik > and [k'>. Geometrical factors of t h e light detection are included in the constant C. The first two terms describe the exponential decay of the eigenstates and the third term the modulation with the modulation depth 21 ct~l 21qk' 12. Since in this two-level system

(b)

[221 Iqk l21qk, 12 = ( Viffho)kk, ) 2 ,

(11)

the emission decay provides n o t only the energy spacings of the two eigenstates by the beat frequency tokk', but also the coupling strength in terms of the spin-orbit matrix element v~tby the beat amplitude. In most cases the interaction of the BO states is relatively weak so that the character o f Ik > is predominantly singlet and that of Ik'> mostly triplet, If, however, a very strong spin-orbit interaction produces a 50: 50 mixture which results in Yk= 7k, = and Iqk 12= ICS~,12, the emission intensity eq. (10) becomes

I~(t)=C{21qkl4exp(-~,t)

[l +cos(tokk, t) ]} . (12)

Under these conditions a complete modulation of the isotropic quantum beat is observed. From the experimental point of view it is evident from eq. (9) that the Fourier width of the laser pulse tOE and the bandwidth of the detector system too de-

.

.

.

.

/\ =4

F=3

(c)

F=2 II I t I

I

t

I

I

I I

I ) t l I I I I

f[ I )

,jvv

I

20

30 Frequency (MHz)

40

Fig. 5. Fourier spectra o f hypertine quantum beats observed in the J' =3, K ' = 1 rotational state of the 6~ band in (a) Earth's magnetic field, in (b) zero field and in (c) a magnetic field of 1.5 G (/~ .LEE). Coherences between the eigenstates Ik, F> and Ik', F> of the four hyperfine states F = 2 , 3, 3, 4 give rise to four quantum beats, two of which are overlapped'in the line at 30.5 MHz [22 l- Each beat is split into 2 F + l componentS~by the magnetic field.

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selected in our example [22 ] is from the rotational state J" = 2, K" = 0 in the electronic and vibrational ground state to the rotational state J, =3, K ' = 1 in the vibronic state 6~ of S~. The spectroscopic notation of this AJ=AK= + 1 transition is ~Ro(2). As will be discussed later, each rotational state is split into four hyperfine components due to the two nonequivalent protons in propynal. The Fourier spectrum in fig. 5b reveals four different beat frequencies which are due to interferences between the hf components of the states Ik, F ) and Ik', F ) . Owing to the selection rule A F = 0 the interference occurs between states with the same total angular momentum F, F = I + I , and therefore the radiation pattern of the quantum beats is isotropic. This type of quantum beats, although without resolved hf components, were the first ones observed in a polyatomic molecule by McDonald and coworkers in their pioneering work [ 11 ]. 4.2. Zeeman-split isotropic quantum beats

In measuring the quantum beats displayed in fig. 5b, a perfect compensation of the Earth's magnetic field was required in order to avoid additional beats due to Zeeman splitting of the hflevels (cf. fig. 5a). When the spectroscopic assignment, i.e. the determination of the total angular momentum F of the individual hyperfine states, is desired, one conveniently achieves this goal by application of a controlled magnetic field ~ that splits each hf beat signal into 2F+ 1 Zeeman components Me [22]. Under these conditions, F is no longer a good quantum number so that for an unpolarized detection of quantum beats the selection rule becomes/IMe= 0. Fig. 5c shows the singlet and triplet hf states Ik, F ) and Ik', F' ) being split into the Zeeman sublevels Ik, F, M r ) and Ik', F', M r ) . Owing to the high resolution only a very weak magnetic field is required, allowing us to treat the field effect by first order perturbation theory. This provides the simple eigenvalue expression for the state Ik, F, M r ) ¢k = ¢k( "~ ----0 ) q" ~BohrgkMFk .~ / h ,

(13)

where the Land6 factor of this state gk = Ictk[2gt given in terms of the BO triplet state g factor. From the energy difference (~k-- (~k' between Ik, F, M r ) and I k ' , F, M F ) and for A M e = 0 we obtain the fie190

15 December 1990

quency splitting of the Zeeman-split isotropic quantum beats &Okk' = Igk--gk' lttBohr~ / h .

(14)

The splittings of the beats shown in fig. 5c are thus proportional to the difference in the g factors of the participating eigenstates. The Land6 factor gt of the pure BO triplet state I1), an important quantity for the description of the molecular structure, is then easily derived with eq. (8) and results in the expression IZXE,gll = 109kk'(gk--gk')l •

(15)

Unless the detuning AE~t of the BO states l i> and I1> is accidentally zero, the value of the Land6 factor gt can also be determined by such quantum beat experiments. However, for the signs of the g factors we need the results of anisotropic Zeeman beat experiments as described in subsection 5.2. Fig. 5c shows the Fourier spectrum of Zeeman-split quantum beats of propynal when a magnetic field of merely 1.5 G is applied. Based on the splitting into 2 F + 1 lines the four states with F = 2 , 3, 3, 4 are identified. In the case of spectral overlap a simple polarization change of the pump laser with respect to the magnetic field axis ~ (II, _1_,or the magic angle 54.7 ° ) facilitates the assignment since this polarization change causes a characteristic change in the intensity pattern of the Me components within an F state. By means of the Zeeman-split quantum beats shown in fig. 5c, an elegant level anticrossing experiment can be carried out which yields directly the magnitude of the spin-orbit matrix elements v~tgiven in eq. (9). By variation of ~ a single Zeeman component M r of the triplet eigenstate Ik' ) is brought into resonance with that of the corresponding singlet state Ik ) . At this field strength the beat frequency reaches a minimum value COkk,( m i n ) = 2ve/h. Since these level anticrossing experiments require variation of ~ over a small low field range (e.g. 0-30 G in the case of propynal) they can be performed without experimental difficulties [ 21 ]. The results of such a level anticrossing experiment are displayed in fig. 6.

t J[t I E N G T H A R T I C L [

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5.1. Hyperfine polarization quantum beats

/ 25

~ 2o

15 I I

lO 0

I

i

5

ll0

Magnetic Field (G) Fig. 6. Zeeman tuning and level anticrossing of the F= 2 quantum beat frequencies shown in fig. 5. At low magnetic field strengths linear Zeemantuning is observed. In the 5 to 10 G range the Mr= - 2 states of the singlet and h'iplet eigenstates Ik) and [k' ) showan anticrossing. Weakinteraction of a third state causes a local anticrossingat 5 G.

An electronically excited state of a closed-shell molecule can be described as a mixture between a pure singlet state and a pure triplet state as given above by eq. (8). The pure triplet state has a magnetic moment and the pure singlet state has a transition dipole moment for the transition to the ground electronic state. Consequently the two mixed states have a non-zero magnetic moment and a non-zero transition dipole moment. If the molecule possesses nuclei with nuclear magnetic moments, then the ifiteraction of the electron and nuclear spins gives rise to magnetic hyperfine structure in both the mixed states. Under these conditions hyperfine polarization quantum beats are observed following coherent excitation of hyperfine components Jk, F ) of the molecular eigenstate Jk) [24]. The intensity I~(t) of the time-resolved emission from the superposition state can be expressed in terms of a geometrical factor which depends on the angle 0 between the vectors of the laser and the detector polarization, and dynamical factors which describe the molecular angular momentum coupling as well as the absorption and emission process [23 ]:

I~(t) o c e x p ( - T t ) [1 +AoP2(cos 0) 5. Anisotropic quantum beats

I f molecular states, split by an internal or external field, are subject to coherent excitation quantum oscillations are observed which show a spatially anisotropic radiation pattern. The symmetry breaking field lifts the degeneracy of the states which differ by the angular m o m e n t u m quantum number ( J or F ) or by the projection (Mj or M r ) onto the field direction. Hence the coupling of the angular momentum of these states to that of the exciting photon gives rise to polarized quantum beats [ 35 ]. Since this type of quantum oscillation vanishes when the emission is integrated over all angles, they can easily be distinguished from isotropic quantum beats such as discussed above.

+ ~ Aj cos(to#t) P2(cos 0 ) ] .

(16)

The decay constant ~,is assumed to be the same for all the hyperfine states of a particular eigenstate, P2(cosO) denotes the second order Legendre polynomial and the sum runs over all possible quantum beats j. This fluorescence decay expression consists of three parts, namely, an unmodulated isotropic part, an unmodulated anisotropic part and a modulated anisotropic part. Again, the molecule propynal with its two nonequivalent hydrogen atoms provides an instructive example for hyperfine polarization quantum beats. In the example chosen [24] the beats originate from a rovibronic state Ik ) with mainly singiet character and a rovibxonic state [k' ) with mainly triplet character (see eq. (8) and fig. 7). The pure triplet ro191

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(a)

010

0'1

A D ( c m -I)

l

)

(b)

L o

io

2'o

go

l~equency(MHz) Fig. 7. Hyperfine polarizationquantum beats observed in the J'= I, K'= I rotationalstateof the 9o~ band. The insertin (a) shows the 'R0(0) transitionin the LIF spectrum, which is split due to spin-orbitinteraction.Excitationof the eigcnstate[k) with predominant singictcharactergives riseto the quantum' beats shown in (a) while excitationof the tripletejgenstatc[k') resultsin quantum beats shown in (b). Note that panel (a) is a scaledimage of panel (b).

tational state, which is described by the quantum number of the rotational angular momentum N and its projection onto the figure axis K, is split into the fine structure levels J = N, N+_ 1 due to spin-spin and spin-rotation interaction [ 36 ]. The nuclear spins of the two non-equivalent protons I~ and 12 interact with the spins of the unpaired electrons and further split the fine structure levels F according to the two-step coupling F~ =J+Ii and F=F1 +12 :(Hund's coupling case b ~ [36] ). Since F is a good quantum number spin-orbit interaction couples singlet and triplet hyperfine levels with the same F. If, as is the case in propynal, the hyperfine interaction neither mixes the fine structure levels nor the hyperfme levels of different Fl, then Ft and ~ are also good quantum numbers. Consequemtly, the hypcrfine levels of the singlet and those o f one fine structure level of the triplet state interact pairwise, i.e. AF= AFI = AJ = 0 , giving rise to two sets ]k, F ) and {k', F ) of four hyperfine levels each, Furthermore, i f the separation 192

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between the eigenstates Ik ) and [k' ) is greater than the laser bandwidth, each state can be selectively pumped and two sets of hyperfine polarization quantum beats can be observed. This situation is shown in the Fourier spectrum of fig. 7 for a singlet and a triplet eigenstate of the 94 band which were excited via the fRo(0) transition. In this case only four of the six quantum beats which arc possible among the four hyperfine levels, are allowed by the selection rule. In both eigcnstates [k) and Ik' ) the same beat pattern is evident and - as expected - the eigenstate with the higher triplet content shows the greater hyperfine splitting. A detailed analysis of these quantum beat spectra provides the coupling mechanism, the hyperfine constants, i.e. the Fermi contact and dipole-dipole constants, and thus a deep insight into the electronic structure of a polyatomic molecule.

5.2. Zeeman and Stark quantum beats In subsection 4.2 we discussed isotropic q u a n t u m beats created in an external magnetic field that splits different molecular cigenStates Ik, F, M F ) and {k', F, M F ) into their Z c c m a n components. Following coherent excitation, interferences arc produced between these eigcnstates subject to A M F = 0 . H o w ever, a different type of q u a n t u m beat can be observed w h e n excitation and detection polarizations

are chosen such that detection of AM~= + 2. coherences becomes possible. These Zeeman beats within an individual eigenstatc Ik ) are of the same nature a s those in atomic spectroscopy or in the spectrosc o p y of molecules in multiplet electronic states. In the case of a linear Zeeman effect, which is appropriate for the small fields applied, the time evolution of the fluorescence intensity is given by [25 ] //7(t) = I p ( 0 ) e x p ( - y t ) [ 1 +A COS(2toLt+ 2~) ] , (17) where o~L=/ZSohrgk~/~ is the Larmor frequency of the excited,eigenstates and ~ the phase angle which depends on the actual geometry of the experimental set up with respect to excitation a n d detection. The Zeeman quantum beats are usually superposed by the much stronger isotropic oscillations but the signal of the latter can bc readily removed by carrying out experiments under different phase angles. Thus,

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subtraction of the signal from a measurement at 0 = 4 5 ° from the signal at - ~ cancels the isotropic part and doubles the signal from the anisotropic beats. Both Zeeman-split isotropic and Zeeman quantum beats provide Land6 factors of zero order BO states such as the pure triplet state, but only Zeeman beat experiments yield the sign of the g factors [251. Quantum beats induced by an external electrical field are in many respects related to Zeeman quantum beats. The interaction of the electrical field ¢ with the permanent electric dipole moment ~ of the molecule splits the Zeeman components of the rotational states and quantum beats can be measured if the excitation and detection polarizations are perpendicular to the electric field. Although Stark quantum beats can be observed in molecules excited to states of any multiplicity, we restrict the following discussion to pure singlet states. The Hamiltonian of a rotating singlet molecule in an electric field is

H=Hrot+He~,

(18)

where / / e l = - ~ ¢ couples the zero field rotational states. The eigenstates in the field are then described in the symmetric top basis IJ K M j ) . Since the electric dipole moment is a polar vector, Zeeman levels having the same absolute Mj value are degenerate. Furthermore, at the low electric field strengths typical for quantum beat experiments, the Stark splittings show a quadratic field dependence and, therefore, 2 J - 1 beat frequencies are in general observed instead of just a single frequency (2to L) in Zeeman experiments. The fluorescence intensity is described by an expression analogous to eq. (17) except that the single beat term is now replaced by a sum over 2 J - 1 terms. Although the beam frequencies ogM~M~=(E(Mj)-E(M'j))/h are no longer given by a simple analytical expression the principal beat pattern can be readily interpreted. As an example we consider excitation of the rotational state J = 2 , K = 1 via the fRo(1 ) transition illustrated in fig. 8. The excitation gives rise to a beat frequency 0920 and another one 090_2= -0920, which due to the IM] degeneracy has the same absolute value but opposite sign. I f the phase angle is 2 0 = 0 ° or 180 ° the beats add up to a single beat at the frequency Ico2l while with 12~1 = 90 ° the beat amplitude vanishes. The third expected frequency o9~_~is zero and, there-

15 December 1990 B~

M 2 1 0 -I -2 :t:2

+1

Fig. 8. Coherences in field-induced quantum beats of a J = 2 state. The laser polarization is perpendicular to the field axis. (a) Linear Zeeman splitting yields 2 J - 1 coherences observed in quantum beats o f a single frequency 2tOL. (b) Quadratic Stark slblitting also gives rise to three coherences, but two coherences oscillate at the same frequency while the third one "oscillates" at frequency zero.

fore, this coherence contributes to the unmodulated anisotropic emission. Stark quantum beats have been used to determine molecular dipole moments [26,37]. If the dipole moment lies along a principal axis of inertia its determination is straight forward by simply choosing a single convenient rotational state for measuring the Stark tuning as a function of 8. In the case the dipole moment does not coincide with one of the principal axes, the Stark tuning of the individual rotational states depends to a different degree on the dipole moment components. Consequently, a set of rotational states has to be selected in order to determine magnitude and orientation of the dipole moment vector in a polyatomic molecule. The complete determination of an excited state dipole moment by the Stark quantum beat technique has been demonstrated with S~ propynal [26 ]. Conventional Doppler-limited Stark spectroscopy merely provides the component /Za of this near-prolate asymmetric rotor molecule due to the fact that the 193

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linear Stark effect is predominant at the high field strengths required to obtairt Stark splittings greater than the Doppler width. The Doppler-free Stark quantum beat method works at low field strengths and provides an accuracy which in the case of propynal has shown to be comparable with microwave spectroscopic measurements of the ground state dipole moment. Apart from dipole moment determination this quantum beat technique can be used to measure energy splittings of quantum states, such as k-doubled or A-split states, which cannot be measured in zero field due to the parity selection rule. In the asymmetry-split states of propynal shown in fig. 9 a coherent excitation is parity-forbidden. The electric field mixes the two states via the dipole component #a. At field strengths as low as 10 V / c m the electric interaction exceeds the asymmetry splitting and linear Stark tuning is in effect. In addition to the Stark quantum beats discussed above, two new strong beats appear which are due to coherences of Mj states of one asymmetry component with M ) states of the other asymmetry component. From these beat frequencies one obtains directly the asymmetry splitting [27 ]. Moreover, since this splitting is deter-

IMI 2

15 December 1990

mined by the rotational constants, three independent asymmetry splittings provide all rotational constants of the molecule in the excited vibronic state. A recent theoretical study gives a detailed account on the full information content of molecular Stark quantum beats [38]. 6. Coherent multi-level excitation

The applications of quantum beats discussed so far were of a spectroscopic nature. They were aimed at the spectroscopist's goal to determine molecular structure parameters and to elucidate molecular state couplings. In this context we considered quantum beats of coherently excited two level systems or of incoherently superposed sets thereof. On the other hand, quantum beat spectroscopy as a time-resolved spectroscopic technique is an invaluable tool for studying most directly the energy flow among excited molecular states [ 39 ]. We applied this method to investigate the dynamics of molecules which are intermediate between the "small molecule case" [ 4042 ] where a single excited state decays radiatively and the "large molecule case" [ 40-42 ] where the excited state decays non-radiatively into a quasi-continuum formed by a dense manifold o f overlapping states. As an extension of eq. (8) the intermediate case is then described by a BO state l i ) coupled to a manifold of N - 1 states { ]1) } which results in a set of N eigenstates

N--I Ik)=Cikli)+ ~ Ctkll).

(19)

I ffi 1

I+)

0

I-) ~

01

t 2

b

10

30 Frequency (MHz)

i 0

i 100 (V/cm)

Fig. 9. Stark quantum beats in closely spaced asymmetry-slalit rotational states. Coherent excitation of Zeeman sublevels of asymmetry-splitrotational levels ( I + ) and I - ) ) is made possible by parity mixingdue to the electric field. In the J' -- 2, K' = 2 rotational states of propynal three types of coherenees are observed (riot) which result in three beat frequencies (left) as, for instance, measured in the vibrationlessSt state of HCCCDO at a field strength of 120 V/era. The difference of the beat frequencies "e" and "a" provide directly the asymmetrysplitting. 194

In the following example Ii ) represents a BO singlet state, ( I1) ) a manifold of BO triplet states, and the initial and final states Ig) and If) are BO singlet states. Therefore, the transition dipole moments ( g l g l l ) and ( f l # l l ) are zero and the time evolution of the fluorescence after coherent excitation of the eigenstates Ik ) becomes in analogy to eq. (10)

Iyt(t)=C(k~ffi, ]Cik'4exp(--Tkt) +

N

~.

21C~kl21C~k ' 12

k=l,k'>k

×exp[ -

(Yk+Tk,)t/2]

COS(tOkk,t ) ) , /

(20)

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where the second term is often referred to as the coherent or fast decay term while the first one is called the incoherent or slow decay term. The fluorescence decay is thus seen to consist of a multiexponential decay with N(N-1 )/2 superposed quantum beats. Furthermore, when the excitation involves different sets (m} of coherently excited states, as is the case in the presence of hyperfine interactions, the total fluorescence is simply the sum of the coherent contributions over all sets i.e. I~q(t)t°t= ~(m~I~(t)". We have studied the dependence of the time-resolved fluorescence la(t) on the number of coherently excited states N in S~ butynal H3C-CmCCHO [28 ]. Without substantially changing the pertinent spectroscopic properties compared to propynal, the methyl group gives rise to an increase of the vibrational degrees of freedom and, in turn, an increase of the number of triplet states interacting with a rovibronic S~ state. Thus, even at low excess vibrational energy, the level density of S~ butynal conforms well to the intermediate case. Increasing the S~ excess energy by selective excitation, the number of interacting triplet states was increased and manifested by an increased number of quantum beats. More importantly, however, was the appearance of a fast decay component at the beginning of the decay as shown in fig. 10. The intensity of this "spike" grew with increasing excess energy, i.e. with increasing number of coherently excited eigenstates. The spike manifests the decay of the coherence in a multilevel system. At t = 0 the laser pulse prepares the quantum beats in phase but shortly after the pulse the quantum beats interfere destructively which results in an initially sharp decay component (second term of eq. (20)). Furthermore, some oscillations come back into phase and recurrence spikes appear superposed on the slow decay component (of. fig. 10a and b). Quantum beats of multi-level systems provide energy level spacings and, therefore, contain information about the level structure e.g. in terms of ergodicity or chaos. An irregular or chaotic level structure exhibits a nearest neighbor spacing distribution which is well described by a Wiener distribution whilst a regular structure is characterized by a Poisson distribution. For Wiener distributed spacings the fluorescence intensity is predicted to show a depression immediately following the fast decay [42,43 ]. This "correlation hole" is the effect of level repulsion in

15 December 1990

(a)

I 0

2

2

4

4 Time (~s)

Fig. 10. Multi-level q u a n t u m beats observed in the S~ state of butynal. These beats are characterized by an initial fast decay and a subsequent slow decay component with superposed recurrences. In the presence of correlation among the eigenstates a "correlation hole" after the fast decay is observed as evidenced in trace b.

an irregular level structure causing small level spacings to disappear. Indeed, such a correlation hole has recently been observed (cf. fig. 10b) in the time-resolved fluorescence of S~ butynal using FTL laser pulses [29]. The presence of correlation among the molecular eigenstates is here most likely due to spinorbit interaction. This observation of a correlation hole in a tithe domain experiment demonstrates that level correlation can be directly observed in the time domain. Although a detailed analysis of the level statistics remains to be done, the results on butynal show that quantum beat spectroscopy may also become a useful tool for elucidating statistical properties of energy levels in polyatomic molecules.

7. Concluding remarks The fact that quantum beats are a single particle effect has two important consequences. First, quantum beat spectroscopy is an essentially Doppler-free technique and second it involves optical preselection 195

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in such a way that only upper sl ates which are dipole connected to a common ground state in absorption and emission give rise to quanlum beats. They measure the energy difference between excited states directly and hence more precisely than conventional spectroscopic methods but they do not provide the exact position of the coherently excited states with respect to the initial state nor their ordering. The latter information may be obtained by other means as has been shown in the case of hyperfine polarization quantum beats [24 ]. Time-resolved spectroscopy yields direct insight into the dynamics of excited molecules. Moreover, the well defined zero time i r quantum beat spectroscopy not only provides the phase of the note and, therefore, additional information on the molecular dynamics [44], but it also opens the possibility to push the limits of resolution beyond the limit of the natural linewidth. This improvement becomes feasible when the emission of the excited molecules is biased in favor of the long-lived species [ 45,46,4 ] as shown in fig. 11. The gain in resolution is in accordance with Heisenberg's uncertainty principle since the natural linewidth is valid only for a nonselected ensemble of excited :molecules decaying by spontaneous emission. In practice, however, improved resolution is gained only at the expense of the signal-to-noise ratio. Monitoring the excited state coherence by spontaneous emission, as described in the reviewed experiments, seems to impose ;timitations on the versatility of the quantum beat technique. Especially, coherence in long-lived meta-stable states, such as excited triplet states or vibrationally excited states in the electronic ground state, is difficult to monitor by this direct method. However, it should be emphasized that any detection technique which keeps the coherently excited states indistinguishable is appropriate. In the past, several different detection methods have been employed in atomic quantum beat spectroscopy and some of them can be utilized for molecules. Most attractive are those which use a delayed second laser pulse. The experiments with two lasers have the advantage that the time resolution is limited only by the duration and the synchronization of the pump and the probe pulses which today can be performed with femtosecond resolution. The delayed probe pulse technique has been successfully 196

15 December 1990

(a)

i

IO

20

30

(b)

i

I0

20 Frequency (MHz)

30

Fig. 11. Example of sub-natural linewidth resolution. (a) Fourier transforraation of the full fluorescence decay results in natural linewidth resolution. (b) Biasing the long-lived fluorescence with a gaussian function (fwhm 5.55 Ixs, maximum at 5.55 ~ts) improves the resolution by a factor of 2.

employed for the detection of coherences by absorption [47 ], stimulated emission [48,49 ], photoionization [50] and by photon echo [51]. We can, therefore, be confident that this variety of techniques will open up time-resolved spectroscopy to a wide range of new applications. Laser spectroscopists will certainly take advantage of both frequency and time domain spectroscopy in order to get a more complete picture of structure and dynamics of polyatomic molecules.

Acknowledgement The support of our work by the Schweizerisches

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Nationalfonds, the Werner Fonds and the Ziircher H o c h s c h u l v e r e i n is gratefully a c k n o w l e d g e d . We t h a n k P.R. W i l l m o t t for critically r e a d i n g t h e manuscript.

References

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