COHERENT EXCITATION AND EVOLUTION OF TWO-PHONON STATES IN MOLECULAR CRYSTALS

HAL Id: jpa-00225080

https://hal.archives-ouvertes.fr/jpa-00225080

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

COHERENT EXCITATION AND EVOLUTION OF

TWO-PHONON STATES IN MOLECULAR

CRYSTALS

F. Vallée, G. Gale, C. Flytzanis

To cite this version:

COHERENT E X C I T A T I O N AND EVOLUTION OF TWO-PHONON STATES I N MOLECULAR CRYSTALS

F. VallBe, G. Gale and C. Flytzanis

Laboratoire drGptique Quantique, EcoZe PoZytechnique, 91128 PaZaiseau Cedex, France

rer la dtat 1

-

Des techniques picosecondes coherentes ont et& utilisees pour mesu- ddpendance en temperature des temps de dBphasage (T2) du trPs faible deux phonons lies 2v2 de l'oxyde nitreux solide et de l'dtat liB com- posite Cv2 + v41} du cristal de chlorure d'ammonium. Ces r6sultats soulignent l'importance de la force de la rdsonance de Fermi pour la dynamique des dtats

1 deux phonons dans les cristaux.

Abstract

-

Picosecond coherent excitation and probing techniques are employed to measure the temperature dependence of the dephasing times (T2) of the very weak 2v2 two-vibron bound states in solid nitrous oxide and of the composite {v2 +

v

~

bound state in crystalline ammonium chloride. These results high-

~

}

light the importance of the strength of Fermi resonance on two-phonon dyna- mics in crystals.

The recent demonstration of coherent excitation of two-phonon states in molecular crystals [l-41 offers some attractive possibilities to selectively study and exploit anharmonic interactions in crystals. Although optical transitions to these states as a general rule are much weaker than to one-phonon states their number is much larger: indeed optically accessible two-phonon states are formed by pairing phonons with opposite wave vectors all over the Brillouin zone in contrast to the one-phonon sta- tes that are restricted in the center of the Brillouin zone. Even so these two-phonon states as a general rule form a continuum over an energy range 2W where

W

is the pho- non branch width and their coherent excitation is in general not feasible. How ver under certain conditions the anharmonic terms of the third and fourth order, he3) and h(4) respectively, in the usual expansion of the lattice hamiltoninan [5-61

cause a spectral condensation into narrow regions[7] leaving behind a residual weaker continuum of the free two-phonon states. Because of their relatively narrow line- width and nonnegligible oscillator strength these local states can be coherently driven with high efficiency [2-31 and their subsequent evolution and decay process observed. Below, we review briefly the main results obtained with the CAHORS techni- que and present some more recent ones in N

₂

0 and NH C1.

4

The above mentionned spectral condensat~on occurs as a consequence of a subtle interplay of the intermole u ar and intramolecular forces. Thus if the strength I? of the, fourth order term hC4j which is mainly intramolecular is larger than the pho- non branch width W a bound two-phonon state splits off the continuum. If a one-pho- non state couples with the two-phonon continuum and is nearly degenerate with such a bound or quasibound two-phonon state then a "~ermi doublet" appears 171 with its two components on either side of a much weaker.residua1 continuum of free two- phonon states [8,9,101.

In the CAHORS technique these "local" states are coherently driven by two time coincident picosecond pulses of frequencies uL and wS respectively with uL-us = Qo,

C7-282 JOURNAL

DE

PHYSIQUE

the "local" compound phonon frequency, and the coherence evolution of this state is followed by delayed antistokes scattering at w + RB of a probe pulse at frequency up. The experimental setup is described elsewhgre [3]; the high sensitivity of our experimental system which allows a dynamic measurement range of

lo9

for a strong Raman scatterer, means that previously inaccessible weak Raman features may now be studied and that the coherent exclusion of narrow Raman lines in a cluttered region of the spectrum may be isolated.

The description of this technique is most conveniently presented in terms of the optically accessible two-phonon coordinate

where q+ and q- are the simple phonon coordinates with wavevector k and - k respec- tively and p is the density matrix operator in the optically accessible tw~-~honon state space. The definitions and properties of this coordinate are given elsewhere

[ I l l .

Ye present below a simple description of the dynamics of the two-phonon states and for simplicity we limit ourselves to the dynamics of the 2v2 state coupled with the v1 state in a crystal with triatomic molecules (CS2, C02, ...) ; this situation actually is quite general andencompasses many other cases with minor changes. In order to include Fermi resonances we assume that the vl single phonon state (state,

1

lo>, coordinate

y1

and frequency wl) and the 2v2 bi-phonon or resonance (state 102>, coordinate Q22 =q2 and frequency R2) interact through the third order anharmonic term

h122 = 8122q1q$ whose only non vanishing matrix element is hf3 = <101hl2102> ; the main effect of the fourth order term h(4) = yq4 is taken into account in the forma-

2

tion of the bound or quasibound two-phonon state but a residual term is still left which can couples the biphonon with the parent free two-phonon states or the low lying intermolecular modes.

The coupling of the above states with the total field E = ELcos(wLt) + EscoswSt

-

in a Raman excitation process, with w

- w S

near resonance, is mediated by the first order Raman tensor of mode 1

.

a$') an3 the second order Raman tensor of order 2

1 - I

a:, through h' =

-

7

ciE2 with ci = ci0 + a l q l

+

;

cii2)qi. The anharmonic interaction between wl and R2 leads to mixing of the wavefunctions and the production of two new states [3] whose frequencies

a+

and R- are given, for small Q2 phonon branch width by a unitary transformation u,

with

cose sine -sin8

case

1 -1

and 0 = - tan [.2@Kwl

-

R$.The coherent amplitudes Q+ and Q- at R+ and R- are dri- 2

ven by Raman interactions A+E2 and A-E' respectively according to the equations

6-

+

r-6-

-

+ Cl2~- = A-E2 with the transformed Raman tensors defined by

the equation also derived in (1).

The solution in time domain of the two equations (la-b) is fairly complex but some insight can be gained by thecr solution in frequency domain. Thus keeping only the term E2 which is near resonance with 52+ and Q- one gets

and similarly for Q-. For

f3m2

-

wl) << 1,(2) reduces to

which is the solution of an harmonic oscillator with constant damping. To a certain entent expression (2) is similar to (3) with an effective damping constant that may strongly depend on frequency. In particular one mode may grow as a consequence of the driving of the other mode. Note that for

y l = y2 one has

r

-

r l

and T' = 0 and the modes amplitudes are completely decoupled harmonic oscillators.

For large coupling B/w -52 >> 1 the w and 52 states are almost completely mixed 1 2

and displaced in energy by approximately1 ?

B.

60th states have comparable Raman strength ; A a A- a a l / n on the order of that of first order transitions.The above

t

situation arlses in C02, the classic example of strong Fermi resonance. For weak coupling and considering wl > Q2, the one phonon state moves up by

8%

- Q

) and the bound state down by the same amount. For the bi-phonon the Raman coupling iecomes

The first term creates two-phonon states directly through the electrical anharmonici- ty and the second indirectly through mechanical anharmonic coupling between wl and

Notice that these two terms may interfere destructively.

02'The amplitudes Q+ and Q- modulate the polarization set up by E and in the probing stage the electromagnetic field of frequency w is scattered off the vibrational overtone shifted in frequency by the compound Pstate frequency, wk = w -0 and

W: = up + Qi the Stokes and antistokes frequencies respectively.

he

~noalinear polarisation source term may be written

in analogy with one-vibron processes.

We hgye studied the coherent relaxation of the 2v2 two-vibron bound state at

C7-284 JOURNAL

DE

PHYSIQUE

molecule NNO or ONN) there are no rigorous Raman selection rules and both the A and F modes should give coherent signals in our experiment, similarly to the observations in spontaneous Raman spectroscopy for the vl one-vibron mode [12].

Fig. 1. Coherent anti-Stokes signal vs probe delay in picoseconds for the 2v2 line of N20 solid at 177 K.

PROBE DELAY (pS)

Fig. 1 shows the observed coherent signal (full circles) plotted on a-logarithmic scale as a function of probe delay in picoseconds for the 2v2 1165 cm mode in solid N20 at 177 K. Due to the weakness of the coherent diffusion the signal around t = 0 is dominated by non-resonant scattering from the cell windows as can be seen from the measurements in an empty cell (open circles

-

dashed line). After the ini- tial fast rise and fall -in this case essentially limited by the system response- the coherent signal exhibits a pronounced dip, near 10 ps probe delay, followed by a broad maximum and subsequent exponential decay with a T2 = 10.9

+

0;Sps.

This behaviour may be explained in terms of the coherent interference of the A and F symmetry components of this line. We may fit the experimental points using the values T (F) = 10.9 ps (determined from the long delay slope), T (A)

-

4

ps, I,(A) = 31

(3)

and the splitting dv = 1.7 cm-l. Although the calculaged curve (full llne, ~ i ~ . ~ l ) is insensitive to the ratio of the aman intensities and the exact value of T2(A) a splitting within 10% of

I

,7 cm-' is required to reproduce the observed oscillation. Only a single, essentially destructive, interference beat is seen because of the large difference between the T2 values of the two closely spaced

A and F components of the 2v2 line.

We have investigated the effect of temperature on 2v down to 77 K. No signifi- cant qualitative change in the shape of the curves is ogserved and quantitatively, only a small increase, of about 30 %, of coherent decay times occurs in the 177-77 K

range. No change in the 1.7 cm-I splitting of the 1165 cm-1 2v2 doublet could be detected in this temperature range.

We may interpret the 2v2 measurement in terms of an intrinsic relaxation mecha- nism. As 2v is only in very weak Fermi resonance with v l , exemplified by its very

2

small Raman line strength, its spectral position is not significantly perturbed and the 1165 cm-I 2v2 doublet is found only about 10 c<' below the top of the v _{+ V2}

processes in solids. We note that a similar result has been observed for the A and

F modes of

both

the 8, and 52- bands of the {v ,2v2) dyad in crystalline C02, gwhere relaxation is always about 1.6 times more elficlent than that of F [13].

Ammonium chloride belongs to the interesting class of crystals whigh exhibit an orderldisorder phase transition. For NH C1 this transition (at T = 242.5 K) is associated with the relative ordering

02

the N H ~ + ions in the cuiic C1- cage. Al- though this crystal is not molecular, the optical vibrations of the N H ~ + ions, like the molecules in a molecular crystal, interact relatively weakly with each other and hence similar considerations to those developed for molecular crystals may be applied. Below TA, in the non-centro-symmetric phase, one observes not only polar two-particle bands but also bound states, enhanced by polariton Fermi resonance with polar internal vibrations of the ammonium ion [14].

We have investigated the temperature dependence of the dephasing time of the

{v

+

v41)

composite bound state at 3070 cm-' which is in strong Fermi resonance wit2 the v3 one-vibron mode.

I Fig. 2. Measured coherent relaxa- tion rates (2/T2) for the {v2 +

v41} composite two-vibron bound state in NH C1 as a function of temperature4 for T<TA (243 K)

T E M P E R A T U R E ( K )

Fig. 2 shows the variation of the measured relaxation rate (2/T ) of {v2 + v

1

between 20 and 180 K. A strong temperature dependence of more tian a factor 410f 7 is observed in this range, in sharp contrast to the results in N20 solid. Although our interpretation of these very new results is not yet definitive we point out the remarkable similarity between the present observations and those for the 52+ line of C02 solid [31 another example of strong Fermi resonance between one and two-phonon states.

JOURNAL

DE

PHYSIQUE

REFERENCES

1

-

C.Plytzanis, G.M. Gale and M.L. Geirnaert, in Applications of Picosecond Spec- troscopy to Chemistry, edited by K.B. Eisenthal (Reidel, Higham, Mass. 1984) p. 205

2

-

M.L. Geirnaert, G.M. Gale and C. Flytzanis, Phys. Rev. Lett. 52, 815 (1984) 3

- G.M. Gale,

P. Guyot-Sionnest, W.Q. Zheng and C.Flytzanis, phys7~ev. LeCt.

14

823 (1984)

4

-

F. VallCe, G.M. Gale and C. Flytzanis to be published in Chemical Physics Letters

5

-

see for instance Lattice Dynamics and Intermolecular Forces edited by S. Cali- fano (Academic Press, N.Y., 1975)

6

- J.H. Reisland, The Physics of Phonons (John Wiley, London, 1973)

7

-

M.H. Cohen and J. Ruvalds, Phys. Rev. Lett.

_-

23, 1378 (1969) 8

- J. Ruvalds and A. Zawadowski, Phys. Rev.

g,

1172 (1970) 9

-

F. Bogani, J. Phys. C11, 1283 and 1269 (1978)

10

-

F. Bogani, and P.R. a v i , J. Chem. Phys.

81,

4991 (1984)

1 1

- G.M. Gale, F. VallCe and C. Flytzanis in Second International Conference on

Time-Resolved Vibrational Spectroscopy, Bayreuth FRG, 1985 (Springer-Verlag) 12 - A. Anderson and T.S. Sun, Chem. Phys. Lett.,

5,

537 (1971)

13 - P. Ranson, R. Ouillon and S. Califano, 5th International Conference on Dynami- cal Processes in Excited States of Solids DPC-85 (Journal de Physique

-

to be published)

14

-

V.S. Gorelik, GIG. Mitin and M.M. Sushinskii, Zh. Eksp. Teor. Fig. 69, 1823