Sound amplification by stimulated emission of phonons using two-level systems in glasses

Sound amplification by stimulated emission of phonons using two-level systems in glasses

ELSEVIER Physica B 219&220 (1996) 235-238 Sound amplification by stimulated emission of phonons using two-level systems in glasses J.-Y. Prieur a'b'...

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ELSEVIER

Physica B 219&220 (1996) 235-238

Sound amplification by stimulated emission of phonons using two-level systems in glasses J.-Y. Prieur a'b' *, M. Devaud b, J. Joffrin a, C. Barre a, M. Stenger a, M. Chapellier a'~ aLaboratoire de Physique des Solides, Bat. 510, Universitb Paris-Sud, 91405 Orsav Cedex, France hLaboratoire d'Acoustique et d'Optique de la Matibre Condensbe, Universitb Pierre et Marie Curie, Tour 13, 4 place Jussieu, 75252 Paris Cedex 5, France cService de Physique de l'Etat Condensb, CE Saclay, 91191, Gif sur Yvette Cedex, France

Abstract We discuss different possibilities to amplify an ultrasonic wave in a glass by inverting the populations of the two-level systems: we show that the only convenient one is the so-called "rapid adiabatic passage". We describe different steps of the procedure needed to apply it to a Tetrasil glass and the experimental results are presented. Emphasis is laid on the differences between our approach and Tucker's experiment. Finally, the actual limitations are discussed and some means to overcome them are proposed.

Glasses have been the subject of many studies for twenty years. It is now well admitted that their lowtemperature properties can be satisfactorily explained within the "two-level system" model (TLS) originally developed by Phillips and Anderson et al. [1,2]. The model postulates the existence of localized entities, the TLS, with two energy levels. The spectral density of their energy splittings is flat and extends to a relatively large value. The exact nature of these two levels is not known yet. Among the low-temperature properties, here we are interested in the interaction between TLS and acoustical waves. It is well known that any TLS can be described as a fictitious 1/2 spin in a fictitious static magnetic field.

*Corresponding address: J.-Y. Prieur, AOMC, Tour 13, Universit~ Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 5, France.

The Hamiltonian H of one TLS with Larmor frequency O~o submitted to an acoustical strain e reads

h(290

H = ---f-az + 7x ecrx.

Here 7x is a coupling constant and ax, 6= are Pauli operators. We have neglected in H a term, ~:a=, since it gives negligible effects at the temperature T we are working with (T < 100 mK). The interaction term leads to the absorption and the emission of one p h o n o n with a simultaneous flip of a TLS from one state to the other. The probability of occurrence of either event depends on the average population of the assembly of TLS for the initial state. The m~ component of the pseudo-magnetization is a measure of the difference of these probabilities. In thermal equilibrium the lower state is more populated than the upper state, mz is negative and absorption dominates stimulated emission; the acoustical wave is attenuated.

0921-4526/96/$15.00 3"', 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 7 0 6 - 7

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J.-E Prieur et al. / Physica B 219&220 (1996) 235 238

Quantitatively, the attenuation coefficient for small amplitude waves is given by :

--eomz rtCv2 b'

with m= = - t a n h

and

,

z

i.- ~ . . . . . . . .

2~T '

%4

-

-~

3

Pz'~

C = --

p v 2"

Here P, p, v, eJ, respectively, represent the spectral density of TLS, mass density of the glass, sound velocity and angular frequency of the wave. If m~ happens to be reversed, attenuation will turn into amplification. Such an inversion of population has already been realized by Tucker [-3] in a ruby crystal where ultrasonic waves are in resonant interaction with true spins when a static magnetic field is used to tune the narrow band spin splitting to the frequency of the wave. In that particular case, there were four levels available (1-4, 1 being the highest). Levels 2 and 3 were coupled via the p h o n o n field, while 1-3 and 2-4 were coupled to an electromagnetic field; moreover, the splittings 1-3 and 2-4 were identical. Therefore, the same field could saturate these two transitions. After saturation, level 2 was more populated than level 3 and p h o n o n amplification resulted. Since this experiment, no other amplification by a similar process was ever made in the acoustic domain. However, phonons have been generated, probably incoherently, by C h a n n i n et al. [-4] in a KC1 crystal doped with Li + ions at 1.3 K. The Li + energy states were pumped by means of a 24 G H z microwave field which coupled resonantly Alg to Tzu and Tzu to Tzg. Of course, sound has been amplified by Hutson and White [5], and many others since, in semiconductor crystals when the current carriers were accelerated up to a point where their velocity exceeds the sound velocity. Population inversion has been obtained in N M R by two methods. The first, known as the "~-pulse technique", consists in applying, during a time interval A t = rc/TB1, a microwave magnetic field B~ in resonance with the TLS, 7 being the gyromagnetic factor. But, in our case, this method is not applicable because there exists a wide distribution of 7~ factors and at given e and At, a pulse will be a ~ pulse only for a small set of TLS. Therefore, no net amplification can be expected. The second technique, the "rapid adiabatic passage" has been used for the first time by Bloembergen et al. [6] and applied many times since. It has also been used in the optical domain by Treacy and Demaria [7] and Loy [-8]. This technique starts by taking a system with a given frequency splitting heoo. Let Z be the quantification axis, i.e. the static magnetic field direction for true spin. We superimpose to the static field Bo, a rotating field B~(t) perpendicular to Z. The rotation frequency is close to COo, the difference Ao) = COo - oo being however larger than

~4 e~ \" l 4

I I

%

X'

fi)

O) 1

n~1

Fig. 1. Evolution in the rotating frame of the effective field direction ~2' and of the magnetization m during different steps of the "'rapid adiabatic passage" (see text). At time t4 the magnetization is reversed in comparison to that at time to

the Rabi precession frequency (ol = 7B1. In the so-called rotating frame, the spin system starts to precess around a field direction f2' very close to that of the static field given by f2' = o)1 e~(t) + Acoe=. The magnetization stays antiparallel to ~2', i.e. almost along - e~. Let us change slowly Am in order to go through the resonance frequency (A~o = 0) and finish with ~ ' = o)1 ex(t) - Ae~ez. If the frequency change is, on the one hand, slow enough for the adiabaticity to be ensured and, on the other hand, fast enough for the relaxation to be ineffective, the magnetization will stay antiparallel to ~ ' during the change and therefore will finish parallel to the static field. The population has been inverted. In N M R [6], Ao) was varied by changing the magnitude of the static field and therefore coo. In the optical domain [8] the use of a Stark field did the same thing. But for glasses, because of the distribution of the coupling coefficient no such field is available. Therefore, instead of tuning Au) by acting on O)o, we did it by changing the field frequency o). In Fig. 1, we show the exact sequence used. From to = 0 to t~, we increase slowly the magnitude of an acoustic pulse of frequency o) slightly lower than the frequency e)o of the wave we want to amplify. When the Rabi frequency o)~ is of the order of Ao), we stop the increase. The precession axis changes from ~{~ to (2'~ which makes about 45 ~ with the Z-axis. In the same time, the magnetization goes from mo to m 1. At time tt we start to move the frequency of the wave towards O)o. f2' goes from f2'~ to f2~ with the magnetization following m2. At time t2 we change very rapidly the acoustic field phase by ~. The (2'-axis changes abruptly from (2~ to ~ +, but the magnetization stays in m2 since it has no

J.-l( Prieur et al. / Ph.vsica B 219&220 (1996) 235 238 time to follow ~2'. m2 is now parallel to ~'. From time t 2 to t3 we decrease the frequency and from t3 to t4 we decrease the field magnitude. 12' ends up at £25, = g2~ and the magnetization in m4; the population has been inverted. The inversion will be effective only if some conditions are fulfilled. First of all, the full sequence must last a time lower than either T~ or T2, the longitudinal and transverse relaxation times. Second, the speed of variation of the angle O(t) between f2'(t) and e~ must be slow enough in order that 101 ~ 1~2'(t)l, i.e. Id)al < Ao)2 in the time intervals [to, t~] or It3, t4] and IAd)] <~ o)2 in It1, t2 ] or It2+, t3]. Both conditions can be compacted in

-80

. . . . . . . . . . . [ • case I • case 2

-82

237 r

- ~-

i

~-84

-

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.

.

.

.



T

.., Bleachin

~

I

!

level •

: . !

~_~6

t

~'-88 i

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i

-90 "J

i

i

-92 -70

-60

-50

-40

-30

-20

P p u m p (riB)

~Ao) + (o~ ~ t4 - to ~ T~, T2.

Since the exact value of oJ1 is not important as far as the above inequality is fulfilled for any time, the distribution of the coupling constant will not matter. The experiment [9] has been made in a Tetrasil glass with size 20 x 4.5 x 4.5 mm 3. Two LiNbO3 transducers, tuned at 340 MHz, were glued on the small faces of the sample. The pump wave was launched at one end and the probe at the other end of the sample just before the arrival of the pump at this end. This configuration is meant to assure a maximum of efficiency for the amplification. The sample was mounted in a dilution refrigerator. Its temperature was decreased to about 20 m K to ensure a large T2 as well as a large T1. The T2 time has been measured by an acoustic pseudo-spin echo experiment and was found to be 5 Bs. The leading and falling edges of the pump pulse were almost exponential versus time and lasted about 1 las, while the duration of either frequency change was 0.5 las. The phase inversion time ( t ~ - t j ) was less than 10ns. The magnitude of the frequency shift has been fixed at about 10 MHz. The probe pulse was detected on the pump transducer and its input power was reduced at the m i n i m u m level possible to detect a signal when there was no pump pulse. To tell if the probe is attenuated or amplified, it is necessary to define a reference level, i.e. the level that the probe would have if there was no TLS in the sample. Therefore, we made a hole-burning experiment which leads for sufficient pump power to a "bleaching" of the TLS. Such an experiment is shown as curve I in Fig 2. Above a - 3 8 dB threshold pump level, the detected probe level becomes independent of the pump. Its - 84 dB m value will be used as a reference. Curve 2, in Fig 2, shows the variations of the detected probe level versus the m a x i m u m pump power when the special sequence described above has been applied. Nothing happens to the probe if the pump level is below - 6 0 dB. Above that pump threshold, the probe level

Fig. 2. Detected probe power versus the maximum pump power in two different cases. Case 1 is a hole-burning experiment (o~ = UJo; no phase shift at time tz). Case 2 corresponds to the full pumping procedure. There is amplification when the probe exceeds the bleaching level.

increases at a much faster rate than in the hole-burning experiment. It reaches the - 8 4 d B m reference for a pump level of - 51 dB. If the pump still increases, the probe exceeds the reference level and therefore the TLS are amplifying. The maximum effect is reached for a power of the pump of - 45 dB. The maximum amplification coefficient measured is 2 dB! Of course, this value should be compared to the maximum expected amplification which is the difference in the output probe levels before applying a bleaching pump pulse and after, i.e. 7 dB. Three reasons might explain the discrepancy. The first, and probably most important reason, is that the probe is not amplified on its whole path. Indeed, a section of the sample is amplifying only after the full passage of the pump. The total duration of the pump pulse implies that, practically in our case, only a half of the sample length is amplifying. A second reason comes from the large distribution of the coupling constants which implies that the adiabatic criteria cannot be fulfilled for every TLS. Therefore, those which escape the criteria stay attenuating. Third, the interaction of the pump with the TLS resonantly coupled with it (those with ~.o0 = ~,)p) has two effects: attenuation and reshaping. The reshaping comes from the dependence of the attenuation on the wave level which has the consequence that the early part of the pump pulse is more attenuated than the latter part. Then during the travel of the pump pulse, the leading edge becomes sharper and the overall magnitude of the pulse decreases. Therefore, the pumping process cannot be optimal for each sample section.

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J.-Y. Prieur et al. /Physica B 219&220 (1996) 235-238

What kind of improvements might be made for further experiments which will check those hypotheses and improve the efficiency of the amplifier? To check the first hypotheses, we design a new amplifier configuration. We make use of transverse waves. The pump wave will propagate along the largest dimension of the sample (Z) and be polarized along the X-direction. The probe will, on the contrary, propagate along the X-direction and be polarized along the Z-direction. The symmetry of the elasticity tensor will ensure that both waves are coupled with the same TLS and with the same coupling coefficient. But the probe wave will have time to make a few back-and-forth travels before that, the amplification process will be stopped by the return of the pump. The third point might be checked merely by saturating the TLS resonantly coupled with the wave before sending the ad hoc pump. Another improvement might also result, if instead of changing by n the phase of the pump wave when it is in resonance with the probe frequency, we continue the upward shift of the frequency as it has been suggested independently by Challis and Clough [10]. Finally, the best improvement will be to destroy the

pump at its arrival at the end of the sample by injecting, for instance, a phase-inverted pulse at this end. The sample will stay amplifier a much more longer time (namely T1).

References [1] W.A. Phillips, J. Low Temp Phys. 7 (1972) 351. [2] P.W. Anderson, B.I. Halperin and C.M. Varma, Philos. Mag. 25 (1972) 1. [3] E.B. Tucker, Phys. Rev. Lett. 6 (1961) 547. [4] D.J. Channin, V. Narayanamurti and R.O. Pohl, Phys. Rev. Lett. 22 (1969) 524. [5] A.R.Hutson, J.H. Mc Fee and D.L. White, Phys. Rev. Lett. 7 (1961) 237. [6] N. Bloembergen, E.M. Purcell and R.V. Pound, Phys. Rev. 73 (1948) 679. [7] E.B. Treacy and A.J. Demaria, Phys. Lett. 29A (1969) 369. [8] M.M.T. Loy, Phys. Rev. Lett. 32 (1974) 814. 1-9] J.-Y. Prieur, R. H6hler, J. Jofl'rin and M. Devaud, Europhys. Lett. 24 (1993) 409. 1-10] L. Challis and S. Clough, Nature 367 (1994) 687.