VOLUME 42& NUMBER 15 PHYSICAL REVIEW LETTERS 9 APRIL 19'79 Backward Echo in Two-Level Systems M. Fujita, H. Nakatsuka, H. Nakanishi, and M. Matsuoka Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received 12 February 1979) The possible three-pulse echoes in two-level systems of solids and gases are discussed with particular emphasis on backward-wave echoes. One example of the backward echoes was realized with the atomic sodium D line. The detection of this echo requires neither optical shutters nor magnetic field. The decay rate of the echo due. to Na-Ar collisions was measured. It is shown that the effect of velocity changing co11isions is negligible compared with that of phase interruptirg ones. We have observed a new photon echo in a two- level system, a backward-wave three-pulse echo in a gas. This is unique to the optical frequency domain. In this method neither optical shutters" nor magnetic field' is required to prevent the ex- citation pulses from saturating a detector. In the first observation of the photon echo in ruby, ' a Kerr-cell shutter was placed before a phototube, and noncollinear (™50 mrad) first and second pulses were used at the sacrifice of a phase-matching condition. Later' a series of three Pockels cells was used which provided more than 10' extinction ratio. In the category of the echoes by pulsed lasers, apart from the Stark or frequency-switching method, ' a weak ax- ial magnetic field was used' to separate the echo by rotation of the echo polarization. In a three- level system, a backward echo was observed us- ing two different fretluencies (trilevel echo). ' In the present experiment, a phase-matching condition is satisfied in the configuration in which the three excitation pulses and the echo, all of a same frequency, are counterpropagating or mak- ing angles with each other. As a result of this configuration, the echo can be polarized perpen- dicularly to one of the excitation pulses which makes the smallest angle to the echo. This meth- od allows a simple and very easy detection of the echo compared with the previous method, togeth- er with some new possibilities for application. We have observed this echo in Na on the O'S„, - O'P„, two-level system. We apply the ordinary two-pulse echo theory by Scully, Stephen, and Burnham' to general three- pulse echoes. Suppose the three excitation puls- es are applied at times ~p (p = l, 2, and 3) in the directions along unit vectors np (We. will set ~, =0 in the following. ) The arrival times tp; at the ith atom at the position r, (tp, ) are determined by the equations ctpt =c&p+np ' rg(tpg). With respect to an observation time t and posi- tion R=Rn, where n is a unit vector, we define the retarded time t; as ct, =ct -B +n ~ r, (t, ). Then the state of the ith atom at t; is expressed by a density operator p;(t;) = exp[-iH„(t, -t„)]U,exp[- iH„-(t„. -t„)]U, exp[-iH„. (t„-t„)]U,p, . x U, t exp[iH„(t„—t„.)]U, t exp[iH „(t„-t„.)]U, t exp[iH„(t; — t„)], where p, is the density operator of the ground state, H« the Hamiltonian of the ith atom, and Up a transformation operator expressing the effect of the pth pulse. If we express the ground and the excited levels as a and 5, respectively, the effect of the transfor- mations V„which leads to the echo component p„ is as follows. The first pulse transforms the ground state into A] p„or [B]pp„each of which has the phase factor exp[+in;(t„—t„)] or exp[- i&a;(t„. — t„. )] at the time f;„, where co; is the eigenfrequency of the ith atom. The second pulse transforms them in- to diagonal elements p» or p„. The third pulse transforms p» and p„ into p„, which gets an addition- al phase factor exp[-ice;(t; -t„)] at the time t, . Depending on the paths [A] or [BJ, we have two terms for the polarization as in the following: (P, (t;))„z, = (n sin8, sin8, sin8, exp[-i~, (t, +t„-t„-t„)]+cc }. . + I p sin8, sin8, sin8, exp[- ie, (t; — t„+t„-t„)]+c.c. ), where the first pair of braces sets off the terms for path [A] and the second those for path [B], and 1979 The American Physical Society