Interaction of superpositions of coherent states of light with two-level atoms

We investigate some of the basic features of the interaction of superpositions of coherent states of light with two-level atoms in the framework of the Jaynes-Cummings model . We compare the behaviour of the system in the case of having a coherent superposition state and a statistical mixture of coherent states as an initial field . We investigate the collapses and revivals of the atomic inversion by studying the evolution of the Q function of the cavity field . We also establish the connection between the purity of the field and the collapses and revivals of the atomic inversion . 1 . Introduction There has been considerable interest in the properties of the so-called superposition states of light (SS) [1-6] . One particularly interesting case is the superposition of two (or more) coherent states [2-6] . Due to the quantum interference, the properties of such a superposition are very different from the properties of the constituent states (coherent states), as well as from the incoherent superposition or statistical mixture (SM) of coherent states . For instance, the superposition exhibits squeezing [5], higher-order squeezing [7], sub-Poissonian photon statistics [2] and oscillations in the photon-number distribution [5, 6], and these properties clearly differentiate the superposition state and the statistical mixture of two coherent states . Because by using coherent states one could produce superpositions of macroscopically distinguishable states (or Schrodinger cat-like states) [8], the problem is of importance for the quantum theory of measurement . There have already been proposed several schemes in order to produce superpositions of coherent states; the nonlinear interaction of the field in a coherent state with a Kerr-like medium can produce an SS [9] . Another possible way would be through the non-unitary evolution of the field either in the interaction of a coherent state with two-level atoms [10, 11] or in quantum non-demolition measurements [12] . The last method would allow us to produce the states known as the even coherent state : [ Permanent address: INAOE, Apdo Postal 51 y 216, Puebla, Pue., Mexico . $ Permanent address : Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 842 28 Bratislava, Czechoslovakia . 0950-0340/92 $3 .00 Q 1992 Taylor & Francis Ltd . Ia>e = A 112(Ia>+I a>), Ae 1 =2[1+exp( 2IaI 2 )], (1) or as the odd coherent state : Ia>o=Ao12(Ia>-I-a>), Ao 1 =2[1-exp(-2IacI 2)] . (2) 1442 A. Vidiella-Barranco et al . The non-classical properties of such superpositions as well as their sensitivity to dissipation have already been discussed in a previous work [5], where we concluded that squeezing has a much slower decay than the oscillations in the photon number distribution, for instance . In this paper we are going to be concerned with the interaction of superpositions of coherent states with matter . This is, in our opinion, another step forward to a better understanding of the consequences of the superposition principle at a `macroscopic' level, because it would allow the utilization of atoms for the field monitoring [13] . We are going to employ one of the simplest possible models, the so called Jaynes-Cummings model (JCM), which consists of a single mode of the quantized electromagnetic field interacting with a two-level atom in a lossless cavity [14] . We are aware of the importance of dissipation in this problem, but we prefer to analyse the ideal case first in order to identify the relevant properties . In spite of its simplicity, the JCM exhibits interesting features, namely, the atomic inversion (the probability of being in the excited state minus the probability of being in the ground state) is very sensitive to the statistics of the initial field . If the field is initially prepared in a number state, the inversion will show the usual Rabi oscillations (also shown when the input field is not quantized) . However, if the input field is prepared in a coherent state, the oscillations are modulated in such a way they exhibit collapses and revivals [15], which have been already experimentally observed [16] . The sensitivity of the model to the photon statistics will be our starting point, because the photon distribution for the SS (which shows strong oscillations) in (1) and (2) is very different from the Poissonian photon distribution for the corresponding SM of two coherent states [5] . We are going to assume that the initial single mode electromagnetic field inside the cavity is in a superposition state of the kind : P=A[Ia><-al+r(I-a><-al)], (3) where A=P +r2 + 2r exp (2a2 )] (-1) with a real . The parameter r can assume the values -1, 0, 1, which corresponds to an odd coherent state (2), a coherent state and an even coherent state (1) respectively. A convenient experimental scheme would be the one proposed in [12], where atoms very detuned from the cavity resonance frequency would produce the SS, followed by immediate injection of atoms in resonance with the cavity field (the case we are going to treat here) . As we know, because the interference terms in (3) have a rapid decay to a SM when we include dissipation, so we want to see how different would be the behaviour of the system if the input state is a statistical mixture of the states la > and I-a>, i .e ., 0=ila><-al, (4) The JCM can be solved exactly (non-perturbatively) in the rotating wave approximation, and this allows us to obtain exact expressions for the expectation values of physical quantities . The first step is to calculate the time-dependent density operator for the system, 13(t), and use it to evaluate the atomic inversion in both cases . We can then use the phase-space approach presented by Eiselt and Risken [17], in order to explain the pattern of collapses and revivals of the Rabi oscillations . It is possible to obtain valuable information directly from the time-dependent quasiprobability distributions [18] . Our analysis will be based on the evolution of the Q function for the cavity field . The Q function is a c-number representation of the Interaction of light with atoms 1443 density operator associated with anti-normally ordered operators, and can be written in a convenient form as : Q(t)=1«Ipf(t)IP> . (5) The evolution of the Q function will be clearly connected to the evolution of the collapses and revivals [17], but one has to be careful in associating the revivals with the collisions of component parts of the distributions . This is because the evolution of the Q function if we start with a SS is very similar to the evolution of the Q function for the SM as an initial state . However, the additional interference structure present in the case of SS will be responsible for the distinction . In order to complement our investigations, we are also going to discuss further aspects of the field evolution . We know that even if the field is initially prepared in a pure state, its subsequent evolution will be very complicated, and during most of the time the field will be in a statistical mixture [19] . The most appropriate indicator of the degree of purity of the field is given by the quantum mechanical entropy [19], defined as [20] : S= -Tr [jf(t) In Af(t)] . (6) It is easy to see from the following property of the density operator describing a pure state, P" =f (n integer), which leads to Tr 32 =1 that the quantum mechanical entropy should be zero for a pure state . Otherwise, if the state is a SM, Tr p2 < 1, and the entropy will be no longer zero . In our problem, however, it will be enough to compute the quantity C = I Trf [/3f ], which can serve as a measure of the purity of the field [10] and has a behaviour similar to the entropy. We want to stress, however, that the entropy gives us the most general measure of the purity of a quantum state simply because it contains all the powers of 5 . Another aspect we would like to discuss briefly is the photon number distribution of the field, defined as : P"(t)= . (7) The complex field evolution is, of course, reflected in the photon distribution, and only at particular times will the photon number distribution resemble that of a pure state. We found that the distribution exhibits a pattern of oscillations (at certain times) very similar to those present when the input field is a coherent state [11] . The paper is organized as follows : in Section 2 we briefly discuss the model and the atomic inversion . In Section 3 we discuss some aspects of the field dynamics, where we show the evolution of the Q function of the field and provide an explanation for the pattern of collapses and revivals found in Section 2 . We conclude the paper by discussing the purity of the state of the field as well as its photon number distribution . 2 . Atomic dynamics 2.1 . The Jaynes-Cummings model The Jaynes-Cummings [14] model is one of the few models that can be exactly solved in physics . It is also one of the most elementary ways of studying the interaction of electromagnetic radiation and matter . It consists of a two-level atom with a ground state Ig> and an excited state le> placed inside a lossless cavity and interacting with a quantized single mode of the electromagnetic field . For the sake of simplicity we are going to consider the cavity field with a frequency w resonant 1444 A. Vidiella-Barranco et al . with the atomic transition frequency [(EE-Ea)/fit=wo], i .e ., w=w0 . The total Hamiltonian in the rotating wave approximation may be written as : H=2hw0cr3 +hw(ata+ 2) +h2(av + +Q_ at) . (8) The atomic operators are simply Q3 = Ie>+sig>) . (14) We will consider some particular values for the parameter s, i .e ., s= ± I (atom in a superposition of ground and excited states) and s = 0 (atom initially in the excited state) . In writing (13), we assumed the atom and field are initially uncorrelated . Then we can proceed-to calculate the time dependent density operator for the system in the atomic basis, inserting (11) and (12) into (9). The result is : At)1 '40'(00 1 A&(001) (1 +S2 ) 1/2 (BPf(0)A t BPf(0)Bt1 where A=Cn+1 is ,S,+1a, B=sC„-iatSn +1 • (16) The operators C and 9 are defined in equations (12) . (9) Interaction of light with atoms 1445 The density operator in (15) will enable us to calculate all the expectation values and quantities we need in our discussion . 2.2. Evolution of the atomic inversion Now we are going to investigate a particular aspect of the atomic dynamics, namely, how the field excites and de-excites the atom . A convenient way of doing that is through the atomic inversion, defined as the probability of the atom being in the excited state minus the probability of being in the ground state, i .e . : W(t) =Tr [,5(t)(I e)<eI Ig)<gi)] . (17) If we insert the density operator (15) in (17) and perform the tracing operation, we obtain : W(t) (1+s2)nY [<nIAPf(0)A tln)-<nlBPr(0)Btln)], (18) where the operators A and B are given in equations (16) . If the field is initially prepared in a SM of states Joe) and I-a), with density operator given by equation (4), the atomic inversion will be : WO) = 2) E PM {cos [2)Lt(n+1) 1 / 2]-s 2 cos [2~,t(n)1/2]}, (19) (1+S n=0 where PM=exp(-a2)a2n/n! is the Poisson distribution, i .e., the photon-number distribution for the initial field (SM) . As we expected, the atomic inversion is the same as if the input field was a coherent state . However, if the field is initially prepared in a superposition of coherent states [equation (3)], the atomic inversion will have the same form as in the former case, Ws(t)= (1 + s2) P~{cos [2At(n+1) 1 2]-s 2 cos [2.1t(n)1I2]}, (20) n=o but with Ps-exp(-a2)a2n[1 + r( -1)"]2 (21) " n![1+r 2 +2rexp(-2a2)]' which is the photon number distribution for the initial field (SS) . We are going to start our analysis taking s=0, i.e., with the atom initially prepared in the upper state . Since the resulting series cannot be analytically performed [14] we will evaluate them numerically . In figure 1 (a) we plot the atomic inversion in function of At for the field initially in a SM (equation (19)), and in figure 1 (b) the atomic inversion for the field initially in an SS (even coherent state), given by equation (20) with r = 1 . The most obvious difference, is that for the SS the revival time will be approximately half of the revival time for the SM . This is an effect due to the interference between the two coherent states in the superposition, and can be understood looking at the photon-number distribution of the initial fields . We know that the collapses are caused by the dephasing of the various terms in the sums (19) or (20) . Thus we can calculate the time in which the revivals will occur by estimating the time that neighbour terms in the sums will be in phase again (for n,: n) [15] : TR[2A(n+ 1) 1 / 2 22(n) 1 / 2) z 2x . (22) 1446 A. Vidiella-Barranco et al .