DTIC S ELECTE. clo3 JUL I by. O TECHNICAL REPORT No OFFICE OF NAVAL RESEARCH. Contract N K-0043

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1 0 OFFICE OF NAVAL RESEARCH Contract N K-0043 O TECHNICAL REPORT No. 105 S Squeezing of Cavity Fields in Cascade Multiphoton Processes I by S Fu-li Li, Xiao-shen Li, D. L. Lin and Thomas F. George Prepared for Publication in Journal of Physics B Departments of Chemistry and Physics State University of New York at Buffalo Buffalo, New York June 1989 Reproduction in whole or in part is permitted for any purpose of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. DTIC S ELECTE JUL clo3

2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE la. REPORT SECURITY CLASSIFICATION Unclassified Form Approved REPORT DOCUMENTATION PAGE OMB No lb. RESTRICTIVE MARKINGS 2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABILITY OF REPORT Approved for public release; distribution 2b. DECLASSIFICATION / DOWNGRADING SCHEDULE unlimited 4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S) UBUFFALO/DC/ 89/TR a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION Depts. Chemistry & Physics (If applicable) State University of New York 6c. ADDRESS (City, State, and ZIPCode) 7b. ADDRESS (City, State, and ZIP Code) Fronczak Hall, Amherst Campus Chemistry Program Buffalo, New York N. Quincy Street Arlington, Virginia Sa. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (If applicable) Office of Naval Research Contract N K c. ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERS Chemistry Program PROGRAM PROJECT TASK WORK UNIT 800 N. Quincy Street ELEMENT NO. NO. NO ACCESSION NO. Arlington, Virginia TITLE (Include Security Classification) Squeezing of Cavity Fields in Cascade Multiphoton Processes 12. PROA UHRS PERSONAL AUTHOR(S)Fu-li Li, Xiao-shen Li, D. L. Lin and Thomas F. George 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year,Month, Day) 15. PAGE COUNT FROM TO June SUPPLEMENTARY NOTATION Prepared for publication in Journal of Physics B 17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) FIELD GROUP SUB-GROUP, MULTILEVEL ATOM - SQUEEZING CAVITY FIELD CASCADE MULTIPHOTON PROCESSES SINGLE-MODE 19. ABSTRACT (Continue on reverse if necessary and identifyk by block number) SATURATION, For a system of a multilevel atom interacting with a single-mode cavity field, squeezing of the field is studied numerically. It is found that the squeezing effect becomes stronger when the atomic level number increases, but tends to saturate when the number exceeds twenty..s 20. DISTRIBUTION /AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION 51 UNCLASSIFIED/UNLIMITED IN SAME AS RPT. 0 DTIC USERS Unclassified 22a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (Include Area Code) 22c. OFFICE SYMBOL Dr. David k. Nelson I DO Form 1473, JUN 86 Previous editions are obsolete. SECURITY CLASSIFICATION OF THIS PAGE UUNCLASSIFIED

3 Journal of Physics B, in press Squeezing of cavity fields in cascade multiphoton processes Fu-li Li Department of Physics Xian Jiao Tong University Xian, Shaanxi, People's Republic of China Xiao-shen Li Center for Theoretical Physics Chinese Center of Advanced Science and Technology (World Laboratory) P. 0. Box 8730 Beijing , People's Republic of China and Shanghai Institute of Metallurgy, Chinese Academy of Sciences Shanghai, People's Republic of China D. L. Lin and Thomas F. George Department of Physics & Astronomy 239 Fronczak Hall State University of New York at Buffalo Buffalo, New York Abstract For a system of a multilevel atom interacting with a single-mode cavity field, squeezing of the field is studied numerically. It is found that the squeezing effect becomes stronger when the atomic level number increases, but tends to saturate when the number exceeds twenty.

4 2 I. Introduction The interaction of an atom with cavity fields has stimulated great interest because many interesting and novel phenomena have been discovered. In addition to quantum collapse and revival (Eberly et al 1980), the atomicfield system has been shown to exhibit a subpoissonian photon distribution (Puri and Agarwa 1986), squeezing (Meystre and Zubairy 1982) and stationary atomic inversion (Savage 1988). Recently, Li and Zhu (1985) and Kozierowski (1986) studied the dynamics of an M-level (M-l)-mode system with the atom having a common upper level. Most recently, the present authors (Li et al 1989) examined the effects of the atomic level number on quantum collapse and revival and photon statistical properties in the interaction of a multilevel cascade atom with single-mode cavity field. In the present paper we study squeezing effects for an M-level atom interacting with a single-mode cavity field. First we present the formalism for the problem. Then we proceed to study fluctuations of the field mode and examine effects of the degenerate multiphoton processes. II. Formulation We consider a cascade atom with M energy levels as shown in Fig. 1. The total Hamiltonian is, in the rotating-wave approximation, M M-1 H - )Mfaa+ oatai + AAt A a +h.c.() i-i i-i where the operator at creates a photon, At creates an atom in the i-th level, and A are the atom-field coupling constants. i Our discussion will be limited,

5 3 for simplicity, to processes involving only a single-photon transition between adjacent atomic levels. We define the total excitation number operator as (Li et al 1989) M N - ata + iaa, (2) i-i which is the sum of the photon number and atomic excitation number operators. * A It can readily be verified that [H,N] - 0, i.e., N is a constant of motion. If we are only interested in the particular case in which the atom is initially in its higher level, then we need only consider that part of the Hilbert space involving state vectors corresponding to N > M, where N is the A eigenvalue of the operator N. This is a consequence of the fact that N is a conserved quantity. Consider the state ji,n> - li>ln> (3) in which there are n photons in the field and the atom is in the i-th state. The subspace corresponding to N - M + n is spanned by the state vectors IM,n>, IM-l,n+l>,..., Ii,n+K-i>,..., ll,n+m-l>, where n is the photon number in the field and can take any positive integer value. Since state vectors in - different subspaces are necessarily orthogonal, the total Hamiltonian can be : 0 diagonalized in every subspace. An arbitrary state in the subspace corresponding to N can be expressed as Avaiiability Codes Avail and/or Dist Special

6 4 M n> - C i,n+m.ili,n+m-i> (4) i-i where the expansion coefficients C i,n+m i satisfy the stationary-state Schr~dinger equation M (H ii 6ii' + Hii E6ii' )Ci',n+M-i' " 0 (5) The matrix elements in (5) are given by Hii - <i,n+m-ilhii,n+m-i> - (n+m-l))(f + i-o (5a) i-i J-i Hig -<i,n+m-iihii',n+m-i'> - Ai,,In+Mi' 6 i,i+l + A ln+m-$'+l 6i, i i' (Sb) Here we have defined the detuning parameters A i " )M " X(COi+l - i ) I i - 1, 2,... M-1 (6) The energy eigenvector Ion > and the corresponding eigenvalue Eno are determined by HI no E no1 nr> (7a)

7 5 > C i n+m-li,n+m-i> (7b) i-i The subscript a in (7a) labels the eigenstates in the subspace in question. If the atom is assumed to be initially in the M-th level, then for all time t > 0 the system can only be found in subspaces with n > M. There are 1 eigenstates of the type (7b) in every subspace. Hence the orthornormality condition is M i-i i- n+m-i Cin+M-i"6, (8) and the completeness relation is CO M n-0 o-l We can now solve the equation of motion for the density matrix. The reason why we use the density matrix instead of the state vector formulation is that it is easier for this to be generalized to treat the complicated problem of a mixed atomic initial state, which we shall study next. Besides, once the Hamiltonian H has been diagonalized, the elements of the density matrix can be obtained in a straightforward fashion without any increase in complexity. In fact, even if one uses the state vectors, one still has to diagonalize H. A density matrix element between energy eigenstates satisfies the equation

8 6 at Pna,n'a' " a n S(t) - (n'na (E )na,n'a' (10) whose solution can be put in the form - (E -E,)t Pno,n'a'(t) - Pna,n'a' (0) e no n a (11) The initial matrix element in (11) is given by Pno,n'a ' (0 ) " PnI (C)M( M nn a,n+m-a (C ' )a',n'+m-a' where C I is the inverse matrix of the transformation defined in (4) and p nn' is the initial density matrix element of the field. The last step is from the inverse transformation of (7b), namely, M In,M>- (C" I)n+M I n> (13) a-l If the initial density matrix is known, the mean values of all dynamical variables of the atom-field system can be calculated. The mean occupation probability of the J-th level is P- Tr(pAtA)P <0 I AtAI10 i i J) L no0 )) n'a' no,n' n'oa j j no " 2 no,(0) C, ca,mj cos((e -E.,)t) (14) na,ai

9 7 It is easily verified that M PC (15) jnna' / j,n+m- Ci j j npl0na,na j-1 n,a,a' j n,a where we have made use of (8) and (9). field, As usual, we introduce the slowly-varying complex amplitudes for the d - i[ae + ate- ] (16a) d - Lai(t[a) at e-i(ot-0 ) (16b) [a These operators satisfy the commutation relation [d 1,d 2 I 1 (17) For simplicity, in this paper we only consider the case of 0-0, for which maximum squeezing is expected, at least for the case where the atom starts in a definite state (Meystre and Zubairy 1982). The corresponding expressions for the fluctuations are then (Ad ) 2 1 [2<ata> <a 2 >e 2Ot + <at 2 > 21Gt <ae i t + ate- iot>2 ] (18a) (Ad )2 - l[2<ata> <a 2 >e 2 it - <at2> '2i Q t 2 4

10 8 + <ae init - at- ift >2 (18b) where <ata> - Tr<pat a> - an'a<ona IataIo na> no' n a' - (n4-m-j) 1 (O) Ca Cjnat cos(e -E )t nana injj,n+-j na na (19) <ae int + ate- -nt > - Trfp(ae it + ate- int) -!n+m-i+il~ 2n+ ~na,(n-i)a'(0 x Cos[( L"' (n+l)a' + O~],(20a) <ae - aten > - Tr[p(ae it- ater - 21 n+-.4-i c ca n+1 ij n+m-i+1,i x p,(0) sin[( na (n+l)ou' - l (20b) <a2 21Wit + f2 e'21qt > r a2 621Gt + 12 e-210t - 2!(n+H-i+1)(n4(-i+2) c +-~ nlu ' ai

11 9 E -E X a (0) Cos[( + 2Q)tl n+m-i+2,i na,(n+2)a, (20c) They satisfy the uncertainty relation Ad.Ad > (21) When either of them satisfy the condition (Ad)2 < I, i - 1, 2 (22) 2. 4 we say that the field is in a squeezed state. For a given n, Eq. (5) can be solved for the energy eigenvalues and eigenfunctions. With the initial density matrix elements determined by (12), we can calculate the field fluctuations and occupation probabilities of any atomic level at any time t > 0. III. Results and discussion For simplicity we assume that A 1-2 "AM-1 - and that the resonant interaction condition is satisfied (LI and Zhu 1985, Kozierowski 1986, Li et al 1989), i.e., a 1 " 0. In our numerical calculations we take ) - 1 and employ 1/A as the unit of time. For the initial conditions, we consider the atom to start from the uppermost level IM> and the field mode to be in a coherent state. Thus we have no n - P nm' -!nn' e- n (23)

12 10 where n - Izi 2 is the initial mean photon number. The relation between the field squeezing and the initial field intensity n is nonlinear. For a given energy level number of a cascade atom, there exists a value of n corresponding to the strongest squeezing or the smallest 2 2 (Ad 1 ) (Zhu et al 1988), which hereafter is referred to as (Adl)min The maximum squeezing corresponding to different level number M for some cases can be seen from Fig. 2. Some of the results are summarized in Table 1, from which it is observed that the field squeezing increases with the increase of the cascade level number. It is also noted that the larger the value of M, the later the field enters squeezed states and the longer it remains squeezed. example, However, on the other hand, we find that there is saturation. For,when M changes from 2 to 3, (Ad) 2 n decreases from to ; i when M changes from 8 to 10, (Ad)i decreases from to ; when changes from 21 to 23, (Ad)i decreases from to Detailed behavior is shown in Fig. 3. In brief, for larger M, (Ad)i n hangesmuc 1 in cmuch less. The variance tends to saturate to some non-zero value. Figure 4 gives an example of the time evolution of the occupation probability for the atom in the uppermost state. Comparing with Fig. 2, we can see that whenever the field enters the squeezed state from its initial coherent state, the probability of the atom lying in the uppermost state is close to its minimum. The opposite situation occurs when the field leaves the squeezed state. Thus we can conclude that squeezing effects have something to do with the absorption process, which is in agreement with the results reported by Shumovsky et al (1987). We have discussed above the effects of the quantities M and n or z on the squeezing. As for the effects of the other quantities Ai and A i, we find

13 11 that increasing detunings A. would weaken the squeezing, and there would be stronger squeezing when the coupling constants A's are equal. Acknovledgments This research was partially supported by the National Science Foundation under Grant CHE , the Office of Naval Research, and the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract F C The United States Government is authorized to copy and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.

14 12 References Eberly J H, Narozhny N B and Sanchez -Mondragon J J 1980 Phys. Rev. Lett Kozierowski M 1986 J. Phys. B 19 L535 Li F L, Lin D L, George T F and Li X S 1989 Phys. Rev. A to be published Li X S and Zhu S Y 1985 Physica A Liu Z D, Zhu S Y and Li X S 1988 J. Mod. Opt Meystre P and Zubairy M S 1982 Phys. Lett. A Puri R R and Agarwal G S 1986 Phys. Rev. A Savage C M 1988 Phys. Rev. Lett Shumovsky A S, Kien F L and Abskenderov E Phys. Lett. A

15 13 Table 1: Maximum squeezing under different conditions M n maximum squeezing % % % % %

16 14 Figure Captions 1. Schematic energy level diagram of the M-level atom. 2. Evolution of the variances (Ad 1 )2 (curve 1) and (Ad 2 )2 (curve 2): (a) M - 2, n - 6; (b) M - 8, n - 10; (c) M - 19, n The minimum variance (Ad 1 )i n versus level number M. 4. Evolution of the highest level occupation probability PM for M - 8 and n - 10.

17 tlam-2 S hw

18 (Ad 1 )

19 (Adi) 2 2(b II I 04

20 1.0(c (Adi)

21 0.1 (Ad )2 1min

22 t

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