Quantum Statistical Behaviors of Interaction of an Atomic Bose Einstein Condensate with Laser

Transcription

1 Commun. Theor. Phys. (Beijing, China) 36 (2001) pp. 9 5 c International Academic Pulishers Vol. 36, No., Octoer 15, 2001 Quantum Statistical Behaviors of Interaction of an Atomic Bose Einstein Condensate with Laser YU Zhao-Xian 1,2 and JIAO Zhi-Yong 2 1 Theoretical Physics Division, Nankai Institute of Mathematics, Nankai University, Tianjin , China 2 Department of Physics, Petroleum University (East China), Dongying , Shandong Province, China (Received Feruary 19, 2001; Revised April 19, 2001) Astract We have investigated quantum statistical ehaviors of photons and atoms in interaction of an atomic Bose Einstein condensate with quantized laser field. When the quantized laser field is initially prepared in a superposition state which exhiits holes in its photon-numer distriution, while the atomic field is initially in a Fock state, it is found that there is energy exchange etween photons and atoms. For the input and output states, the photons and atoms may exhiit the su-poissonian distriution. The input and output laser fields may exhiit quadrature squeezing, ut for the atomic field, only the output state exhiits quadrature squeezing. It is shown that there exists the violation of the Cauchy Schwartz inequality, which means that the correlation etween photons and atoms is nonclassical. PACS numers: 2.50.Ct, 2.50.Dv, Fi Key words: atomic Bose Einstein condensate, su-poissonian distriution, quadrature squeezing, Cauchy Schwartz inequality 1 Introduction Recently, in a remarkale experiment Anderson et al. [1] have produced the Bose Einstein condensate (BEC) of magnetically trapped 87 R gas y cooling it to temperatures in the nanokelvin range. In other experiments the BEC of trapped lithium [2] and sodium [3] vapor is oserved. This reakthrough opens up new opportunities for studying the optical properties, [,5] statistical properties, [6 9] phase properties, [10,11] tunneling effect, [12 18] and atomic Cooper pairs. [19,20] In this paper, we will investigate quantum statistical ehaviors of photons and atoms in interaction of 7 Li-atomic Bose Einstein condensate with laser when the quantized laser field is initially prepared in a superposition state which exhiits holes in its photonnumer distriution, while the atomic field is initially in a Fock state. This paper is organized as follows. In Sec. 2, the model is riefly introduced. In Sec. 3, the energy exchange etween photons and atoms is discussed. In Sec., we study the su-poissonian distriutions of photons and atoms. In Sec. 5, quadrature squeezings for photons and atoms are investigated. In Sec. 6, the correlation etween photons and atoms is discussed. At last, a conclusion is given in Sec Basic Model and Solution We consider an atomic system of lithium-7 atoms in the ground hyperfine sulevels F = 1, M F = ±1, 0, denoted y g ±,0, respectively. The sulevels g 0 and g + are supposed to e depopulated initially. We assume that N = atoms are accumulated in the state g 0 to form a Bose condensate. Two co-propagating travelingwave light fields are employed to excite the atomic condensate via two-photon excitation. The one light wave is assumed to e strong enough to e treated as a classical field and the other one is a weak quantized field. It is shown [21] that if the quantized light field is initially prepared in a certain nonclassical state, a nonclassical condensate can e generated in the sulevel g. We assume that the two applied light waves are laser pulses with the same temporal envelop f(γ L t), and have forms of the plane wave packets propagating in the k L direction with the center frequency ω and polarizations σ + and σ, respectively. In the rotating-wave approximation, the Hamiltonian of this system reads [21] H = h ɛ n ( g+,n g+,n + g,n g,n ) + h [ (ɛ n + δ j ) j,n j,n + h Ω j e iϕ1 f(γ L t) j,n 1 g+,n2 n 1 e i k L r n 2 n j,n j,n 1,n 2 + ] k L r n 2 + h.c., (1) j,n 1,n 2 g j e iϕ2 f(γ L t)a j,n 1 g,n2 n 1 e i The project supported in part y National Natural Science Foundation of China

2 50 YU Zhao-Xian and JIAO Zhi-Yong Vol. 36 where i,n and i,n are atomic creation and annihilation operators with the atomic internal structure and the center-of-motion eing the states i = g ±, j and Fock states n of virational energies ɛ n, respectively. ϕ i (i = 1, 2) are the phases of the applied fields. The weak quantized laser is descried y operators a and a, Ω j and g j are the coupling constants, and δ j = ω j ω is the laser detunings with ω eing the laser frequency and hω j the energy of the intermediate quantum state j. If we selecting ɛ n = 0 and taking an adiaatically eliminated intermediate-state atomic field operators j,n, then, equations of motion of condensated-atomic field and the weak quantized laser field are given y i ] t g+,0 = f 2 (γ L t) [K 1 g+,0 + R a g,0, (2) Nc i [ t a = f 2 K2 ] (γ L t) g,0 N a g,0 + R g,0 g+,0, (3) c Nc where the ac Stack shifts and the two-photon coupling constant are given y K 1 = j K 2 = j R = j Ω j Ω j /δ j, g j g j N c /δ j, g j Ω j Nc e i(ϕ1 ϕ2 θ) /δ j, () where N c is the atom numer in the condensation at the sulevel g, 0, usually, the condensate is characterized y a coherent state β with g,0 β = N c e iθ β. In analogy to the treatment of the linear coupling of two quantized light fields, [22] one has a(τ) e ikτ = a(0) cos Rτ i(0) sin Rτ, (τ) e ikτ = (0) cos Rτ ia(0) sin Rτ, (5) where we have let τ = t f 2 (γ L t )dt, R = R e iθ, (6) K 1 = K 2 = K, (τ) = g+,0 (τ). (7) In what follows, we consider photons referring to photons in the weak quantized laser field and atoms to atoms in the condensate at the sulevel g +, 0. 3 Energy Exchange Between Atoms and Photons Considering a superposition state denoted as ψ(ξ, φ), ψ(ξ, φ) = η( ξ α 1 + e iφ 1 ξ α 2 ), (8) where η is a normalization constant η = [1 + 2 ξ(1 ξ) Re( e iφ α 1 α 2 )] 1/2. (9) It is shown that state ψ(ξ, φ) exhiits holes in its photonnumer distriution. [23] Note that when ξ 1 (ξ 0), one has that ψ(ξ, φ) α 1 ( ψ(ξ, φ) α 2 ), where α j is a coherent state, j = 1, 2. Hence, the field state in Eq. (8) interpolates etween two aritrary coherent states α 1 and α 2. In Ref. [23], Baseia et al. assumed that α j = r e iθj, j = 1, 2, and showed that, y setting ξ = 1/2 and φ + n θ = (2m + 1)π (m = 1, 2, 3,...), a hole is urned at the component n = N if choosing φ = (1 N/N 0 )π, θ = θ 2 θ 1 = π/n 0, (10) where N 0 is an integer. Assume that the laser field is initially prepared in the state ψ( 1 2, φ) while the atomic field is initially in a Fock state, i.e., the initial state of photon-atom system is in ψ(0) = ψ( 1 2, φ) m. Then, the numers of photons and atoms evolve in the following way N a (τ) = 2A 2 r 2 cos 2 Rτ[1 + e r2 +r 2 cos θ cos(φ θ r 2 sin θ)] + 2mA 2 sin 2 Rτ[1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)], (11) N (τ) = 2A 2 r 2 sin 2 Rτ[1 + e r2 +r 2 cos θ cos(φ θ r 2 sin θ)] + 2mA 2 cos 2 Rτ[1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)], (12) where A = [1 + Re( e iφ α 1 α 2 )] 1/2 is the normalization constant. Equations (11) and (12) denote that there exists energy exchange etween photons and atoms in the time evolution due to the following term N a (τ) N (τ) = 2A 2 r 2 cos 2Rτ[1 + e r2 +r 2 cos θ cos(φ θ r 2 sin θ)] 2mA 2 cos 2Rτ[1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)]. (13) In particular, if the laser pulse area is selected to satisfy 2Rτ 0 = (n )π (n = 0, 1, 2,...), we have N a(τ) = N (τ), which means that the out-state photon numer equals the out-state atom numer in the condensate. This implies that there is no energy exchange etween photons and atoms.

3 No. Quantum Statistical Behaviors of Interaction of an Atomic Bose Einstein Condensate with Laser 51 where Furthermore, the output variances are given y ( N a (τ)) 2 = N 2 a(τ) N a (τ) 2, (1) ( N (τ)) 2 = N 2 (τ) N (τ) 2, (15) N 2 a(τ) = 2A 2 r 2 cos Rτ{r e r2 +r 2 cos θ [cos(φ θ r 2 sin θ) + r 2 cos(φ 2 θ r 2 sin θ)]} + 2m 2 A 2 sin Rτ[1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)] + 2A 2 cos 2 Rτ sin 2 Rτ[(m + 1)r 2 + m + (m + 1)r 2 e r2 +r 2 cos θ cos(φ θ r 2 sin θ) + m e r2 +r 2 cos θ cos(φ r 2 sin θ)], (16) N 2 (τ) = 2A 2 r 2 sin Rτ{r e r2 +r 2 cos θ [cos(φ θ r 2 sin θ) + r 2 cos(φ 2 θ r 2 sin θ)]} + 2m 2 A 2 cos Rτ[1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)] + 2A 2 cos 2 Rτ sin 2 Rτ[(m + 1)r 2 + m + (m + 1)r 2 e r2 +r 2 cos θ cos(φ θ r 2 sin θ) + m e r2 +r 2 cos θ cos(φ r 2 sin θ)]. (17) It is easy to find that the output variance of photons equals that of atoms under the condition Rτ 0 = (n + 1 )π. According to aove discussions, we see that the numer of atoms in the condensate can e controlled y properly selecting the pulse area of the laser. Su-Poissonian Distriutions of Photons and Atoms It is well known that the su-poissonian distriution of photon is a nonclassical effect. [2] A question naturally arises, do the atoms exhiit this quantum effect? In what follows, we will answer this question. Following Mandel, [25] the Q parameters for photons and atoms are defined y, respectively, Q a (τ) = ( N a(τ)) 2 N a (τ) 1, Q (τ) = ( N (τ)) 2 N (τ) 1. (18) The su-poissonian photon (atom) statistics exists whenever 1 Q a() < 0. When Q a() > 0, the state is called super-poissonian while the state with Q a() = 0 is called Poissonian. Below, we calculate Q a() when the initial state of photon-atom system is in ψ(0) = ψ( 1 2, φ) m. Sustituting Eqs (11), (12) and (1) (17) into Eq. (18), we have the following results for the photons: (i) If taking θ = π/2, and oeying the inequality 2 sin(r 2 φ) < e r2, then Q a (0) in < 0; (ii) If taking θ = π, and satisfying cos φ < 0, we also have Q a (0) in < 0. Cases (i) and (ii) indicate that the initial photons are in the su-poissonian distriutions. But, if the laser pulse area is selected to satisfy Rτ 0 = (n )π, we have Q a(τ 0 ) out = 1, which means that the out-state photons are in the su-poissonian distriution. Similarly, for the atoms, we find that: (i) If taking θ = π 2 (or θ = π), then Q (0) in = 1, which indicates that the initial atoms are in the su-poissonian distriutions; (ii) If selecting the laser pulse area to satisfy Rτ 0 = (n )π, we find that when θ = π/2 and 2 sin(r 2 φ) < e r2, Q out < 0. This conclusion also holds for the case of θ = π and cos φ < 0. The aove discussions imply that the out-state atoms can also e in the su-poissonian distriution. 5 Quadrature Squeezings of Laser and Atomic Fields The quadratures for the laser field and for the atomic field can e defined y, respectively, X a = 1 2 (a + a ), Y a = 1 2i (a a ); X = 1 2 ( + ), Y = 1 2i ( ). (19) For the laser field, the degree of the squeezing may e descried y the parameters [26] S 1a (τ) = 2 N a + 2 Re a 2 (τ) ( Re a(τ) ) 2, (20) S 2a (τ) = 2 N a 2 Re a 2 (τ) ( Im a(τ) ) 2. (21) The laser field exhiits quadrature squeezing if S 1a or S 2a is in the range ( 1, 0). Similar expressions are suitale for the atomic field. Below, we investigate quadrature squeezings for the laser field and for the atomic field when the initial state of the photon-atom system is in ψ(0) = ψ( 1 2, φ) m.

4 52 YU Zhao-Xian and JIAO Zhi-Yong Vol. 36 It is easy to get that a(τ) = A 2 r cos Rτ{ e i(θ1 Kτ) + e i(θ2 Kτ) + e r2 +r 2 cos θ [ e i(φ Kτ+θ2 r2 sin θ) + e i(φ+kτ θ1 r2 sin θ) ]}, (22) a 2 (τ) = A 2 r 2 cos 2 Rτ{ e i(2θ1 2Kτ) + e i(2θ2 2Kτ) + e r2 +r 2 cos θ [ e i(φ 2Kτ+2θ2 r2 sin θ) + e i(φ+2kτ 2θ1+r2 sin θ) ]}. (23) Sustituting Eqs (22) and (23) into Eq. (20), we find that when taking θ = π/2 and satisfying the inequality 1 + e r2 [cos(r 2 φ) sin(r 2 φ) sin(r 2 φ) sin 2θ 1 ] 1 2 e 2r2 sin(2r 2 2φ)(1 + sin 2θ 1 ) 2 cos 2( θ 1 + π or taking θ = π and oeying the following inequality )[ 2 ( e r cos φ r 2 + π )] 2 < 0, (2) cos 2 θ 1 < ( e 2r2 + cos φ) e 2r e 2r2 cos φ + e r2, (25) we have S 1a (0) in < 0, which means that the input laser field may exhiit the X a component squeezing. Similarly, for the input laser field, the Y a component squeezing appears under the following condition, respectively, i.e., or θ = π, cos 2 θ 1 > (1 + e 2r2 cos φ)/(1 + 2 e 2r2 cos φ + e r2 ), (26) θ = π 2, 1 + [cos(r 2 φ) sin(r 2 φ) + sin(r 2 φ) sin 2θ e r2 1 ] + e 2r2 sin(2r 2 2φ) 2 [sin 2θ 1 1] 2 sin 2( θ 1 + π )[ 2 ( e r cos r 2 φ π )] 2 < 0. (27) On the other hand, for the atomic field, we have (τ) = A 2 r sin Rτ{ e i(θ1 Kτ π/2) + e i(θ2 Kτ π/2) + e r2 +r 2 cos θ [ e i(φ Kτ+θ2 r2 sin θ π/2) + e i(r2 sin θ φ Kτ+θ 1 π/2) ]}, (28) 2 (τ) = sin 2 RτA 2 r 2 { e i(2θ1 2Kτ) + e i(2θ2 2Kτ) + e r2 +r 2 cos θ Sustituting Eqs (12), (28) and (29) into the following expressions we see that the input squeezing parameters are simply given y [ e i(φ 2Kτ+2θ2 r2 sin θ) + e i(r2 sin θ φ 2Kτ+2θ 1) ]}. (29) S 1 (τ) = 2 N (τ) + 2 Re 2 (τ) ( Re (τ) ) 2, (30) S 2 (τ) = 2 N (τ) 2 Re 2 (τ) ( Im (τ) ) 2, (31) S 1 (0) in = S 2 (0) in = ma 2 [1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)] 0, (32) which means no atomic field squeezing occurs in the input state. However, if we select the laser field to oey Kτ 0 = 2nπ, then, the X component squeezing happens under the following condition, respectively, θ = π 2, m + m cos(r 2 φ) + r 2 sin 2 Rτ e r2 0 e r2 sin 2θ 1 sin(r 2 r 2 sin 2 Rτ 0 φ) 2[1 + e r2 cos(r 2 φ)] { ( sin θ 1 + cos θ e r2 sin φ r 2 + π ) ( cos θ 1 + π )} 2 < 0, (33)

5 No. Quantum Statistical Behaviors of Interaction of an Atomic Bose Einstein Condensate with Laser 53 or θ = π, cos 2θ 1 > (1 + e 2r2 cos φ)(m + m e 2r2 cos φ) r 2 sin 2 Rτ 0 e r2 sin 2 φ r 2 sin 2 Rτ 0 (1 + 2 e 2r2 cos φ + e r2 ) Similarly, the Y component squeezing appears at the following condition, respectively, or θ = π 2, θ = π,. (3) m + m cos(r 2 φ) + r 2 sin 2 Rτ e r2 0 e r2 cos 2θ 1 cos(r 2 2r 2 sin 2 Rτ 0 φ) 1 + e r2 cos(r 2 φ) { 2 (θ 2 cos 1 + π ) ( + e r2 cos θ 1 + π ) ( cos φ r 2 + π )} 2 < 0, (35) cos 2θ 1 < r2 sin 2 Rτ 0 sin 2 φ e r 2 (m + m e 2r 2 cos φ) r 2 sin 2. (36) Rτ 0 (1 + 2 e 2r2 cos φ + e r2 ) According to the aove discussions, we conclude that the input and output laser fields may exhiit quadrature squeezing. But for the atomic field, only the output state exhiits quadrature squeezing. 6 Photon-Atom Correlation It is well known that the correlation etween the laser field and the atomic field can e descried y the second-order cross-correlation function Q a = g (2) a (0) 1, g(2) a (0) = N a(τ)n (τ) N a (τ) N (τ). (37) The function Q a vanishes for uncorrelated states, it is positive for correlated states and negative for anti-correlated states. For a system consisting of two osonic modes (say a and ), the Cauchy Schwartz inequality (CSI) reads [27] (g (2) a (0))2 g (2) a (0)g (2) (0), (38) where g a (2) (0) and g (2) (0) are the second-order zero-time correlation functions which are related to Mandel s Q parameters y g a (2) (0) = 1 + Q a(τ) N a (τ), g(2) (0) = 1 + Q (τ) (39) N (τ) with Q a (τ) and Q (τ) defined y Eq. (18). It is shown that the violation of the CSI can e accompanied y violation of Bell s inequality. [28] If the inequality (38) is violated, the correlation etween the two modes is called nonclassical, which can e characterized y I 0 (τ) = [ g(2) a (0)g (2) (0)] 1/2 g (2) a (0) 1, (0) which is negative if the inequality (38) is violated. We now compute the Q a and study the CSI for the photon-atom system. The initial state of the system is supposed in ψ(0) = ψ( 1 2, φ) m. Notice that For τ = 0, we have N a (τ)n (τ) = sin 2 Rτ cos 2 Rτ[(a (τ)a(τ)) 2 + ( (τ)(τ)) 2 2a (τ)a(τ) (τ)(τ) + a 2 (τ) 2 (τ) + a 2 (τ) 2 (τ) a (τ)a(τ) (τ)(τ)] + (cos Rτ + sin Rτ)a (τ)a(τ) (τ)(τ). (1) N a (0)N (0) = 2mA 2 r 2 [1 + e r2 +r 2 cos θ cos(φ θ r 2 sin θ)]. (2) Sustituting Eqs (11), (12) and (2) into Eq. (37), we find that the in-state Q in a = 0, which means that the laser field and the atomic field are initially uncorrelated. If properly selecting the input laser pulse to satisfy Rτ 0 = (n )π, the out-state Q out a = 0, which means that the uncorrelation etween the laser field and the atomic field happens. On the other hand, under the condition Rτ 0 = (n )π, one has I0 out (m 1)[1 + e (τ 0 ) = r2 +r 2 cos θ cos(φ 2 θ r 2 sin θ)][1 + e r2 +r 2 cos θ cos(φ r 2 sin θ)] 1. (3) m[1 + e r2 +r 2 cos θ cos(φ θ r 2 sin θ)] 2

6 5 YU Zhao-Xian and JIAO Zhi-Yong Vol. 36 It is ovious that when m = 1, I out 0 (τ 0 ) = 1 < 0, which means that there exists the violation of the CSI. We now consider the influence of θ on the CSI. (i) When taking θ = π/2, and satisfying 1 [ m ( 2 e 2r 1) e 2r2 + e r2 sin(r 2 φ) 2 1 ] sin(r 2 φ) < 0, () m e r2 the quantity I out 0 (τ 0 ) < 0. (ii) when taking θ = π, I out 0 (τ 0 ) = m 1 m 1 < 0. (5) Therefore, we conclude that when taking θ = π 2 or π, there exists the violation of the CSI. 7 Conclusions In this paper, we have investigated quantum statistical ehaviors of photons and atoms in interaction of 7 Li-atomic Bose Einstein condensate with quantized laser field. When the quantized laser field is initially prepared in a superposition state which exhiits holes in its photonnumer distriution, while the atomic field is initially in a Fock state, it is shown that there exists energy exchange etween photons and atoms. For the input and output states, the photons and atoms may exhiit the su- Poissonian distriution. The input and output laser fields may exhiit quadrature squeezing; ut for the atomic field, only the output state exhiits quadrature squeezing. It is revealed that there is the violation of the Cauchy Schwartz inequality, which means that the correlation etween photons and atoms is nonclassical. It is found that quantum statistical properties of atoms in the condensate can e controlled y properly selecting the pulse area of the laser field. This will e useful to find the physical application of atomic Bose Einstein condensation in the future. Acknowledgment The authors acknowledge the referee for constructive suggestions. References [1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269 (1995) 198. [2] C.C. Bradley, C.A. Sackett, J.J. Tollet and R.G. Hulet, Phys. Rev. Lett. 75 (1995) [3] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J.van Druten, D.S. Durfee, D.M. Kurn and W. Ketterle, Phys. Rev. Lett. 75 (1995) [] H.D. Polizer, Phys. Rev. A55 (1997) [5] L. You, M. Lewenstein and J. Copper, Phys. Rev. A51 (1995) 712. [6] S. Grossman and M. Holthans, Phys. Rev. A52 (1995) 188. [7] E. Timmermans, P. Tommasini and K. Huang, Phys. Rev. A55 (1997) 365. [8] T.T. Chou, C.N. Yang and L.H. Yu, Phys. Rev. A55 (1997) [9] L.M. KUANG, Commun. Theor. Phys. (Beijing, China) 30 (1998) 161. [10] J. Ruistekoski and D.F. Walls, Phys. Rev. A55 (1997) [11] Y. Castin and J. Daliard, Phys. Rev. A55 (1997) 330. [12] M.W. Jack, M.J. Collet and D.F. Walls, Phys. Rev. A55 (1997) [13] L.M. Kuang and Z.W. Ouyang, Phys. Rev. A61 (2000) [1] H.J. Wang, X.X. Yi and X.W. Ba, Phys. Rev. A62 (2000) [15] F.K. Adullaev and R.A. Kraenkel, Phys. Lett. A272 (2000) 395. [16] I. Marino, S. Raghavan, S. Fantoni, S.R. Shenoy and A. Smerzi, Phys. Rev. A60 (1999) 87. [17] C.P. Sun, quant-ph/ (2000). [18] Z.X. YU and Z.Y. JIAO, Commun. Theor. Phys. (Beijing, China) 36 (2001) 20. [19] F. Weig and W. Zwerger, Europhys. Lett. 9 (2000) 282. [20] P. Torma and P. Zoller, Phys. Rev. Lett. 85 (2000) 87. [21] H. Zeng, W. Zhang and F. Lin, Phys. Rev. A52 (1995) [22] P.L. Knight and L. Allen, Concepts of Quantum Optics, A. Wheaton Co. Ltd (1983). [23] B. Baseia, M.H.Y. Moussa and V.S. Bagnato, Phys. Lett. A20 (1998) 277; B. Baseia and C.M.A. Dantas, Phys. Lett. A253 (1999) 123. [2] R. Loudon, Rep. Pro. Phys. 3 (1980) 913. [25] L. Mandel, Opt. Lett. (1979) 205. [26] L.M. Kuang, X. Chen and M.L. Ge, Phys. Rev. A52 (1995) [27] G.S. Agarwal, J. Opt. Soc. Am. B5 (1988) 190. [28] M.D. Reid and D.F. Walls, Phys. Rev. A3 (1986) 1260.