Commun. Theor. Phys. (Beiing, China) 41 (2004) pp. 935 940 c International Academic Publishers Vol. 41, No. 6, June 15, 2004 A New Kind o -Quantum Nonlinear Coherent States: Their Generation and Physical Meaning WANG Ji-Suo, 1,2,3 LIU Tang-Kun, 3,4 FENG Jian, 2 SUN Jin-Zuo, 1 and ZHAN Ming-Sheng 3 1 Department o Physics, Yantai University, Yantai 264005, China 2 Department o Physics, Liaocheng University, Liaocheng 252059, China 3 State Key Laboratory o Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute o Physics and Mathematics, the Chinese Academy o Sciences, Wuhan 430071, China 4 Department o Physics, Hubei Normal University, Huangshi 435002, China (Received July 29, 2003; Revised September 28, 2003) Abstract A new ind o -quantum nonlinear coherent states, i.e., the eigenstates o the -th power ˆB ( 3) o the generalized annihilation operator ˆB â 1 o -oscillators, are obtained and their properties are discussed. The ( ˆN) completeness o the states is investigated. An alternative method to construct them is proposed. It is shown that these states may orm a complete Hilbert space, and all o them can be generated by a linear superposition o Roy-type nonlinear coherent states. Physically, they can be generated by a linear superposition o the time-dependent Roy-type nonlinear coherent states at dierent instants. PACS numbers: 42.50.Dv, 03.65.-w, 03.65.Ca Key words: orthonormalized eigenstate, -quantum nonlinear coherent states, completeness, physical meaning 1 Introduction During a ew last years, there has been growing interest in the nonlinear coherent states (NLCS) called - coherent states, [1 5] which are eigenstates o the annihilation operator  â( ˆN) o -oscillators, where ( ˆN) is an operator-valued unction o the boson number operator ˆN. A class o -coherent states can be realized physically as the stationary states o the centre-o-mass motion o a trapped ion. [1] The -coherent states exhibit nonclassical eatures such as squeezing and sel-splitting. The P -representation o density matrix in the NLCS case was investigated by Fan et al. [6] The phase properties o the NLCS were studied. [7] The even and odd NLCS, which are two orthonormalized eigenstates o the square (â( ˆN)) 2 o the operator  â( ˆN), were constructed and their nonclassical eects were studied. [8,9] Based on this wor, the orthonormalized eigenstates o the -th power (â( ˆN)) ( 1) were constructed and their properties were discussed by Liu. [10] Recently the -quantum nonlinear coherent state (KNLCS) has been introduced, whose generation schemes as well as various nonclassical properties such as multi-peaed number distribution, sel-splitting, antibunching, and squeezing, have been studied in detail in Re. [10] [13]. A harmonically trapped two-level ion driven properly by two laser beams, one in resonance with the ion s electronic transition and the other detuned to the -th lower sideband, is shown to realize the KNLCS in the steady regime or suitably chosen control parameters (see, e.g., Res. [11] and [14]). Thereore, it is very signiicant to study some quantum statistical properties o the KNLCS. In practice, the eigenstates o the operator (â( ˆN)) is a ind o KNLCS, and quantum statistical properties o the states were investigated by us. [15] Recently, a new ind o NLCS was constructed by Roy et al. [16] (reerred to as Roy-type NLCS hereater). This ind o NLCS is the eigenstates o the operator ˆB â 1. Based on the wor, the even and odd Roytype NLCS were deined by us, [17] which are two orthonor- ( ˆN) malized eigenstates o the operator ˆB 2. Naturally, in this paper, it is very desirable to study the orthonormalized eigenstates o the -th power ˆB ( 3) o the operator ˆB. The paper is organized as ollows. In Sec. 2, the orthonormalized eigenstates o the operator ˆB are obtained. In Sec. 3, the mathematical properties o the states are discussed. In Sec. 4, an alternative method to construct them is proposed. Their physical meaning is explored in Sec. 5. 2 Orthonormalized Eigenstates o Operator ˆB For convenience o reerence and completeness, in this section we begin with some related results or the NLCS [2] and the Roy-type NLCS. [16] We notice that the generalized annihilation (creation) operator associated with NLCS is given by  â( ˆN),  ( ˆN)â, ˆN â â, (1) The proect supported by National Natural Science Foundation o China under Grant No. 10074072 and the Natural Science Foundation o Shandong Province o China under Grant No. Y2002A05
936 WANG Ji-Suo, LIU Tang-Kun, FENG Jian, SUN Jin-Zuo, and ZHAN Ming-Sheng Vol. 41 where â and â are standard harmonic oscillator creation and annihilation operators and (x) is a reasonably wellbehaved real unction, called the nonlinearity unction. From the relations (1), it ollows that Â, Â, and ˆN satisy the ollowing closed algebraic relations: [ ˆN, Â] Â, [ ˆN,  ] Â, [Â,  ] 2 ( ˆN)( ˆN + 1) 2 ( ˆN 1) ˆN. (2) Clearly, the nature o the nonlinear algebra depends on the choice o the nonlinearity unction ( ˆN). For ( ˆN) 1 we regain the Heisenberg algebra. NLCS α, are then deined as right eigenstates o the generalized annihilation operator Â,[1,2]  α, α α,. (3) In the number state basis, α, is given by α n α, C n, (n)! { α 2n } 1/2 C, (4) [(n)!] 2 where α is a complex number, and (n)! (n)(n 1) (1)(0), (0) 1. (5) The canonical conugate o the generalized annihilation and creation operators  and  are given by [16] ˆB â 1 ( ˆN), ˆB 1 ( ˆN)â. (6) Thus Â, ˆB and their conugates satisy the ollowing algebras, [16] [Â, ˆB ] 1, [ ˆB,  ] 1. (7) In the number state basis, the Roy-type NLCS [16] are deined as the right eigenstates o the new generalized annihilation operator ˆB, β, N β n (n)! n, { β 2n [(n)!] 2 } 1/2 N, (8) where β is an arbitrary complex number. Now, let us consider the ollowing states β n+ (n + )! ψ (β, ) C n +, (9) (n + )! where is a positive integer (here and henceorth 3; we do not indicate it in the ollowing), 0, 1, 2,..., 1. C are normalized actors. With ˆB operating on ψ (β, ), we obtain ˆB ψ (β, ) β C β n+ (n + )! (n + )! n + β ψ (β, ). (10) As a result, the states o Eq. (9) are all the eigenstates o the operator ˆB with the same eigenvalue β. It is easy to chec that, or the same value o, these states are orthogonal to one another with respect to the subscript : ψ i (β, ) ψ (β, ) 0, i, 0, 1, 2,..., 1, i. (11) Let x β 2. We simply suppose C to be a real number. Using the normalized conditions o the states given by Eq. (9), we have [ C [B (x, )] 1/2 x n+ [(n + )!] 2 ] 1/2. (12) (n + )! From Eq. (12) it ollows that x n [(n)!] 2 B (x, ) N 2 e (x). (13) 3 Mathematical Properties o Eigenstates o Operator ˆB Firstly, it is seen that the eigenstates o the operator ˆB contain the complex parameter β. When β taes dierent values, the internal product o every eigenstate does not equal zero, i.e., ψ (β, ) ψ (β, ) [B ( β 2, )B ( β 2, )] 1/2 (β β ) n+ [(n + )!] 2 (n + )! [B ( β 2, )B ( β 2, )] 1/2 B (β β, ) 0 (i β β ). (14) This means that in the β maniold, each o the eigenstates o the operator ˆB is not orthogonal by itsel. This property is the same as that o the normal coherent states.
No. 6 A New Kind o -Quantum Nonlinear Coherent States: Their Generation and Physical Meaning 937 Secondly, in the space consisting o the eigenstates o the operator ˆB, each o the eigenstates can be generated by the annihilation operator ˆB. For example, i the operator ˆB is used successively on ψ 0 (β, ), we have ˆB i ψ 0 (β, ) β i B 1/2 0 ( β 2, )B 1/2 i ( β 2, ) ψ i (β, ), i 1, 2,...,. (15) That is, under the action o ˆB, the eigenstate ψ 0 (β, ) may be transormed in turn as ψ 0 (β, ) ψ (β, ) ψ 2 (β, ) ψ 1 (β, ) ψ 0 (β, ). Thereore, the operator ˆB plays the role o a rotation operator among the eigenstates o the operator ˆB. The inal question that concerns us is whether the states given by Eq. (9) could construct a complete Hilbert space, i.e., whether they could be used as a representation. In order to construct the completeness ormula o the states, we use the density operator method. [18] We deine the density operator (i.e., a density matrix) o the state n + ρ P (n + ) n + n +, (16) where P (n + ) P (n +, β)d 2 β is the probability distribution o the (n + )-th state n + in the state ψ (β, ), in which Thus we have P (n +, β) n + ψ (β, ) 2 1 β 2(n+) [(n + )!] 2 B ( β 2. (17), ) (n + )! P 1 (n + ) n + n +. Thereore, the completeness ormula o the states given by Eq. (9) can be written as The proo o Eq. (18) is given as d 2 β ψ (β, ) ψ (β, ) m0 2π d 2 β ψ (β, ) ψ (β, ) 1. (18) m0 (m + )!(n + )! (m + )!(n + )! rdr (r2 ) n+ [(n + )!] 2 B (r 2 n + n +, )(n + )! P (n + ) n + n + d 2 β βm+ β (n+) B ( β 2 m + n +, ) P 1 (m + ) m + m + P (n + ) n + n + n n 1, (19) where β r exp(iθ), d 2 β rdrdθ. Thereore, the linear combination o the states may orm a complete representation. They can be used as a representation. For example, in this representation, the Roy-type NLCS β, [16] [see Eq. (8)] may be expressed as β, N 1/2 B 1/2 ( β 2, ) ψ (β, ). (20) It is interesting to note that when (n) 1, the states given by Eq. (9) become orthonormalized eigenstates o the high powers o the annihilation operator â o the usual harmonic oscillator. [19,20] 4 Generation o the Eigenstates o the Operator ˆB According to Eq. (20), we consider the ollowing Roy-type NLCS: β l, β e i2πl/, e 1/2 ( β 2 β n (n)! ) e i(2π/)nl n (l 0, 1, 2,..., 1). (21) The Roy-type NLCS are discretely distributed with an equal interval o angle along a circle around the origin o the β-plane. The inner product o the two states o Eq. (21) is β l, β l, e 1 ( β 2 )e ( β 2 e i2π(l l)/ ), (l, l 0, 1, 2,..., 1). (22)
938 WANG Ji-Suo, LIU Tang-Kun, FENG Jian, SUN Jin-Zuo, and ZHAN Ming-Sheng Vol. 41 Consider a linear transormation S such that where β, ϕ S β,, (23) β 0, β 1, β,, ϕ ϕ 0 ϕ 1 ϕ. (24) S is a matrix that maes ϕ orthonormal, and ϕ ϕ δ. The above requirement leads to a set o algebraic equations or S i, e 1 ( β 2 )e ( β 2 e i(2π/)(l l) )SlS l δ. (25) l 0 The solution o Eq. (25), S i, can be ound as ollows. By virtue o the relation e ( β 2 e ±i(2π/)(l l) ) e i(2π/)l e i(2π/)l e ( β 2 e ±i(2π/)l ) e i(2π/)l, (26) l 0 the matrix elements o S that satisy Eq. (25) are given by l 0 S l 1 1 e1/2 ( β )[ 2 e ( β 2 e i(2π/)l ) e i(2π/)l] 1/2 e i(2π/)l l 0 1 e1/2 ( β 2 )B 1/2 ( β 2, ) e i(2π/)l (, l 0, 1, 2,..., 1). (27) From Eqs. (23) and (27), we obtain orthonormalized states, ϕ 1 B 1/2 ( β 2, )e 1/2 ( β 2 ) e i(2π/)l β e i(2π/)l, ( 0, 1,..., 1), (28) which are ust what we want. By maing use o the relation it can be proved that According to Eq. (28), or 2, we obtain e i(2π/)lt 0 (t 1, 2,..., 1), (29) ϕ ψ (β, ), ( 0, 1,..., 1). (30) ϕ 0 2 1 2 B 1/2 0 ( β 2, )e 1/2 ( β 2 )( β, + β, ), (31) ϕ 1 2 1 2 B 1/2 1 ( β 2, )e 1/2 ( β 2 )( β, β, ), (32) which are ust the so-called even and odd Roy-type NLCS deined by us. [17] The ϕ ( 0, 1,..., 1) in Eq. (28) are exactly the orthonormalized eigenstates o the operator ˆB obtained in Sec. 2, but reconstructed here by a dierent method. From the above reconstruction, we come to an important conclusion that any orthonormalized eigenstates o the operator ˆB can be generated rom a linear superposition o Roy-type NLCS β e i(2π/)l, (l 0, 1, 2,..., 1), which have the same amplitude but dierent phases. Then, rom Eq. (28), one can ind the connection between the Roy-type NLCS and these orthonormalized eigenstates o the operator ˆB. In practice, the eigenstates o the operator ˆB is a new ind o KNLCS. 5 Physical Meaning o the Eigenstates o Operator ˆB In this section, we shall explore the physical meaning o the orthonormalized eigenstates o the operator ˆB by constructing them rom the time-dependent Roy-type NLCS generated rom a time-dependent Schrödinger equation. Suppose a system evolves according to the time-dependent Schrödinger equation i h χ(t) Ĥ χ(t). (33) t I the system is initially (t 0) in a Roy-type NLCS β, and the Hamiltonian is independent o time, then at time t the system reaches the state χ(t) exp ( ī ) hĥt β,. (34)
No. 6 A New Kind o -Quantum Nonlinear Coherent States: Their Generation and Physical Meaning 939 Choosing Ĥ hω ˆN, we have Thereore, at the instant the system is in the state t l 2π ω χ(t) β e iωt,. (35) l (l 0, 1, 2,..., 1), (36) χ(t l ) β e i(2π/)l,. (37) Now let us consider a linear superposition o the above time-dependent Roy-type NLCS at dierent instants, φ i Cl i β e i(2π/)l,. (38) Suitably choosing the expansion coeicients, we can construct the orthonormalized eigenstates. The inner product o the two states o Eq. (38) is where and φ i φ (Cl i ) C l βz, l βz l, C i M C (i, 0, 1, 2,..., 1), (39) l 0 C M C 0 C 1 C z exp( i2π/), (40), C i (C0 i C1 i C) i, (41) β β β βz β βz βz β βz βz βz βz βz β βz... βz βz βz. (42) Note that the symbol is suppressed in the expression o the matrix elements o M, i.e. βz l βz l βz l, βz l, (l, l 0, 1, 2,..., 1). (43) The matrix elements o M read M l,l e 1 ( β 2 )e ( β 2 z l l ) (l, l 0, 1, 2,..., ), (44) where the unction e (x) is given by Eq. (13). Because M is Hermitian, its eigenstates with dierent eigenvalues must be orthogonal to one another. Suppose that C i and C are its two eigenstates. It ollows that M C i λ i C i, M C λ C, (45) where 1 z C z 2, (46) z () λ e 1 ( β 2 ) e ( β 2 z)z l l. (47) Thereore the orthonormal relation reads C i C δ i. (48) Replacing the expansion coeicients in Eq. (48) by the column vector (46) and considering the normalization condition o the states (38), we get φ (λ ) 1/2 z l βzl,, ( 0, 1, 2,..., 1). (49) By virtue o Eqs. (45) and (48), it is easy to prove that the inner product o two states o Eq. (49) is 1 φ i φ C i M C λ i /λ δ i δ i, (50) λ i λ which indicates that the states (49) orm an orthonormalized set. The physical meaning o the states φ in Eq. (49) has been made clearer. The states φ can be generated by a linear superposition o the time-dependent Roy-type NLCS β e iωt l, (l 0, 1, 2,..., 1) at dierent instants, while the superposition coeicients are
940 WANG Ji-Suo, LIU Tang-Kun, FENG Jian, SUN Jin-Zuo, and ZHAN Ming-Sheng Vol. 41 C l exp( i2πl/). In addition, it can be proved that φ 0 ψ 0 (β, ), φ l ψ l (β, ), (l 1, 2,..., 1). (51) Thereore, the states φ given by Eq. (49) are exactly the eigenstates o the operator ˆB in Eq. (9). Obviously, the above discussion includes the limiting case o (n) 1 investigated by us. [19,20] In act, the method or construction o the eigenstates o the operator ˆB in this section is somewhat dierent rom that in section 4. 6 Conclusions and Discussions In this paper, we have derived the orthonormalized eigenstates o the -th power ˆB ( 3) o the generalized annihilation operator ˆB ( ˆB â 1 ) o -oscillators, ( ˆN) discussed their mathematical properties and given some applications o the result. It is shown that the linear combination o the states may orm a complete Hilbert space. We have proposed an alternative method to construct these eigenstates o the operator ˆB, and come to an important conclusion that all o them can be expressed as a linear superposition o Roy-type NLCS that have the same amplitude but dierent phases. We have also explored their physical meaning. It is shown that they can be generated by a linear superposition o the timedependent Roy-type NLCS at dierent instants. It is well nown that a class o NLCS can be realized physically as the stationary states o the center-omass motion o a trapped ion. [1] These experimental requirements can be ulilled using available trapped ion techniques. [21 23] Similarly, a harmonically trapped twolevel ion driven properly by two laser beams, one in resonance with the ion s electronic transition and the other detuned to the -th lower sideband, is shown to realize the KNLCS in the steady regime or suitably chosen control parameters. [11,14] In act, the eigenstates o the operator ˆB constructed by this paper is a new ind o KNLCS. This ind o KNLCS will reduce to the Roy-type NLCS deined by Eq. (8) when 1. Thereore, we construct this new ind o KNLCS and study some properties o them, which are very signiicant. The construction o this ind o KNLCS provides a means to manipulate the quantum state. This would give relevant motivation or more thorough studies in the uture. In addition, it is interesting to note that when (n) 1, the orthonormalized eigenstates o the operator ˆB become the states investigated by us in Res. [19] and [20]. Thereore, the above discussion includes the limiting case o (n) 1 investigated by us. [19,20] Reerences [1] R.L. de Matos Filho and W. Vogel, Phys. Rev. A54 (1996) 4560. [2] V.I. Man o, G. Marmo, et al., Phys. Scr. 55 (1997) 528. [3] B. Roy and P. Roy, J. Opt B: Quantum Semiclass. Opt. 1 (1999) 341. [4] A. Aniello, V. Man o, et al., J. Opt B: Quantum Semiclass. Opt. 2 (2000) 718. [5] S. Sivaumar, J. Opt B: Quantum Semiclass. Opt. 2 (2000) R61. [6] H.Y. Fan and H.L. Cheng, Commun. Theor. Phys. (Beiing, China) 37 (2002) 655. [7] T.Q. Song and Y.J. Zhu, Commun. Theor. Phys. (Beiing, China) 38 (2002) 606. [8] S. Mancini, Phys. Lett. A233 (1997) 291. [9] S. Sivaumar, Phys. Lett. A250 (1998) 257. [10] X.M. Liu, J. Phys. A32 (1999) 8685. [11] V.I. Man o, G. Marmo, et al., Phys. Rev. A62 (2000) 053407. [12] N.B. An, Chin. J. Phys. 39 (2001) 594. [13] N.B. An, Phys. Lett. A284 (2001) 72. [14] N.B. An, quant-ph/0103077 (2001). [15] J.S. Wang, J. Feng, T.K. Liu, and M.S. Zhan, J. Phys. B35 (2002) 2411. [16] B. Roy and P. Roy, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) 65. [17] J.S. Wang, J. Feng, T.K. Liu, and M.S. Zhan, Acta Phys. Sin. 51 (2002) 2059 (in Chinese). [18] S.R. Hao, Acta Phys. Sin. 42 (1993) 1057 (in Chinese). [19] J.Z. Sun, J.S. Wang, and C.K. Wang, Phys. Rev. A44 (1991) 3369. [20] J.Z. Sun, J.S. Wang, and C.K. Wang, Phys. Rev. A46 (1992) 1700. [21] C. Monroe, D.M. Meeho, et al., Science 272 (1996) 1131. [22] C. Monroe, D.M. Meeho, et al., Phys. Rev. Lett. 75 (1995) 4011. [23] F. Diedrich, J.C. Bergquist, et al., Phys. Rev. Lett. 62 (1989) 403.