Bose Description of Pauli Spin Operators and Related Coherent States

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1 Commun. Theor. Phys. (Beijing, China) 43 (5) pp. 7 c International Academic Publishers Vol. 43, No., January 5, 5 Bose Description of Pauli Spin Operators and Related Coherent States JIANG Nian-Quan,, FAN Hong-Yi,,3 and LU Hai-Liang 3 Department of Physics, Wenzhou Normal College, Wenzhou 357, China Department of Material Science and Engineering, University of Science and Technology of China, Hefei 36, China 3 Department of Physics, Shanghai Jiao Tong University, Shanghai 3, China (Received December, 3) Abstract Using both the fermionic-like and the bosonic-like properties of the Pauli spin operators σ +, σ, and σ z we discuss the derivation of Bose description of the Pauli spin operators originally proposed by Shigefumi Naka, and deduce another new bosonic representation of Pauli operators. The related coherent states, which are nonlinear coherent state and coherent spin states for two spins, respectively, are constructed. PACS numbers: 3.65.Ca Key words: Bose description of Pauli spin operators, nonlinear coherent state, coherent spin state Introduction The purpose of this paper is to discuss the singlemode Bose description of Pauli spin operators, as one can see shortly later, that this description is different from the Schwinger two-mode Bose realization of angular momentum, while the former is only valid for spin /, the latter applies for any values of angular momentum. The Bose description of Pauli spin operators was originally proposed by Shigefumi Naka. It can also be applied to the Fermi operators, since the quantization of the anti-commuting variables by using anti-commutators generates the elements of Clifford algebra such as Pauli matrices. To see the exact resemblance between Fermi operators and the Pauli spin operators more clearly, we use the matrix representation of the Fermi operators. Let f and f be fermion annihilation and creation operators respectively, satisfying anti-commutative relations {f, f } +, f, f, () and let be the fermion vacuum state, then f, f, f. One can see when using the matrix representation of and as, ( ) ( ),, () ( ) f, ( ) f, (3) so the representations of f and f are just the Pauli matrices. Therefore, the Bose description of the Pauli spin operators is equivalent to the Bose description of Fermi operators. In Ref. Shigefumi Naka firstly proposed the representation of the anti-commuting operators (or pseudospin operator) in the Hilbert space from the simple observation such that dividing the Hilbert space into two subspaces, the anti-commuting operators can be realized as the operators having matrix elements between these subspaces. By setting n, n, (n + )! (a ) n+ n +, (n)! (a ) n n, (4) in Fock space, where n (a n / n!) is the Fock state, and noting that cos πn Shigefumi Naka identified f cos (πn/) N + a n n, (5) n n n +, n f a cos (πn/) N + n + n, (6) n though without presenting a manifest derivation. Later the bosonization structure of fermions is used in Ref. 3. Note that in treating spin / system (or two-level atoms) the traditional notation is that σ flips from the upper level (denoted as ( ) ) to the lower level s/ ( ), while σ s / + flips from lower to upper, ) ) ( σ ( ) σ + s/ s / ( ) the standard Pauli matrices are ( σ + ( ), σ, s /, (7) s/ ( ). (8) The project supported by National Natural Science Foundation of China under Grant No. 7557

2 8 JIANG Nian-Quan, FAN Hong-Yi, and LU Hai-Liang Vol. 43 In Sec. we shall show how to directly employ both the bosonic-like property and fermionic-like property of Pauli spin operators σ i to derive Shigefumi Naka s description. In Sec. 3, to go a step further, we make a supplement to this description, which means that another Bose representation of Pauli spin operators can be found. In Secs. 4 and 5 the related coherent states, which are nonlinear coherent state and coherent spin states for two spins, respectively, are constructed. In particular, we can use the Bose representation of Pauli spin to introduce two spins SU() coherent state with j /, which seems to be new. Derivation of Shigefumi Naka s Bose Operator Description of Pauli Spin Pauli spin operators σ i obeys the anticommutative relation {σ i, σ j } + δ ij, i, j,, 3. (9) Introducing σ ± (σ ± iσ ), σ + σ, () Louisell, 4 summarizes the following commutative relations (bosonic-like property), σ ±, σ ±σ z, σ ±, σ iσ z, σ ±, σ z σ ±, σ +, σ σ z, σ z σ 3 () as well as the anti-commutative relations (fermionic-like property) {σ ±, σ } + I, {σ ±, σ } + ±i, {σ ±, σ z } +, {σ +, σ } + I. () We can deduce the representation (6) by the following method. Without loss of generality, we assume that the Pauli spin operators can be expressed as σ g(n)a, σ + a g(n), (3) where g(n) is an operator-valued function of number operator N a a. It follows from a, a, and that g(n)a ag(n ), a g(n) g(n )a, (4) σ σ + (N + )g (N), σ + σ a g (N)a Ng (N ). (5) Substituting Eq. (5) into the anti-commutative relation () we see (N + )g (N) + Ng (N ) I. (6) Because the eigenvalues of N are,,,..., the solution to Eq. (6) is since (N + )g (N) cos l πn, l,,..., (7) cos l πn π(n ) + cosl I. (8) It then follows from Eqs. (7) and (3) that g(n) cos l πn N +, σ From Eqs. (3), (4), and (9) we see σ cosl (πn/) N + N + cos l πn a. (9) a cosl (πn/) a N + cosl (πn/) N + cos l π(n + )/ N + a. () When l, with the use of the completeness relation of Fock state n n n, () we see n cos πn ( + cos πn) ( + cos πn) n n n n n, () n and using n a n + n +, σ n n +, (3) n which is just the Shigefumi Naka s representation. l k, k,, 3..., we have cos k πn sin k πn n n, n For n + n +. (4) n Let us recall the even- and odd-coherent state representations, 5 z e e z / z n n, n! so z o e z / n n n z n+ (n + )! n +, (5) n n cos k πn π z ee z, n + n + sin k πn n π z oo z. (6) 3 Another Bose Operator Representation of Pauli Spin From Eq. () we see when l, cos πn n cos πn n n

3 No. Bose Description of Pauli Spin Operators and Related Coherent States 9 sin πn ( ) n n n, n n sin πn n n ( ) n n + n +. (7) n It then follows from Eqs. (4) and (7) that when l k +, cos k+ πn ( ) n n n n n sin k+ πn n ( ) n n n, n n ( ) n n + n +. (8) n In this case, using Eqs. (3), (9), and (8) we have σ ( ) n n n +, σ + n ( ) n n + n, (9) n which is another Bose operator realization of Pauli spin operators. Due to ( ) N/ z e e z / ( ) n z n n n! iz e, n ( ) (N )/ z o e z / n ( ) n z n+ (n + )! n + iz o. (3 ) From Eqs. (6), (8), and (3), the corresponding coherent representation of Eq. (8) is cos k+ πn π iz ee z, sin k+ πn π iz oo z. (3) For l being both even and odd, from Eqs. (9) and/or (3) we have σ + σ n + n +, σ σ + n n n, (3) n it then follows from Eq. (6) that σ z σ +, σ ( n + n + n n ) n sin πn cos πn cos πn e iπn + e iπn ( ) N+. (33) Using the technique of integral within an ordered product of operators 6 we know its normally ordered form is d z σ z π ( )N z z π z z π : exp z za + z a a a : : e a a :. (34) For the new representation we can check its fermion-like property. Using Eqs. (9) and (33), we see {σ, σ z } + ( ) 3n+ + ( ) 3n+ n n + n. (35) Its bosonic-like property can also be checked, i.e., σ, σ z ( ) 3n+ ( ) 3n+ n n + n ( ) n n n + σ. (36) n 4 Eigenvector of σ as a Nonlinear Coherent State In Refs. 7 the nonlinear coherent state is defined as eigenstate of g(n)a, g(n)a z g z z g. (37) From relation (4), g(n)a, g(n ) a, (38) the nonlinear coherent state z g can be constructed directly as z z g exp g(n ) a. (39) Since σ is now in the form g(n)a (see Eqs. (3)), from Eqs. (9) and (39) the eigenvector of σ can be discussed in the context of nonlinear coherent states and be formally expressed as N ξ g exp cos l π(n )/ a ξ. (4) Thus and because σ, σ ξ g ξ ξ g, (4) σ (σ ξ g ) ξ ξ g, (4) therefore ξ, so ξ must be a Grassmann number. Expanding the right-hand side of Eq. (4) we obtain N ξ g exp cos l π(n )/ a ξ + ξ. (43)

4 JIANG Nian-Quan, FAN Hong-Yi, and LU Hai-Liang Vol. 43 Thus equation (4) is also a fermionic coherent state. It is Ohnuki and Kashiwa who first employed Grassmann numbers in constructing fermionic coherent states. 5 Coherent Spin States in Bosonic Representation It has been known for long that the disentangling of the SU() coset element is where exp(ξσ + ξ σ ) exp(τσ + ) expσ z ln( + τ ) exp( τ σ ), (44) ξ θ e iφ, τ e iφ tan θ. (45) Using the Bose operator representation (), (3), and (33) we construct the state exp(ξσ + ξ σ ) n exp(τσ + ) expσ z ln( + τ ) exp( τ σ ) n where τ n obeys the eigenvector equation ( + τ ) exp(τσ + ) n ( + τ )( n + τ n + ) τ n, (46) (σ + τ σ + ) τ n τ τ n. (47) When n, τ is just the Radcliffe coherent spin state. 3 5 For two spins, it is not difficult to see that σ σ, σ + σ +, and (σ + σ + σ + σ )/ form an SU() algebra, 6 σ σ, σ + σ + (σ +σ + σ + σ ), (48) where σ σ, (σ + σ + σ + σ ) σ σ, (49) σ + σ +, (σ + σ + σ + σ ) σ + σ +, (5) σ + σ + σ + σ (σ z I + I σ z ). (5) The proof is shown in Appendix. Hence we can express exp(ξσ + σ + ξ σ σ ) in terms of Bose operators in two-mode Fock space exp(ξσ + σ + ξ σ σ ) exp(τσ + σ + ) exp 4 (σ z I + I σ z ) ln( + τ ) exp( τ σ σ ). (5) Using the Bose representation (3) we have exp(ξσ + σ + ξ σ σ ), which can be proved to obey the eigenvector equations (, + τ, ) τ, (53) + τ (σ σ + τ σ + σ + ) τ τ τ, (54) and σ σ + τ 4 (σ z I + I σ z ) τ τ τ, (55) which tells us that τ is an SU() coherent state with j /, (note (/4)(σ z I +I σ z ),, ). Moreover, we can show exp(ξσ + σ + ξ σ σ ),,, (56) exp(ξσ + σ + ξ σ σ ),,, (57) exp(ξσ + σ + ξ τ σ σ ), exp(τσ + σ + ) + τ,, + τ + τ (, τ, ). (58) Equations (53) and (58) are two entangled states. In summary, using both the fermionic-like and the bosonic-like properties of the Pauli spin operators we not only have shown how to derive the Shigefumi Naka s Bose operator description of the Pauli spin operators, but also derived another new bosonic representation. The eigenvector of σ in this representation is a nonlinear coherent state with the eigenvalues being the Grassmann numbers. The coherent spin states of exp(ξσ + σ + ξ σ σ ), is also derived. Needless to say, the bosonic realization of spin operators (or named pseudospin) makes the calculation of expectation values of any operator function f(σ + σ +, σ σ, 4 (σ z I + I σ z )) in any two-mode quantum states (say the two-mode squeezed state) possible.

5 No. Bose Description of Pauli Spin Operators and Related Coherent States Appendix To prove Eq. (48) we notify σ σ σ σ ( ) ( ), and σ + σ +, (A) so σ σ, σ + σ +. (A3) On the other hand σ + σ I I σ + σ I I, (σ z I + I σ z ) +, Thus equation (48) is proved. (A) (A4) (A5) References J. Schwinger, Quantum Theory of Angular Momentum, Academic Press, New York (965) p. 9. Shigefumi Naka, Prog. Theor. Phys. 59 (978) 7. 3 Ruan Tu-Nan, Jing Si-Cong, and Wang An-Ming, Euro. Phys. Lett. 3 (993) Y. Ohnuki and T. Kashiwa, Prog. Theor. Phys. 6 (978) W.H. Louisell, Qauntum Statistical Properties of Radiation, John Wiley, New York (973). 6 M. Hillery, Phys. Rev. A36 (987) 3796; T.M. Seligman and K.B. Wolf, Phys. Scr. 55 (997) For a review, see Hong-Yi Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (3) R R7. 8 Hong-Yi Fan and Hai-Ling Cheng, Phys. Lett. A85 () S. Sivakumar, J. Phys. A: Math. Gen. 9 (996) S. Mancini, Phys. Lett. A33 (997) 9. V.I. Manko, G. Marmo, E.C.G. Sudarshan, and F. Zaccaria, in Proc. IV Wigner Symp., ed. Atakishiev, Guadalajara, Mexico, July 995, World Scientific, Singapore (996) p. 4. F.A. Brezin, The Mothod of Second Quantization, Academic Press, New York (966). 3 F.T. Arechi, et al., Phys. Rev. A6 (97). 4 Hong-Yi Fan and Jun-Hua Chen, Eur. Phys. J. D3 (3) M. Radcliffe, J. Phys. A: Gen. Phys. 4 (97) Fan Hong-Yi, Commun. Theor. Phys. (Beijing, China) (989) 59.