One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schr'odinger Equation and Solution of Bogoliubov Equation in These Systems

Here Q is a square matrix, and the minus(plus) sign corresponds to the self-(de)focusing case. In the context of BEC, the minus(plus) sign represents the system of attractive(repulsive) bosons. Subsequently, integrable BEC systems are found for every integer spin-n. 3) The energy functional of integrable spinor BECs are given by 3)

where ψ m is a condensate wavefunction with magnetic quantum number m, ρ = n m=-n |ψ m | 2 is a number density, and Θ = n m=-n (-1) m ψ m ψ -m is a singlet pair amplitude. The coefficient 2 n-1 can be eliminated by a choice of normalization, but we keep it for later convenience.

In this short note, we show that the above integrable systems (2) are all mapped to MNLSE, and solve the Bogoliubov equation of these systems. While the integrability itself has been already proved by using a different Lax pair, 3) the mapping to MNLSE has the following advantages: (i) The explicit expression for N -soliton solution is already given under both vanishing 4) and nonvanishing 5) boundary conditions. (ii) With the aid of the theory of squared eigenfunction(or squared Jost function), 6,7) solutions of Bogoliubov equation can be obtained. Since there exist various kinds of low-energy excitations in spinor BEC systems, 8) it would be an interesting future work to solve the scattering problem of low-energy excitations in the presence of an external potential 9) by using the solution given in this short note.

Mapping to MNLSE-We first write the mapping ma- * E-mail address: takahashi@vortex.c.u-tokyo.ac.jp trix and its (anti)symmetrization, and next explain how to find it. The mapping from spin-n integrable spinor BEC to 2 n -dimensional MNLSE is given by

Here

(for i = 0, we define A ⊗0 := 1.) The energy functional which yields Eq. ( 1) is

Substituting the matrix (3) to this functional, one can obtain the functional (2). Furthermore, we can (anti)symmetrize the above matrix by the unitary matrix

The matrix Q = QV then satisfies QT = (-1)

2 n(n-1) Q. We briefly summarize how to construct the matrix (3). In the theory of Lie algebra, 10) writing the elements of Cartan subalgebra as H 1 , . . . , H r (r: rank of this algebra) and raising or lowering operators as E α (which changes weight from µ to µ + α), an irreducible representation D is determined from the following relations:

Here µ = (µ 1 , . . . , µ r ) is a weight vector, and N α,µ is a representation-dependent constant. The irreducible tensor operator 11) corresponding to this representation, D, is characterized by the following commutation relations:

Now, let us consider the algebra so(2n + 1). We construct the tensor operators corresponding to 2n + 1dimensional fundamental representation (we write it as 2n+1 -it is unitary equivalent to the definition of the algebra itself.). As for the matrix representation of generators, H i s and E α s, we use the 2 n -dimensional spinor representation. Therefore tensor operators also become 2 n -dimensional matrices. In this representation, generators are represented by the n-fold tensor product of Pauli matrices. 10) One can determine the matrix elements of T µ,2n+1 (µ = 0, ±e 1 , . . . , ±e n ) through the commutation relations ( 8) and (9), and obtain the matrices appearing in Eq. ( 3).

Bogoliubov equation -Next, we solve the Bogoliubov equation of these integrable systems. Since we are interested in an application to the scattering problem of collective excitations, 9) we discuss the problem of non-vanishing boundary condition, 5) and consider MNLSE of the self-defocusing case with a chemical potential term:

The Bogoliubov equation, though its standard derivation is diagonalization of second-quantized Hamiltonian in mean field approximation, is easily obtained by substituting Q = Q + δQ in Eq. ( 10) and discarding the higher order terms of δQ. Rewriting (δQ, -δQ † ) = (U, V ), one obtains

ISM of MNLSE is formulated through the following extended Zakharov-Shabat type eigenvalue problem: 4,5) ∂ ∂x

Here λ is a spectral parameter, and f and g, which are called Jost function, are n-component vectors when

Eq. (10). Assume that Q is (anti)symmetric: Q = εQ T (ε = ±1). We can then show by a straightforward calculation that if ( f 1 , g 1 ) T and ( f 2 , g 2 ) T are the solutions of Eqs. ( 12) and ( 13) with the same λ, (U, V ) = ( f 1 f T 2 , -ε g 1 g T 2 ) satisfies Eqs. (11). Thus, the squared Jost function gives a solution of Bogoliubov equation.

Example -A

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