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Wang Ji-Suo, Meng Xiang-Guo, Fan Hong-Yi. Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem. Chinese Physics B, 2019, 28(10): 100301  
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Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem
Wang Ji-Suo1, †, Meng Xiang-Guo2, Fan Hong-Yi3
Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, College of Physics and Engineering, Qufu Normal University, Qufu 273165, China
Shandong Provincial Key Laboratory of Optical Communication Science and Technology, School of Physical Science and Information Engineering, Liaocheng University, Liaocheng 252059, China
Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China

 

† Corresponding author. E-mail: jswang@qfnu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11347026) and the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2016AM03 and ZR2017MA011).

Abstract

We study how can an angular momentum coherent state keeps its form-invariant during time evolution governed by the Hamiltonian . We discuss this topic in the context of boson realization of . By employing the entangled state representation and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function , which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of is totally determined by the parameters involved in the Hamiltonian.

PACS: 03.65.-w;03.65.Ud;42.50.Dv
Keyword:invariance of angular momentum coherent state;entangled state representation;new binomial theorem;time evolution
1. Introduction

Coherent state theory has been widely used in quantum optics and laser physics. The optical coherent state with the amplitude α is defined as[1–4]

where a and are respectively boson annihilation and creation operators, . It has been common knowledge that the Hamiltonian
can keep form-invariant for coherent state, i.e., if the initial state is a coherent state, it will evolve into still a coherent state. In this work we search for which kind of Hamiltonian can maintain the form of angular momentum coherent state (AMCS), the AMCS is defined as[5–9]
where and are raising and lowering operators of angular momentum, respectively, obeying the following commutative relations
Jz has its eigenstate
and is the lowest weight state annihilated by , i.e., . Using the disentangling formula
where and . Equation (3) becomes
We shall begin with the boson realization of AMCS to proceed our discussion. We then introduce the entangled state representation and derive a new binomial theorem involving two-subscript Hermite polynomials to go on. It turns out that the wave function of AMCS in the entangled state representation is classified as single-subscript Hermite polynomials . Based on this result and the recursive property of we find that the Hamiltonian can maintain the form-invariance of AMCS , and the value of is totally determined by f(t), , and g(t).

The rest of this paper is arranged as follows. In Section 2 we reform the angular momentum coherent state in boson realization and show that it is an entangled state. In Section 3 we set up the entangled state representation and introduce the two-subscript Hermite polynomials. In Section 4 we deduce the generalized binomial formula which involves two-subscript Hermite polynomials. In Section 5 we obtain the wave function of in the entangled state representation. In Section 6 we examine the form-invariance of AMCS during time evolution and determine the value of .

2. Angular momentum coherent state in boson realization

Using Schwinger boson realization of angular momentum operators

the eigenstate of Jz is
where is two-mode vacuum state in Fock space, , , thus
and we can reform the AMCS in Eq. (7) as
which exhibits quantum entanglement between the two modes, this enlighten us to introduce the entangled state representation to study .

3. The entangled state representation and the two-subscript Hermite polynomials

In Refs. [10] and [11], the unnormalized continuous-variable entangled state , which is the common eigenvector of and , noting , has been constructed as follows:

which takes the two-subscript Hermite polynomials as its expansion function in the two-mode Fock space
here can be either defined through its generating function[12,13]
or by its power-series expansion
Note that is not a direct product of two independent single-subscript Hermite polynomials. have their own uses in studying quantum optics and light propagation modes in quadratic graded-index media. From Eq. (13) we see
An important insight in is its completeness relation

4. A generalized binomial formula involved two-subscript Hermite polynomials

From Eqs. (11) and (16) we see the wave function

We are challenged to make the summation in Eq. (18), which involves . Instead of directly doing this, we try to prove a generalized binomial formula by considering
where is in antinormal ordering form, noting that a and are permuted within the antinormal ordering symbol .[14,15] Using Eq. (14) we have
Comparing on both sides of Eq. (20) we have the operator identity
where denotes normal ordering.[16–19] Thus
By making up
noting on the two sides of Eq. (23) we see
therefore equation (22) becomes
Since the two sides of Eq. (25) are both in antinormal ordering, so we obtain a new generalized binomial formula
its right-hand side appears as a single-subscript Hermite polynomial. Further, let
Equation (18) becomes
here denotes the single-subscript Hermite polynomial of order 2j. Hence the wave function of angular momentum coherent state in the entangled state representation is classifies as . This brings convenience for further studying .

5. Time evolution of AMCS

As an application of the above theory, we consider the system described by the Hamiltonian

and supposing it initially is in angular momentum coherent state , we wish that it still evolves into an angular momentum coherent state, which means , then the Schrödinger equation is
Inserting the completeness relation of entangled states in Eq. (17)
and using the following relations
we have
Since we assume is an angular momentum coherent state, using Eq. (28) the above equation (34) turns to
Noting χ in Eq. (27), so
and using the property of (Refs. [6] and [20])
we can respectively express the terms in Eq. (35) as
The reverse relation of is
Thus equation (38) becomes
while equation (39) takes the form
Then the eigenfunction equation is
Since
and the Hermite polynomials of different orders are mutual orthogonal, comparing the two sides of Eq. (44) we finally obtain
and
Because the eigenvalue equation cannot depend on the entangled state parameter ζ, then cannot depend on χ either, we have the value of τ, that is
which is totally determined by the parameters in the Hamiltonian (29). Moreover, we know the eigenvalues are

In summary, we have studied how an angular momentum coherent state keeps its form-invariant during time evolution governed by the Hamiltonian . By employing the entangled state representation and deriving a new binomial theorem involving two-subscript Hermite polynomials we have derived the wave function , which is a single-subscript Hermite polynomial. Based on this result the time evolution of angular momentum coherent state is examined, and the value of is determined. It should be noted that our idea can be generalized to many other quantum entangled systems: using recursion relations of new binomial theorem involving special functions to convert differential operation to algebraic operation.

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