^{†}Corresponding author. Email: fhym@ustc.edu.cn
^{*}Project supported by the National Natural Science Foundation of China (Grant No. 11175113), the Fundamental Research Funds for the Central Universities of China (Grant No. WK2060140013), and the Natural Science Foundation of Jiangsu Higher Education Institution of China (Grant No. 14KJD140001).
By virtue of the operatorHermitepolynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are presented.
In quantum mechanics and mathematical physics, the special function, the Hermite polynomial, is of fundamental importance. Physically, it is the eigenfunction of the harmonic oscillator.^{[1]} In addition, by introducing a convenient complex form of the α th twodimensional fractional Fourier transform operation, it is found that the bivariate Hermite polynomial is a new eigenmode that propagates in quadratic graded index media.^{[2]} Mathematically, the singlevariable Hermite polynomial H_{n}(x) can be defined in terms of its generating function,
In Ref. [3], by combining the operatorHermitepolynomials (OHP) method^{[4]} and the technique of integration within an ordered product (IWOP) of operators, ^{[5]} we derived the generating functions of even and odd Hermite polynomials
when l = 0, 1. Since H_{0}(x) = 1, H_{1}(x) = 2x, we have
and these formulas are not only accumulated in mathematical handbooks but can also be useful in constructing some optical quantum states.^{[3]} Furthermore, by introducing the Hermitepolynomialoperator H_{n}(X), where X is the coordinate operator (or the quadrature operator in quantum optics theory), and combining the IWOP technique, some new operator identities about quantum squeezing are derived, which are useful for studying the squeezed number state.^{[6]} On the other hand, by replacing the arguments of the special function by quantum mechanical operators, a binomial theorem involving Hermite polynomials and a negativebinomial theorem involving Laguerre polynomials have been derived. These two theorems will have essential applications in quantum optics calculations, and the method is concise and helpful in deducing many operator identities, which may become a new branch in mathematical physics theory.^{[7]} Using the entangled state representation, a twomode squeezed number state can be converted into a Hermite polynomial excited squeezed vacuum state.^{[8]} The generalized photonadded coherent state is obtained by repeatedly acting the combination of Bose creation and annihilation operations on the coherent state. It is found that it can be regarded as a Hermite excited coherent state due to its normalization factor related to Hermite polynomials.^{[9]}
For the bivariate Hermite polynomials H_{m, n}(x, y) (note that H_{m, n} is not a direct product of two independent singlevariable Hermite polynomials)^{[10]}
with the generating function in the form of
The bivariate Hermite polynomials H_{m, n} have their own applications in studying quantum optics. For example, in reference to the Weyl ordering
where X and P are coordinate and momentum operator, respectively. In Ref. [11], we examined the operators’ parameterized ordering and its classical correspondence, and found the fundamental functionoperator correspondence
and its complementary relation
where the symbol : : denotes normal ordering.
In Ref. [12] we derived
and
These two equations are the generating functions of H_{n, 2m} (x, y) and the more complicated one is
where A = 1 – 4st.
When k = l = 0, we have
In this work we shall derive some new generating function formula of the product of bivariate Hermite polynomials (PBHP), and we shall employ the OHP method to realize our goal. By the OHP method in this work, we aim to replace the classical Hermite polynomials H_{m, n}(x, y) by the operator ⋮ H_{m, n}(a^{† }, a)⋮ first, where a^{† } and a are the Bose creation and annihilation operator, respectively, obeying [a, a^{† }] = 1, ⋮ ⋮ denotes antinormal ordering. By using
we can derive some operator identities in definite operator ordering (proof is given in Appendix A). Then we convert them back to the classical case. We also present some applications of these new generating function formulas of PBHP in quantum optics. Next, we shall briefly review the OHP method and exhibit its convenience and simplicity.
One of the fundamental operator identities in the OHP method is
where
Again by using^{[13]}
we have
Converting X back to x, we obtain
Thus, the merit of using the OHP method is fully displayed.
Now we prove the following oneindexsummation of the product of two bivariate Hermite polynomials:
In this case, the OHP method we use is to replace H_{m, n}(x′ , y′ ) by ⋮ H_{m, n}(a^{† }, a)⋮ .
Proof Rewrite Eq. (5) as
By comparing the same power term of t^{n}/n! on the two sides we conclude
Then we employ the OHP method to consider the summation
where ⋮ H_{m, n}(a^{† }, a)⋮ is in the antinormal ordering form. Noting that a and a^{† } are permuted within the antinormal ordering symbol ⋮ ⋮ and using Eqs. (13) and (21) we can reexpress Eq. (22) as
Then we need to change the righthand side of Eq. (23) into antinormal ordering for comparing with the lefthand side. Using the identity^{[14]} which is responsible for converting an operator ρ (a, a^{† }) into its antinormally ordered form
where  β 〉 = exp(−  β  ^{2}/2 + β a^{† })  0〉 is a coherent state, 〈 −  β 〉 = e^{– 2 β  2}, we try to put (y – sa^{† })^{n} (a – sx)^{n} into its antinormal ordering
where in the last step we have used the integration formula
Substituting Eq. (25) into Eq. (23) we obtain
Since both sides in Eq. (18) are in antinormal ordering and a^{† }, a are commutable within ⋮ ⋮ , we can replace a^{† } → x′ , a → y′ , leading to Eq. (19).
Based on the relation to the Laguerre polynomials
where
we can obtain
Similarly, we can prove
By using the generating function of Laguerre polynomials
we can further make summation
This formula can be used to derive
On the other hand, using Eq. (13) it is equal to
we obtain a new operator identity which links normal ordering to antinormal ordering
In particular, when x = y = 0,
Next we examine the summation
To proceed, using the OHP method and Eq. (13) we instead consider
Using the binomial theorem regarding the bivariate Hemite polynomials^{[15]}
we have
Thus, equation (29) becomes
Since the both sides are in antinormal ordering, we have
This is another type of generating function of bivariate Hermite polynomials.
As an application of Eq. (30), we consider the twomode photonsubtracted squeezed vacuum state a^{m}b^{m}S_{2}(λ ) 00〉 , where S_{2}(λ ) = exp [λ (a^{† }b^{† } – ab)] is the twomode squeezing operator, with λ being a real squeezing parameter. The twomode squeezed state is^{[15]}
Using
its derivation is shown in Appendix A. From Eq. (30), we obtain
Thus, the twomode photonsubtracted squeezed vacuum state can be expressed as
Therefore,  λ 〉 _{n} is equivalent to Laguerre polynomial excitation on the twomode squeezed vacuum state. In Ref. [16], we calculated its normalization factor
where P_{m}(x) is the Legendre polynomial
Using Eqs. (46) and (47) as well as the coherent state’ s completeness relation
so
which is a new integration formula.
As an application of Eq. (33), we obtain
which changes the antinormally ordered form e^{sab} e^{ta† b† } into its normal ordering, or
Then we turn to an application of Eq. (42) in quantum optics. We consider the following state vector which is generated by operating H_{l, k} (ξ – b^{† }, ξ * – a^{† }) on the entangled state  ξ 〉 , where
which takes the bivariate Hermite polynomials H_{m, n}(ξ , ξ * ) as its expansion function in the twomode Fock space
 ξ 〉 obeys^{[18]}
so
In comparison with Eq. (42) we see
where
It follows the Fock space wave function of H_{l, k} (ξ – b^{† }, ξ * – a^{† })  ξ 〉 that
In summary, by virtue of the OHP method and the IWOP technique, we have derived some new generating function formulas about the bivariate Hermite polynomials. Their applications in studying quantum optics are presented as well.
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