Fock representation for two-mode quantum phase operators’ eigenstates

The quantum phase description of Noh, Fougères and Mandel (NFM) leads to a two-mode theory of phase in the strong local oscillator limit. We show that the explicit form of the eigenstates of the sine and the cosine phase operators in two-mode Fock space can be constructed. This brings great convenience to the NFM quantum phase approach. Recently, phase operators and phase measurements in quantum optics have attracted the attention of physicists [1]. In [2], NFM proposed an operational quantum phase description, which means that the Hermitian phase operators can be defined in an operational way. Their theory is based on an eight-port homodyne-detection scheme. Then, in [3], Freyberger, Heni and Schleich (FHS) further showed that in the limit of a strong local oscillator, the NFM description contains an essential two-mode basis which leads to the simultaneously measurable operator pair Ĉ = x̂ √ x̂2 + p̂2 (1) and Ŝ = p̂ √ x̂2 + p̂2 (2) where x̂ = x̂1 + x̂10 and p̂ = p̂1 − p̂10 commute with each other and x̂i = 1 √ 2 (âi + ↠i ) p̂i = 1 √ 2i (âi − ↠i ) i = 1, 10 . (3) where (â1, â10) are the two input mode operators defined in [3] for the eight-port interferometer in the operator description by NFM and they can be any arbitrary states. In this paper, we adopt the definitions used in [3] for convenience and use them as a starting point for our calculation. The simultaneous eigenstates of x̂ and p̂ and their phase space are also considered in [3] as x̂|x, p〉 = x|x, p〉 p̂|x, p〉 = p|x, p〉 . (4) 1355-5111/96/030393+05$19.50 c © 1996 IOP Publishing Ltd 393 394 Letter to the Editor These eigenstates compose a complete set ∫ ∫ dx dp|x, p〉〈x, p| = 1 . (5) With this condition, the phase operator pair can be described in the phase space as Ĉ = ∫ ∫ dx dp x √ x2 + p2 |x, p〉〈x, p| (6) Ŝ = ∫ ∫ dx dp p √ x2 + p2 |x, p〉〈x, p| (7) with [Ĉ, Ŝ] = 0. Although the representation of |x, p〉 in terms of the quadrature states and of the number states of modes 1 and 10 is discussed in [3], we still think that it is necessary to derive the explicit form of |x, p〉 in the two-mode Fock space. With an explicit form of |x, p〉, many calculations can be made much easier and apparent, as will be seen in later sections. The paper is arranged as follows. In section 2, we derive the explicit form of |x, p〉 ≡ |ξ1, ξ2〉 ≡ |ξ〉 with ξ = 1 √2 (x + ip) and, then, employ the newly developed technique of integration within an ordered product (IWOP) of operators [4, 5] to prove the completeness relation of such a constructed eigenstate (5). As one can see, our approach is much simpler and more direct than the previous one adopted in [3]. In section 3, we deduce the normally ordered form of Ĉ and Ŝ by virtue of the IWOP technique. To compare our results with the single-mode phase description proposed by Paul in [6], we further analyse our state |ξ〉 in section 4. We show that the normally ordered forms of Ĉ and Ŝ reduce to normally ordered cosine and sine Paul phase operators, respectively. We now show that the explicit form of |x, p〉 in a two-mode Fock space takes the form of |x, p〉 = exp[− 2 |ξ |2 + ξa† 1 + ξ ∗a† 10 − a† 1a 10]|0 0〉 ≡ |ξ〉 (8) where |0 0〉 is the two-mode vacuum state. ξ = ξ1 + iξ2 = 1 √2 (x + ip) is a complex number. ξ2 is the parameter corresponding to the input mode â10. Actually, by operating â1 and â10 on |x, p〉, respectively, we have â1|ξ〉 = (ξ − ↠10)|ξ〉 â10|ξ〉 = (ξ∗ − ↠1)|ξ〉 . (9) After combining them, we obtain (â1 + â10 + ↠1 + ↠10)|ξ〉 = √ 2x|ξ〉 or (x̂1 + x̂10)|ξ〉 = x|ξ〉 and (10) (â1 − â10 − ↠1 + ↠10)|ξ〉 = i √ 2p|ξ〉 or (p̂1 − p̂10)|ξ〉 = p|ξ〉 . Thus, |ξ〉 is a state such that the real and imaginary parts of ξ correspond to the eigenvalues of x̂1 + x̂10 and p̂1 − p̂10, respectively. With the use of the normally ordered product form |0 0〉〈0 0| = : e−â 1 â1−â 10â10 : (11) and the mathematical formula ∫ d2ξ π eλ|ξ | 2+f ξ+gξ∗ = − λ e−fg/λ Letter to the Editor 395 as well as the IWOP technique [4, 5], we can immediately perform the following integration: ∫ d2ξ π |ξ〉〈ξ | = ∫ d2ξ π : exp{−|ξ |2 + ξ(↠1 + â10) + ξ ∗(â1 + ↠10) − ↠1â 10 − â1â10 − ↠1â1 − ↠10â10} : =: exp{(↠1 + â10)(â 10 + â1) − (↠1 + â10)(â1 + ↠10)} : = 1 . (12) This approach is, as one may conclude, more concise than the derivation of equations (B.16) and (B.17) in [3]. Besides, we can easily see that 〈ξ ′|(x̂1 + x̂10)|ξ〉 = x ′〈ξ ′ | ξ〉 = x〈ξ ′ | ξ〉 and 〈ξ ′|(p̂1 − p̂10)|ξ〉 = p′〈ξ ′ | ξ〉 = p〈ξ ′ | ξ〉 . (13) Thus, 〈ξ ′ | ξ〉 = πδ(ξ1 − ξ ′ 1)δ(ξ2 − ξ ′ 2) (14) which is the orthogonal condition for the eigenstates. The overlap between the two-mode coherent state 〈z1z2| and |ξ〉 is 〈z1z2 | ξ〉 = exp [ −|ξ | 2 2 + ξz∗ 1 + ξ ∗z∗ 2 − z∗ 1z 2 − 1 2 (|z1| + |z2|) ] . At this point, we state that the common eigenstates of the relative coordinate (x̂1 − x̂10) and total momentum (p̂1 + p̂10) of two particles (their commutative property was first pointed out by Einstein et al [7]) are obtained in [8]. Since we have got the explicit form of the eigenstate |ξ〉, we are now able to perform the following integration with the IWOP: ∫ d2ξ π ξ |ξ | |ξ〉〈ξ | = ∫ ∞