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Chinese Journal of Quantum Electronics

Dual eigenvectors of the Susskind-Glogower phase operators for a two-mode field

Xuefen Xu

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

We find that there exist dual eigenvectors |(?),q〉_* of the phase state-vectors |(?),q〉of the two-mode Susskind-Glogower phase operator e~(iφ).By contour integration we show that |(?),q〉_* is the common eigenvector of e~(-iφ)and the photon number difference operator.The properties of |(?),q〉_* are studied,it turns out that |(?),q〉_* and |(?),q〉can compose a completeness relation in contour integration form.A phase state representation for two-mode field is thus presented.

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We find that there exist dual eigenvectors |(?),q〉* of the phase state-vectors |(?),q〉of the two-mode Susskind-Glogower phase operator e~(iφ).By contour integration we show that |(?),q〉* is the common eigenvector of e~(-iφ)and the photon number difference operator.The properties of |(?),q〉* are studied,it turns out that |(?),q〉* and |(?),q〉can compose a completeness relation in contour integration form.A phase state representation for two-mode field is thus presented.
By contour integration we show that the phase operator ${\mathit{e}\mathit{\^{}}}^{\mathrm{\ensuremath{-}}\mathit{i}\mathrm{\ensuremath{\varphi}}}$ (one of the pair of Susskind-Glogower operators) also possesses eigenkets |\ensuremath{\gamma}〉∗{\mathrm{〉}}_{\mathrm{*}}〉∗​, which are the dual vector of ${\mathit{e}\mathit{\^{}}}^{\mathit{i}\mathrm{\ensuremath{\varphi}}}$'s eigenkets. The properties of |\ensuremath{\gamma}〉∗{\mathrm{〉}}_{\mathrm{*}}〉∗​ are studied and we see that |\ensuremath{\gamma}〉∗{\mathrm{〉}}_{\mathrm{*}}〉∗​ and ${\mathit{e}\mathit{\^{}}}^{\mathit{i}\mathrm{\ensuremath{\varphi}}}$'s eigenkets can also make up a phase-state representation. \textcopyright{} 1996 The American Physical Society.
By noticing that although single-mode Susskind–Glogower phase operator does not commute with photon number operator, two-mode nonlinear phase operator commutes with number-difference operator. We reveal that the common eigenstate <η| of two-particles' relative coordinate and total momentum involves the entanglement between phase and amplitude. We also reveal entanglement between number difference and amplitude inherent in the state |D,|η|>, where [Formula: see text] is a deductive state of |η>.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^{\hat{E}} = e^{i \hatϕ}E^=eiϕ^​, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^[{\hat{E}}, {\hat{n}}] = {\hat{E}}[E^,n^]=E^; consequently, the phase ϕ^\hatϕϕ^​ satisfies the canonical commutation relation [n^,ϕ^]=i[{\hat{n}}, {\hatϕ}] = i[n^,ϕ^​]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.