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Unified treatment of heterodyne detection: the Shapiro-Wagner and Caves frameworks

Giulio Landolfi, G. Ruggeri, Giulio Soliani

Published Feb 3, 2005• 1 citation
OpenAlex

Abstract

A comparative study is performed of two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator, called Shapiro-Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator $\\hat \\psi$ which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where $[\\hat \\psi, \\hat \\psi^{\\dag}] =0$, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by $[\\hat \\psi, \\hat \\psi^{\\dag}] \\neq 0$, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.

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UNIFIED TREATMENT OF HETERODYNE DETECTION: THE SHAPIRO–WAGNER AND CAVES FRAMEWORKS

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In the context of the readout scheme for gravitational-wave detectors, the "double balanced homodyne detection" proposed in [K.~Nakamura and M.-K.~Fujimoto, arXiv:1709.01697.] is discussed in detail. This double balanced homodyne detection enables us to measure the expectation values of the photon creation and annihilation operators. Although it has been said that the operator $\hat{b}_θ:=\cosθ\hat{b}_{1}+\sinθ\hat{b}_{2}$ can be measured through the homodyne detection in literature, we first show that the expectation value of the operator $\hat{b}_θ$ cannot be measured as the linear combination of the upper- and lower-sidebands from the output of the balanced homodyne detection. Here, the operators $\hat{b}_{1}$ and $\hat{b}_{2}$ are the amplitude and phase quadrature in the two-photon formulation, respectively. On the other hand, it is shown that the above double balanced homodyne detection enables us to measure the expectation value of the operator $\hat{b}_θ$ if we can appropriately prepare the complex amplitude of the coherent state from the local oscillator. It is also shown that the interferometer set up of the eight-port homodyne detection realizes our idea of the double balanced homodyne detection. We also evaluate the noise-spectral density of the gravitational-wave detectors when our double balanced homodyne detection is applied as their readout scheme. Some requirements for the coherent state from the local oscillator to realize the double balanced homodyne detection are also discussed.

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IEEE Journal of Quantum Electronics•Sep 1, 1974•0 citations
A comparative study is performed of two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator, called Shapiro-Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator hatpsi\\hat \\psihatpsi which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where [hatpsi,hatpsidag]=0[\\hat \\psi, \\hat \\psi^{\\dag}] =0[hatpsi,hatpsidag]=0, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [hatpsi,hatpsidag]neq0[\\hat \\psi, \\hat \\psi^{\\dag}] \\neq 0[hatpsi,hatpsidag]neq0, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.
A comparative study is performed on two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator called Shapiro–Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator [Formula: see text] which allows a unified formulation of both cases. For the Shapiro–Wagner scheme, where [Formula: see text], quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [Formula: see text], within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.
In the context of the readout scheme for gravitational-wave detectors, the "double balanced homodyne detection" proposed in [K.~Nakamura and M.-K.~Fujimoto, arXiv:1709.01697.] is discussed in detail. This double balanced homodyne detection enables us to measure the expectation values of the photon creation and annihilation operators. Although it has been said that the operator b^θ:=cos⁡θb^1+sin⁡θb^2\hat{b}_θ:=\cosθ\hat{b}_{1}+\sinθ\hat{b}_{2}b^θ​:=cosθb^1​+sinθb^2​ can be measured through the homodyne detection in literature, we first show that the expectation value of the operator b^θ\hat{b}_θb^θ​ cannot be measured as the linear combination of the upper- and lower-sidebands from the output of the balanced homodyne detection. Here, the operators b^1\hat{b}_{1}b^1​ and b^2\hat{b}_{2}b^2​ are the amplitude and phase quadrature in the two-photon formulation, respectively. On the other hand, it is shown that the above double balanced homodyne detection enables us to measure the expectation value of the operator b^θ\hat{b}_θb^θ​ if we can appropriately prepare the complex amplitude of the coherent state from the local oscillator. It is also shown that the interferometer set up of the eight-port homodyne detection realizes our idea of the double balanced homodyne detection. We also evaluate the noise-spectral density of the gravitational-wave detectors when our double balanced homodyne detection is applied as their readout scheme. Some requirements for the coherent state from the local oscillator to realize the double balanced homodyne detection are also discussed.
Un des problemes theoriques les plus importants des communications optiques quantiques a detection heterodyne est la definition d'un operateur obeissant aux lois de la mecanique quantique. Le present memoire est consacre a l'analyse des differentes approches de cet operateur. La premiere partie est consacree a l'expose des divers criteres permettant d'optimiser les performances d'un recepteur: probabilite d'erreur, capacite, debit critique. Les differents recepteurs (direct, homodyne et heterodyne) sont passes en revue et il est montre qu'ils ne sont tous que sous-optimaux. Dans la deuxieme partie, la detection directe (comptage du nombre de photons et mesure de l'intervalle de temps separant deux photons) pour un canal binaire est etudiee en detail dans deux cas: un processus de poisson issu d'un signal sans memoire dans un canal faiblement bruite; des processus de poisson composes (groupement et antigroupement) dans un canal sans bruit. Il est montre qu'une description approfondie de la lumiere comporte necessairement des aspects non classiques. La derniere partie est consacree a un expose complet des resultats recents demontres pour l'operateur de phase et il est montre comment l'extension non-standard de l'espace de hilbert de l'optique quantique permet la definition d'un operateur de phase auto-adjoint
A relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed. A phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the RNS representation space. The phase operator is expressed as a unitary exponential operator. The properties of the relative number operator, the phase operator, and their eigenstates are investigated in detail. The phase variable has a maximum uncertainty in any stationary state. Also, a time operator as a dynamical variable can be defined in the RNS representation space. The RNS representation is closely related to the Liouville space formulation and to thermofield dynamics. The RNS representation is shown to be a suitable method for investigating the Josephson junction with ultrasmall capacitance. A basic formulation of number-phase quantization in the Josephson junction is given in terms of the RNS representation.
The quantum limits on simultaneous phase and squared-amplitude measurements made via optical heterodyne detection on a single-mode radiation field are established. The analysis proceeds from a fully quantum mechanical treatment of heterodyning with ideal photon detectors. A high mean field uncertainty principle is proven for simultaneous phase and squared-amplitude observations under the condition that the signal and image band states are independent, and the image band has zero mean. Operator representations are developed which show that no such principle applies when arbitrary signal/image band dependence is permitted, although the mean observations are no longer functions of the signal field alone. A multimode two-photon coherent state illustrating this behavior at finite energy is exhibited. Potential applications for the resulting improved accuracy measurements are briefly described.