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Physical Review A

Measurement of amplitude probability distributions for photon-number-operator eigenstates

Bernard Yurke, David Stoler

Published Aug 1, 1987• 59 citations
Publisher

Abstract

A homodyne detector measures a field-amplitude component of the incoming signal. If the incoming signal is in an $n$-photon eigenstate the homodyne detector's output probability distribution exhibits $n$ fringes. Here the degree to which the visibility of these fringes is degraded by a homodyne detector with less than unit quantum efficiency is evaluated. An experiment employing conjugate pairs of photons generated via a parametric down conversion or four-wave mixing is proposed by which these fringes could be observed.

DOI
10.1103/physreva.36.1955

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Measuring amplitude probability distribution for photon number-operator eigenstates

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In parametric downconversion, pump photons are converted into pairs of signal and idler photons. A photodetector counting photons in the idler beam can be used to gate a homodyne detector in the signal beam so that it is on only when an n-photon wave packet enters its input. The probability distribution for the charge delivered by the homodyne detector has the same form as the position probability distribution of a harmonic oscillator in its nth energy eigenstate. In particular, the probability distribution exhibits fringes. We have calculated the visibility of these fringes for the case when both the photodetector and the homodyne detector have less than unit quantum efficiency. By reducing the photon flux in the signal and idler beam sufficiently, the effect of idler photodetector loss on the fringe visibility can be made negligible. The degree to which the fringes have been degraded then depends only on the inefficiency of the homodyne detector. For the case n = 1, the fringe is still visible when the homodyne detector efficiency is 0.5. Hence the proposed experiment should be feasible with existing technology.

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A homodyne detector measures a field-amplitude component of the incoming signal. If the incoming signal is in an nnn-photon eigenstate the homodyne detector's output probability distribution exhibits nnn fringes. Here the degree to which the visibility of these fringes is degraded by a homodyne detector with less than unit quantum efficiency is evaluated. An experiment employing conjugate pairs of photons generated via a parametric down conversion or four-wave mixing is proposed by which these fringes could be observed.
In parametric downconversion, pump photons are converted into pairs of signal and idler photons. A photodetector counting photons in the idler beam can be used to gate a homodyne detector in the signal beam so that it is on only when an n-photon wave packet enters its input. The probability distribution for the charge delivered by the homodyne detector has the same form as the position probability distribution of a harmonic oscillator in its nth energy eigenstate. In particular, the probability distribution exhibits fringes. We have calculated the visibility of these fringes for the case when both the photodetector and the homodyne detector have less than unit quantum efficiency. By reducing the photon flux in the signal and idler beam sufficiently, the effect of idler photodetector loss on the fringe visibility can be made negligible. The degree to which the fringes have been degraded then depends only on the inefficiency of the homodyne detector. For the case n = 1, the fringe is still visible when the homodyne detector efficiency is 0.5. Hence the proposed experiment should be feasible with existing technology.
Homodyne detection is the most effective detection scheme employed in quantum optics to characterize quantum states. It is based on mixing at a beam splitter the signal to be measured with a coherent state, called the “local oscillator,” and on evaluating the difference of the photocurrents of two photodiodes measuring the outputs of the beam splitter. If the local oscillator is much more intense than the field to be measured, the homodyne signal is proportional to the signal-field quadratures. If the local oscillator is less intense, the photodiodes can be replaced with photon-number-resolving detectors, which have a smaller dynamics but can measure the light statistics. The resulting new homodyne-like detector acquires a hybrid nature, being it capable of yielding information on both the particle-like (statistics) and wave-like (phase) properties of light signals. The scheme has been tested in the measurement of the quadratures of coherent states, bracket states and phase-averaged coherent states at different intensities of the local oscillator.
A squeezed state of the electromagnetic field has less noise in one of its two amplitude components than that of a coherent state. This property makes squeezed states potentially useful in low noise and precision measurement. Squeezed states may be generated by, among other things, degenerate parametric amplifiers and four-wave mixers. The reduced noise in one of the amplitude components is most easily observed via a homodyne detector. A number of experimental efforts to generate and detect squeezed states are under way. Here, in describing experimental techniques, the experiments' carried out by Slusher et al. are emphasized. A number of techniques were employed to successfully generate squeezed light. An atomic beam of sodium provided the optical nonlinearity in a cavity-backward four-wave mixer. By employing pairs of Fabry-Perot modes spaced symmetrically about the pump frequency, squeezed light was generated in nondegenerate conditions where Rayleigh scattering of pump light could be avoided. To observe squeezing, a number of nonidealities such as phase jitter and detector inefficiencies also had to be overcome.
Optical homodyne detection has been widely used in continuous-variable (CV) quantum information processing for measuring field quadrature. In this paper we explore the possibility of operating a conjugate homodyne detection system in ``photon counting'' mode to implement discrete-variable (DV) quantum key distribution (QKD). A conjugate homodyne detection system, which consists of a beam splitter followed by two optical homodyne detectors, can simultaneously measure a pair of conjugate quadratures XXX and PPP of the incoming quantum state. In classical electrodynamics, X2+P2{X}^{2}+{P}^{2}X2+P2 is proportional to the energy (the photon number) of the input light. In quantum optics, XXX and PPP do not commute and thus the above photon-number measurement is intrinsically noisy. This implies that a blind application of standard security proofs of QKD could result in pessimistic performance. We overcome this obstacle by taking advantage of two special features of the proposed detection scheme. First, the fundamental detection noise associated with vacuum fluctuations cannot be manipulated by an external adversary. Second, the ability to reconstruct the photon number distribution at the receiver's end can place additional constraints on possible attacks from the adversary. As an example, we study the security of the BB84 QKD using conjugate homodyne detection and evaluate its performance through numerical simulations. This study may open the door to a family of QKD protocols, complementary to the well-established DV-QKD based on single-photon detection and CV-QKD based on coherent detection.
We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state 1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+/-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space.
Optical homodyne detection is examined in view of the joint probability distribution. The usual view is that the relative phase between independent laser fields is localized by photon-number measurements in interference experiments such as homodyne detection. This is why operationally coherent states for laser fields are used in the description of homodyne detection and optical quantum-state tomography. Here, we elucidate these situations by considering the joint probability distribution and the invariance of homodyne detection under the phase transformation of optical fields.