Operational approach for reconstruction of quantum distributions
in a preamplified homodyne-detection scheme
Marcelo A. Marchiolli,* Salomon S. Mizrahi,
²
and Victor V. Dodonov
‡
Departamento de Fı ´sica, Universidade Federal de Sa ˜ o Carlos, Via Washington Luiz km 235, 13565-905 Sa ˜ o Carlos, SP, Brazil
Received 28 April 1997
We use the operational theory of homodyne detection, introduced by Banaszek and Wo ´dkiewicz Phys. Rev.
A 55, 3117 1997, to express a generic quantum phase space quasiprobability distribution in the sense of
Cahill and Glauber of an electromagnetic field in terms of the photocount moments. We adapt the method of
degenerate parametric preamplification of a signal, within the operational theory, in order to overcome the
drawback of nonunit efficiency of the detector. Simulations show that as the gain parameter of the amplifica-
tion is increased, the quasiprobability distribution goes closer to the original one, even for an efficiency lower
than 0.5. We also express the mean value of an arbitrary observable in terms of the same photocount moments.
S1050-29479710111-1
PACS numbers: 42.50.Dv, 03.65.Bz, 42.65.Ky
I. INTRODUCTION
After the laboratory verification 1,2 of the method of
optical homodyne tomography OHT, proposed for the first
time in 3,4, various aspects of the problem of quantum
state reconstruction from experimental statistical data were
investigated in detail 5–15. However, a controversy still
exists about the influence of the detector efficiency on the
quality of state reconstruction. As was shown in 5,8,a
straightforward application of a usual homodyne detection
scheme becomes impossible if the quantum efficiency is
less than 0.5. On the other hand, preamplifying the signal
before homodyning, one obtains a scale factor the amplifier
gain that permits one to overcome the drawback of a low
efficiency detector. In fact, this procedure allows, in prin-
ciple, the reconstruction of a quantum phase space qua-
siprobability distribution QPD for arbitrarily small values
of 6,9,10,16–18.
The primary set of experimental data used in the OHT
method is some classical continuous in principle probabil-
ity distribution function, associated with the rotated quadra-
ture components of the field mode. However, there exist
other schemes of quantum state reconstruction, based from
the beginning on the discrete photocount statistics, in par-
ticular, on the measured moments of all orders 16–22. Our
aim here is to consider the reconstruction of the signal field
quasiprobability functions in terms of moments related to the
photon statistics measured by a nonideal detector, in the
framework of the operational theory of homodyne detection
introduced in 23. In this theory a clear distinction is made
between the so-called operational observables, which are
Hermitian operators already incorporating the effects of a
nonideal photodetector, and the intrinsic quantum observ-
ables, which represent the signal field, thus, independent of
the measuring device. In the present paper we modify the
operational approach, as originally proposed, by incorporat-
ing into the formalism the effects of signal preamplification
before measurement. Considering a =0.4 detector effi-
ciency, we simulate the reconstruction of the original signal
QPD from an ‘‘experimental’’ statistic function SF and il-
lustrate this procedure plotting sequences of QPD’s for sev-
eral values of the amplification gain parameter. We also ob-
tain the mean value of an arbitrary observable in terms of the
operational moments when the signal is preamplified, and
discuss the limit of high gains.
This paper is organized as follows. In Sec. II we give a
brief review of the operational theory of homodyne detec-
tion. In Sec. III we discuss the QPD reconstruction with a
nonideal apparatus ( 1) but amplifying the signal before
homodyning, and present two illustrative examples. Using
the same method as Sec. III, in Sec. IV we show that the
mean values of observables associated to the signal prior to
amplification can be ‘‘homodyned,’’ hence, they are ex-
pressed in terms of the experimental SF or in terms of the
operational moments. Section V is devoted to a summary
and conclusions and finally, the Appendixes contain deriva-
tions of expressions used in the text.
II. ELEMENTS FROM OPERATIONAL THEORY
OF HOMODYNE DETECTION
The OHT is based on the measurements of an observable
represented by the rotated field quadrature operator intrinsic
observable
X
=
e
i
a
²
+e
-i
a
2
=Q cos +P sin 0 2 ,
1
written in terms of the creation and annihilation signal field
operators ( Q and P are the effective ‘‘coordinate’’ and ‘‘mo-
mentum’’ operators and the parameter , which corresponds
to the phase of a local oscillator LO. The statistical prop-
erties of X
can be extracted from the generating operator
*Electronic address: pmar@iris.ufscar.br
²
Electronic address: salomon@power.ufscar.br
‡
Electronic address: vdodonov@power.ufscar.br
PHYSICAL REVIEW A NOVEMBER 1997 VOLUME 56, NUMBER 5
56 1050-2947/97/565/42789/$10.00 4278 © 1997 The American Physical Society