Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths

Haus, Hermann A.

Kaertner, Franz X.

English

NASA Technical Reports Server

N95- 13910
ON THE THEORY OF QUANTUM MEASUREMENT
Hermann A. Haus
Fraxtz X. K_rtner
Department of Electrical Engineering and Computer Science
Research Laboratory of Electronics
Massachusetts Institute of Teehnologzt
Cambridge, MA 0_189
Abstract
Many so called paradoxes of quantum mechanics are clarified when the measurement
equipment is treated as a quantized system. Every measurement involves nonlinear processes.
Selfconsistent formulations of nonlinear quantum optics are relatively simple. Hence optical
measurements, such as the quantum nondemolition (QND) measurement of photon number,
are particularly well suited for such a treatment. It shows that the so called %ollapse of the
wave function" is not needed for the interpretation of the measurement process. Coherence
of the density matrix of the signal is progressively reduced with increasing accuracy of the
photon number determination. If the QND measurement is incorporated into the double slit
experiment, the contrast ratio of the fringes is found to decrease with increasing information
on the photon number in one of the two paths.
1 Introduction
The Theory of Quantum Measurement has a long and venerable history. Many of the original
discussions of the founders of quantum mechanics are contained in the reprint volume of Wheeler
and Zurek[Z]. Yet, inspite of its long history, the issues raised in these well known discussions
have not been fully settled.
In this paper we attempt to make a modest contribution to this weighty problem. In doing so
we are guided by a quote of Niels Bohr which reads: "... one sometimes speaks of "disturbance
of phenomena by observation" or "creation of physical attributes to atomic objects by measure-
ment." Such phrases, however, are apt to cause confusion, since words like phenomena and obser-
vation, just as attributes and measurements, are here used in a way incompatible with common
language and practical definition. On the lines of objective description, [I advocate using] the
word phenomenon to refer only to observations obtained under circumstances whose description
includes an account of the whole experimental arrangement. In such terminology, the observa-
tional problem in quantum physics is deprived of any special intricacy and we are, moreover, di-
rectly reminded that every atomic phenomenon is closed in the sense that its observation is based
on registrations obtained by means of suitable amplification devices with irreversible functioning
such as, for example, permanent marks on a photographic plate, caused by the penetration of
pA__ INTE_ITIONALLYB"EANK, 107
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electrons into the emulsion" [Ref. 1, p. 3]. We have underlined the words that we consider
particularly worthy of note. Bohr requires a description of the whole experimental arrangement.
Further, if one is to state the outcome of the experiment in classical language, large amplification
is required.
At the risk of making statements that may be considered even more controversial by the ad-
herents of the Einsteinian school, we should like to strengthen Bohr's quote by saying: "Physical
reality cannot be formulated until the measurement equipment used to determine the observables
is specified and treated as a quantum system. The large gain of the measurement equipment
provides the classical interface at the output of the measurement apparatus."
Much of the controversy involving quantum measurements is the consequence of the fact that
it is very difficult to describe well the measuring equipment_ according to our interpretation, to
describe it quantum mechanically.
In quantum optics we have made great progress in describing optical components quantum
mechanically. The theory has been well tested experimentally. The squeezing by a parametric
amplifier is well understood theoretically and amply confirmed experimentally[2-6]. Less exten-
sively explored, yet also tested, is the self-phase modulation and squeezing in optical fibers via
the optical Kerr effect [_-9]. Hence it appears natural to use the well tested quantum description
of optical devices to construct a measurement apparatus and test some of the predictions of
quantum mechanics using such a measurement apparatus. This is the main objective of this
paper. We start by describing a Quantum Nondemolition Measurement of the photon number
of a signal via a nonlinear Mach-Zehnder interferometer. We follow the development of the
composite wave function of the signal and measurement apparatus to the output. We shall see
that the photon number in the signal can be determined with a negligible probability of error
if the gain of the measurement apparatus is large enough. Further, when this is the case, the
density matrix of the signal, obtained by tracing over the (Hilbert) coordinates of the measure-
ment equipment, is diagonalized. Finally, since the probability of error of measuring a particular
photon number approaches zero, each measurement, and not the whole ensemble, can be in-
terpreted as yielding an interpretable result. This corresponds to the yon Neumann projection
operator interpretation. However, when the gain is not very large, the signal density matrix
does not decohere, it is not diagonalized. This is consistent with Bohr's dictum that we can put
the measurement results into classical language only if the gain of the measurement equipment
is very large.
When no measurement is performed, and the signal and "measurement" beams are passed
on into a second nonlinear Mach-Zehnder interferometer with a Kerr coefficient of opposite
sign, the entire action of the first intefferometer can be undone; the wave functions emerge
disentangled! This confirms the reversibility of quantum mechanics.
We conclude with the double slit experiment. We put a nonlinear Mach-Zehnder mea-
surement apparatus in each of the two light beams. As the accuracy of the photon number
determination is systematically increased, the contrast of the interference fringes decreases ac-
cordingly.
108
2 The Quantum Nondemolition Measurement
Figure 1 shows a nonlinear Mach-Zehnder interferometer. The signal beam _n at one frequency
and the probe beam b_ at another frequency enter a Kerr medium through a dichroic mirror. At
the end of the Kerr medium they are again separated by another dichroic mirror. A portion of the
probe beam has been passed on directly for interference. Classically, the Kerr medium produces
a phase shift on the probe beam that can be measured giving an indication of the intensity of
the signal beam. Quantum mechanically, the process is described by the Hamiltonian of the
Kerr medium [1°]
•- ^t ^ _t_
= n_aeaso o (1)
where _ is a factor proportional to the Kerr coefficient; _, is the annihilation operator of the
signal photons, b that of the probe photons. They obey the usual commutation relations:
[a,,a]] = 1 (2)
: 1 (3)
It should be noted that the Hamiltonian (1) does not account for a self-phase shift. This has
been left out for convenience. A medium resonant at the sum frequency of signal and probe
would be described by such a simplified Hamiltonian.
The two portions of the probe beam are combined by a beam splitter with the Hamiltonian:
= hM[bt_ -I- 8t/_] C4)
As usual, one may consider the wave packets to evolve in time as they propagate along the
system. If the beam splitter is 50/50, the parameter M must be chosen
- (5)
v 0 4
where l is the length of the medium and v e is the group velocity, l/vg is the travel time.
From the known Hamiltonian one may determine the evolution of the wave function I¢), ll3)10)
of the three input ports. They are products at the input, and become entangled at the output.
We denote the output annihilation operators by ] and _. The balanced photodetector measures
the expectation values of the difference current operator ] = it] _ Ot_ and its moments[ TM.
(]) = I 1' sin(, ala) , l l'ala. (6)
The expectation value traced over the Hilbert space of the probe yields the sine of the signal
photon operator. If the sine function can be expanded to first order, it becomes the photon
operator. The mean square fluctuations follow from the second moment and are [u ]
(IMI') = I 1' (7)
if the signal is in a photon number state. This is shot noise since 1/312 is the photon number in
the probe beam.
109
v/
n
v
/
Dichroic Mirrors
Kerr Medium
kin
==..=
v
_in
Io)
Measurement
Subsystem
_tout
\
(
Signal
Subsystem
FIG. 1. Schematic of nonlinear Mach-Zehnder interferometer and balanced de-
tector.
/ I---1 \
IV>s/ ", /
_ v S
_ v S
/
X
FIG. 2. Two nonlinear Mach-Zehnder interferometers with media of equal and
opposite Kerr coefficients.
110
The probability of error follows from the mean square fluctuations (7) that approach gaus-
sians in the large photon number limit{Ill:
1 ,¢,,,2, 8
iDerror _'_ 25/--'re-'re TM pI /V (8)
If [_/_[2 >> 1, the probability of error can be made arbitrarily small. The physical meaning of
this quantitity can be fathomed as follows. _lfl[ 2 is the phase shift due to the probe photons,
itself is the phase shift due to one photon. The geometric mean of these two products has to
be made very large. If we used fiber interferometers, these operating parameters are not easily
achieved. Here, however, we are not concerned with the practical realization of the measurement
apparatus, but only with the theoretical conclusions that can be drawn from it. In particular,
we find that the probability of error can be made arbitrarily small, for ]_/_1 = 10, it is 10 -e. This
:_eans that each measurement has vanishing error probability. Hence one may interpret every
measurement, and not only the ensemble, as yielding a definite result. This is analogous to the
von Neumann projection postulate which interprets a measurement as projecting the state into
an eigenstate. Pursuing this interpretation further, we can say that a measurement with the
Mach-Zehnder interferometer at large gain projects the signal into a photon state.
3 The Density Matrix
The trace of the density matrix over the measurement system part at the output of the signal-
measurement system of Fig. 1 can be evaluated for a signal wave function[Ill:
= chin) (9)
n
It is
r_=O lr/,!
(10)
O0 2 2
m,t=O error probability
In the limit of large gain, the density matrix traced over the measurement equipment becomes
diagonal at the same rate as the probability of error approaches zero (note the exponential
factor!). Hence, again, we see that the signal acquires a classical (decohered) appearance when
the gain of the measurement system (l_#t) is made very large.
4 Reversibility
If one does not perform a measurement on the probe beam, but reintroduces it in the second
Mach-Zehnder as shown in Fig. 2, which has a Kerr coefficient of opposite sign, one can
disentangle entirely the wave functions. This shows, of course, the reversibility of quantum
111
mechanics if no measurement intervenes in the process. Of course, no measurement could have
been undertaken, because the probe beam was completely recycled. This brings us back to
the act of measurement. A measurement is an irreversible process that prevents recycling.
Indeed, in the present example the probe beam is passed into a balanced detector in which it
is absorbed. Only then can one apply the homodyne photon detection formula to evaluate the
current operator statistics.
5 Tracing, Decoherence and the Act of Measurement
The density matrix of the signal system becomes diagonal in the signal Hilbert space when traced
over the probe space. Tracing is a mathematical operation which, according to the postulates of
quantum mechanics, evaluates expectation values. In the context of the derivation of the signal
density matrix, the reduced density matrix can be interpreted as a "Gedankenexperiment" on
the density matrix of the signal after passage through the Mach-Zehnder. Accompanied by the
statement that the signal and probe systems would never be combined again, the entanglement
that in fact exists between the two systems could never be reversed. In this sense, the reversibility
of quantum mechanics is broken. In an actual measurement, of course, the apparatus works on
the probe subspace, causes partial or total decoherence in that space, and leads "de facto" to
an irreversible action.
6 Two Slit Experiment
Finally, let us look at the "two-slit" interference experiment of Fig. 3. The two slits are here
replaced by the two arms of an interferometer. A phase shifter in one of the arms changes the
phase of the superimposed beams. If the two beams were perfectly coherent, the intensity at
the detector would have to show perfect extinction. However, we mount two QND apparati in
each of the arms to ascertain the number of photons passing through them individually. The
gain of the apparati can be adjusted, thus changing the accuracy of the measurement of the
photon number passing through each arm. One can then compute the expectation value of the
contrast and finds it to be [11] (see Fig. 4)
(i) = e -I_BI_/4cos # (11)
Thus, a similar exponential factor as the one that appears in the error probability determines
the extinction of the contrast. The factor is squared, because two measurements are being
performed. Here again we find that the transition between the behavior of the photon as a wave
and that of a particle is a continuous one. The accuracy of the determination of the photon
number determines how much the photon behaves as a particle.
112
_in
l
10)
18)
A.
bm
_in
E7
FIG. 3. An interferometer representing two-slit interference and attached QND
measurement apparati.
1 0
0.5 -25
oi
/ _
([)= e-]_13] /4 cos e 2 oC_0
FIG. 4. Expectation value of detector current versus phase and error probability
of photon number determination.
113
7 Conclusion
We started with the postulate that a proper formulation of a quantum measurement has to
quantize the measuring apparatus as well. The quantum formalisms developed for optical com-
ponents enable one to do a full quantum analysis of an optical measurement apparatus. The
measurement apparatus of photon number with infinite gain yields results that can be described
in classical language: photons behave as particles (since we chose a particle measurement appa-
ratus). When the gain is not infinite, the behavior is more duplicitous, it is not what one would
call the behavior of a classical particle. This confirms Bohr's statement that it is necessary
to have large gain to obtain measurement results that can be put into classical language. We
also found that a measurement with infinite gain is equivalent to a projection operation on the
signal.
If no measurement is undertaken, the entanglement of the signal and probe states can be
fully undone by an inverse apparatus.
Finally, the "double-slit" experiment can also be described in terms of partial knowledge of
the photon number in each of the paths. If the knowledge is only partial, there can still be
interference of the two beams.
8 Acknowledgments
This work was supported by the Office of Naval Research, Grant N00014-92-J-1302.
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"_ 115



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