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Physica B 151 (1988) 339-348
North-Holland, Amsterdam
TESTING SCHR6DINGERIS PARADOX WITH A MICHELSON INTERFEROMETER
Evan Harris WALKER
U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland, USA
E.C. MAY, S.J.P. SPOTTISWOODE and T. PIANTANIDA
SRI International, Menlo Park, California, USA
The Schr6dinger paradox points out that quantum mechanics predicts a linear superposition of states even for
macroscopic objects prior to measurement. However, at the macroscopic level of ordinary objects it has not been possible
to maintain the phase correlations needed to demonstrate or disprove the reality of such a superposition of states as
opposed to the mixture of states. Without such a quantum "signature", this paradoxical prediction of quantum theory
would seem to have no testable consequences. State vector collapse in that case becomes indistinguishable from a
stochastic ensemble description.
The experiment described here provides a means for testing Schr6dingers' paradox. A Michelson interferometer is used
to test for the presence of state superposition of a pair of shutters that are placed along the two optical arms of the
interferometer and driven by a beta decay source so that either the first shutter is open and the second closed or vice versa.
The shutters take on the role of the cat in the Schr6dinger paradox.
The experiment that we discuss here has been carried out at SRI International. Under the conditions of the experiment,
the results remove the possibility of the existence of macroscopic superposition prior to observation.
1. Introduction
The Schr6dinger paradox is among the oldest
of the puzzles surrounding the interpretation of
quantum mechanics. Like the Einstein-
Podolsky-Rosen (EPR) paradox it has engen-
dered a great deal of speculation about our basic
understanding of physical reality. Also like the
EPR paradox, Schr6dinger proposed his paradox
to point out that the statistical interpretation of
quantum theory must at some level contain a
flaw, since it implies the reality of a linear super-
position of states even at the macroscopic level -
before observation. Moreover, just as in the case
of the EPR paradox, the Schr6dinger paradox
has long been thought of as an untestable con-
sequence of quantum theory, since it relates to
the state of a macrosystern just before observa-
tion, a state that we know must approach asym-
ptotically to that given by classical mechanics.
That is to say, we know that the usual interfer-
ence effects by which we distinguish the presence
of state superposition in atomic processes can be
shown to be too small to observe in the case of
macroscopic systems. But the development of
Bell's theorem showed us how to test the
paradoxical implications of quantum mechanics
that had been pointed out by Einstein, Podolsky
and Rosen in 1935. Within the limits of our
experimental setup, we have now done the same
for the Schr6dinger paradox.
There are important reasons for doing this
experiment. All of us are quite aware of the fact
that the existence of a linear superposition of
states at the macroscopic level is quite counter-
intuitive. Nevertheless, no experiment has ever
been done that has yielded results contrary to
the literal application of quantum theory, The
absence of superposition at the macrolevel prior
to observation has not been experimentally de-
monstrated - and in fact it has generally been
thought that such a test was not feasible. This
has led to the developernent of various interpre-
tations of quantum mechanics having to do with
the macroscopic reality of quantum states.
A second reason for carrying out an experi-
ment of the present type is that it represents an
efficient way to search for the nature of and
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340 E.H. Walker et al. / Testing Schr6dinger's paradox with Michelson interferometer
cause of state vector collapse. We all know that
the machinery that effects state vector collapse,
whatever that phrase actually means, must be
somewhere between the thing observed and the
observer. Indeed, this observer-observed dich-
otomy has become a rather commonly used
phrase in discussions of the measurement prob-
lem. But the gap between these two covers a lot
of territory. Moreover, we also know that the
perponderant opinion is that the transition from
the pure state to the mixed state probably takes
place at a level above that of the largest coher-
ence that exists for the system being observed.
But to focus all our attention at that level at this
stage of the game when we still know so little
about what causes state vector collapse may not
be an efficient way to explore the physics invol-
ved. It may be a more efficient strategy to carry
out experiments looking for the existence of
state superposition at various levels between the
atomic level and that of the macroscopic world.
Most experiments in this field are designed to
examine a cut in von Neumann's chain between
the observer and the observed just above the
level of the basic atomic interaction itself. Our
experiment goes to the opposite extreme to look
for state superposition immediately prior to ob-
servation at the macroscopic level.
By doing this we are able to deal experimen-
tally with what has come to be a quite wide-
spread and popular conception of what quantum
mechanics has to say about physical reality. The
Schr6dinger paradox has been used to imply an
actual "observer-observed" dichotomy exists as
a fundamental aspect of physical reality, and to
imply that the observer creates his own reality in
the act of observation. It has been used to raise
such questions as the "Wigner's friend" paradox
and even to promote speculation that by our
observation we may be creating the Big Bang of
the universe. If our experiment does nothing
more than lay such speculation to rest it will
have been more than worthwhile.
At the other extreme, however, we should
recognize the possibility that it is through this
doorway that some phenomena, heretofore not
dealt with by science, may be approached. The
existence of consciousness as a phenomenology
that lies beyond what we as physicists mean by
distance, mass, electric charge and the other
constructs of our physical equations cannot be
denied. Consciousness surely arises out of some-
thing that goes on in the brain of each of us, but
yet lies beyond its usual description as a physical
object no matter how complex. It may be that if
quantum mechanics does require the observer as
an essential and irreducible aspect of physical
reality, then we may find its proper scientific
description to be bound up with an understand-
ing of how state vector collapse comes about.
Whatever the likelihood that we will find evi-
dence for this in the experiment we discuss here',
it would seem to be worth the effort to look.
The Schr6dinger paradox arises because the
prescription for writing the general state vector
for any system requires that one can sum all
possible component states for an unmeasured or
unobserved system irrespective of the scale of
the system to be observed. As a consequence
according to Schr6dinger, a cat placed in a box
rigged to release a tranquilizer (out of difference
to the SPCA -and this writer) if a beta decay
occurs in a specified interval of time, or not if the
beta decay does not occur, must be represented
by a state vector that is the sum of both possible
outcomes before a measurement is made on the
system. Using Dirac's notation this would give
for the combined state
fl = I q1A) + 10S)
where the subscripts A and S refer to the awake
and sleep states respectively. Although this is
generally regarded to be a preposterous conclu-
sion, there exist no experiments that violate this
or any of the basic premises of quantum mech-
anics. A definitive experiment that would de-
monstrate that such a superposition of states
does not exist would be a significant if not a
surprising achievement, On the other hand, since
there exist no examples of a violation of the
principles of quantum mechanics, it must be
considered a viable possibility that quantum
mechanics is valid here as well.
It is usually thought that one of two possible
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E.H. Walker et al. / Testing Schr6dinger's paradox with Michelson interferometer 341
influences causes state vector collapse. One of
these is that something happens during the trans-
ition from the microscopic realm of the system to
the macroscopic realm. Efforts so far to formu-
late such a suggestion have been unsuccessful -
in fact all such proposals that have been reduced
to a mathematical prescription have proved to be
wrong because they have predicted results at
variances with experiments already conducted.
Of course we know that on measurement state
vector collapse will occur-or will have occur-
red. If we open the box, we will see the cat
ACOUSTO-
OPTIC
MODULATOR
He -Ne LASER BEAM
SOURCE ATTENUATOR
BEAM
SPLITTER
PHOTOGRAPHIC
EMULSION
ACOUSTO
OPTIC
MODULATOR
either awake or asleep. Some few scientists have
suggested that the act of conscious observation
causes state vector collapse. Wigner has pointed
out that such is a peculiar implication of quan-
tum mechanics, but he has made no effort to
formulate what this would mean, if indeed he
takes this possibility seriously. Wheeler has poin-
ted out that Bohr specifically "rejected the term
4consciousness' in describing the elemental act of
observation ... he emphasized that no measure-
ment is a measurement until it is 'brought to a
close by an irreversible act of amplification and
CESIUM
PHOTOMUL71PLIER SOURCE
3 7
-01CE
AMPLIFIER NAL
PULSE SCINTILLATOR
SHAPER
i4==zz-) -rL-nm
BISTABLE MULTIVIBRT ~R
B DELAY CIRCIUTS
-L---FLr
<~ _J ----- LFL
SOMHz
RF
SWITCH
8OMHz
80MHZ
RF
SWITCH
MIRROR
Fig. 1. Experimental arrangement for testing the Schr6dinger paradox using a Michelson interferometer. The cesium 137 gamma
source provides the random quantum event that triggers the bistable multivibrator (flip-flop) circuit controlling the AO cells so
that in any given state one cell is always on while the other is always off.
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342 E.H. Walker et al. / Testing Schr6dinger's paradox with Michelson interferometer
until the result is 'communicable in plain lan-
guage."' These ideas have not been reduced to
mathematical formulation, have not been de-
rived from the Schr6dinger equation (note that
the Ehrenfest theorem does not cover the case
considered here; it does not show that any reduc-
tion in the number of states exists in the transi-
tion to the microscopic, only that certain kinds of
macroscopic processes approach the classical in
the limit), and the latter prescription is clearly
anthropomorphic - nothing more.
We have carried out an experiment that not
only tests the Schr6dinger paradox, but has the
potential through modest modifications to range
the entire gamut of possibilities in order to estab-
lish exactly where and how state vector collapse
takes place. The experiment makes use of
Michelson interferometer (a Mach-Zehnder
interferometer could equally be used) in which
two shutters, acousto-optical (AO) cells, are
placed one in each arm of the interferometer.
These AO cells are driven by a quantum mech-
anical process, specifically, a cesium 137 gamma
source driving a bistable multivibrator (flip-flop)
circuit in such a way as to gate one or the other
of the two possible paths in the interferometer.
Thus, knowing the state of the quantum process
driving the AO cells we would know that either
path 1 was open while path 2 was closed or vice
versa. Since we do not know the quantum state
driving the AO cells, however, the system must
be in both states - that is, in the linear superposi-
tion of states. Fig. 1 shows the layout of the
experiment.
Although we have replaced Schr6dinger's cat
with the more manageable AID cells, it is easy to
see - as in figs. 2 and 3 - that this is a realization
of the Schr6dinger paradox in which we have
MIRROR
VI/l/,//1/1/4
MIRROR
Jim -41--
F=
He-Ne LASER M
SOURCE SPLITTER
PHOTOGRAPHIC
EMULSION
Fig. 2. Schematic showing that the experiment of fig. I is in fact a variant of the Schr6dinger paradox arrangement. Here the cat
in Schr6dinger's thought experiment is shown awake and sitting in the way of arm 1 of the interferometer.
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E.H. Walker et al. Testing Schr6dinger's paradox with Michelson interferometer
MIRROR
MIRROR
He-Ne LASER BEAM
7B.E'A.~
SOURCE SPLITTER
11
PHOTOGRAPHIC
EMULSION
343
Fig. 3. In this figure we see the state in which the beta source has caused the release of the tranquilizer gas - causing the cat to
fall asleep in the way of arm 2 of the interferometer.
added an interferometer to test the existence of a
superposition of pure states or simply the pres-
ence of one or the other unknown state of a
mixture of states. Now let us look at the equa-
tions appropriate to the problem.
observation probability p,
p
( 9*1 ( 4111 + A 2* ( 02)( Al 1411 ) + P2102
1911' + I P2 12 + 9 02 ( 01 102
+ A *2 161 ( 412 101 (2)
2. The cat and the correlation of states
Let us now look at why we should not ordi-
narily expect to observe any effect with large
objects in the first place. The system described
by 191) = 101) + 102) for which an observation
operator A(x) in configuration space would yield
A14,I) = 011411) and A1412) =02102) yields for an
Because our object is large however, the phase
factors entering into 41, and 4/, will vary rapidly,
so rapidly that the terms ( 0, 102 ) and ( 4/214,1 )
are for macroscopic objects zero. As a con-
sequence, Eq. (2) reduces to P =I 91 1'+ 192 1'
which is indistinguishable from the classical
probabilities for the system.
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344 E.H. Walker et al. Testing Schr6dinger's paradox with Michelson interferometer
3. Michelson interferometer
Assume the arrangement shown in fig. 1, but
in which both AO cells are always on. This gives
us then the standard Michelson interferometer
with a CW laser source. Using the subscripts 1
and 2 for the two arms of the device, we write
for the state of a single photon I go ) = 101 ) +
102 ~. For a configuration space photon absorp-
tion operator A (x) satisfying A (x) 01 ) = 8, 1 (k, )
and A(x)l '02) = 92102) obviously we will have
for the probability p,,, po = ('P~ I AA I TO), so that
= 1,61 12 + 12 + p p2 (0'102)
P
0 192
+ P*2131(021 01~ 1 (3)
decay and of the arm of the interferometer in
which the gate is located, while the photon re-
presentation as before depends on the arm of the
interferometer, position and time t. We have in
general I IF) G) 0 1 (P ). This gives us
IV,) = GI(B, arm l)I(k#l, t))
+ GAB, arm 2)102(X2' 0) (5)
The parameter B has two states which we desig-
nate "on" (or "+") and "off" (or "-") for arm
number one of the interferometer and for the
second arm, "off" (or "-") and "on" (or "+")
respectively, as determined by the logic of the
switching circuit. Eq. (5) becomes
which is formally what we found in eq. (2).
However, for photons in an interferometer,
terms like (01102) contribute significantly. We
will return to this later. The point here, how-
ever, is that it is the presence of these cross
terms that lead to the interference effects we
observe with an interferometer.
Since we are using a constant wave (CW) laser
for our source, the photon is represented by
.01 = al ei(kxl-W11) (4)
where x, is the path length for arm number one
in the interferometer, tj is the time of the mea-
surement, k and w are the wave number and
angular frequency and a, is a normalization fac-
tor. If (01 1 (k.) and
(0210,~ are averaged over
complete cycles OfX,1X21 tj and t2l these terms
will vanish. With equivalent paths for the two
arms of the interferometer, 01 2 = -62 2 2 So
that we can simply define ~O = 2,6
4. Michelson interferometer test of the
Schr6dinger paradox
Now let us look at the the complete problem
as shown in fig. 1 in which the state of the
quantum mechanical system depends on the cou-
pled gamma decay-photon system. The gates,
G, are functions of a parameter B of the gamma
IIP,) = G+101) + G-102)+ G-I(kl) + G+102) .
1 2 1 2
(6)
As before we write A(x)j(k,) etc. We
therefore have
A11PI) = G+Pll(kl) + G-92102)+ G_jGjjol~
1 2 1
+ G+02102) (7)
2
The detection probability function p, is then
p, = ( W, I AA I IF, )
= ((G + plkl + G_ 16202 +G-P,.Ol
1 2 1
+ G+P202)1(G+Alol + G_ P202
2 1 2
+ G-0101
G+9202)) (8)
1 + 2
which gives
I p1 12 +12
A IGI +fl*1P2G+'G_((kl 101)
1 2
* 113112 G+*G- + P*p2G+*G+((Al (k2)
I 1 1 1 2
* P*A,G-*G+ (021 (k, ) + 1/32 12 IG -12
12 2
* P*13,G-*Gl 2 2
2 2 - (02101 ) + 1132 G_*G+
* 1011'G-*G+ + P*P2G_*G_ ((k, 102)
+1,8112 1-12 1 1 1 2
IGI +P*1P2G_*G+~Ol 102~
1 2
+ P*13,G+*G+(021 (k, ) + 10212G+*G-
2 2 1 12 2+12.2 (9)
+ P*j8,G+*G_(02I0l)+Ifl2 IG
2 2 1 2
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E.H. Walker et al. / Testing Schr6dinger's paradox with Michelson interferometer 345
Since the component states such as 0, and02 in
the linear superposition are the same functions
as for the single states alone, we have s ly
!7,_=G_=G_*=G_*=0, and IGiT
t
hat 2 2
I G +I = 1. Eq. (9) reduces to
2
Ip'12 + 12 + '0*p2((kj 102)
PI 102 1
+ 10*2 91 ~ 021 '01 ) 1 (10)
which is the same result as in eq. (3) for the
Michelson (or Mach-Zehnder) interferometer
result without AO cells.
5. Representation of the photon
Since the laser is not pulsed and since we can
assume the switching rate of the AO cells to be
much lower than the frequency of the photon,
we can write simply
probabilities represent a measurement over a
time that is long with respect to the photon
frequency. Of course, for thin films we have
simply
P0, = a2[2 + e ik(x2-xl) + eik(XI _X2) (14)
Therefore, in the absence of any formalism that
would prescribe state vector collapse below the
macroscopic level, our calculation predicts the
presence of interference fringes in the present
experiment despite its counterintuitiveness.
Therefore, a failure to detect a robust interfer-
ence pattern will show that the experimental
results. are in disagreement with our theoretical
prediction.
The converse outcome holds equally remark-
able significance. The occurrence of interference
fringes would mean that the linear superposition
of state holds before observation even on the
macroscopic scale.
A01 ~ Pla, e'(kxj-wr) (11)
and
A02= j62a2e'(kX2_W0 (12)
where we have introduced the factors a, and a2
as normalization factors for the particular condi-
tions of the experimental arrangement and de-
tection interval. For essentially identical arms in
the interferometer, P,a, ~ 62a2, so that we can
write
po' = IF I AA I IF
Xj+AX
= a 2 2Ax + eik(x2-xl' dx?
I I
X1
X2+AX
+ eik(xl -xi) dx' (13)
f 211
X2
where Ax is the thickness of the photographic
film layer and where we have incorporated the
time interval of the measurement in our normali-
zation factor a. The primes indicate that the
6. The apparatus
The apparatus consists of a simple Michelson
interferometer with optical switches in the relay
arms. A schematic diagram of the arrangement is
shown in fig. 1.
The polarized output beam from a 6328 A CW
helium-neon siule mode laser is attenuated by
a factor of 10- so as to produce a beam of
1.3 x 10-14 W, approximately 4.17 X 104 photons
per second intensity. The attenuation is achieved
by a combination of the deflected beam intensity
reduction, neutral density filters and a polarizer.
The light incident on the beam splitting is polar-
ized with the electric vector perpendicular to the
plane of fig. 1.
The light is passed to a beam splitter which
produces beams of nearly equal intensity. Each
arm of the interferometer contains an acousto-
optical modulator consisting of a Te02 crystal
coupled to an ultrasonic piezoelectric transducer.
When no input voltage is applied to the trans-
ducer, light travels through the crystal unde-
viated. With an 8OMHz rf signal applied to the
transducer, the resulting acoustic waves in the
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346 E.H. Walker et al. Testing Schr6dinger's paradox with Michelson interferometer
crystal diffract the light beam by twice the Bragg
angle, which is 5.9 mrad for the devices used. It
is this diffracted beam that is used to obtain
interference in the interferometer. Turning the
transducer off turns the AO cell off. These
acousto-optical devices exhibit a rise and fall
time of 120 ns when switching a beam of
0.75 mm diameter. Most of the power (80%)
goes into the first-order diffracted beam, which
leaves the device at an angle of 11.8 mrad to the
zero-order undiffracted beam, while smaller
amounts of power are deposited into the higher
orders. As stated, it is the first-order diffracted
beams that are reflected off the interferometer
mirrors and back through the modulators where
a second diffraction occurs. Only the first-order
beams are shown in fig. I for clarity. The zero-
order and higher-order diffacted light is absor-
bed by various beam stops. Thus the acousto-
optical modulators act to chop the light in the
interferometer arms with a contrast ratio
adequate to assure that none of the photons
reaching the film will have passed through an
"off" shutter.
The switched beams from the interferometer
arms are recombinated in the beam splitter and
are deposited on a high speed photographic
emulsion. The beam divergence of the laser and
the geometry of the apparatus are chosen so as
to result in a 2 mm diameter image on the film
with approximately four linear fringes visible in
this area.
The acousto-optical modulators are driven by
switched 80 MHz oscillators which are in turn
gated on and off by the outputs of a bistable
multivibrator, or flip-flop, so that one modulator
is on while the other is off, A delay of 150 ns is
introduced into the gating signals applied to the
rf drivers so that one shutter does not start to
open until the other has closed, thus ensuring
that at no time are both interferometer arms
open. The flip-flop is clocked by pulses from a
photomultiplier, which have been suitably am-
plified, shaped and discriminated. The photo-
multiplier looks into a sodium iodide scintillator
crystal. With a cesium 137 source of approxi-
mately 30 ~Xi placed 2 cm from the scintillator,
the pulses which clock the flip-flop have a mean
repetition rate of 118 kHz.
With an average photon rate of 4.17 x 104 S-A
emerging from the attenuator, there is an aver-
age of less than one photon in each arm of the
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Fig. 4. Photograph of the laboratory layout.
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E.H. Walker ef al. / Testing Schr6dinger's paradox with Michelson interferometer 347
interferometer during each period for which the
acousto-optic modulator in the arm is open. The
resulting low intensity image is recorded on
Kodak 2415 technical pan film hypersensitised in
forming gas prior to exposure. A high speed
Polaroid film has also been used. A photograph
of the laboratory setup is shown in fig. 4.
7. Test runs
In order to verify that the apparatus properly
discriminates between the two possible ex-
perimental outcomes, test runs were made with
the logic circuit connected so that both shutters
would either be on or off to make certain that
the apparatus could give interference patterns.
This run gives us a reference interference pattern
that we can use to judge the results of our
experimental run. The average shutter rate in
this case was the same as for the final experimen-
tal run. The resulting photograph is shown in
Fig. 5.
8. Experimental results and conclusions
Fig. 6 shows the experimental outcome for a
photon rate of 4.17 x 104 s-'. The figure speaks
for itself. Thee is no interference pattern present
in the figure. The figure clearly shows the abs-
ence of the interference fringes predicted by our
formalism.
This absence of interference fringes in the
experimental run is a result that, although expec-
ted on the basis of commonsense, we neverthe-
less interpret as a prima facie case that quantum
theory may be violated. The result is important
because it provides a starting point for us in our
search for the cause and experimental meaning
of state vector collapse. These results are also
important because they are related to questions
about the Schr6dinger paradox.
We do not yet know just how and where state
vector collapse occurs -nor do we know what
this even means. Our experiment does not solve
or remove the measurement problem. If any-
thing, it deepens the problem. We must find out
Fig. 5. Test run in which both shutters are either open or closed at the same time to assure a conventional interference pattern.
Photon rate was 4.17 x 10' s ', less than one photon in each arm of the apparatus at any moment.
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348 E.H. Walker et al. Testing Schrodinger's paradox with Michelson interferometer
Fig. 6. Final experimental run. No interference pattern was obtained. This figure clearly shows the absence of interference.
the mechanics of state vector collapse. The pre-
sent experiment provides a basic plan for future
experiments to search out how state vector col-
lapse occurs and to give us a clear understanding
of just what state vector collapse entails. It gives
us the too] we need to find just where to cut von
Neumann's chain.
Acknowledgements
We wish to acknowledge the helpful discus-
sions about this experiment held with C.O.
Allcy, Y.H. Shih, and George Hinds of the
Department of Physics of the University of
Maryland.
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