Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
E.H. Walker et al.
I
Testing Schrodinger's paradox with Michelson interferometer
342
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 Schrodinger 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 Schrodinger 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
MIRROR
EZZZZZZ3
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 Schrodinger’s cat
with the more manageable AO cells,
it is easy to
see - as in figs. 2 and 3 - that this is a realization
of the
Schrodinger paradox
in which we have
PHOTOGRAPHIC
EMULSION
)
J
Fig. 2. Schematic showing that the experiment of fig.
is in fact a variant of the Schrodinger paradox arrangement. Here the cat
in Schrodinger’s thought experiment
is shown awake and sitting in the way of arm
1 of the interferometer.
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
E.H. Walker el
al.
f
Testing Schrodinger' s paradox with Michelson interferometer
343
He-Ne LASER
SOURCE
MIRROR
PHOTOGRAPHIC
EMULSION
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.
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 |^) = |*Ai) + 1^)
for which an observation
operator A(x) in configuration space would yield
=
and A\ip2 ) = /32 |t/»2 ) yields for an
observation probability p,
p = (V\A
fA\V)
= Ot<^l + 02<fc)(A l«M + ftlfc))
= l/3 1
|
2 +
l|82
|
2 + /3t/32 (^1 |^2 )
+
•
(2)
Because our object
is large however, the phase
factors entering into
</*, and
if>2 will vary rapidly,
so rapidly that the terms
( ^
1/»2 ) and
{ i//2
i/f,
)
are
for
macroscopic
objects
zero. As
a
con-
sequence,
Eq.
(2)
reduces
to p = |/3,|
2 + |/32
|
which
is
indistinguishable
from
the
classical
probabilities for the system.
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
EH. Walker el al.
I
Testing Schrodinger’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
and 2 for the two arms of the device, we write
for
the
state of a
single photon
|lP0 ) =
|</>i) +
| <t>2 ). For a configuration space photon absorp-
tion operator A(x)
satisfying A(x)\<t> l ) =
and
A(x)\<j>2 ) = /?2
1 02 )
obviously we
will have
for the probability p0 , p0 = (VjjAtAllPj,), so that
po=i/3 1 r+i^r+)8*
j82 <</> 1 id>2 )
+ PtPMAtr)
,
(3 )
which
is
formally what we
found
in
eq.
(2).
However,
for
photons
in
an
interferometer,
terms like
( </>j
|
</>2 )
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
^ =
(4)
where x,
is the path length for arm number one
in the interferometer,
t
is the time of the mea-
surement, k and
to
are
the wave number and
angular frequency and a 1
is a normalization fac-
tor.
If
(</»il</>2 ) and
( <£2 1</>!> are averaged over
complete cycles of
, x2 ,
tx and
t2 . these terms
will
vanish. With equivalent paths
for the two
arms of the interferometer,
| /3 2
2 =
| j82
|
2 = /3
2 so
that we can simply define p0 = 2(3
4. Michelson interferometer test of the
Schrodinger paradox
Now let us look at the the complete problem
as shown
in
fig.
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
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
I
interferometer, position and time
t. We have in
general
|lP) = |G}®|<P). This gives us
)
1^) = G,(B, arm l)|<£i(x,, t))
+ G2 (B, arm 2)\<f>2 (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
l*i> = g; i^> + g 2
1 </>2 ) + cr i<m + g 2
+
i</.2 )
.
(6 )
As before we write A(x)\(f)1 ) = /3J <(>x ),
etc. We
therefore have
A|^> =G^M + G 2-/32 \</>2 ) + Gj-ftl^)
+ G 2 /32 \<t>2 )
.
(7)
The detection probability function p 1
is then
Pi = (%\A
1A\%)
=
+ G 2 /?2 </>2 + Gj"^!^
+ G 2 (32 <f>2 )\(G^ (3^ + G 2 (32 <f>2
+ G 1-/3
1 </>1 + G 2
+
/32 </.2 )),
(8)
which gives
Pi = \h\
2\Gt\
2 + PlfrG^G;
+
| (3 ,
|
:
G 1"*
G
r + (3 *(32G ;<*
G
+
<
|
</>2 )
+ I3*2 ^G;*G ; <«^2
| <Aj> + |)32
|
2 |G 2
+
i8*/3
1G 2-*Gr(^2 |^) + |/32
|
2G 2-*G 2
+
+ l^i|
2Gr*G!
+ + (3*(32G~*G 2
<</>!
|</>2 )
+ Mg;\
2 + /3^2Gr*G 2
+ (^ \<f>2 )
*
+ KhGfGKh |*,> +
\(32
\
2G +
2 *G~
+ ^2^G 2 *G]“ (</>2
| </>j) + \(32 \
2\G 2
\
.
(9)
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
E.H. Walker el al.
/
Testing Schrodinger's paradox with Michelson interferometer
Since the component states such as
</>, and
ti>2 in
the
linear superposition are the same functions
as
for the
single
states
alone, we have simply
that G“ = GJ = G”* = G 2
~* = 0, and
|G,
+
| =
|G 2
|
2 = 1. Eq.
(9) reduces to
Pi = l0l| 2+ Iftf + ^lft^ll^)
+ fiMh\h)>
(io)
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
Ah = ft a ,
(11)
and
A(f>2 = ft« 2
,
(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, ftuj = fto2 , so that we can
write
P
'
0 =y\A'A\yy
Xi + Ax
= «
2 [2Ajc +
f
e
ik(*2~xi) dx;
*1
x2 + Ax
+
f
e
ik(Xl ~*2> dx'],
(13)
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
345
probabilities
represent
a measurement
over
a
time
that
is
long
with
respect
to
the
photon
frequency. Of
course,
for
thin
films we have
simply
pi = a
[2 +
+ e
i*(*’"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.
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 single 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 10
4 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 Te0 2
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 80 MHz rf signal applied to the
transducer,
the
resulting
acoustic waves
in the
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
E.H. Walker el
al.
i
Testing Schrodinger' s paradox with Michelson interferometer
346
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.
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 p,Ci 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 10
s
emerging from the attenuator, there
is an aver-
age of less than one photon
in each arm of the
Fig.
4. Photograph of the laboratory layout.
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
E.H. Walker et
al.
I
Testing Schrodinger'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 10
s
_l
. 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 Schrodinger 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.
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1
348
E.H. Walker et al
l
Testing Schrodingers paradox with Michelson interferometer
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 tool 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.
Alley,
Y.H.
Shih,
and
George
Hinds
of
the
Department
of
Physics
of
the
University
of
Maryland.
Approved For Release 2000/08/08
: CIA-RDP96-00789R002200240001-1