What does th E and the M stand for in the equasion E=MC squared mean or stand for?

- Introduction: The Problem
- The First Solution -- Simple interference
- Step 2 -- the Quantum Zeno Effect
- Combined Bliss
- Work in Progress
- Other Schemes
- Hint of Things to Come

The problem we wish to solve is how to *optically* detect the presence
of something, but with no photons hitting it. The something could be many
things, like a hand, a detector, an ultrasensitive piece of film, or a single
atom. The two physicists who brought this topic to light in 1993, Elitzur and
Vaidman (EV), considered a "superbomb", which, if it possessed the
trigger/detonator element, would explode whenever hit by even a single photon.
Some of the bombs possess the element, and some do not. (The former are the
so-called "good" bombs, the latter are the "bad" bombs [please do not get on my
case about which ones are *really* the good ones!]). The goal now is, given
a supply of these bombs in sealed crates, find out which ones are the good
bombs. (Also, we aren't allow to shake the crates, or otherwise risk disturbing
the bombs.)

A detective limited to the realm of classical physics is in trouble. He can
go into a completely darkened room, and pry off the lid of the crate. Then what?
If there really is no light at all -- if no photons at all hit the trigger
element -- then he gets no information. If, on the other hand, a single photon
hits the element, well then by definition there is a loud explosion, and the
detective knows that this *was* a good bomb. There seems to be no way to
find the good bombs without always exploding them.

Enter quantum mechanics. The first solution to the problem posed by EV was also suggested by them. Namely, we can use the complementary wave-particle nature of light (or, indeed, of any quantum). Consider a Mach-Zehnder inteferometer, as shown below (left). It is composed of two perfect mirrors, and two 50-50 beam splitters. The upper and lower path lengths are set to be exactly equal. Under these conditions, there is complete destructive interference to the upper exit port -- any incident light always exits to the other port. Put differently, the probability is 0 for an incident photon to reach detector D_dark (hence the name), while the probability to reach detector D_light is 1 (ditto).

What happens when there is an object -- like the EV "superbomb" -- in one
of the paths? (See above, right.) Now we can use our classical notions to
understand the possibilities. There is a 50% chance that a photon will take
the path containing the object, resulting in an explosion. But if this didn't
happen (also 50% likely), then the photon must have taken the other path. At
the second beam splitter, there is no longer interference, since there is only
one way to get there. Therefore, the photon again makes a random choice. There
is thus a 25% (= 0.5 x 0.5) chance that the photon goes to D_light; such a
result gives no information (but also does not disturb the bomb), since it
would have happened in the absence of the bomb. There is also a 25% chance
that the photon is detected at D_dark. This *never* happened in the
absence of the bomb -- whenever D_dark goes "click", we know that there was an
object in the interferometer. And since we only send in a single photon, and
it shows up at D_dark, it could not have interacted with the object in the
interferometer (which object, even if it does not actually explode, is assumed
to be completely non-transmitting).

As alluded to above, the reason this works is the wave-particle duality of
quanta. When the interferometer is empty, the two possible ways to get to a
detector are indistinguishable, and we get interference -- the photon behaves
like a wave. On the other hand, when one of the paths is blocked, there is
only one way to reach a detector, so there is no interference -- the photon
behaves like a particle, in that it can only show up at the bomb *or* at
D_light *or* at D_dark, but never at more than one of these.

It will be helpful below to define a figure of merit for these systems,
that will tell us what fraction of our measurements can be interaction-free.
(Simply taking the probability of D_dark firing is too restrictive, because we
could always take the photons that leave toward D_light and recycle them.)
This is given by P(D_dark)/(P(D_dark) + P(boom)), where we are defining D_dark
as the probability of an interaction-*free* measurement, and P(boom) as
the probability of an interaction-*full* measurement.

Now, it turns out that by adjusting the reflectivities of the beam splitters in the interferometer, it is possible to achieve P(D_dark) = P(boom) (well, almost -- see the theoretical plot of the efficiency below, as well as our experimental data). For this case, the efficiency is 1/2. (Note however, that in this limit, both P(D_dark) and P(boom) actually approach 0, while P(D_light) --> 1.) The question immediately arose, is 50% the best we can do -- are we destined to be blown up half the time?

The answer, amazingly enough, is No. We can in fact make the chance that the bomb blows up as small as we like (in principle, anyway). To understand the method, we need to consider another rather peculiar quantum mechanical phenomenon, the so-called "Quantum Zeno effect". First discussed in 1977 by Misra and Sudarshan, the QZE involves using repeated quantum measurements to inhibit the evolution of a quantum system. It relies on the quantum "projection postulate", which basically states that, for any measurement made on a quantum system, only certain answers are possible, and that the resulting quantum system is then in a state determined by the obtained results. This is easiest to understand with a particular example.

Consider a series of N polarization rotators, each of which rotates the
polarization of light by an angle 90°/N. Therefore, after passing through all
N of them, an initially horizontally-polarized photon will be vertically
polarized. That is, it will have zero chance of passing through a horizontal
*polarizer* and being detected:

However, if we intersperse a series of horizontal polarizers, one at each
stage, then the outcome is quite different. For concreteness, consider the
case of six cycles, so that the rotation angle at each stage is 15°. At the
first polarizer, the photon has only a small chance of being absorbed -->
6.7% = sin^{2}(15°). If it is *not* absorbed, then by the
projection postulate, the photon must be horizontally polarized. The identical
process happens at every stage. For the case of N=6, the chance that the
photon was transmitted through all 6 polarizers is simply
(cos^{2}(15°))^{6}, which is about 2/3. Note that without the
interspersed polarizers, we never saw light at the detector. Hence, whenever
we get a "click" at the detector, we know that the extra polarizers were
inserted. Moreover, if we perform the experiment with a single photon input,
and it shows up at the final detector, then of course it could not have been
absorbed by any of the extra polarizers. And as we let the number of stages N
become large (at the same time reducing the angle of polarization rotation
accordingly), then the probability that the photon is absorbed vanishes -- the
photon is always transmitted!

To demonstrate this phenomenon, we did not use six rotators and six polarizers, mostly because we did not have so many elements. Instead, we used a single rotator and a single polarizing beam splitter, but arranged our light to pass through these six times, as shown below:

What we found, in agreement with the above predictions, is that when the polarizing beam splitter (which transmits horizontal polarized light, and reflects vertical polarized light) was not in place, the detector D essentially never fired; whereas when the polarizing beam splitter was in place, the detector D fired 2/3's of the time.

We see that we have come half-way to our goal from either side. Using the
EV simple disrupted interferometer, we can detect the presence of any object
(i.e., an opaque object), but only up to 50% of the time interaction-free. And
with the quantum Zeno technique we have shown that we can detect the presence
of a *polarizing* object better than 50% of the time. By making a hybrid
of the two schemes, one can achieve Nirvana -- detection of an opaque object
with an arbitrarily small chance of it absorbing a photon. While there are
many methods to do this, we will present here only the one most closely
connected with the previous discussions.

Consider the system shown at the right: We again have a cycled photon, as in the previous demonstration of the Zeno effect. It makes N cycles in the system before being let out, and its polarization analyzed. At each cycle, again there is a rotation of the polarization by 90°/N, so that at the end, the initially horizontally-polarized photon is vertically polarized. The key difference to the previous experiment is the inclusion of the polarization Mach-Zehnder interferometer. Instead of normal beam splitters, it uses two polarizing beam splitters, which transmit horizontal and reflect vertical polarized light. The two arms of the interferometer are set up to have equal lengths. Therefore, any incident light is split into the horizontal and vertical components (the former taking the "high road", the latter taking the "low road"), which are then added up again to reform the original polarization state.

This is only true if the arms of the interferometer are unblocked. If,
instead, we have an object in the lower path, then the evolution is totally
different. Again taking the case of N=6 for concreteness, during the first
cycle there is only a 6.7% chance that the photon takes the lower path (and is
absorbed). If this does *not* happen, then the photon wavefunction is
"collapsed" into the upper path -- the photon is again completely horizontally
polarized. The same thing happens at every cycle, until on the Nth cycle the
photon is allowed to leave. If it has successfully survived (i.e., not been
absorbed) every cycle, then the photon is definitely horizontally polarized.
Remember that without the object, the photon was definitely *vertically*
polarized. By measuring the final state of the photon's polarization, we can
tell whether or not an object was blocking the lower path. And by using a
large number of cycles, we can make the probability that the photon is at some
point absorbed by the object arbitrarily small. This is the essence of
interaction-free measurement.

The current setup used at LANL to demonstrate interaction-free measurements is shown to the right. A very weak pulse (less than 1 photon) is produced from a 670nm laser diode. The pulse is coupled by a 4% reflector into our system, with an initial Horizontal polarization. The pulse is cycled in the system by the 88%-reflecting mirror, its polarization rotated each cycle by a small amount (determined by the orientation of the rotator [a quarter waveplate]).

The interferometer at the top is of the Michelson variety, except that it
uses a *polarizing* beamsplitter (which reflects vertical and transmits
horizontal polarization). In the absence of any object in the interferometer,
it has absolutely no effect on any input polarization state, simply breaking
it into the horizontal and vertical components, and adding them right back up
in phase. Therefore, over the course of N cycles, the originally H-polarized
light will undergo a stepwise rotation to V-polarization.

If, on the other hand, there is an object blocking the
vertical-polarization path, then we have the inhibition of rotation described
above. The *non*-detection of the photon in the vertical path "collapses"
the wavefunction, so that the photon lies solely in the horizontal path. After
N cycles the photon still has horizontal polarization.

Finally, to determine the polarization, we must remove the photon from the system. In a more advanced system currently under construction, the photon will be "switched out" of the system after exactly the desired number of cycles. For the present experiment, however, the photon is allowed to randomly leak out. By knowing when the photon was sent in, and the time it takes to make a given cycle, we can electronically examine only those photons that spent exactly N cycles in the system.

A time-resolved image of photons escaping the interferometer. Vertical axis shows the number of photons and horizontal axis shows time (relative to when the pulse was first sent into the system). The entire horizontal axis shows a duration of about 50 ns.

Theoretical curves of the probability of making an interaction-free measurement, and the complementary probability of the object absorbing the photon. Last year's data from Innsbruck is shown, as is the most recent LANL data, demonstrating interaction-free measurements nearly 70% of the time.

For simplicity, we have concentrated thus far on only a few simple systems.
However, the techniques of interaction-free measurement can in fact be applied
to any 2-level system. As another example, consider a series of connected,
perfect Mach-Zehnder interferometers whose beamsplitters' reflectivities are
R=cos^{2}(pi/(2N)) where N is the number of beamsplitters (see figure
below in (a)).

In this case a photon initially incident on the bottom half of the interferometer chain will gradually "slosh" into the upper half. In fact, with R as given above, after N interferometers the photon will be entirely in the upper half. That is, the photon will, with certainty, exit the string of N interferometers via the "up" output port.

When an object is placed in the top portion of the interferometers, however,
this inhibits the evolution of the light -- it gets "trapped" in the lower
half (as in part (b), above). For N interferometers, the photon will now have
a probability P=[cos^{2}(pi/(2N))]^{N} to emerge at the down
port, whereas this probability was 0 when there was no object.

Just as we were able in the quantum Zeno example to convert a string of elements
into a cycle involving the same element again and again, here we can do the
same thing, though the result might look a little different. Consider now
that we have two identical cavities, which are weakly coupled by a mirror
of reflectivity R=cos^{2}(pi/(2N)). If we start out a photon in the
one of the cavities, say the left one, then with time it will "slosh" into
the other one. After N/2 cycles it will be equally likely to be in either
cavity, and after N cycles it will only be in the right cavity. (If we were
to allow it to continue cycling, it would eventually -- after another N cycles
-- slosh back into the initial, left cavity.) Note that this sloshing back
and forth is completely analogous to the coupled oscillations of two weakly
coupled pendula.

On the other hand, if there is an absorbing object in the right cavity, this
will prevent the "sloshing" from occuring. At each cycle there will be a small
probability that the photon will be absorbed by the object (simply given by
the coupling transmittivity T = 1 - R). But if this *doesn't* happen,
the wavefunction of the photon is "collapsed" back into the initial cavity
and we start all over again. Thus, at the end of the N cycles, the photon
has a probability R^{N} of still being in the left cavity (and a probability
1 - R^{N} that it was absorbed by the object). As we let the number
of cycles N go to infinity (easier said on a Webpage than done in reality),
the photon *always* stays in the initial cavity.

Curiously, there is a related method of doing interaction-free measurements,
suggested to us by, amongst others, Profs. Yamamoto (at Stanford University)
and Haroche (at the Ecole Normal Superieure in Paris), which relies on a *single*
cavity. The point is to consider a very high finesse cavity which is tuned
to allow the wavelength of the incident light to pass (note that this means
that the incident light must have a bandwidth less than the cavity bandwidth).
If the cavity is empty, then the light will simply pass right through (but
only after the resonance interference effect has had time to build up in the
cavity, which is basically the same as the cavity ring-down time). But if
there is an absorber in the cavity, then the incident light will basically
be reflected by the first high-reflectivity mirror. By seeing whether the
light was transmitted or reflected, we can see whether the cavity was empty
or not.

Before actually publishing the details of these ideas (and hence subjecting them to "peer review"), I am hesitant to say too much. However, the reader who has made it this far deserves to know what lies ahead. It is only natural, once one has considered interaction-free measurements of a "normal" object, to start considering modifications. For example, what happens if the object is semi-transparent? It turns out that the above-described schemes do not work as well (after all, they rely on a "collapse" which in turn depends on the complete non-absorption of a photon). However, there are ways to improve them, and perhaps even to use them to make ultra-sensitive measurements of optical density.

Another very interesting topic is that of a quantum object, i.e., one that can be in a superposition of being "there" and "not there". One such example is an atom in an atom interferometer, which simultaneously exists in both arms. Another is the recent separated-ion demonstration by Wineland et al., in which a single ion in a trap is made to coexist at two separated points in space. If such systems are evaluated using the interaction-free measurement schemes, then the two sub-systems -- the quantum object and the interrogating light -- become entangled. In fact, although we have not discussed it at all here, for sufficiently large N, the interaction-free measurement methods even work for multiple-photon states, even for dim classical pulses. Therefore, combining such an input with a quantum object, one is able to transfer the quantum superposition of the latter onto the former. In other words, one could make superpositions of "bunches" of photons; for example, one could prepare a pulse of light with an average of 20 photons in it, all of whom were horizontal, or all of whom were vertical, and yet until a measurement was made, none of them would have a definite polarization. Such a peculiar state of affairs would be a modest example of a Schroedinger cat.

Finally, another avenue we are currently exploring is to extend the above
techniques to allow two-dimensional imaging of an object. As a simple example,
one could then make an *in situ* movie of a Bose-Einstein condensate
without blowing it up, since very few of the photons actually end up being
absorbed by the ultra-cold condensate atoms. (Practically speaking, however,
the latest far-off-resonance methods of Ketterle et al. are likely to be more....practical.)

In conclusion, it is not altogether clear what wonderful applications the principles of interaction-free measurements will have, but it is certain that some very interesting physics is yet to be uncovered.

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