BECs, lasers, and other clocks.
Some remarks on time measurement (&c.)...
• BECs versus lasers
• Do Bose-Einstein condensates have a macroscopic phase?
• Do lasers have a definite phase?
• What does have a phase?
• How can it be measured?
• Does any of this make a difference?
• Interference as measurement
• Elitzur & Vaidman's "bomb"
• Hardy's Paradox
• Some brief thoughts about time measurement
• The tunneling-time problem
• The Larmor times
11 Nov 2003
FIRST:
Thomas's allegory for interfering BECs
But actually:
The surprise is that even if the
classes do have well-counted numbers
of students, interference may still be
observed.
How can this be?
Two complementary pictures
Studying Bose Condensates as a whole
Studying individual atoms
Each interference experiment is a single
measurement – an approximate measurement of
relative phase.
Atoms coming from two sources with no relative
phase [EQUIVALENTLY: atoms whose origin
could be determined (i.e., by counting how many
remain in each BEC) ] do not interfere – the
Each pair of condensates begins with an
unknown relative phase, but the interferogram
forces it to collapse to a particular value.
expected atom number aa does not depend on
position.
All the individual atoms "agree" as to the phase
of the interferogram; they are really part of a
single measurement.
N-1N atoms
(no phase)
atoms
N-1
atoms
N atoms
(no phase)
Pairs of atoms, on the other hand, can interfere à
la HOM. In other words, aa a a
may exhibit dependence on relative position.
These correlations mean that fringes can build
up; atoms are more likely to appear 1 period
apart than 1/2 a period apart.
measured phase
Two indistinguishable paths (RR&TT) to 2-atom detection
Where did the phase come from?
• Once atoms started leaking out, an interference pattern
formed, with a previously unpredictable phase.
• It's the measurement itself (as in the quantum eraser) which
generated this coherence.
• Originally, one could certainly have counted atoms, and
measured their momenta to discern which cloud each came
from.
• Only after detecting an atom in such a way that it's
impossible to tell which cloud it came from do the atom
numbers of the two clouds become entangled, giving rise to
coherence.
• As soon as one atom is detected, there is some coherence
(relative phase between neighboring atom numbers), but it
has been shown that it builds up more and more as more
atoms are detected.
Funny realisation
• Even though photon number isn't conserved, energy is.
• All these arguments about being able to tell in principle how many atoms
were in each cloud also apply to being able to tell how much energy is
stored in each of two lasers.
• Even if laser beams are not coherent states, but fixed-photon-number
states, interference would still occur.
• Lasers don't have "spontaneous" phases, in this picture – but the relative
phase between different lasers gets fixed as soon as the beams interfere
with each other. As soon as you try to measure a laser's phase, there's no
way you can tell whether or not it was defined before you measured it!
• Non-uniqueness of density-matrix expansions (see next slide)...
Does a radio transmitter have a phase, or is that also only relative?
Does anything in the universe? How would we know?
The density matrix of a laser
You can only measure phase via a
reference
• Direct detection measures aa, particle number.
• A field (ae-iwt + h.c.) is a + a or a – a... like X and P. To measure this
operator, one needs to put it inside a von Neumann Hamiltonian. But it
doesn't obey conservation of number (or energy)!
• Fields and phases are always measured by beating against another
oscillator which already has a phase (i.e., an uncertain number). To
observe interference, one must be unsure whether any given particle
came from the system or the local oscillator.
(How to measure time if you don't already have a clock?!)
• Compare "superselection rules" – superpositions of different charge
states aren't supposed to exist.
(What about different mass, or energy states?)
But in fact, we can't tell whether they exist or not.
• Since number is conserved, relative uncertainty is produced by letting
systems interact in such a way that only their total number is known.
• [Note current controversies about superselection etc in quantum info-how is it possible to establish a shared reference for any coordinate?]
Another example of interference
as measurement:
Interaction-free measurements
"Interaction-Free Measurements"
(AKA: The Elitzur-Vaidman bomb experiment)
.....what else does interference allow us to "measure"?
Problem:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
If detector D fires, you can say with certainty that the bomb was blocking the
path – although at the same time, you know that no particle encountered the bomb.
Did the bomb disturb the "phase of the vacuum"?
What do you mean, interaction-free?
Measurement, by definition, makes some quantity certain.
This may change the state, and (as we know so well), disturb conjugate variables.
How can we measure where the bomb is without disturbing its momentum (for
example)?
But if we disturbed its momentum, where did the momentum go? What exactly
did the bomb interact with, if not our particle?
It destroyed the relative phase between two parts of the particle's wave function.
Hardy's Paradox
C+
D+
D-
BS2+
C-
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
Outcome Prob
D+
e- was
D+ and
C- in
1/16
D- e+ was in
D- and C+ 1/16
C+ and ?C- 9/16
D+DD+ and D- 1/16
But
… if they4/16
were
Explosion
both in, they should
have annihilated!
What does this mean?
Common conclusion:
We've got to be careful about how we interpret these
"interaction-free measurements."
You're not always free to reason classically about what would
have happened if you had measured something other than what
you actually did.
(You decide whether or not you buy this...
we'll come back to it in a few weeks.)
Introduction to tunneling times
MOTIVATION:
(1) background for some later topics...
(2) how does one actually measure time?
(recall there is no operator for time.)
What you measure depends on how you measure it.
• How long does it take a particle to tunnel through a forbidden region?
• Classically: time diverges as energy approaches barrier height.
• "Semi"classically: kinetic energy negative in tunneling regime; velocity
imaginary?
• Wave mechanics: this imaginary momentum indicates an evanescent
(rather
than propagating) wave. No phase is accumulated...
vanishing group
delay?
• Odd predictions first made in the 1930s and 1950s (MacColl, Wigner,
Eisenbud), but largely ignored until 1980s, with tunneling devices.
• This was the motivation for us to apply Hong-Ou-Mandel interference to
measurements: to measure the single-photon tunneling time.
time-
Two Hong-Ou-Mandel dips
How can this be?
n1 n2
.......
Very little light is transmitted
through a tunnel barrier (a
quarter-wave-stack dielectric
mirror, in our experiment).
But how that's all classical waves...
how fast did a given photon travel?
Larmor Clock (Baz', Rybachenko,
and later Büttiker)
z
y
e-
ex
B
x
f = wT
But in fact:
=
x
z
z
+
fz = wTz
+
-z
-z
f = wTy
Which is "the" tunneling time?
Ty? Tz?
Tx2 = Ty2 + Tz2 ?
Disturbing feature... Ty is still nearly insensitive to d, and often < d/c.
Büttiker therefore preferred Tx... which also turns out < d/c, but rarely!
Too many tunneling times!
Various "times":
group delay
"dwell time"
Büttiker-Landauer time
(critical frequency of oscillating barrier)
Larmor times (three different ones!)
et cetera...
Questions which seem unambiguous classically may have multiple
answers in QM – in other words, different measurements which all
yield "the time" classically need not yield the same thing in the
quantum regime.
In particular: in addition to affecting a pointer, the particle itself
may be affected by it.
A few things to note:
• This -m˚B interaction is a von Neumann measurement of B (which in turn stands
in for whether or not the particle is in the region of interest)
• Since Bz couples to sz , the pointer is the conjugate variable (precession of the
spin about z) –– Note that this measurement is thus just another interference
effect, as the precession angle f is the phase difference accumulated between 
and .
Some references
BEC interference: Andrews et al., Science 275 , 637-641 (1997)
Interference of different sources: Magyar G and Mandel L 1963 Nature 198 255
Klaus Mølmer, Phys Rev A 55, 3195 (1997) -- "Optical coherence: a convenient fiction"
Implications for quantum info: T. Rudolph & B.C. Sanders, PRL 87, 077903 (2001)
Wright, Wong, Collett, Tan, Walls, PRA 56, 591 (1997): BEC interference theory.
H. Wiseman, quant-ph/0303116,
"Optical coherence and teleportation: Why a laser is a clock, not a quantum channel "
Interaction-free measurements:
Elitzur and Vaidman Foundations Physics 23, 987 (1993).
Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich PRL 74, 4763(1995)
Tunneling times et cetera:
Hauge and Støvneng, Rev. Mod. Phys. 61, 917 (1989)
Büttiker and Landauer, PRL 49, 1739 (1982)
Büttiker, Phys. Rev. B 27, 6178 (1983)
Steinberg, Kwiat, & Chiao, PRL 71, 708 (1993)
Steinberg, PRL 74, 2405 (1995)