Interference as measurement
(quantum states of light, BEC, etc.)
• The phase of a field (light or matter)
• Why don't single photons interfere with each other?
- Quantum states of the electromagnetic field
- Heterodyne measurements
• Why do single photons interfere?
• Do Bose-Einstein condensates have a macroscopic phase?
• Do lasers have a definite phase?
• Does any of this make a difference?
• Interference as measurement
• Elitzur & Vaidman's "bomb"
• Hardy's Paradox
4 Nov 2003
Remember: a photon has no phase
Two photons meeting at a beam-splitter don't interfere; their relative phase is undefined.
|1>
|1>
n1
n2
The cross terms change n1 and n2; unless these
numbers are uncertain, these terms vanish.
Optical phase actually refers (roughly) to the quantum phase difference
between the amplitude to have n photons and the amplitude to have n+1.
The overall quantum phase of the state is completely different (and unobservable).
Each em mode is a harmonic
oscillator:
Complex field amplitude E = |E| eif e-iwt
= E1 cos wt + E2 sin wt
Cf. harmonic oscillator, x(t) = |x0| eif e-iwt
= x(0) cos wt + p(0) sin wt
Energy = |E1|2 + |E2|2
or x2 + p2
(mw2x2/2 + p2/2m, of course)
Quantized energy = (n+1/2) hw.
Equally spaced energy levels correspond to different numbers of photons.
A wave packet about to oscillate
xx
p
A wave packet in oscillation
xx
p
A wave packet in oscillation
xx
p
Orbit in phase space
E2 (think p)
E1 (think x)
Exactly the classical phase-space trajectory, with
an extra circle of uncertainty.
A low-momentum wave packet
xx
p
A low-momentum wave packet
xx
Behaves like a collection of particles starting at rest at different positions...
p
A well-localized wave packet
xx
p
A well-localized wave packet
xx
Behaves like a collection of particles starting at x=0 with different momenta
p
"Just right" (the ground state)
xx
Perfect balance:
As much force to the left as to the right, and
as much momentum to the left as to the right.
Just enough binding force to counteract the spreading of the wavepacket.
TIME-INDEPENDENT (i.e., no phase)
p
Also "just right" (1st excited state)
xx
Larger amplitude oscillations; higher energy.
But still equally likely to be on the left or the right;
or to have momentum to the left or to the right.
Bigger amplitude, still undefined phase (still time-independent).
p
Phase-space picture
complex E = |E| eif e-iwt
= E1 cos wt + E2 sin wt
E2 (think p)
|E|
f
The vacuum isn't empty:
E1 (think x)
minimum phase-space area,
as for X and P of a particle.
E2 (think p)
E2 (think p)
E1 (think x)
A phase-space distribution
for a 1-photon state
E1 (think x)
A "coherent" or "quasiclassical" state.
A wave packet about to oscillate
xx
The quasiclassical or "coherent" state: follows a classical orbit,
maintaining the same minimum-uncertainty product in x and p
as in the ground state.
pp
The coherent state
Electric field of a coherent state
E
t
(nice pictures stolen from Gerd Breitenbach's web page)
How does one measure these things?
With a radio-frequency field, of course, one measures E with an antenna, in real time.
Not possible at optical frequencies - we can only detect power/energy/photons.
Homodyning/heterodyning: interfere a signal with a strong oscillator.
ELO
ES
|Es + |ELO| eif |2
|ELO|2 + 2 |ELO| cos f Re Es +
+ 2 |ELO| sin f Im Es + ...
By varying phase of local oscillator, can measure different quadratures
(Re Es , Im Es , or anything in between) just by measuring intensity.
Note: not so different from RF; your oscilloscope has a local oscillator too.
E
E2 (think p)
E1 (think x)
E1
E2
E2
E2 (think p)
E1 (think x)
n
E2 (think p)
E1 (think x)
Back to Bohr & Einstein
(and Young and Taylor,...)
So, does a single photon exhibit
interference, or not?!
|1>
Entangled state
|01> + |10>; no
definite number
in either arm.
Compare EPR: [x1-x2 , p1+p2]
= [x1,p1] - [x2,p2]
=
0
;
though x and p of each are incompatible,
x1-x2 is compatible with p1+p2.
Dirac: two photons never interfere with each other;
a photon only interferes with itself.
Bose-Einstein Condensation
• At low enough temperatures, all bosons
"condense" into the lowest-energy state.
• People like to say they act like a single, coherent
quantum wave.
• In fact, the formalism says they become a coherent
state, like the field of a laser -- the number of
particles in the condensate becomes uncertain, and
a phase is spontaneously acquired.
• Naturally one of the first goals once atomic BEC
was achieved was to demonstrate this phase, by
interfering Bose condensates.
But wait a second...
• As people here know, atoms interfere "all
the time" without being in a BEC.
• Of course, the interfering paths must
originate at the same source, à la Dirac.
• Contrast Magyar/Mandel – two different
lasers can interfere (as can different radio
stations).
Interference of two BECs
BEC 1
BEC 2
When an atom is detected, I can't tell
which BEC it came from, so I observe
interference.
Distinguishing information?
N atoms
N-1 atoms
Uncertain atom number
• Around 1996, heated arguments about whether or
not the number of atoms could be quantummechanically uncertain.
• Condensed-matter theorists said yes: the theory
works for superfluid Helium, so why not?
• Atomic-physics experimentalists said no: unlike
photon number, atom number is conserved. I put
some number of atoms in my vacuum chamber
(even if I can't quite count them all), and each
atom goes somewhere...
So is there interference?
(I forgot to say: of course the experimentalists were right.
Sort of.)
Sneaky, sneaky argument:
Of course I can't see interference, because interference measures
the phase difference f, which is completely uncertain.
So what?? If a particle has completely uncertain position and I measure
X, I still get an answer don't I? Only the answer is random.
Once I measure the phase difference f, if I measure it again, I'll
get the same answer.
Every time I detect an atom, its position relative to the hypothetical
two-slit pattern gives me more information about f.
After many atoms are detected, the phase has been rather well measured,
and subsequent atoms continue the pattern.
Andrews et al., Science 275 , 637-641 (1997)
Each time the experiment is performed, an interference pattern
is observed, but each time it has a different, random phase.
(If the two condensates are formed "independently"; more interesting issues...)
Collapse Picture
(cf. Munro et al.)
N
N
If an atom is detected at the midpoint (or any
other peak of the pattern), it had equal, in-phase
amplitudes to come from either BEC.
This collapses the state of the system:
( )( )
N
N-1
N-1
N
+
Note the similarity with the one-photon interferometer: now the
BECs are left in a state like |0,1> + |1,0> in which atom number is uncertain.
Spontaneous coherence
So, where does this phase come
from?
• 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: a mixture of coherent
states with random phase f turns out to be exactly the same density matrix
as a mixture of (Poisson-distributed) number states n, with no phase at all.
• Ongoing arguments...
Superselection rules.... shared reference frames... shared local oscillators... synchronized
lasers...
Does a radio transmitted have a phase, or is that also only relative?
Does anything in the universe? How would we know?
You can only measure phase via a
reference
• Direct detection measures aa, particle number.
• A field is a + a or a – a... to measure this operator, one needs to put it
inside a von Neumann Hamiltonian. But it doesn't obey conservation
of number!
• 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.
• Compare "superselection rules" – superpositions of different charge
states aren't supposed to exist.
(What about different mass, or energy states?)
• 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.]
"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?
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.
More complete conclusion:
Up to you...
Some references
Visit http://home.t-online.de/home/gerd.breitenbach/gallery/
for info & pictures about squeezed-states and their measurement.
There exist uncountable textbooks & review articles on quantized light;
see for instance Loudon; Walls & Milburn; Scully & Zubairy.
R.J. Glauber, Phys Rev 130, 2529 (1963);
Proc. 1963 Les Houches summer school on quantum electronics
J.F. Clauser, PRA 6, 49 (1972): how to prove that photons are necessary?
Grangier, Roger, Aspect, Europhys. Lett. 1, 173 (1986): single-photon interference
Andrews et al., Science 275 , 637-641 (1997): Ketterle's BEC interference experiment
Magyar G and Mandel L 1963 Nature 198 255 : interference of different sources
Klaus Mølmer, Phys Rev A 55, 3195 (1997) -- "Optical coherence: a convenient fiction"
T. Rudolph & B.C. Sanders, PRL 87, 077903 (2001): more on optical coherence
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 "