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 phaseTwo 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 differencebetween 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| ei e-it

= E1 cos t + E2 sin t

Cf. harmonic oscillator, x(t) = |x0| ei e-it

= x(0) cos t + p(0) sin t

Energy = |E1|2 + |E2|2

or x2 + p2 (m2x2/2 + p2/2m, of course)

Quantized energy = (n+1/2) hw.Equally spaced energy levels correspond to different numbers of photons.

x px

A wave packet about to oscillate

x px

A wave packet in oscillation

x px

A wave packet in oscillation

E1 (think x)

E2 (think p)

Orbit in phase space

Exactly the classical phase-space trajectory, with an extra circle of uncertainty.

x px

A low-momentum wave packet

x px

A low-momentum wave packet

Behaves like a collection of particles starting at rest at different positions...

x px

A well-localized wave packet

x px

A well-localized wave packet

Behaves like a collection of particles starting at x=0 with different momenta

x px

"Just right" (the ground state)

Perfect balance:As much force to the left as to the right, andas 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)

x px

Also "just right" (1st excited state)

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).

Phase-space picturecomplex E = |E| ei e-it

= E1 cos t + E2 sin t

|E|

E1 (think x)

E2 (think p)

The vacuum isn't empty:

E1 (think x)

E2 (think p)

A phase-space distributionfor a 1-photon state

E1 (think x)

E2 (think p)

A "coherent" or "quasiclassical" state.

minimum phase-space area,as for X and P of a particle.

x px

A wave packet about to oscillate

The quasiclassical or "coherent" state: follows a classical orbit,maintaining the same minimum-uncertainty product in x and pas in the ground state.

p

The coherent state

(nice pictures stolen from Gerd Breitenbach's web page)

Electric field of a coherent stateE

t

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.

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.

|Es + |ELO| ei |2 |ELO|2 + 2 |ELO| cos Re Es +

+ 2 |ELO| sin Im Es + ...

ELO

ES

E

E1

E2

E1 (think x)

E2 (think p)

E2

n

E1 (think x)

E2 (think p)

E1 (think x)

E2 (think p)

Back to Bohr & Einstein(and Young and Taylor,...)

So, does a single photon exhibitinterference, or not?!

|1>Entangled state|01> + |10>; no definite numberin 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 tellwhich BEC it came from, so I observeinterference.

Distinguishing information?

N atoms N-1 atoms

Uncertain atom number

• Around 1996, heated arguments about whether or not the number of atoms could be quantum-mechanically 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 measuresthe phase difference , which is completely uncertain.

Once I measure the phase difference , if I measure it again, I'llget the same answer.

Every time I detect an atom, its position relative to the hypotheticaltwo-slit pattern gives me more information about .

After many atoms are detected, the phase has been rather well measured,and subsequent atoms continue the pattern.

So what?? If a particle has completely uncertain position and I measureX, I still get an answer don't I? Only the answer is random.

Andrews et al., Science 275 , 637-641 (1997)

Each time the experiment is performed, an interference patternis 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.)

If an atom is detected at the midpoint (or anyother peak of the pattern), it had equal, in-phaseamplitudes to come from either BEC.

This collapses the state of the system:

N-1 NN N-1

+) )((Note the similarity with the one-photon interferometer: now theBECs are left in a state like |0,1> + |1,0> in which atom number is uncertain.

NN

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... synchronizedlasers...

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 aa, 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.]

Problem:Consider a collection of bombs so sensitive that

a collision with any single particle (photon, electron, etc.)is guarranteed to trigger it.

Suppose that certain of the bombs are defective,but differ in their behaviour in no way other than thatthey will not blow up when triggered.

Is there any way to identify the working bombs (orsome of them) without blowing them up?

"Interaction-Free Measurements"(AKA: The Elitzur-Vaidman bomb experiment) .....what else does interference allow us to "measure"?

BS1

BS2

DC

Bomb absent:Only detector C fires

Bomb present:"boom!" 1/2 C 1/4 D 1/4

If detector D fires, you can say with certainty that the bomb was blocking thepath – 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 exactlydid the bomb interact with, if not our particle?

BS1-

e-

BS2-

O-

C-D-

I-

BS1+

BS2+

I+

e+

O+

D+C+

W

Outcome Prob

D+ and C- 1/16

D- and C+ 1/16

C+ and C- 9/16

D+ and D- 1/16

Explosion 4/16

Hardy’s Paradox

D- e+ was in

D+D- ?

But … if they wereboth in, they shouldhave annihilated!

D+ e- was in

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.

Andrews et al., Science 275 , 637-641 (1997): Ketterle's BEC interference experiment

Klaus Mølmer, Phys Rev A 55, 3195 (1997) -- "Optical coherence: a convenient fiction"

R.J. Glauber, Phys Rev 130, 2529 (1963);Proc. 1963 Les Houches summer school on quantum electronics

T. Rudolph & B.C. Sanders, PRL 87, 077903 (2001): more on optical coherence

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

Magyar G and Mandel L 1963 Nature 198 255 : interference of different sources

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 "