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 theclasses do have well-counted numbersof students, interference may still beobserved. How can this be?

Two complementary picturesStudying Bose Condensates as a whole Studying individual atoms

Each interference experiment is a single measurement – an approximate measurement of relative phase.

Each pair of condensates begins with an unknown relative phase, but the interferogram forces it to collapse to a particular value.

All the individual atoms "agree" as to the phase of the interferogram; they are really part of a single measurement.

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 expected

atom number aa does not depend on position.

Pairs of atoms, on the other hand, can

interfere à la HOM. In other words, aa

a amay 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.

N atoms(no phase)

N atoms(no phase)

measured phase

N-1 atoms

N-1 atoms

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 aa, particle number.• A field (ae-it + 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 interferenceas measurement:

Interaction-free measurements

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?

It destroyed the relative phase between two parts of the particle's wave function.

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.

(You decide whether or not you buy this...we'll come back to it in a few weeks.)

Introduction to tunneling times

• 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 time-measurements: to measure the single-photon tunneling time.

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.

Two Hong-Ou-Mandel dips

How can this be?

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

n1n2 .......

Very little light is transmittedthrough a tunnel barrier (aquarter-wave-stack dielectricmirror, in our experiment).

But how that's all classical waves...how fast did a given photon travel?

Larmor Clock (Baz', Rybachenko, and later Büttiker)

B

e- e-

Tx

+

z

-z

But in fact: =

x+

z

-z

Which is "the" tunneling time?

Ty? Tz? Tx2 = Ty

2 + 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!

z

x

y

Ty

zTz

Too many tunneling times!

Questions which seem unambiguous classically may have multiple answers in QM – in other words, different measurements which allyield "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 -˚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 z , 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 is the phase difference accumulated between and .

Various "times": group delay"dwell time"Büttiker-Landauer time

(critical frequency of oscillating barrier)Larmor times (three different ones!)

et cetera...

Some references BEC interference: Andrews et al., Science 275 , 637-641 (1997)

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)

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

Interaction-free measurements:Elitzur and Vaidman Foundations Physics 23, 987 (1993). Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich PRL 74, 4763(1995)

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 "

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)