BECs, lasers, and other clocks. Some remarks on time measurement (&c.)...

1. 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

2. 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?

3. N-1 atoms N-1 atoms N atoms (no phase) N atoms (no phase) measured phase Two indistinguishable paths (RR&TT) to 2-atom detection Two complementary pictures Studying Bose Condensates as a whole Studying individual atoms 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 aa does not depend on position. 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. 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.

4. 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. Where did the phase come from?

5. 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)... Funny realisation Does a radio transmitter have a phase, or is that also only relative? Does anything in the universe? How would we know?

6. The density matrix of a laser

7. 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?] You can only measure phase via a reference

8. Interaction-free measurements Another example of interferenceas measurement:

9. "Interaction-Free Measurements" D C BS2 BS1 (AKA: The Elitzur-Vaidman bomb experiment) .....what else does interference allow us to "measure"? 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 that they will not blow up when triggered. Is there any way to identify the working bombs (or some of them) without blowing them up? 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 the path – although at the same time, you know that no particle encountered the bomb. Did the bomb disturb the "phase of the vacuum"?

10. 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.

11. Hardy's Paradox D+ D- C+ C- BS2+ BS2- I+ I- O- O+ W BS1+ BS1- e- e+ D+ e- was in D- e+ was in D+D- ? But … if they were both in, they should have annihilated!

12. 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.)

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

14. Two Hong-Ou-Mandel dips

15. n1 n2 ....... Very little light is transmitted through a tunnel barrier (a quarter-wave-stack dielectric mirror, in our experiment). How can this be?

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

17. z y e- x B x f = wT e- z z But in fact: + = + fz = wTz -z -z x f = wTy Larmor Clock (Baz', Rybachenko, and later Büttiker) 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!

18. 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 .

19. 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)