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Introduction to tunneling times and to weak measurements • How does one actually measure time ? (recall: there is no operator for time) • 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. • How does one discuss subensembles in quantum mechanics? • Weak measurement • How can the spin of a spin-1/2 particle be found to be 100? • How can a particle be in two places at once? • Where is a particle when it's in the forbidden region? 18 Nov 2003 How Long Does Tunneling Take? We frequently calculate the tunneling rate, e.g., in a two-well system. But how long is actually spent in the forbidden region? Classically, time diverges as E approaches V0; the "semiclassical" time (whatever it means) behaves the same way... Since the 1930s, group-velocity calculations yielded strange results: evanescent waves pick up no phase, so no delay is accumulated inside the barrier? 1980s: Büttiker & Landauer and others propose many other times. What's the speed of a photon? Can tunneling really be nearly instantaneous? Group-delay prediction saturates to a finite value as barrier thickness grows. For thick enough barriers, it would then be superluminal ( < d/c). Recall that the Hong-Ou-Mandel interferometer can be used to compare arrival times of single-photon wavepacket peaks. We used one to check the delay time for a photon tunneling through a barrier. tunnel barrier The results 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? Interaction Times • Büttiker and Landauer: "no law guarrantees that a peak turns into a peak." • Ask instead how long the particle interacted with something in the barrier region • (More relevant to condensed-matter systems anyway) 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. Okay -- so let's consider specific measurements. What is this measurement? 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 . • We want to know the outcome of this von Neumann measurement only for those cases where the particle is transmitted. • "Being transmitted" doesn't commute with "being under the barrier"; is it valid to even ask such post-selected questions? If so, how can you do so without first collapsing the particle to be under the barrier? • Note: this Larmor precession could not determine for certain whether or not the particle had been in the field, or for how long; only on a large ensemble can the precession angle be measured to better accuracy than 180o . Predicting the past ? Standard recipe of quantum mechanics: 1. Prepare a state |i> (by measuring a particle to be in that state; see 4) 2. Let Schrödinger do his magic: |i> |f>=U(t) |i>, deterministically 3. Upon a measurement, |f> some result |n> , randomly 4. Forget |i>, and return to step 2, starting with |n> as new state. Aharonov’s objection (as I read it): No one has ever seen any evidence for step 3 as a real process; we don’t even know how to define a measurement. Step 2 is time-reversible, like classical mechanics. Why must I describe the particle, between two measurements (1 & 4) based on the result of the first, propagated forward, rather than on that of the latter, propagated backward? Conditional measurements (Aharonov, Albert, and Vaidman) AAV, PRL 60, 1351 ('88) Prepare a particle in |i> …try to "measure" some observable A… postselect the particle to be in |f> i i Measurement of A f f Does <A> depend more on i or f, or equally on both? Clever answer: both, as Schrödinger time-reversible. Conventional answer: i, because of collapse. Reconciliation: measure A "weakly." Poor resolution, but little disturbance. "weak values" A (von Neumann) Quantum Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx System-pointer coupling x x Well-resolved states System and pointer become entangled Decoherence / "collapse" Large back-action A Weak Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx x System-pointer coupling x Poor resolution on each shot. Negligible back-action (system & pointer separable) Strong: Weak: Bayesian Approach to Weak Values Aw = f Ai f i Note: this is the same result you get from actually performing the QM calculation (see A&V). Very rare events may be very strange as well. Ritchie, Story, & Hulet 1991 Weak measurement & tunneling times Conditional probability distributions A problem... These expressions can be complex. Much like early tunneling-time expressions derived via Feynman path integrals, et cetera. A solution... Conditional P(x) for tunneling What does this mean practically? QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Predicting the past... A+B B+C What are the odds that the particle was in a given box (e.g., box B)? It had to be in B, with 100% certainty. Consider some redefinitions... In QM, there's no difference between a box and any other state (e.g., a superposition of boxes). What if A is really X + Y and C is really X - Y? Then we conclude that if you prepare in (X + Y) + B and postselect in (X - Y) + B, you know the particle was in B. But this is the same as preparing (B + Y) + X and postselecting (B - Y) + X, which means you also know the particle was in X. If P(B) = 1 and P(X) = 1, where was the particle really? The 3-box problem Prepare a particle in a symmetric superposition of three boxes: A+B+C. Look to find it in this other superposition: A+B-C. Ask: between preparation and detection, what was the probability that it was in A? B? C? Aw = f Ai f i PA = < |A><A| >wk = (1/3) / (1/3) = 1 PB = < |B><B| >wk = (1/3) / (1/3) = 1 PC = < |C><C|>wk = (-1/3) / (1/3) = -1. Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed? [Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)] Remember that test charge... ee- e- e- Aharonov's N shutters PRA 67, 42107 ('03) Some references 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) Weak measurements: Aharonov & Vaidman, PRA 41, 11 (1991) Aharonov, Albert, & Vaidman, PRL 60, 1351 (1988) Ritchie, Story, & Hulet, PRL 66, 1107 (1991) Wiseman, PRA 65, 032111 Brunner et al., quant-ph/0306108 Resch and Steinberg, quant-ph/0310113 The 3-box problem: Aharonov et al J Phys A 24, 2315 ('91); PRA 67, 42107 ('03) Resch, Lundeen, & Steinberg, quant-ph/0310091