Download Seventh lecture, 18.11.03 (Tunneling times and introduction to weak

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