•
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 V
0
; 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?
n
1 n
2 .......
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
e
x
B x f = w
T
But in fact: x
= z
+
-z z
+
-z f z
= w
T z f = w
T y
Which is "the" tunneling time?
T y
? T z
? T x
2 = T y
2 + T z
2 ?
Disturbing feature... T y is still nearly insensitive to d, and often < d/c.
Büttiker therefore preferred T x
... 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 B 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 f is the phase difference accumulated between
 and

.
couples to s z
• 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 180 o .

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
H int
=gAp x
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
H int
=gAp x
System-pointer coupling x x
Poor resolution on each shot.
Negligible back-action (system & pointer separable)
Strong:
Weak:

Bayesian Approach to Weak Values
A
= w f A i f i
Note: this is the same result you get from actually performing the QM calculation (see A&V).

Ritchie, Story, & Hulet 1991
Very rare events may be very strange as well.

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...
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?
A
= w f A i f i
P
A
P
B
= < |A><A| > wk
= < |B><B| > wk
= (1/3) / (1/3) = 1
= (1/3) / (1/3) = 1
P
C
= < |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...
e e 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