Let's Make a Quantum Deal!

Let's Make a Quantum Deal!

• The 3-box problem• Another case where airtight classical reasoning yields seemingly contradictory information• Experimental consequences of this information• Actual experiment!

• Weak measurements shed light on Hardy's Paradox as well• "Weak probabilities" obey all the constraints we expected in that example.• There is no contradiction, because if negative "probabilities" are accepted, all the joint probabilities can be constructed.

• Which-path experiments• Recall the debate: is momentum disturbed or not? How could one determine whether or not momentum had changed?• Weak measurement predictions share some of the properties claimed by Scully et al. and some of those claimed by Walls et al.• More negative probabilities needed...

• Conclusion• These "probabilities" are real, measurable things...• ...but what in the world are they?

25 Nov 2003(some material thanks to Kevin Resch,

Reza Mir, Howard Wiseman,...)

Weak measurements: from the 3-box problem to Hardy's Paradox to the which-path debate

Recall principle of weak measurements...

Hint=gApx

System-pointercoupling

x

Initial State of Pointer

x

Final Pointer Readout

By using a pointer with a big uncertainty, one canprevent entanglement ("collapse").

By the same token, no single event provides much information...

x x

x

Initial State of Pointer

x

Final Pointer Readout

x x

But after many trials, the centre can be determinedto arbitrarily good precision...

Predicting the past...

A+B

What are the odds that the particlewas in a given box (e.g., box B)?

B+C

A+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?

A + B= X+B+Y

B + C =X+B-Y

X Y

A redefinition of the redefinition...

X + B'= X+B+Y

X + C' =X+B-Y

X Y

So: the very same logic leads us to conclude theparticle was definitely in box X.

What does this mean?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?

But back up: is there any physical sense in which this is true?What if you try to observe where the particle is?

The 3-box problem: weak msmtsPrepare 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?

Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed?

PA = < |A><A| >wk = (1/3) / (1/3) = 1PB = < |B><B| >wk = (1/3) / (1/3) = 1

PC = < |C><C|>wk = (-1/3) / (1/3) = 1.if

iAfA =w

[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

Remember that test charge...

e-

e-

e-

e-

Aharonov's N shutters

PRA 67, 42107 ('03)

The implementation – A 3-path interferometer

(Resch et al., quant-ph/0310091)

Diode Laser

CCDCamera

MS, A

MS, C

Spatial Filter: 25um PH, a 5cm and a 1” lens

BS1, PBS

BS2, PBSBS3, 50/50

BS4, 50/50

Screen

GP C

GP B

GP A/2

/2

PD/2

The pointer...

• Use transverse position of each photon as pointer

• Weak measurements can be performed by tilting a glass optical flat, where effective

gtFlat

xint pAAgH =

Mode A

cf. Ritchie et al., PRL 68, 1107 ('91).

The position of each photon is uncertain to within the beam waist...a small shift does not provide any photon with distinguishing info.But after many photons arrive, the shift of the beam may be measured.

0.4

0.6

0.8

1

1.2

1.4

100120140160180200220

Intensity (arbitrary units)

Pixel Number

Rails A and B (no shift)

Rail C(pos. shift)

A+B–C(neg. shift!)

A negative weak valuePerform weak msmt on rail C.

Post-select either A, B, C, or A+B–C.

Compare "pointer states" (vertical profiles).

[There exists a natural optical explanation for this classical effect – this is left as an exercise!]

Data for PA, PB, and PC...

-2

-1

0

1

2

-3 -2 -1 0 1 2 3Post-selected state displacement

(Units of RMS Width)

Displacement of Individual Rail (Units of RMS Width)

Rail C

Rails A and B

WEAK STRONGSTRONG

Is the particle "really" in 2 places at once?

• If PA and PB are both 1, what is PAB?• For AAV’s approach, one would need an

interaction of the formxint pBBAAgH =

OR: STUDY CORRELATIONS OF PA & PB...

- if PA and PB always move together, thenthe uncertainty in their difference never changes.

- if PA and PB both move, but never together,then (PA - PB) must increase.

Practical Measurement of PAB

yBxAint pBBgpAAgH +=

We have shown that the real part of PABW can be extracted from such correlationmeasurements:

( ) )BRe(Ptgg

xy2PRe BW

*AW2

BAABW −=

Use two pointers (the two transverse directions)and couple to both A and B; then use theircorrelations to draw conclusions about PAB.

Non-repeatable data which happen to look the way we want them to...

no correlations(PAB = 1)

exact calculation

anticorrelatedparticle model

And a final note...

The result should have been obvious...

|A><A| |B><B|= |A><A|B><B|

is identically zero becauseA and B are orthogonal.

Even in a weak-measurement sense, a particlecan never be found in two orthogonal states atthe same time.

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 ParadoxD- e+ was in

D+D- both were in?

But … if they wereboth in, they should

have annihilated!

BS1-

e-

BS2-

O-

C-D-

I-

BS1+

BS2+

I+

e+

O+

D+C+

W

Probabilities e- in e- out

e+ in 1

e+ out 0

1 0

0 1

1 1

But what can we say about where the particles were or weren't, once D+ & D– fire?

Upcoming experiment: demonstrate that "weakmeasurements" (à la Aharonov + Vaidman) willbear out these predictions.

PROBLEM SOLVED!(?)

Quantum Eraser(Scully, Englert, Walther)

Suppose we perform a which-path measurement using amicroscopic pointer, z.B., a single photon deposited intoa cavity. Is this really irreversible, as Bohr would have allmeasurements? Is it sufficient to destroy interference? Canthe information be “erased,” restoring interference?

Which-path measurements destroy interference (modify p-distrib!)

How is complementarity enforced?

The fringe pattern (momentum distribution) is clearly changed –yet every moment of the momentum distribution remains the same.

The debate since then...

Why the ambiguity?

Weak measurements to the rescue!

To find the probability of a given momentum transfer,measure the weak probability of each possible initialmomentum, conditioned on the final momentum observed at the screen...

Convoluted implementation...

Glass plate in focalplane measures P(pi) weakly (shiftingphotons along y)

Half-half-waveplatein image plane measurespath strongly

CCD in Fourier plane measures<y> for each position x; thisdetermines <P(pi)>wk for eachfinal momentum pf.

Calibration of the weak measurement

A few distributions P(pi | pf)

Note: not delta-functions; i.e., momentum may have changed.Of course, these "probabilities" aren't always positive, etc etc...

EXPERIMENT THEORY

(finite width due to finitewidth of measuring plate)

The distribution of the integrated momentum-transfer

EXPERIMENT

THEORY

Note: the distribution extends well beyond h/d.

On the other hand, all its momentsare (at least in theory, so far) 0.

CONCLUSIONS• Weak-measurement theory can predict the output of meas-urements without specific reference to the measurement technique.

• They are consistent with the surprising but seemingly airtight conclusions classical logic yields for the 3-box problem and for Hardy's Paradox.

• They also shed light on tunneling times, on the debate over which-path measurements, and so forth.

• Of course, they are merely a new way of describing predictions already implicit in QM anyway.

• And the price to pay is accepting very strange (negative, complex, too big, too small) weak values for observables (inc. probabilities).

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 et al, PRL 60, 1351 (1988)Ritchie, Story, & Hulet, PRL 66, 1107 (1991)Wiseman, PRA 65, 032111Brunner et al., quant-ph/0306108Resch 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

Hardy's Paradox:Hardy, PRL 68, 2981 (1992)Aharonov et al, PLA 301, 130 (2001).

Which-path debate:Scully et al, Nature 351, 111(1991)Storey et al, Nature 367 (1994) etcWiseman & Harrison, N 377,584 (1995)Wiseman, PLA 311, 285 (2003)