Unusual Interactions of Pre- and
Post-Selected Particles
1
Y A K I R A H A R O N O V 1, E L I A H U C O H E N 1*
1School
of Physics and Astronomy, Tel Aviv University, Tel-Aviv 69978, Israel
*[email protected]
ICFP 2012, Greece
14.06.12
ABL
2
 In their 1964 paper Aharonov, Bergmann and
Lebowitz introduced a time symmetric quantum
theory.
 By performing both pre- and postselection (  (t ') and
(t '') respectively) they were able to form a symmetric
formula for the probability of measuring the
eigenvalue cj of the observable c:
P(cj ) 
(t '') cj cj  (t ')
 (t '') c
i
i
ci  (t ')
TSVF
3
 This idea was later widened to a new formalism of
quantum mechanics: the Two-State-Vector
Formalism (TSVF).
 The TSVF suggests that in every moment,
probabilities are determined by two state vectors
which evolved (one from the past and one from the
future) towards the present.
 This is a hidden variables theory, in that it completes
quantum mechanics, but a very subtle one as we
shall see.
Strong Measurement
4
?
efficient detectors
(very low momentum uncertainty)
Stern-Gerlach magnet
Weak Measurement - I
5
inefficient detectors
(high momentum uncertainty)
?
?
Stern-Gerlach magnet
Why Weak Measurement?
6
s
ns
s


0
s
ns
ns
[ i ,  j ]  2i ijk k
?
n
?
Weak Measurement - II
7
 The Weak Measurement can be described by the Hamiltonian:
H (t ) 

N
g (t ) AsPd
 In order to get blurred results we choose a pointer with zero
expectation and
 

N
standard deviation.
 In that way, when measuring a single spin we get most results

  , but when summing up the N/2↑
within the wide range
N
results, most of them appear in the narrow range 
N / 2  N / 2
agreeing with the strong results when choosing    .
Weak Value
8
 The “Weak Value” for a pre- and postselected (PPS)
ensemble:
 fin A  in
A
w

 fin  in
 It can be shown that when measuring weakly a PPS
ensemble, the pointer is displaced by this value:
 fin (Qd )  e
 (i / ) A
w
Pd
 in (Qd )   in (Qd  A w )
No counterfactuals!
The Weak Interaction
9
 We generalize the concept of weak measurement to
the broader “weak interaction”.
 It can be shown that the Hamiltonian
H (1, 2)  H1 (1)  H 2 (2)  V (1, 2)  H 0 (1, 2)  V (1, 2) , when
particle 1 is pre- and post- selected, results, to first
order in  , in the weak interaction :
VW (2)  
 20 V  10
 2 1
Which Path - I
10
Can we outsmart the “which path” uncertainty?
Interference
0%
100%
Which Path - II
11
Can we outsmart the “which path” uncertainty?
50%
50%
50%
Which Path - III
12
Can we outsmart the “which path” uncertainty?
50%
50%
50%
Which Path - IV
13
Can we outsmart the “which path” uncertainty?
Interference
0%
100%
50%
Which Path Measurement Followed by
Interference
14
50%
50%
Lf
Rf
L2
R2
100%
λ2/2
L1
- λ2/2
R1
50%
50%
L0
L
1
w

R f  L1 L 0
R f L0

0.5
1 ,
0.5
A Quantum Shell Game
15
 in
1

(A  B  C
3
i  i i
A
 fin
W
1
B
W
1
C
W
 1
1

(A  B C
3
Aharonov Y., Rohrlich D., “Quantum Paradoxes”, Wiley-VCH (2004)
Tunneling
16
p2
H
  ( x)
2m
 in ( x)   e
 x
 fin ( x)  (  )
2
Where   
1/ 4
2
e
 ( x  x0 ) / 2 2
/ m , x0   3m /
 Ek  
2
2
2m
Aharonov Y., Rohrlich D., “Quantum Paradoxes”, Wiley-VCH (2004)
2
The Correspondence Principle
17
 Every quantum system is described by quantum numbers.
 When they become large, the system approaches its
classical limit.
 This correspondence was first described by Bohr in the
1920’ regarding the atom, but has a broader meaning.
 For example, the appropriate quantum number for the
1
E

mA2 2
classical energy of an oscillator
1 m A2 1 1
n
 
 
 4.74 1033 
 2
2
2 2
E
2
High excitations
Challenging The Correspondence Principle
18
 Let
1
H  ( x2  p2 )
2
1/ 4
2



exp[

(
x

x
)
/ 2)]
 We Pre- and Post-select : i
0
1/ 4
2
and  f   exp[( x  x0 ) / 2)] where x0  1 ,
-x0
+x0
 The weak values can be calculated to be:
x  0 and pw  ix0 .
w
[
x
i
, p]   0
x0
x0
.
Challenging The Correspondence Principle
19
 We argue that using the idea of weak interaction, this
weird result gets a very clear physical meaning.
2
 When interacting with another oscillator  t  exp( p )
through Hint   p1 p2 g (t ) , it changes its momentum
rather then its position:
exp(i   p1 p2 g (t )dt ) exp( p2 2 )  exp( x0 p2 ) exp(  p2 2 ) 
 exp( x0 p2 ) exp( p2 2 )  exp[( p2   x0 / 2) 2 ]exp( 2 x0 2 / 4)
Summary
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 Weak measurements enable us to see and feel the TSVF.
 They also present the uniqueness of quantum mechanics.
 By using them we overcome the uncertainty principle in
a subtle way and enjoy both which-path measurement
and interference.
 Weak values, as strange as they are, have physical
meaning:


In case of many measurements followed by proper postselections:
Weak interaction or deviation of the measuring device.
Otherwise: An error due to the noise of the measurement device.
 In that way we avoid counterfactuals and obtain
determinism in retrospect.
Acknowledgements
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 Dr. Avshalom C. Elitzur
 Paz Beniamini
 Doron Grossman
 Shay Ben-Moshe
Questions