Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system Lev Vaidman...
-
Upload
lorena-knight -
Category
Documents
-
view
215 -
download
0
Transcript of Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system Lev Vaidman...
Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system
Lev Vaidman
12.7.2015
Conditioned evolution
Weak values and weak measurements
Evolution of pre- and post-selected system
Plan
Two state-vector
Past of a quantum particle
3-box paradox
Correlations of uncorrelated pre- and post-selected particles
( )tUnitary evolution
no click
Non-unitary evolution
no click
Non-unitary evolution
no click
Non-unitary evolution
Collapse of the wave function
What is the evolution conditioned on nondetection?
no click
no click( ) ?t
What is the evolution conditioned on nondetection?
What was time evolution before the particle was detected, given that it was detected?
What is the evolution conditioned on detection?
What is the evolution conditioned on detection?
What was the interaction Hamiltonian for (weak) interaction with other systems?
What is the evolution conditioned on detection?
What is the evolution conditioned on detection?
What was the interaction Hamiltonian for (weak) interaction with other systems?
What was the interaction Hamiltonian for (weak) interaction with other systems?
Where were the pre- and post-selected photons?
B
A
Asking photons where have they been
POWER SPECTRUM
fB0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
UP DOWNSIGNAL I I
A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Phys. Rev. Lett. 111, 240402 (2013)
POWER SPECTRUM
fA fB0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
B
A
Asking photons where have they been
[ ]f Hz
UP DOWNSIGNAL I I Photons were on the paths they could pass
B
A
Asking photons where have they been
POWER SPECTRUM
fA fB0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
UP DOWNSIGNAL I I Photons were on the paths they could pass
B
A
Asking photons where have they been
POWER SPECTRUM
fA fB0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
UP DOWNSIGNAL I I Photons were on the paths they could pass
C
F
E
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f HzB
A
Asking photons where have they been
Photons were on the paths they could pass
Asking photons where have they been
B
C
A
F
E
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
Photons were on the paths they could not pass!
How to explain this?
The two-state vector formalism
t
P 1
1t
2t
P 1
The two-state vector
t
t
P 1
1t
2t
P 1
The two-state vector
t
t
1t P 1
t
2t P 1
t
t
t
t
t
t
t
The two-state vector is a complete description of a system at time t
t
2t P 1
1t P 1
?
3tThe two-state vector is what we can say now ( )about the pre- and post-selected system at time t
3t
So, what can we say?
t
P 1
1t
2t
P 1
The Aharonov-Bergmann-Lebowitz (ABL) formula:
?C
2
2
PProb( )
Pi
C c
C ci
C c
described by the two-state vector:
Strong measurements performed on a pre- and post-selected system
t
P 1
1t
2t
P 1
?C
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected systemdescribed at time t by the two-state vector
w
CC
w wwA B A B
w wwAB A B
Weak value of a variable C of a pre- and post-selected systemdescribed at time t by the two-state vector
The outcomes of weak measurements are weak values
2 2
x yy x
y x
wy x y x
t
1tx
?
1x
1y y
2t
w
CC
2x y
The weak value
t
2t
1t
3t
w
CC
If the pre- and post-selected system is coupled to other systems through C, then its coupling at time t is described (completely) by the weak value wC
intˆ ˆH gCB
ˆ ˆCig Bdte ˆ ˆ1 Cig Bdt 1C
ig Bdt
ˆ
wigC Bdte
intˆ
wH g BC intˆ ˆH gCB
ˆwigC Bdte
Effective non-Hermitian Hamiltonian
y z x
n z wy x
i
1tx
1x
1y y
2t
1
N
nn
gH
N
1y x x
n x wy x
1y y x
n y wy x
1
N
eff nn w
gH
N
x y zg i
eff x y zH g i
Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, and L. Vaidman, PRL 77, 983-987 (1996)
Asking photons where have they been
B
C
A
F
E
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
Photons were on the paths they could not pass!
How to explain this?
B
A
The two-state vector formalism explanation
B
A
The two-state vector formalism explanation
B
APOWER SPECTRUM
fA fB0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
The two-state vector formalism explanation
B
C
A
F
E
D
The two-state vector formalism explanation
B
C
A
F
E
D
The two-state vector formalism explanation
B
C
A
F
E
D
The two-state vector formalism explanation
B
C
A
F
E
D
The two-state vector formalism explanation
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
w
CC
) 0A w (P ) 0B w (P ) 0C w (P
) 0E w (P ) 0F w (P
B
C
A
F
E
D
The two-state vector formalism explanation
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
w
CC
) 0A w (P ) 0B w (P ) 0C w (P
) 0E w (P ) 0F w (P
B
C
A
F
E
D
The two-state vector formalism explanation
1
3A B C
B
C
A
F
E
D
The two-state vector formalism explanation
1
3A B C
B
C
A
F
E
D
The two-state vector formalism explanation
1
3A B C
1
3A B C
) 1AA w
P
(P
) 1BB w
P
(P
) 1CC w
P
(P
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f Hz
K.J. Resch, , J.S. Lundeen, , A.M. Steinberg , PLA 324, 125 (2004)
Experimental realization of the quantum box problem
t
2t
1t
3t
Where is the ball?
?
1
3A B C
1
3A B C
A B C
Aharonov and Vaidman, JPA 24, 2315 (1991)
Aharon and Vaidman, PRA 77, 052310 (2008)
The 3-boxes paradox Vaidman, Found. Phys. 29, 865 (1999)
The three box paradox
t
2t
1
3A B C
1t 1
3A B C
3t
It is in
A B C
Aalways !
The three box paradox
t
2t
1t
3t
It is always in
B
1
3A B C
1
3A B C
A B C
Two useful theorems:
1 1A A w P P
The three box paradox
1 1B B w P P
1 1A B C A B C w P P P P P P
1A B Cw w w P P P
1C w P
t
2t
1t
1
3A B C
1
3A B C
A B C
Prob( ) 1i w iC c C c For dichotomic variables:
Prob( ) 1w i iC cC c
Correlation between separable pre- and post-selected particles
( , )corr A B AB A B
t
1t 1x
1y 2t
1x
1y
Aharonov and Cohen, arXiv:1504.03797
A B
A z wi B z w
i
( , )A z B z A z B z A z B zcorr 0A z B z
?A z B z
A z B z A z B zw w w i i 1 1A z B z
( , ) 1A z B zcorr
A z B z A z B z
Failure of the product rule for pre- and post-selected particles
t
1t
1x 2t 1y A B
1A y
1
2
1B x 1A y B x
,A a B b AB ab
Pre- and post-selected quantum systems are described best by two-state vector and weak values of observables
w
CC
Evolution of systems coupled to pre- and post-selected quantum systems is described by non-Hermitian Hamiltonians
Conclusions
B
C
A
F
E
D
The one-state vector formalism explanation
C
F
E
POWER SPECTRUM
fA fB fC fE fF0
280 290 300 310 320 330
1
0.8
0.6
0.4
0.2
[ ]f HzB
A
Photons:Wheeler is right!
jj C cj
w i
c
C c
P
2
2Prob( ) 1 , 0i
j
j
C c
i C c
C cj
C c j i
PP
P
ii C ci
C c P
Prob( ) 1i w iC c C c
For dichotomic variables Prob( ) 1w i iC cC c
1 21 21
C c C cw
c cC c
P P2 1
I - C c C c P P
1 1
1 1 2 1C c C cc c c
P P1 1
1 21 1C c C cc c
P P1 1C c
P
w
CC
2
2Prob( )
i
C c
C ci
C c
P
P
Connection between strong and weak measurements
-i jC c C c
j i
P I P I
Pointer probability distribution
? t
1tx
1x
1y y
2t
1.4w !
strong
weak
Weak measurement of
Pre-selection 1x
2x y
int ( ) MDH g t P 2
22( )Q
MDin Q e
Post-selection 1y
The outcomes of weak measurements are weak values
Pointer probability distribution
Weak Measurement of
t
1t
20
1x i
i
1i x
20 particles pre-selected 1x 20 particles post-selected 1y
1i y 20
1i
iy
20
1
1
20 ii
20
1
1
20 ii
Robust weak measurement on a pre- and post-selected single system
The system of 20 particles
20
1
11.4
20 ii w
!
strong
weak
2t