Download Quantum Disentanglement Eraser

Quantum Disentanglement Eraser
M. Suhail Zubairy
(with G. S. Agarwal and M. O. Scully)
Department of Physics, Texas A&M University, College Station, TX 77843
Quantum Eraser
Marlan O. Scully
Texas A&M University
Girish S. Agarwal
Herbert Walther
M. Suhail Zubairy
Institute for
Quantum Studies
COMPLEMENTARITY
(N. BOHR, 1927)
• Two observables are “COMPLEMENTARY” if precise knowledge of
one of them implies that all possible outcomes of measuring the other
one are equally probable
– POSITION-MOMENTUM
– SPIN COMPONENTS
– POLARIZATION
• TRADITIONALLY
Complementarity in quantum mechanics is associated with
“Heisenberg’s uncertainty relations”
[ x, p]  i  xp  
• However it is a more general concept!!!
Scully, Englert, Walther, Nature 351, 111 (1991).
Newsweek, June 19, 1995, p. 68
Erasing Knowledge!
As Thomas Young taught us two
Hundred years ago, photons interfere.
But now we know that:
Knowledge of path (1 or 2) is the reason
why interference is lost. Its as if the photon
knows it is being watched.
But now we discover that:
Erasing the knowledge of photon path
brings interference back.
“No wonder Einstein was confused.”
Photon correlation experiment
• Light impinging on atoms at sites 1 and 2.
Scattered photons γ1 and γ2 produce
interference pattern on screen.
• Two-level atoms are excited by laser pulse and
emit γ photons in the a → b transition (Fig. b).
• Atom-scattered field system:
 
1
b1b2 (  1   2 )
2
• The state vector for the scattered photon from
the ith atom:
g k e ik ri
i  
1k
k ( k   )  i / 2
______________________________
M. O. Scully and K. Druhl, PRA 25, 2208 (1982)
1
• Correlation function for the scattered field:
2
1
()
G (r , t )    1   2 E (r , t ) 0
2
1
 0 E (  ) (r , t )  1  0 E (  ) (r , t )  2
2
1
2
 1 (r , t )  2 (r , t )
2
(1)
2
• This is just the interference pattern associated with a
Young’s double-slit experiment generalized to the
present scattering problem. Note that when the γ1 and
γ2 photons arrive at the detector at the ‘same time’,
interference fringes are present.
• Three-level atoms excited by a pulse l1
from |c> → |a> followed by emission of
γ-photons in the |a> → |b> transition
(Fig. c).
• State of the coupled atom-field system:
1 
1
 b1c2  1  c1b2  2
2

• Field correlation function:




1
2
2
1 (r , t )  2 (r , t )  1* (r , t )2 (r , t ) b1c2 c1b2  c.c.
2
1
2
2
 1 (r , t )  2 (r , t )
2
G (1) (r , t ) 2 

• Which path information available - No
fringes
• Can we erase the information (memory)
locked in our atoms and thus recover
fringes?
• Four-level system: a second pulse l2
takes atoms from |b> → |b’>. Decay from
|b’> → |c> results in emission of Φphotons.
• The second laser pulse l2 , resonant with
|b>→ |b’> transition, transfers 100
percent of the population from |b> to |b’>
(second laser pulse - π pulse).
• State of the system after interacting with
the l2 pulse is
2 

1
b1' c2  1  c1b2'  2
2

• The ith atom decays to the |c> state via
the emission of |Φi> photon. State vector
after Φ-emission:
3 
1
c1c2  1 1  2 2
2

•
•
•
•
•
Scattered photons γ and γ result from a → b transition.
Decay of atoms from b′→ c results in Φ photon emission
Elliptical cavities reflect Φ photons onto a common photodetector.
Electrooptic shutter transmits Φ photons only when switch is open.
Choice of switch position determines whether we emphasize
particle (shutter open) or wave (shutter
• closed) nature of γ photon.
• “Delayed choice” quantum eraser!!!
U. Mohrhoff, Am. J. Phys. 64, 1468 (1996)
“Delayed choice” quantum eraser experimental demonstration a
• Pair of entangled photons is emitted from either atom A or atom B by
atomic cascade emission.
• ‘Clicks’ at D3 or D4 provide which path information (No interference
fringes!!)
• ‘Clicks’ at D1 or D2 erase the which path information (Fringes!!)
• absence or restoration of interference can be arranged via an
appropriately contrived photon correlation experiment.
_______________________________________________
a
Kim, Yu, Kulik, Shih, and Scully, PRL 84, 1 (2000)
Experimental considerations
• Distance LA, LB between atoms A, B and detector D0 << distance
between atoms A,B and the beam splitter BSA and BSB where the
which path or both paths choice is made randomly by photon 2
• When photon 1 triggers D0, photon 2 is still on its way to BSA, BSB
• After registering of photon 1 at D0, we look at the subsequent
detection events at D1, D2, D3, D4 with appropriate time delay
• Joint detection events at D0 and Di must have resulted from the
same photon pair
• Interference pattern as a function of D0’ s position for joint counting
rates R01 and R02
• No interference pattern for R03 and R04
Experimental setup a
• The delayed choice to
observe either wave or
particle behavior of the
signal photon is made
randomly by the idler
photon about 7.7 ns
after the detection of
the signal photon
a
Kim, Yu, Kulik, Shih, and Scully, PRL 84, 1 (2000)
Experimental results a
a
Kim, Yu, Kulik, Shih, and Scully, PRL 84, 1 (2000)
U. Mohrhoff, Am. J. Phys. 67, 330 (1999)
Double-slit experiment with atoms
• In the absence of laser-cavity system:  (r )  1  (r )   (r ) i
1
2
2
r is the center-of-mass coordinate and i denotes the
internal state of the atom.
• The probability density for particles on the screen:
• Fringes!!
P( R) 



1
2
2
 1   2   1* 2   2* 1 i i
2
Micromaser Which-Path Detector
•
State of the correlated atomic beam-maser system:  (r ) 
•
Probability density at the screen:
P( R) 
•


1
2
2
 1   2   1* 2 1102 0112   2* 1 0112 1102
2
Because <1102|0112> vanishes,
P( R) 
•
No fringes!!
1
 1 (r ) 1102  2 (r ) 0112  b
2

1
2
2
1   2
2

 b b
Quantum Eraser a
•
Is it possible to retrieve the
coherent interference crossterms by removing (‘erasing’)
the which-path information
contained in the detectors?
• The answer is yes, but how
can that be? The atom is now
far removed from the
micromaser cavities and so
there can be no thought of any
physical influence on the
atom’s center-of-mass wave
function.
a
Scully, Englert and Walther, Nature 351, 111 (1991)
•
•
 (r ) 
1
 1 (r ) 1102  2 (r ) 0112  b d
2
After absorbing a photon, the detector atom, initially in state |d> would be excited to state |e>.
1
 (r )    (r )    (r )   b d
2
with
  (r ) 
•
•
Detector produces
1
 1 (r )  2 (r );   1  1102  0112
2
2

1
  (r ) 0102 e   (r )  d  b
2
i.e., the symmetric interaction couples only to the symmetric radiation state |+>; the
antisymmetric state |-> remains unchanged.
 (r ) 
•
Atomic probability density at the
screen:

1 *
  ( R)  ( R)   * ( R)  ( R)
2
P( R) 

• No interference fringes if the final
state of the detector is unknown!!
• Probability density Pe(R) for finding
both the detector excited and the
2
Pe ( R)    ( R)
atom at R on the screen:

Fringes → solid lines!!
• Probability density Pd(R) for finding
both the detector deexcited and the
atom at R on the screen:
P ( R)  
e
Antifringes → broken line!!


 
1
2
2
 1 ( R)   2 ( R)  Re  1* ( R) 2 ( R)
2


( R)
2
 
1
2
2
 1 ( R)   2 ( R)  Re  1* ( R) 2 ( R)
2


Quantum disentanglement erasersa
• Involves at least three-subsystems A, B, T.
• Entangled state of the AB subsystem:
1
 AB   00 AB  11 AB 
2
• Wave function of whole system:
1
 ABT   00 AB 0 T  11 AB 1 T 
• State of the AB subsystem:
2


1
00
2
AB
00  11
AB
11

• Entanglement of subsystem AB is lost!
• However if one erases the tag information, then the entanglement is
restored.
• Thus entanglement of any two particles that do not interact (directly
or indirectly) never disappears but is encoded in the ancilla of the
system. A projective measurement that seems to destroy such
entanglement could always in principle be erased by uitable
manipulation of the ancilla.
aR.
Garisto and L. Hardy, PRA 60, 827 (1999)
• Entangled state of the AB subsystem:
 AB 

1
00
2
AB
 11
AB

• Wave function of whole system:
• Define

 ABT
1
 00
2
T 
1
0 T 1T
2

AB
0 T  11
AB
1T


• Thus
 ABT 
1 
 1

00
 T
2
 2

 11
AB
AB
  


 1
00
T
 2
 11
AB
• Measurement of the tagging qubit realizes the
entangled state.
AB


•
•
AB system is given by the polarization, T is given by the path of particle 1.
1
At t0
t 
h s s 0 p  h s s 1 p
1
2

1 2
1
1 2
1


1
 h h  h h 
2
•
After passage through polarizing beam splitter (PBS)
•
If we measure the spin of photons at this point, we obtain mixed state
t 
•

1
h
2

 h s s 0 p
0
s1s2
No entanglement!!
1 2
1
To reversibly erase the tagging information at t = 2, we perform the reverse of the
operation of t=1.
Entanglement is restored!!
Cavity QED Implementation
• Consider cavities A and B with |0> state and
an atom 1 in excited state |a> passes
through the two cavities
• After passage through cavity A with
interaction time corresponding to π/2 pulse:
1 

1
0
2
A

a 1 1 A b 1 0
B
• After passage through cavity B with
interaction time corresponding to π pulse:
1 

1
0
2
A
1B1
A
0
B
b
1
Entangled state!!!
• Atom 2 (tagging qubit) now passes through
cavity A
• Atom 2 has dispersive coupling
with cavity A,
ca  v A  
• Effective Hamiltonian:

Η eff  (g 2 / ) aa  c c  a  a a a

• Initially atom 2 is in state
a
 b / 2
• After passage through cavity A,
a quantum phase gate is made
3  e
1

2
-iΗ eff 
2
  1
g 2

 1

i
 0 A1B  1A0 B   a2  0 A1B  e 1A0 B  ,  
 b2 


 2

  2
_____________________________________
A. Rauschenbeutal et. al PRL 83, 5166 (1999).


• Pass atom through classical field with
(1 / 2 ) a2  b2   a2
(1 / 2 ) a2  b2   b2
• Resulting state
(with η=π):
4 

1
0 A1B a 21 A 0 B b
2
2
b
• Entanglement between
cavities A and B is
controlled by atom 2!!
1
• Initial state:
1 

1
0
2
A
1B1
A
0
B
b
3
• After passage through
cavity A:
1
b 1  a 0 0
 
2
3
2
B
3
B
A
• Phase shift:
3 

1
b 3 1 B  ei (t ) a 3 0
2
B
0
A
• After passage through cavity B:


1
b3 ,1B  a3 ,0 B  ei (t ) ( a3 ,0 B  b3 ,1B ) 0 A
2
1
 b3 ,1B 1  ei (t )  a3 ,0 B 1  ei (t ) 0 A .
2
4 










1
 4 (t )  b3 ,1B 1  ei (t )  a3 ,0 B 1  ei (t ) 0 A .
2
Detection probabilities:
1
1
i ( t ) 2
Pa3  1  e
 1  cos  (t ) 
4
2
1
Pb3  1  cos  (t ) 
2
Haroche et. al, Nature (2000)
Quantum Eraser
• Initial state:
2' 
• After passage through cavity A:

1
b 31B a 2 a 3 0 B b
2

2
0
b
1' 
1
0 A1B a 21 A 0 B b
2
3' 
1
b 3 1 B a 2  ei (t ) a 3 0 B b
2
• Phase shift:

2
2
A
3
.
.
0
A
.
• After passage through cavity B:
4' 





1
b3 ,1B a2  b2 ei (t )  a3 ,0 B a2  b2 ei (t ) 0 A .
2
• Detection probabilities:
1
1

P  Tr  a  b e  a  b e

4
2
i ( t )
a
2
Pb 
2
 i ( t )
2
2
1
2
• Restoration of fringes:
a2  (1 / 2 ) a2  b2
5' 
1
b2  (1 / 2 ) a2  b2

 a ,0  a 1  e  b 1  e  b ,1  a 1  e  b 1  e 
i ( t )
2 2
1
Pa 
2

{
3
2
i ( t )
i ( t )
2
1  cos (t )
for a2
1  cos (t )
for b2
3
1
Pb 
2
{
B
2
i ( t )
2
1  cos (t )
for a2
1  cos (t )
for b2
Quantum teleportation
• Initial state is an entangled
state between cavities A
and B along with the tagged
qubit T:
2' 

1
0
2
A
1
B
a 21
A
0
B
b
2
b
1
.
• We want to teleport the state of qubit C:
 C  c0 0  c1 1
to cavity B
• State of combined system ABCT is
 ABCT 
1

[ c

AC
2
2 2
 1 
2 2

c0 1  c1 0
0  c1 1 B   AC
0
[ c

AC
2
0


c0 1  c1 0
0  c1 1 B   AC




c0 1  c1 0


c
0

c
1


AC
0
1
AC
B
B

B

c0 0  c1 1 B   AC c0 1  c1 0
  AC
where

 AC

1
 01  10 ;  AC  1  00  11 ;  2  1  a  b
2
2
2

2

• A Bell-basis measurement of  AC
reduces the BT
state to
1
1
c 0 0  c 1 1 
  c 0      c 1      
2
BT
0
B
2
2
1
B
2
2
]
2
0
B 2
1
B 2
B
]
B
Induced coherence without induced
emission
• Recall we produced:
3  e
-iΗ eff 
2
1   1
g 2

 1

i
 0 A1B  1A0B   a2  0 A1B  e 1A0 B ,  

 b2

2   2

 2



• Interference terms are only
partially erased in the reduced
two-cavity density matrix ρAB,
given by
 AB




1
1
  0 A1B 0 A1B  1A0 B 1A0 B   0 A1B 1A0 B e i  1  c.c.
2
2
• Probabilities for finding the atom 3 in the excited and
ground states:
1  cos   cos   
Pa3  1 
2
2
 , Pb3  1  Pa3 .
• For η≠π, we have the control of the interferences in
unconditional measurements on atom 2.
• Visibility of the fringes is equal to |sin(η/2)|.
Brian Greene in The Fabric of the Cosmos (2004)
These experiments are a magnificent
affront to our conventional notions of
space and time. . . . . . . . . .For
a few days after I learned of these
experiments, I remember feeling
elated. I felt I'd been given a glimpse
into a veiled side of reality.
Table of Contents
•
•
•
•
•
•
•
•
•
•
•
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•
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•
•
•
Quantum Disentanglement Eraser
Quantum Eraser
Complementarity (Bohr)
Erasing Knowledge
Photon Correlation Experiment
Correlation Function
Three-Level Atom
Can we erase?
Particle or Wave
Restoration of Interference (Mohrhoff)
Delayed Choice
Experimental Considerations
Experimental Set-up
Experimental Results
Objectivity, retrocausation (Mohrhoff II)
Double-Slit Experiment
Micromaser Which-Path Detector
Quantum Eraser
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Interference Fringes
Atomic Probability
Quantum Disentanglement
Entangled State
AB System
Cavity QED
Eraser Field
Classical Field
Initial State
Detection Probabilities
Quantum Eraser
After Passage
Quantum Teleportation
ABCT
Induced coherence
Probabilities
Fabric of the Cosmos