Unusual Interferometry via
Translational-Internal
Entanglement
Nir Bar-Gill
Michal Kolar
Tomas Opatrny
Gershon Kurizki
Outline

Interferometry


Which-way and phase information
Disorder

Momentum localization
Interferometric Information


Use a TIE particle in a Mach-Zehnder
interferometer
Compared to standard particles


Information stored in 4D Hilbert space
Both which-way and phase information can be
extracted
Standard MZI with which-way detector
a reminder



Insert a which way detector with
distinguishability D
The state in the interferometer
is measured
The probability to reach the
output detectors
P 

1
1  (1  D)(1  D) cos 
2
D

V

Thus, the visibility of the interference pattern is
V  1  D 2 , and thus V 2  D 2  1
Interferometer for TranslationallyInternally Entangled Particles

Initial state:
1
 k1 1  k2 2
 in 
2

2, k2
1, k1
 2 k12
 2 k 22
E 
 1 
 2
2m
2m


No which-way detector
Phases of internal states
acquired according to path
length (due to entanglement
with different momentum states)
TIE with
birefringent photons /
two-level atoms
 WW marker travels with the particle
Output Detectors




Mean path lengths LA,LB
are known
We choose internal state
projection basis at output
accordingly
The click probability at the detectors
following an  projection:
1

P  PA  PB  PAB 
8
1 2
1 2
PA  A
PB  1  B
2
2
PAB  A
1
The interference pattern is I   P  P  PAB
4




( Click: A  B)
TIE Interferometry

Standard
Visibility not good measure
Interference
(interference pattern not
sinusoidal)
1
DTIE
TIE Interference
0.9
Phase sensitivity:
PAB
S


Standard case:


Smax=V
S2+D2 1
0.8
0.7
Information Measures

0.6
0.5
0.4
0.3
0.2

TIE case (e.g. k1/k2=3):


S =1/3, D 1
S2+D2>1
0.1
0
1
1.5
2
2.5
 /
3
3.5
4
Momentum Localization using TIE

Couple random momentum states of an atom
using resonant two-photon processes
atom
Coupling beams
(regular or random angles)

n, kn
3, k3
g , k0
1, k1
Create diagonal order or disorder using TIE


Order: i, j  Ei  Ek  E j  Ek
Disorder: random ki → Ei  Ek  E j  Ek
i
j
i
i j
j
2, k2
Hamiltonian of Random Momenta

Hamiltonian with random diagonal elements, and
controllable coupling strengths
 E0

J
H 


J


J
E1
0
0
J 

0 0
 0

0 En 

Order :
E n  E0
Disorder :
E n  [   / 2   / 2]
Through the change of coupling strength J, we can
access the


weak disorder regime - J  1
Strong disorder regime - J  1
Localization via Multi-State Coupling
Due to coupling between ground state and all
other states – N-dimensional system

Localization occurs for certain (finite)

Momentum Distribution for Ordered and Disordered Systems
Interference Pattern for  /J=50
V = 98%
1
0.8
Intensity [arb. units]
Disordered,
Ordered,
for /Delta/J=10
allfor
/J
Disordered,
Ordered,
all
 /J
Disordered,
/J=50
/J=1
Disordered,  /J=1
Disordered, /Delta/J=10
0.9
Disordered,  /J=50
0.6
0.04
0.02
0
0.5
-2
-1.5
-1
0.3
0.2
0.1
0
-0.5
0
0.5
x [arb. units]
1
1.5
2
-3
x 10
Interference Pattern for  /J=1
V = 69%
0.4
1
2
3
4
5
6
7
8
Momentum states ki
i
9
10
11
Intensity [arb. units]
Occupation
0.7

J
0.04
0.02
0
-2
-1.5
-1
-0.5
0
0.5
x [arb. units]
1
1.5
2
-3
x 10
Interference Fringe Visibility:
Measure of momentum localization
Passing the atom through an interferometer
(light grating) can measure momentum
distribution
Visibility of Interference Fringes as a function of  /J
100
95
90
Visibility

85
80
75
70
65
0
10
20
30
40
50
 /J
60
70
80
90
100
Conclusion

Interferometry



TIE gives both which-path and phase information
Does not obey standard complementarity
Momentum Localization via TIE



“Anderson-like” disorder in momentum space
Strength of disorder controlled by strength of
interactions
Localization measured through TOF images or
visibility of interference pattern
The End