Download Towards Heisenberg Limit in Magnetometry with

Heisenberg Limited
Sagnac Interferometry
AZIZ KOLKIRAN, G.S. AGARWAL
Oklahoma State University,
2007 APS March Meeting,
A33 Focus Session: Quantum Limited Measurements,
Monday March 5th,2007, 9:48-10:00am
Sagnac phase shift.
2 R
t1 
c  R
t2
t1


2 R
t2 
c  R
4 R 2 
4 R 2
t  2

2 2
c R 
c2
8
4 LR
  t 
A 
c
c
Sagnac interferometer with classical fields.
Beam splitter
transmission and
reflection coefficients
r  i / 2  r ,
t  1/ 2  t 
E1  r rEin e it2  t 2 Ein e it1  Ein e it2 ei / 2i sin( / 2)
E2  rtEin eit1  t rEin eit2  Ein e it2 ei / 2i cos( / 2)
1
I1 | E1 | | Ein | sin ( / 2)  | Ein |2 (1  cos( ))
2
2
2
2
I 2 | E2 |2 | Ein |2 cos 2 ( / 2) 
1
| Ein |2 (1  cos( ))
2
Single-photon Sagnac interferometer*
*G Bertocchi, O Alibart, D B Ostrowsky, S Tanzilli and P Baldi, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 1011–1016
Can we improve the resolution by using quantum states of light?
 b1  1 1 i  1 0  1 1 i   a1 
 



 a 
i 
b
i
1
0
e
i
1
2

 2
 2 
 2
I1  b1 b1   (1/ 2)[1  cos( )]
†
|10  sin( / 2) |10  cos( / 2) | 01
I 2  b2 b2   (1/ 2)[1  cos( )]
†
|11 
1
sin( )( | 20 | 02 )  cos( ) |11
† †
I12  b1 b2 b1b2   (1/ 2)[1  cos(2 )]
2
experimentally employed by single photons coming from a spontaneous down-converter
Bertocchi G. et al, “Single photon Sagnac interferometry”, J. Phys. B 39, 1011 (2006)
Use of entangled photon pairs from down-conversion.
a1
pump
NL
a2

1
i
n
|  
(

e
tanh
r
)
| n1 | n 2

cosh r n 0
Fig. 4
input state from the down-conversion process:
r is the interaction parameter (a.k.a. squeezing parameter)
of entangled pair production from vacuum
r  L | Ep |  2
†
†
I1  b1 b1   sinh 2 r  b2 b2   I 2
2
tanh
r
P2 | 11| U |   |2 
[1  cos(2 )]
2
2cosh r
Can we improve the resolution further by using
multi-photon entanglement?
Fig. 5
tanh 4 r
1
P4 
TT
R
R
1 2 1 2 [11  12cos(2 )  9cos(4 )]
2
cosh r
8
Ti , Ri ' s are transmission
and reflection
probabilities at the
beam splitters
Ri  1  Ti
Can we turn the fringes into ones with equal height**?
Fig. 6
tanh 4 r 9 2
P4 
T1 R1T2 R2 [1  cos(4 )]
2
cosh r 8
**see
also T. Nagata, et al. Science 316, 726-729, (2007).
4-photon interference using asymmetric beam splitters*
The experiment
B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou
Optics Letters, Vol. 32, Issue 10, pp. 1320-1322 (2007).
How does this work?
classical field
|
phase shifter
i
| e 
For two-photon coincidence detection (Fig. 4),
the state, after entering and completing the
path, evolves into the following:
For four-photon coincidence detection (Fig. 5
and 6), the state, after entering and completing
the path, evolves into the following:
Non-classical
Fock state
phase shifter
| n
ei n  | n
| 20 | 02
| 20  ei 2 | 02
|11 

2
2
3
| 22 
4
 | 40  ei 4 | 04 
1 i 2
e | 22

 
2
4


Only this part contributes to
the detection in the setup given
by Fig. 6
This term leads to
unequal fringes in the
detection setup given
by Fig. 5
All interferometric schemes using the maximally correlated entangled states show
the phase sensitivity equal to 1/N, which is the Heisenberg limit.
Conclusions
•Use of PDC light in SI leads to 2- and 4-fold
increase in the sensitivity depending on the detection
mechanism.
•We think that the experiment should be feasible,
because single photons have already been used from a
down-converter in SI and in many experiments 2- and
4-photon interference effects have been observed.
•If successful, it could be a stimulating application in
the field of quantum metrology.
Opt. Express 15, 6798-6808 (2007).
Current work in progress
Quantum Interferometric Optical Lithography:
Exploiting Entanglement to Beat the Diffraction Limit
1  sinh 2 G
Visibility 
1  5sinh 2 G
20% at large gain
G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, Phys. Rev. Lett. 86, 1389 (2001),
comment on A.N. Boto, et al., Phys. Rev. Lett. 85, 2733 (2000).
Quantum Imaging and Sensing Using Coherent Beam Stimulated
Parametric Down conversion
| 0 
| 0 
SETUP: Using an input from non-degenerate stimulated parametric down-conversion for the determination
of phase via photon-photon correlations.
RESULTS:
Stimulated emission
enhanced visibility of two-photon counts
for various phases of the coherent field
with respect to the gain factor g. The
pump phase 
is fixed at  . The
modulus of the coherent field |  0 | is
chosen such that the coincidences
coming from SPDC and the coherent
fields are equal to each other. The
dashed line shows the visibility for the
case
of
photons
produced
by
spontaneous
parametric
downconversion.
Aziz Kolkiran and G S Agarwal, in preparation
Future work
Interaction free measurements (non-distortion quantum
interrogation) using entangled photons
How to optically detect the presence of something without photons hitting it.
two-level
atom
25% chance of detecting the bomb without explosion. By using
different beam splitter reflectivity's R, this can be made 50%.
Elitzur A. C. and Vaidman L. (1993). Quantum mechanical interaction-free measurements. Found. Phys. 23, 987-97.
Future work
Interaction free measurements (non-distortion quantum
interrogation) using entangled photons
Entangled source
| 
| 
object (three level atom)