Heisenberg Limited Sagnac InterferometryHeisenberg Limited Sagnac Interferometry

AZIZ KOLKIRAN, G.S. AGARWALOklahoma State University, 2007 APS March Meeting,

A33 Focus Session: Quantum Limited Measurements,Monday March 5th,2007, 9:48-10:00am

1

2 Rt

c R

2

2 Rt

c R

2 2

2 2 2 2

4 4R Rt

c R c

8 4 LRt A

c c

Sagnac phase shift.Sagnac phase shift.

1t

2t

2 1 22 / 21 sin( / 2)i t i t i t i

in in inE r rE e t E e E e e i 1 2 2 / 2

2 cos( / 2)i t i t i t iin in inE rtE e t rE e E e e i

/ 2 ,

1/ 2

r i r

t t

Beam splitter transmission and

reflection coefficients

2 2 2 21 1

1| | | | sin ( / 2) | | (1 cos( ))

2in inI E E E

2 2 2 22 2

1| | | | cos ( / 2) | | (1 cos( ))

2in inI E E E

Sagnac interferometer with classical fields.Sagnac interferometer with classical fields.

Single-photon Sagnac interferometer*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

1 1

2 2

1 1 0 11 1

1 0 12 2i

b ai i

b ai e i

|10 sin( / 2) |10 cos( / 2) | 01

1|11 sin( )( | 20 | 02 ) cos( ) |11

2

experimentally employed by single photons coming from a spontaneous down-converterBertocchi G. et al, “Single photon Sagnac interferometry”, J. Phys. B 39, 1011 (2006)

†1 1 1 (1/ 2)[1 cos( )]I b b

† †12 1 2 1 2 (1/ 2)[1 cos(2 )]I b b b b

†2 2 2 (1/ 2)[1 cos( )]I b b

Can we improve the resolution by using quantum states of light?Can we improve the resolution by using quantum states of light?

1 20

1| ( tanh ) | |

coshi n

n

e r n nr

† †21 1 1 2 2 2sinhI b b r b b I

22

2 2

tanh| 11| | | [1 cos(2 )]

2cosh

rP U

r

22

2 2

tanh| 11| | | [1 cos(2 )]

2cosh

rP U

r

Use of entangled photon pairs from down-conversion.Use of entangled photon pairs from down-conversion.

r is the interaction parameter (a.k.a. squeezing parameter) of entangled pair production from vacuum

input state from the down-conversion process:Fig. 4

NL

pump

2| |pr L E

1a

2a

4

4 1 2 1 22

tanh 1[11 12cos(2 ) 9cos(4 )]

cosh 8

rP TT R R

r

4

4 1 2 1 22

tanh 1[11 12cos(2 ) 9cos(4 )]

cosh 8

rP TT R R

r

Can we improve the resolution further by using Can we improve the resolution further by using multi-photon entanglement?multi-photon entanglement?

, 'i iT R s are transmission and reflection probabilities at the beam splitters

1i iR T

Fig. 5

42

4 1 1 2 22

tanh 9[1 cos(4 )]

cosh 8

rP T R T R

r

42

4 1 1 2 22

tanh 9[1 cos(4 )]

cosh 8

rP T R T R

r

Can we turn the fringes into ones with equal heightCan we turn the fringes into ones with equal height****??

**see also T. Nagata, et al. Science 316, 726-729, (2007).

Fig. 6

4-photon interference using asymmetric beam splitters*4-photon interference using asymmetric beam splitters*

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).

The experiment

How does this work?How does this work?

2| 20 | 02 | 20 | 02|11

2 2

ie

243 | 40 | 04 1

| 22 | 224 2 4

iie

e

All interferometric schemes using the maximally correlated entangled states show the phase sensitivity equal to 1/N, which is the Heisenberg limit.

| ie

phase shifter

| n |i ne n classical field

Non-classical Fock state

|

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:

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

phase shifter

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 progressCurrent work in progress

Quantum Interferometric Optical Lithography: Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction LimitExploiting Entanglement to Beat the Diffraction Limit

2

2

1 sinh

1 5sinh

GVisibility

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 Quantum Imaging and Sensing Using Coherent Beam Stimulated Parametric Down conversionParametric 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 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 down-conversion.

0| |

Aziz Kolkiran and G S Agarwal, in preparation

Future workFuture work

Interaction free measurements (non-distortion quantumInteraction free measurements (non-distortion quantuminterrogation) using entangled photonsinterrogation) using entangled photons

How to optically detect the presence of something without photons hitting it.

Elitzur A. C. and Vaidman L. (1993). Quantum mechanical interaction-free measurements. Found. Phys. 23, 987-97.

25% chance of detecting the bomb without explosion. By using different beam splitter reflectivity's R, this can be made 50%.

two-level atom

Future workFuture work

Interaction free measurements (non-distortion quantumInteraction free measurements (non-distortion quantuminterrogation) using entangled photonsinterrogation) using entangled photons

| Entangled source

|

object (three level atom)