US/Japan WorkshopBreckenridge, CO, USA, Aug. 23-25, 2006

Schrödinger cat and EPR statewith quantum optics

Akira FurusawaDepartment of Applied Physics

University of TokyoCREST, JST

A. Furusawa Univ. of TokyoT. Aoki, H. Yonezawa, K. Wakui, H. Takahashi, Y. Takeno, J. Yoshikawa, T. Kajiya, N. Lee, M. Yukawa, Y. Miwa, H. Uchigaito,J. S. Neergaard-Nielsen (NBI), N. Takei (ERATO)A. Huck (Erlangen)

M. Sasaki NICTM. Fujiwara, M. Takeoka, J. Hayase, A. Kitagawa, K. Tsujino

M. Ban Hitachi

S. L. Braunstein, P. van Loock, U. L. Andersen

Collaborators

Quantum opticsannihilation operator a

quantum complex amplitude

ˆ ˆ ˆa q ip= +q: cosine componentp: sine component

ˆ ˆ[ , ]2iq p =

†ˆ ˆ[ , ] 1a a = 12

⎛ ⎞=⎜ ⎟⎝ ⎠h

ˆ ˆ[ , ]2ix p =

x: positionp: momentum

Photon-number units

α

p

time2/(pi) * exp(2*(-(x-x0)**2-(y-y0)**2))

-6-4

-20

24

6 -6-4

-20

24

6-0.6-0.4-0.2

00.20.40.6

xp

W(x, p)

Coherent states

x

Rotatingframe

Time evolution

Wigner function

αMinimum uncertainty state

Laser

2

2

0 !

n

ne n

n

α αα∞−

=

= ∑

a α α α=

Wigner function

Squeezed vacuumMinimum uncertainty state

( ) ( )

( )

2 † 2ˆ ˆ2

0

ˆ 0 0

2 !1 tanh 22 !cosh

r a a

nn

n

S r e

nr n

nr

−

∞

=

=

= ∑

( ) ( )† †ˆ ˆˆ ˆ ˆcosh sinh

ˆ ˆr r

S r a S r a r a r

e x ie p−

= −

= +

( )

2

2

2

0

2

0

!

!

n

n

n

n

e nn

e nn

α

α

αα

αα

∞−

=

∞−

=

=

−− =

∑

∑

22n 1

2

n 0e 2n 1

(2n 1)!

α αα α+∞−

=

− − = ++

∑

Schrödinger cat state

S. Lloyd, S.L. BraunsteinPRL 82, 1784 (1999)

Quantum informationprocessing

Unitary transformation

ψ ϕU ˆ

ˆ

Hi t

U

e

ϕ ψ

ψ−

=

= h

Arbitrary Hamiltonians （ polynomials of ）

ˆ ˆ,x p

( )†ˆ ˆ ˆ ˆ,x p i a aα α∗ −2 2 †ˆ ˆ ˆ ˆx p a a+

( )†2 2ˆˆ ˆ ˆ ˆ ˆxp px i a a+ −

( )† †1 2 1 2 1 2 1 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆp x x p i a a a a− −

( ) ( )2 22 2 †ˆ ˆ ˆ ˆx p a a+

(2)χ(3)χ

Beam splittersDisplace in phase spacePhase shifters

Squeezers

Kerr effect

ˆ ˆ ˆa x ip= +Gaussian operations

Non-Gaussian operations

Toward universal QIP

Quantum teleportation of non-Gaussian states

Schrödinger cat state

Time domain EPR correlation

A non-Gaussian input state

Resource for quantum teleportation

catψ α α− −

Output -Quantum teleportationInput

Output -Classical teleportation

Schrödinger cat1 ( 1.5 1.5 )2

− −

S. L. Braunstein & H. J. Kimble, PRL 80, 869 (1998).

Creation of Schrödinger cat statewith photon subtraction

K. Wakui, H. Takahashi, A. Furusawa, & M. Sasaki, CQIQCII-2006

( )

2

2

2

0

2

0

!

!

n

n

n

n

e nn

e nn

α

α

αα

αα

∞−

=

∞−

=

=

−− =

∑

∑

22n 1

2

n 0e 2n 1

(2n 1)!

α αα α+∞−

=

− − = ++

∑

Schrödinger cat state

Photon subtraction

even photons

KNbO3

Pulsed light: A. Ourjoumtsev et al., Science 312, 83 (2006).CW light: J. S. Neergaard-Nielsen et al., quant-ph/0602198.

OPO

( )0,0 0.026W = −

LO LO

odd photons

α α− −conditional homodynetomography

phasescan

• KNbO3

• Type-I non-critical phase-matching• Output coupler :

Optical Parametric Oscillator

430nm

KNbO3

Ti:S Squeezedvacuum

SHG

860nm

x

p

T 15%≅

Photon subtraction

Squeezed state

Squeezed state

Photon subtraction

Best result with KNbO3 without any correction

Photon subtraction

α α− −

even photons

odd photons

KNbO3PPKTP

OPO

conditional homodyne tomography

One of the results with PPKTP

without any correction

( )0,0 0.043W = −

W(0,0) = -0.075W(0,0) = W(0,0) = --0.0750.075

W(0,0) = -0.055W(0,0) = W(0,0) = --0.0550.055 W(0,0) = -0.043W(0,0) = W(0,0) = --0.0430.043

20mW20mW20mW

40mW40mW40mW

Pump Power: 10mWPump Power: 10mWPump Power: 10mW

30mW30mW30mW

W(0,0) = -0.059W(0,0) = W(0,0) = --0.0590.059

Time domain Einstein-Podolsky-Rosen(EPR) correlation

N. Takei, N. Lee, D. Moriyama, J. S. Neergaard-Nielsen, & A. Furusawa, quant-ph/0607091

Time-domain EPR correlation

( )Ax t

( )Bx t

( )Ap t

( )Bp t

x measurements

A AA( , )x p B BB( , )x p

[ ]A B A Bˆ ˆ ˆ ˆ, 0x x p p− + =

A BEPR dx x x∝ ∫

A B

A B

00

x xp p

− =+ =

Simultaneous eigenstates of ˆ ˆ ˆ ˆ( ) & ( )A B A Bx x p p− +

p measurements EPR beams in quantum optics

Mode matching to photon counting

• Ordinary teleportation experiment: side band

freq.

ΔΩΔΩ

+Ω−Ω 0

Ω Ω

• Broad band

freq.

ΔΩ

2ΔΩ

2ΔΩ

− 0

TΔ ≈Time resolution 1/bandwidth

cavity bandwidth

Generation of EPR beams

1/2R

Alice

Bob

x

p

(0) (0)1 2

B

(0) (0)1 2

B

ˆ ˆˆ2

ˆ ˆˆ2

r r

r r

e x e xx

e p e pp

−

−

−=

−=

(0)A B 1

(0)A B 2

ˆ ˆ ˆ2

ˆ ˆ ˆ2

r

r

x x e x

p p e pr

−

−

− =

+ = → ∞

Squeezed vacuum

(0) (0)1 2

A

(0) (0)1 2

A

ˆ ˆˆ2

ˆ ˆˆ2

r r

r r

e x e xx

e p e pp

−

−

+=

+=

“EPR noise” “EPR correlation”

０

( )HBS A B A B

† †A B A B A B

2A B

0

ˆ ˆˆ ( ) ( ) 0 0

ˆ ˆ ˆ ˆexp 0 0

1 n

n

B S r S r

r a a a a

q q n n∞

=

− ⊗

⎡ ⎤= − ⊗⎣ ⎦

= − ⊗∑tanhq r=

2A B

0

1 n

n

q q n n∞

=

− ⊗∑

r → ∞

A B A B0n

n n dx x x∞ ∞

−∞=

=∑ ∫

x

p

p

x

Generation of EPR beams

Experimental setup

430nm

860nm

Ti:S

Doubler

OPO1

OPO2

Squeezedvacuum

LO

Bob

50%R

Optical Parametric Oscillator• KNbO3• Type-I non-critical

phase-matching• Output coupler : ～13%

ADC PC

Alice

LO

LO

ADC

Alice

Bob

50%R

ADC PC

LO

LO

Cavity lock

Cavity lock

Probe

Probe

Experimental setup

ADC

x or p

x or p

Alice

Bob

50%R

LO

LOlock

lock

lock

Cavity lock

Cavity lock

Probe

Probe

Experimental setup

ADC PC

ADC

Alice

Bob

50%R

LO

LOhold

hold

hold

Cavity lock

Cavity lock

Probe

Probe

Experimental setup

ADC PC

ADC

Alice

Bob

50%R

LO

LOhold

hold

hold

Cavity lock

Cavity lock

Probe

Probe

Experimental setup

ADC PC

ADC

Alice

Bob

50%R

LO

LOhold

hold

hold

Cavity lock

Cavity lock

Probe

Probe

Experimental setup

ADC PC

ADC

50MS/sec for 2msec (100000pts)Quadrature values =10pts average (averaged for 200nsec) 5kHzHPF

AliceBob

p measurements

x measurementsTime domain EPR correlation

2A Bˆ ˆ[ ( ) ] 3dBp pΔ + ≈ −

p correlationx correlation

2ˆ ˆ[ ( ) ] 3dBA Bx xΔ − ≈ −

PPKTP

No BLIIRA!

Trying to get more squeezing

-7dB squeezing

7.2 0.2 dB− ±

S. Suzuki, H. Yonezawa, F. Kannari, M. Sasaki, & A. Furusawa, APL 89, 061116 (2006).

Pump power dependence of squeezing

Theoretical squeezing level calculated from G+ and lossestaking account of the phase fluctuation of the LO

3.9θ = o%

2 2cos sinS S ASθ θ′ = +% %

RMS

Requirements for high-level squeezing

-14dB -12dB

-10dB

intra-cavity lossof OPO fluctuation of LO phase

squeezing

0.0063.9

Lθ

== o%

For -7.2dB

Requirements for high-level squeezing

-14dB -12dB

-10dB

intra-cavity lossof OPO fluctuation of LO phase

squeezing

0.0042.0

Lθ

== o%

Present

We should have -9dB!!

-12

-9

-6

-3

0

3

6

9

12

15

18

0.00 0.02 0.04 0.06 0.08 0.108.3 0.2 dB− ±

0.0042.0

Lθ

== o%

Present

Time domain EPR correlation

Non-Gaussian states

Quantum teleportation of non-Gaussian states

Near future

Schrödinger cat state