Quantum effects in the Collective Atomic Recoil Lasing (CARL)

N. Piovella, M.M. Cola, L. Volpe,Dipartimento di Fisica, Università di Milano,

via Celoria 16, Milano (ITALY)G.R.M. Robb

SUPA, Department of Physics, Univ. of Strathclyde,Glasgow (SCOTLAND)

R.BonifacioINFN-Milano (ITALY) & CBPF - Rio de Janeiro (BRASIL)

CONTENTS:

•Classical regime of CARL & Superradiance (SR)•Quantum description of CARL & SR in optical cavity

and in free space•Quantum regime of CARL & SR•Transverse self-focusing force in CARL•Atom-atom & atom-photon entanglement in CARL

Collective Atomic Recoil Lasing (CARL)

Collective Atomic Recoil Lasing = Optical gain + bunching

laser pump

cold atomsscattered field

λ /2

t>0

t=0optical potential

Particles self-organize to form compact bunches ~ λ/2which radiate coherently as N2.

coldatoms

A1

A2

R=1

R R A2inA1

in

A2out A1

out

BIDIRECTIONAL RING CAVITY:only one mode in each direction interacts with atoms

(Tubingen experiment)

CARL model:• strong pump >> scattered field• pump is far detuned from atomic resonance, (Δ0>>γ) so atoms

remain in their ground state• adiabatically eliminate internal atomic degrees of freedom

( )∑=

θ−+θ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−=N

1j

ii2zj jj eaaegim2

pH h

[ ] [ ]

V2dgggkz2

1a,aip,z

0

2

10

p1jj

'jj'zjj

εω

=⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

Ω==θ

=δ= +

h

h Ωp : pump Rabi frequency

Δ0=ω −ω0: pump detuning

R. Bonifacio, L. De Salvo, NIMA 341 (1994) 360

CLASSICAL REGIME of CARL:• Operators (zj ,pzj) and a are considered as classical variables

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

Δ+=

+−=

ω=θ

∑=

θ−

θ

aiegdtda

.)c.cae(gdtdp

p2dtd

N

1j

i

ij

jrj

j

j

uningdetprobepump

bunchingeN

b

frequencyrecoilm

)k(

unitsrecoilinmomentumk

pp

pump

N

j

i

r

zjj

j

−ω−ω=Δ

=

=ω

=

∑=

θ−

1

2

12

22

hh

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

δ+=τ

+−=τ

=τ

θ

θ

AibddA

.)c.cAe(dpd

pdd

jij

jj

ρωΔ

=δ

ρ=

ρ=

ρω=τ

r

jj

r

2

NaA

pp

t2

3/2

r2Ng

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=ρ

CARL parameter

CARL instability• exponential growth of scattered intensity

and bunching• saturation at A~1 (Nphoton ~ρ N μ N4/3)

∑=

θ−=N

1j

i jeN1b

b~0 b~0.8

time

bunching:

Atoms behave as coupled pendula in a self-consistent potential:

)cos(|A|2)(V ϕ+θ=ϕ+θ

τ∝ 32 e|A|

SUPERRADIANT REGIMEIn free space:

(EXACT PROPAGATION)

In optical cavity:

(MEAN FIELD APPROXIMATION)

( )

N1,]K/2exp[|A|

NNNKb)(A

2K1Kfor

t2

23photon

rcav

∝στ∝

∝ρ∝⇒ρ∝≈τ

ρω>>

KAbddA

.)c.cAe(dd

ji2j

2

−=τ

+−=τ

θ θ

cavcav

r

cav

LcTK

2KK

=

ρω=

bAA

.)c.cAe( ji2j

2

=ζ∂

∂+

τ∂∂

+−=τ∂

θ∂ θ

( ))c/z(2 rρω=ζ

self-similar solution:

( ) ⎥⎦⎤

⎢⎣⎡ τ⋅∝

>>

ζ−τζζ=τζ

3/2a

2

a

1

Ltcosexp|A|

:c/Ltfor))((A),(A

SUPERRADIANT REGIME : SELF-SIMILAR SOLUTION

0 5 10 15 20 25 300.00

0.02

0.04

0.06

0.08

0.10

|A1|2

y

)(y

)y(),()y(A),(A

j1j

1

ζ−τζ=

θ=ζτθ

ζ=ζτ

∑=

θ−

θ

=+

+−=θ

N

1j

i1

1

i12

j12

j1

j1

eN1A

dydA

2y

.)c.ceA(dy

d

SUPERRADIANT EMISSION (EXACT PROPAGATION)

QUANTUM EFFECTS ARE EXPECTED WHEN THEATOMS ARE INITIALLY COLDER THAN Trecoil

momentum spread σp≤ ћk T< Trecoil i.e. a BEC !

two possible regimes:

k2)k2(p maxz hh >>ρ≈

CLASSICAL REGIME:

rNg1 ω>>⇒>>ρ

QUANTUM REGIME:

1NNphoton >>ρ≈

rNg1 ω<⇒<ρ

NNphoton = k2pz h=

pump laser

QUANTUM MODEL FOR CARL

( )

∫π

θ−

θ

−Δ+θΨθ=

−−θ∂ψ∂

ω−=Ψ=∂Ψ∂

2

0cav

i2

i2

2

r

a)Ki(e|)t,(|dgNdtda

.c.haegiHt

i hhh

( )∑=

θ−+θ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−=N

1j

ii2zj jj eaaegim2

pH h

( )

∫π

θ−

θ

−δ+τθΨθ=τ

Ψ−ρ−θ∂ψ∂

ρ−=

τ∂Ψ∂

2

0

2

2

2

21

A)Ki(e|),(|dddA

.c.hAeii

i

i the quantum CARLmodel depends on ρ !

N. Piovella, M. Gatelli & R. Bonifacio, Opt. Comm. 194 (2001) 167N. Piovella, M. Cola & R. Bonifacio, PRA 67 (2003) 01387

Momentum representation:

discrete values of momentum : pz= n (2hk) , n=0,±1,..

n=1pz n=0

n=-1

∑∞

−∞=

θτ=τθΨn

inn e)(c),(

( )

A)Ki(ccddA

cAAcc2in

ddc

n

*1nn

1n*

1nn

2n

−δ+=τ

−ρ−ρ

−=τ

∑∞

−∞=−

+−

|cn|2 = probability to find anatom with pz =n(2ћk)

2ћk

- 1 5 - 1 0 - 5 0 5 1 00 .0 0

0 .0 5

0 .1 0

0 .1 5

( b )

n

p n

0 1 0 2 0 3 0 4 0 5 01 0 - 9

1 0 - 7

1 0 - 5

1 0 - 3

1 0 - 1

1 0 1

κ = 0 , ρ = 1 0

( a )

τ

|A|2

‘Good-cavity limit’ K=0classical regime ρ>>1

at the peak, the number of momentum states occupied is ~ ρ

ρω<<

ω>>

rcav

r

2KNg

initial state:c0=1 (all atoms with pz=0)

ρ

θ−− τ+τ∝τθΨ i10 e)(c)(c),(

only TWO momentum states : pz =0 and pz =-2ћk

n=0

n=-1

the condensate behaves as a two-level system(i.e. a laser !)

‘Good-cavity limit’ K=0quantum regime ρ<1

quantum CARL equations reduce to Maxwell-Bloch equations !

0 100 2000

2

4

6

8

10

τ

|A|2

0 100 200

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

τ

<p>

for K=0 train of sech2 pulses (area=2π)

n=0

n=-1

BEC behaves like an unstable pendulum

0p =

k2p h−=

SechN

)2/(sinpA

sinK

2 ϕ−=

ρ=ϕϕρ=ϕ+ϕ

&

&&&

0 250 500

0,0000

0,0005

0,0010

0,0015

0,0020

0,0025

τ

|A|2

when photons escape fast from atoms (large K)..

SEQUENTIAL SUPERRADIANT SCATTERING:atoms recoil by 2ћk, emitting a SR pulse

QUANTUM SUPERRADIANT REGIME:

n=0

n=-2n=-1 BEC behaves as an overdamped pendulum:

ϕρ

≈ϕ sinK

&

n=0n=-1

n=-2n=-3

etc..

0 250 500

-4

-2

0

τ

<p>

cavsrrcav K

NgGK2

=>ω>>

CLASSICAL & QUANTUM SUPERRADIANT REGIME :

cavsrrcav K

NgGK2

=>ω>>

QUANTUM LIMIT (sequential two-level SR)

CLASSICAL LIMIT ( CARL SR)

rcav

srcav KNgGK ω>>=>

2

Quantum Superradiant CARL has been observedin 1999 by Ketterle at MIT and at LENS in 2002

normalfluorescence

SUPERfluorescence

E

E

n=0 n=1 n=2 n=3n=0 n=1 n=2 n=0 n=1

Production of an elongated 87Rb BEC in a Ioffe-Pritchard magnetic trap

Laser pulse during first expansion of the condensate

Absorption imaging of the momentum components of the cloud

Experimental values:

Δ = 13 GHzw = 750 μmP = 13 mW

The LENS experiment

laser beam kw,

BEC

absorption imaging

trap

g

L. Fallani et al., PRA (2006)

The LENS experiment

pump light

Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse.

TRANSVERSE EFFECTS ANDSELF-FOCUSING IN CARL

[ ]

a)Ki(egNaZic

ta

.c.ce)t,x(agim2ppH

cavi2

R

i22

z

−Δ+=∇−∂∂

−−+

=

θ−⊥

θ⊥

⊥ rh

( ) 0N

|a|dtpd 2

<⎟⎟⎠

⎞⎜⎜⎝

⎛ ∇Δ−φ= ⊥⊥

r&h

r

0,0 =Δ>φ&

self-focusing force proportional tothe field intensity gradient and to the phase shift

SELF-FOCUSING IN QUANTUM REGIME

θ−− τ+τ∝τθΨ i10 e)(c)(c),(

KAccciDAcAc~icic

Accic

0*10

2'x

*0

*11

2'x

1

102

'x0

−+∇=τ∂

∂

−Δ+∇η=τ∂

∂

−∇η=τ∂

∂

−

−−−

−

NaA

NgKK

tNgx'x

NgZcD

Ng

cav

R

==

=τσ

=

=ω

=η ⊥

2r

2

Rrec

rec m2Ng~,4Z,

vcD,

mkv

σ=ω

ω−Δ=Δ

λπσ

=η== ⊥hh

N. Piovella et al. , Laser Physics 1 (2007)

GOOD-CAVITY REGIME (K=0, D=0) and η=0.1

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

ENER

GY

t

(a)

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

(b)

ATO

MIC

PR

OFI

LE

x'

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

(c)

ATO

MIC

PR

OFI

LE

x'-15 -10 -5 0 5 10 15

0.0

0.2

0.4

0.6

(d)

ATO

MIC

PR

OFI

LE

x'

(neglecting radiation diffraction..)

(b)

(c) (d)

tNg x/σ

x/σx/σ

GOOD-CAVITY REGIME (K=0, D=0) and η=0.1

radiation intensity atomic density

x/σ

tNg tNg

x/σ

GOOD-CAVITY REGIME:effects of diffraction and cavity damping

atomic density

η=0.02, D=10η , K=0

atomic density

η=0.02, D=0 , K=10η

tNg tNg

x/σ x/σ

SUPERRADIANT REGIME (K>>D>1)

12

012

'x1

02

102

'x0

c|c|c'i't

c

c|c|c'i't

c

−−−

−

+∇η=∂

∂

−∇η=∂∂

NgKcav >>

cavR LZ >> 0*1cc

K1A −≈

( )1K >>

( )DK >>

G'Gt't ⊥ω=η=

cav

2

KNgG = (SR gain) (in free space Kcav~c/La)

⎟⎠⎞

⎜⎝⎛

σ=ω⊥ 2m2

h

SUPERRADIANT REGIME

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

ENER

GY

t'

(a)

-3 -2 -1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

(b)

DEN

SITY

x'0 20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

EN

ER

GY

t'

(a)

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

(b)

DE

NS

ITY

x'

ADIABATIC LIMIT

K>>D (ZR>>L)

η’=0.005

NON ADIABATIC LIMIT

K=D=1 (L=ZR)

η’=0.002

x/σ x/σGt Gt

Entanglement in CARL

a=a3

c0=√N

c1=a2

c-1=a1

a=a3

pump

BEC a=a3 pump

( )[ ] aa.c.hccaigccnHn

1nnnnr2 ++

++++ Δ−−+ω= ∑

( ) ( ) ( )[ ].c.haaaigaaaaH 32111r22r −++ω−Δ−ω+Δ= ++++

CARL Hamiltonian

linear approximation: ti3

ti12,1 eaa,eca Δ−Δ± == m

N. Piovella, M. Cola and R. Bonifacio, PRA 67 (2003) 013817

three-mode entanglement:

∑∞

=

φ+φ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛++

=Ψ0

2

1

22

1

3

1

32

1111

n,m

)mn(i/n/m

m,n,nm!n!m)!nm(e

NN

NN

N)t( ( )iii aaN +=

(N3=N1-N2)

in classical regime: N1 ~ N2 >> N3

∑∞

=

φ⎟⎟⎠

⎞⎜⎜⎝

⎛++

=Ψ0

2

1

2

1

011

1n

in/n

,n,neN

NN

)t(

in quantum regime: N1 ~ N3 >> N2

∑∞

=

φ⎟⎟⎠

⎞⎜⎜⎝

⎛++

=Ψ0

2

1

3

1

011

1m

im/m

m,,meN

NN

)t(

atom-atom entanglement

atom-photon entanglement

[ ] { } ρκ+ρ+γ+ρ−=τρ ˆ]a[Lˆ]a[L]a[Lˆ,Hi

dˆd

321 22

Entanglement is robust againstdecoherence (γ) and cavity damping (κ)

We have solved the MASTER EQUATION transforming it in an equation for

a three-mode WIGNER FUNCTION ( )321 ααα ,,W

Then, we have calculated numerically the COVARIANCE MATRIX C (6x6)

ρω=κ

ρωΓ

=γr

cav

r

edecoherenc K,22

M.M. Cola, M.G.A. Paris and N, Piovella PRA 70 (2004) 043809

Entanglement and separability criteria havebeen applied to the covariance matrix C(τ)

ηi>0 : separability of the i-th mode from the other two modes

Classical regime ρ=100 and Δ=0

Classical regime ρ=100 and Δ=0

Quantum regime ρ=0.2 and Δ=ωr

QuQuantum AAtoms in Milan

Nicola Piovella, Luca Volpe, Mary M. ColaUniversity of Milan

Gordon R. M. Robb, SUPA, Univ. df StrathclydeGlasgow

Rodolfo Bonifacio, CBPF, Rio & INFN

For our publications visit www.mi.infn.it\quiquoqua