Quantum Phase Transitionsin Optical Cavity QED

Felipe DimerHoward CarmichaelSPBenoit Estienne (Paris)Sarah Morrison (Innsbruck)

• Single-mode Dicke model– equilibrium phase transition– T=0 quantum phase transition

• Proposed realisation in optical cavity QEDDimer, Estienne, Parkins & Carmichael, PRA 75, 013804 (2007)

– Raman transition scheme– open system dynamics – non-equilibrium phase transition– monitoring the system: cavity output field– critical behaviour of quantum entanglement

• Other possibilities for effective spin systems

Outline

• N two-level atoms at fixed positions in a cavity of volume V(constant coupling strength)

• Inter-atomic separations large ⇒ neglect direct interactionsbetween atoms

• However, the atoms interact with the same radiation field ⇒ they cannot be treated as independent, must be treated as a

single quantum systemDicke, Phys. Rev. 93, 99 (1954)

Dicke Model

The Single-Mode Dicke Model

• N two-level atoms coupled identically to a single EM field mode

• Coupling constant

• Collective atomic operators!

HDicke

=" a+a +"

0Jz

+#

Na + a+( ) J$ + J +( )

!

" #N

V

!

J" = 0

i

i=1

N

# 1i, J

z=1

21i1i" 0

i0i( )

i=1

N

#

!

0

!

1

Hepp & Lieb, Phys. Rev. A 8, 2517 (1973)Hioe, Phys. Rev. A 8, 1440 (1973)Carmichael, Gardiner & Walls, Phys. Lett. 46A, 47 (1973)

• Phase transition to superradiant state for

!

" > "c

=##

0

2, T < T

c where

##0

4"2= tanh

#0

2kBTc

$

% &

'

( )

Phase Transition in the Dicke Model

!

Thermodynamic limit

N,V "#, N V finite

“Order Parameters” (T=0)

(Dashed lines: finite atom number, N=1,2,3,6,10)

Mean atomicinversion(ground state)

Mean photonnumber

But … no equilibrium phase transition with A2 termincluded

Emary & Brandes, Phys. Rev. E 67, 066203 (2003)

• Holstein-Primakoff representation of angular momentumoperators

• Large-N expansion of HDicke

→ Hnormal, HSR quadratic in (a,a+,b,b+) → diagonalise (Bogoliubov transformation) → excitation energies

!

J" = N " b

+b( ) b, J

z= b+

b "N

2, b,b

+[ ] =1

Dicke Model Quantum Phase Transition (T=0)

!

HDicke

=" a+a +"

0Jz

+#

Na + a+( ) J$ + J +( )

!

" ="0

=1, #c

= 0.5( ) Excitation Energies

!

" < "c

: J# $ N b

" > "c

: a$ a ±%, b$ b ± & (coherent displacements)

then expand in N

(i.e. linearisation about semiclassical amplitudes)

a+a = %

2, J

z= &

2#N

2

Note: Derivation of {Hnormal, HSR}

!

" x,y( )2

!

" "c

= 0.4

!

" "c

=1.0

!

" "c

=1.2

!

" "c

=1.4

Transition fromlocalised stateto delocalised“Schrödinger Cat”state

Note: Ground State “Wave Function”

!

"g ~ # $N 2x

+ $# N 2x

where

Jx ±N 2x

= ±N 2 ±N 2x

(N = 10 atoms)

Critical behaviour of atom-field and atom-atomquantum entanglement at transition

Lambert, Emary & Brandes, Phys. Rev. Lett. 92, 073602 (2004)

Entanglement properties

Issues to confront:• To date, λ << {ω, ω0} in cavity QED experiments• Atomic spontaneous emission, cavity mode losses• And the A2 issue

Our approach:• Raman scheme, {ω, ω0}∝{level shifts, Raman detunings},

λ ∝ Raman transition rate• Open-system dynamics

⇒ non-equilibrium (dynamical) quantum phase transition

Possible Realisation?

Dimer, Estienne, Parkins & Carmichael, PRA 75, 013804 (2007)

Possible Realisation in Optical Cavity QED

• N atoms identically coupled to single optical (ring) cavity mode

• Lasers + cavity field drive two distinct Raman transitionsbetween stable ground states |0> and |1>

Model: Adiabatic elimination of atomic excited states

!

H = "cav

+1

2N

gr2

# r

+gs2

# s

$

% &

'

( )

*

+ ,

-

. / a

+a +

gr2

# r

0gs2

# s

$

% &

'

( ) a

+aJz

+1r

2

4# r

01s

2

4# s

+ 2 " $

% &

'

( ) Jz

+gr1r

2# r

aJ+ + a+

J0( ) +

gs1s

2# s

a+J

+ + aJ0( )

Effective Hamiltonian (rotating frame)

Choose

!

gs2

" s

=gr2

" r

,gr#r

2" r

=gs#s

2" s

then …

!

"cav

=#cav$

1

2#

Ls+#

Lr( )

% " =#1$

1

2#

Ls$#

Lr( )

Effective (Dissipative) Dicke Model

!

˙ " = #i H,"[ ] +$ 2a"a+ # a+a" # "a+

a( )

!

H =" a+a +"

0Jz+

#

Na + a+( ) J$ + J +( )

!

" = #cav

+Ngr

2

$ r

, "0

= % # , & =Ngr'r

2$ r

Master equation for atom-field density operator ρ :

where

with

!

" tunable" such that " ~ "0~ #

Potential Experimental Parameters?

• Ring cavity / many atoms (e.g., Tübingen, Hamburg, 85Rb)

• Strong coupling CQED / few atoms (e.g., Georgia Tech, 87Rb)!

gi 2" # 50 kHz, $ 2" # 20 kHz, N #106

!

"i

#i

$ 0.005 %&

2'$ N ( 0.125 kHz $125 kHz

!

gi 2" # 30MHz, $ 2" # 2MHz, N #100

!

"i

#i

$ 0.05 %&

2'$ N ( 0.75 MHz $ 7.5 MHz

e.g., ring cavity + 87Rb + magnetic field

Holstein-Primakoff Analysis (N∞) : Normal Phase

!

˙ " = #i H(1)

,"[ ] +$ 2a"a+ # a+a" # "a+

a( ) with

H(1) =%a+

a +%0b+b + & a + a+( ) b + b+( )

for

& < &c

=1

2

%0

%$ 2 +% 2( ) κ = 0.1, 0.2, 0.5

Holstein-Primakoff Analysis (N∞) : Superradiant Phase

!

˙ " = #i H(2)

,"[ ] +$ 2c"c + # c +c" # "c +

c( ) with

H2( ) =%c +

c +%0

2µ1+ µ( )d+

d +%0 1#µ( ) 3 + µ( )

8µ 1+ µ( )d

+ + d( )2

+ &µ2

1+ µc

+ + c( ) d+ + d( ), µ =&c

2

&2

for

& > &c

=1

2

%0

%$ 2 +% 2( )

!

a" c ±# , b" d m $

Field and Atomic Amplitudes α and β

!

" = ±#$

0

2#c

2

N

41%

#c

4

#4&

' (

)

* + 1+ i

,

$

&

' (

)

* + , - = m

N

21%

#c

2

#2&

' (

)

* +

Spectra of the Light Emitted from the Cavity

Cavity fluorescence Probe transmission

Eigenvalues of the linearised model

Probe transmission spectra (ω = ω0 = 1, κ = 0.2)

Energyeigenvalues

Homodyne detection (quadrature fluctuation spectra)

λ↓

~λc

entanglement

Atom-field entanglement

!

"u( )2

+ "v( )2

<1

!

Xa

" =1

2ae

# i" + a+ei"( ), X

b

$ =1

2be

# i$ + b+ei$( )

u = Xa

" + Xb

$, v = X

a

" +% 2 # Xb

$ +% 2

Gaussian continuous variable state: quadrature/EPR operators

Possible to deduce fromcavity output field

Other possibilities for effective spin systems

• Two cavity modes + off-resonant Raman transitions• Effective spin-spin interactions:

(Lipkin-Meshkov-Glick model)

• λ » dissipative rates possible

• 1st or 2nd order quantum phase transitions

!

Heff

= "2hJz "2#

NJx2 + $Jy

2( ), "1< $ <1

Example:(“antiferromagnetic”)

!

Heff

= "2hJz"

2#

NJx

2, # < 0

!

˙ " = #i Heff

,"[ ] +4$

a

N2J

x"J

x# J

x

2" # "Jx

2( ) +$b

N2J#"J+ # J+J#" # "J+J#( )

!

" = #1, $a

= 0.01, $b

= 0.2

1st-order quantum phase transition

Probe transmission spectrum

!

Jz

N 2

h=0.8

h=0.01

!

C" =1#4

N$J"

2 #4

N2J"

2

> 0, J" = Jx sin" + Jy cos"

!

CR

=max"C"

Bipartite entanglement criterion / spin variances

Time-dependence of entanglement, CR(t)

Note: Cavity output field bout∝J– so CR can bededuced from measurable correlation functions

Summary

• Proposed realisation of Dicke model in cavity QED for study of(non-equilibrium) quantum phase transition

• Well-defined cavity output provides measurablesignatures/properties of the phase transition

• Other effective spin models possible

Further possibilities …

• Finite-N systems– small → large quantum noise– entangled state preparation

and characterisation– measurement back-action

• Combination with optical-lattice many-body systems(long-range + short-range interactions)

• Disordered systems