quantum phase transitions in optical cavity...
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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