The Theory of Effective Hamiltonians for Detuned Systems
Universität Ulm, 18 November 2005
Daniel F. V. JAMESDepartment of Physics, University of Toronto,60, St. George St., Toronto, Ontario M5S 1A7, CANADAEmail: [email protected]
2/19
Detuned Systems• Example: Two level system, detuned field
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detuning Δ
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S€
D
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ˆ H I = hΩ2
D S e−iΔt + h.a.
• Interaction Picture Hamiltonian:
• BUT: we know what really happens is the A.C. Stark shift, i.e.:
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ˆ H eff = −hΩ 2
4ΔD D − S S( )
• Is there a systematic way to get Heff from HI (preferably
without all that tedious mucking about with adiabatic elimination)?
3/19
Time Averaged Dynamics: Definitions• Unitary time evolution operator
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ψ t( ) = ˆ U t,t0( ) ψ t0( )
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1 2 3
Filter Function (real valued)
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ih ∂∂t
ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( ) (1)
• Time-Averaged evolution operator
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ˆ U t,t0( ) = f t − ′ t ( ) ˆ U ′ t ,t0( )d ′ t −∞
∞
∫ (2)
• Define the effective Hamiltonian by:
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ih ∂∂t
ˆ U t,t0( ) = ˆ H eff t( ) ˆ U t,t0( ) (3)
4/19
General Expression I
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ih ∂∂t
ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( )
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⇒ ih ∂∂t
ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( )
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⇒ ˆ H eff t( ) ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( ) (4)
• Use a perturbative series for U and Heff:
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ˆ U t, t0( ) = λn ˆ V n t( )n=0
∞
∑ ; ˆ V n+1 t( ) = 1ih
ˆ H I ′ t ( ) ˆ V n ′ t ( )d ′ t t
∫ ; ˆ V 0 t( ) = ˆ I
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ˆ H I t( ) ˆ V n t( ) = ˆ W n− p t( ) ˆ V p t( )p=0
n
∑ (5)
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ˆ H eff t( ) = λn ˆ W n t( )n=0
∞
∑
5/19
General Expression II
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ˆ H I t( ) ˆ V n t( ) = ˆ W n− p t( ) ˆ V p t( )p=0
n
∑ (5)
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n = 0 : ˆ W 0 t( ) = ˆ H I t( ) (6a)
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n =1: ˆ H I t( ) ˆ V 1 t( ) = ˆ W 1 t( ) + ˆ W 0 t( ) ˆ V 1 t( )
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⇒ ˆ W 1 t( ) = ˆ H I t( ) ˆ V 1 t( ) − ˆ H I t( ) ˆ V 1 t( ) (6b)
etc...
6/19
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ˆ H eff t( ) = ˆ H I t( ) + ˆ H I t( ) ˆ V 1 t( ) − ˆ H I t( ) ˆ V 1 t( ) + ...
• Hamiltonians have to be Hermitian!
where
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HP ˆ A { } = 12
ˆ A + ˆ A †( )
• This is easy to fix:
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ˆ H eff t( ) =HP ˆ H I t( ) + ˆ H I t( ) ˆ V 1 t( ) − ˆ H I t( ) ˆ V 1 t( ) +...{ } (7)
• This can be justified by deriving a master equation:– excluded part of the frequency domain takes role of reservoir;– Lindblat equation with unitary part given by (7);– Neglect dephasing effects.
What’s wrong with this result?
7/19
General Expression III
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• Definition of a real averaging process implies:
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ˆ H I t( )†
= ˆ H I t( ) ˆ V 1 t( ) = 1ih
ˆ H I ′ t ( )d ′ t t
∫ ⇒ ˆ V 1 t( )†
= − ˆ V 1 t( )
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and so, (AT BLOODY LAST):
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ˆ H eff t( ) = ˆ H I t( ) + 12
ˆ H I t( ), ˆ V 1 t( )[ ] − ˆ H I t( ), ˆ V 1 t( )[ ] +...( ) (8)
• Result is independent of lower limit in integral for V1(t).
• This is NOT a perturabtive theory.-YES, we have used perturbation theory with reckless abandon, BUT -Solving Schrödinger’s equation with this Hamiltonian gives a result that involves all orders of the perturbation parameter
• Also applies statistical averages over a stationary ensemble.
8/19
9/19
Harmonic Hamiltonians + Low Pass Filter• Suppose we have a Hamiltonian made up of a sum of harmonic terms:
• And the time averaging has the effect of removing all frequencies ≥ min{m}, so that
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ˆ H I t( ) = 0
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ˆ V 1 t( ) = 0
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⎫ ⎬ ⎭ on the hole, looks rather boring
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ˆ H I t( ) = ˆ h me−iωmt + h.a.m∑ ωm > 0( ) (9a)
important special case:
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ˆ V 1 t( ) = 1ih
ˆ H I ′ t ( )d ′ t t
∫ = 1hωm
ˆ h me−iωmt − ˆ h m† eiωmt
( )m∑ (9b)
10/19
€
12hn
ˆ h m , ˆ h n[ ]e−i ωm+ωn( )t − ˆ h m , ˆ h n†
[ ]e−i ωm−ωn( )t
{m,n∑
+ ˆ h m† , ˆ h n[ ]e
i ωm−ωn( )t − ˆ h m† , ˆ h n
†[ ]e
i ωm+ωn( )t}
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ˆ H eff t( ) = ˆ H I t( ) + 12
ˆ H I t( ), ˆ V 1 t( )[ ] − ˆ H I t( ), ˆ V 1 t( )[ ] +...( )Eq.(8):
0 0 0
0
0
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ˆ H eff t( ) = 1hωmnm,n
∑ ˆ h m† , ˆ h n[ ]e
i ωm−ωn( )t (10)
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1mn
= 12
1ωm
+ 1ωn
⎛ ⎝ ⎜
⎞ ⎠ ⎟where:
Ref: D. F. V. James, Fortschritte der Physik 48, 823-837 (2000); Related results: Average Hamiltonians (NMR); C. Cohen-Tannoudji J Dupont-Roc and G. Grynberg, Atom-Photon Interactions (Wiley, 1992), pp. 38-48.
11/19
Example 1: AC Stark Shifts
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detuning Δ
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S€
D
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ˆ H I = hΩ2
D S e−iΔt + h.a.
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ˆ H eff = −hΩ02
4ΔD D − S S( )
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ˆ h 1 = hΩ2
D S ; ω1 = Δi.e.:
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ˆ h 1†, ˆ h 1[ ] = h2 Ω 2
4S D , D S[ ] ; ω11 ≡ ω1 = Δ
12/19Raman Transitions
A.C. Stark shifts (again!)
Example 2: Raman Processes
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Δ1
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S
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′ S €
P
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Ω1
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Δ2
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Ω2
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ˆ H I = hΩ12
P S e−iΔ1t
+ hΩ22
P ′ S e−iΔ2t + h.a.
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=−hΩ1
2
4Δ1P P − S S( ) − hΩ2
2
4Δ2P P − ′ S ′ S ( )
+ hΩ1*Ω2
4ΔS ′ S ei Δ1−Δ2( )t + h.a.
⎛
⎝ ⎜
⎞
⎠ ⎟
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ˆ h 1
1 2 4 3 4
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ˆ h 2
1 2 4 3 4
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ˆ H eff = 1hω1
ˆ h 1†, ˆ h 1[ ] + 1
hω2
ˆ h 2†, ˆ h 2[ ] + 1
hω12
ˆ h 1†, ˆ h 2[ ]e
i ω1−ω2( )t + h.a. ⎛ ⎝ ⎜
⎞ ⎠ ⎟
13/19
Wales’s Grand Slam, 20055th February 2005 Wales 11 - 9 England12th February 2005 Italy 8 - 38 Wales26th February 2005 France 18 - 24 Wales13th March 2005 Scotland 22 - 46 Wales19th March 2005 Wales 32 - 20 Ireland
14/19
“job security factor”: D.F.V. James, Appl. Phys. B 66, 181 (1998).
Example 3: Quantum A.C. Stark Shift
C. d’Helon and G. Milburn, Phys. Rev. A 54, 5141 (1996); S. Schneider et al., J. Mod Opt. 47, 499 (2000); F. Schmidt-Kaler et al, Europhys. Lett. 65, 587 (2004).
one trapped ion
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S€
D
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ˆ z t( )
laser
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ˆ H I t( ) = hΩ2
D S eikz ˆ z t( )−iΔt + h.a.
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{
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kz ˆ z t( ) = ηN
smp ˆ a pe−iωpt + ˆ a p
†eiωpt( )
p=1(all modes)
N
∑
15/19
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eikz ˆ z t( ) ≈ 1+ ikz ˆ z t( )• Lamb-Dicke approximation:
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ˆ h 1 = hΩ2
D S• “carrier” term
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1 =Δ
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ˆ h 2 = iηhΩ2 N
smp D S ˆ a p
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2 =Δ+p• red sideband:
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ˆ h 3 = iηhΩ2 N
smp D S ˆ a p
†• blue sideband:
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3 =Δ−p
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Heff = − hΩ 2
4Δ1+ 2η 2
Nsm
p( )
2 Δ2
Δ2 −ωp2 np + 1
2( ) ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ D D − S S( )
• low pass filter excludes oscillations at p, hence:
16/19
What about two ions?
big-ass laser
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ˆ H I t( ) = hΩ2
D S 1eikz ˆ z 1 t( ) + D S 2eikz ˆ z 2 t( )( )e
−iΔt + h.a.
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ˆ z 1 t( )
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ˆ z 2 t( )
17/19
new term: wasn’t there for single ion
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ˆ h 1 = hΩ2
ˆ J (+), ˆ h 2 = iηhΩ2 N
ˆ J (+) ˆ a p , ˆ h 3 = iηhΩ2 N
ˆ J (+) ˆ a p†
• “carrier”, red and blue sideband terms:
• nearly resonant with the CM (p=1) mode
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smp=1 =1( )
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ˆ J (+) = D S 1 + D S 2( )• Define a collective spin operator
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1Δ+1( )
ˆ J (−) ˆ a 1†, ˆ J (+) ˆ a 1[ ] + 1
Δ −ω1( )ˆ J (−) ˆ a 1, ˆ J (+) ˆ a 1
†[ ] =
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2ΔΔ2 −1
2( )
ˆ n 1 + 12( ) ˆ J (−), ˆ J (+)
[ ] − 2ω1
Δ2 −ω12
( )ˆ J (−), ˆ J (+){ }
18/19
Couples the two ions: VERY INTERESTING!!!
Quantum A.C. Stark shift again: BORING!
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Heff = −hΩ 2
4Δ1+ 2η 2Δ2
N Δ2 −ω12
( )n1 + 1
2( ) ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ D D m − S S m( )m∑
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−hΩ 2h21
2 Δ2 −12
( )ˆ J (−), ˆ J (+){ }
Hence the effective Hamilton is
• Add another laser (with negative detuning): Quantum A.C. Stark shifts cancel, but coupling term is doubled: Mølmer-Sørensen gåtë
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ˆ J (−), ˆ J (+){ } = 2 ˆ I + ˆ σ x(1) ˆ σ x
(2) + ˆ σ y(1) ˆ σ y
(2)• Take a closer butchers at the coupling term and it looks like spin-spin coupling: Quantum Simulations
19/19
Conclusions
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ˆ H eff t( ) = 1hωmnm,n
∑ ˆ h m† , ˆ h n[ ]e
i ωm−ωn( )t
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1mn
= 12
1ωm
+ 1ωn
⎛ ⎝ ⎜
⎞ ⎠ ⎟where:
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ˆ H I t( ) = ˆ h me−iωmt + h.a.m∑
• The time-averaged dynamics of a system with a harmonic Hamiltonian of the form:
Is described by an effective Hamiltonian given by:
• Quantum Simulations are a lot easier than Porras and Cirac said.
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