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: dfvj@physics.utoronto.ca

2/19

Detuned Systems• Example: Two level system, detuned field

€

detuning Δ

€

S€

D

€

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.:

€

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

€

ψ t( ) = ˆ U t,t0( ) ψ t0( )

€

1 2 3

Filter Function (real valued)

€

ih ∂∂t

ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( ) (1)

• Time-Averaged evolution operator

€

ˆ U t,t0( ) = f t − ′ t ( ) ˆ U ′ t ,t0( )d ′ t −∞

∞

∫ (2)

• Define the effective Hamiltonian by:

€

ih ∂∂t

ˆ U t,t0( ) = ˆ H eff t( ) ˆ U t,t0( ) (3)

4/19

General Expression I

€

ih ∂∂t

ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( )

€

⇒ ih ∂∂t

ˆ U t,t0( ) = ˆ H I t( ) ˆ U t,t0( )

€

⇒ 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:

€

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

€

ˆ H I t( ) ˆ V n t( ) = ˆ W n− p t( ) ˆ V p t( )p=0

n

∑ (5)

€

ˆ H eff t( ) = λn ˆ W n t( )n=0

∞

∑

5/19

General Expression II

€

ˆ H I t( ) ˆ V n t( ) = ˆ W n− p t( ) ˆ V p t( )p=0

n

∑ (5)

€

n = 0 : ˆ W 0 t( ) = ˆ H I t( ) (6a)

€

n =1: ˆ H I t( ) ˆ V 1 t( ) = ˆ W 1 t( ) + ˆ W 0 t( ) ˆ V 1 t( )

€

⇒ ˆ W 1 t( ) = ˆ H I t( ) ˆ V 1 t( ) − ˆ H I t( ) ˆ V 1 t( ) (6b)

etc...

6/19

€

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

€

HP ˆ A { } = 12

ˆ A + ˆ A †( )

• This is easy to fix:

€

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

€

• Definition of a real averaging process implies:

€

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( )

€

and so, (AT BLOODY LAST):

€

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

€

ˆ H I t( ) = 0

€

ˆ V 1 t( ) = 0

€

⎫ ⎬ ⎭ on the hole, looks rather boring

€

ˆ H I t( ) = ˆ h me−iωmt + h.a.m∑ ωm > 0( ) (9a)

important special case:

€

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}

€

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

€

ˆ H eff t( ) = 1hωmnm,n

∑ ˆ h m† , ˆ h n[ ]e

i ωm−ωn( )t (10)

€

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

€

detuning Δ

€

S€

D

€

ˆ H I = hΩ2

D S e−iΔt + h.a.

€

ˆ H eff = −hΩ02

4ΔD D − S S( )

€

ˆ h 1 = hΩ2

D S ; ω1 = Δi.e.:

€

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

€

Δ1

€

S

€

′ S €

P

€

Ω1

€

Δ2

€

Ω2

€

ˆ H I = hΩ12

P S e−iΔ1t

+ hΩ22

P ′ S e−iΔ2t + h.a.

€

=−hΩ1

2

4Δ1P P − S S( ) − hΩ2

2

4Δ2P P − ′ S ′ S ( )

+ hΩ1*Ω2

4ΔS ′ S ei Δ1−Δ2( )t + h.a.

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ˆ h 1

1 2 4 3 4

€

ˆ h 2

1 2 4 3 4

€

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

€

S€

D

€

ˆ z t( )

laser

€

ˆ H I t( ) = hΩ2

D S eikz ˆ z t( )−iΔt + h.a.

€

{

€

kz ˆ z t( ) = ηN

smp ˆ a pe−iωpt + ˆ a p

†eiωpt( )

p=1(all modes)

N

∑

15/19

€

eikz ˆ z t( ) ≈ 1+ ikz ˆ z t( )• Lamb-Dicke approximation:

€

ˆ h 1 = hΩ2

D S• “carrier” term

€

1 =Δ

€

ˆ h 2 = iηhΩ2 N

smp D S ˆ a p

€

2 =Δ+p• red sideband:

€

ˆ h 3 = iηhΩ2 N

smp D S ˆ a p

†• blue sideband:

€

3 =Δ−p

€

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

€

ˆ H I t( ) = hΩ2

D S 1eikz ˆ z 1 t( ) + D S 2eikz ˆ z 2 t( )( )e

−iΔt + h.a.

€

ˆ z 1 t( )

€

ˆ z 2 t( )

17/19

new term: wasn’t there for single ion

€

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

€

smp=1 =1( )

€

ˆ J (+) = D S 1 + D S 2( )• Define a collective spin operator

€

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

†[ ] =

€

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!

€

Heff = −hΩ 2

4Δ1+ 2η 2Δ2

N Δ2 −ω12

( )n1 + 1

2( ) ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ D D m − S S m( )m∑

€

−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

€

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

€

ˆ H eff t( ) = 1hωmnm,n

∑ ˆ h m† , ˆ h n[ ]e

i ωm−ωn( )t

€

1mn

= 12

1ωm

+ 1ωn

⎛ ⎝ ⎜

⎞ ⎠ ⎟where:

€

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.