The Theory of Effective Hamiltonians for Detuned Systems

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.