On the Controllability of spin-spring quantum systems

On the Controllability of spin-spring quantum systems

Pierre Rouchon École des Mines de Paris, pierre.rouchon@ensmp.fr

Sao Paulo, December 2006

A group around Paris investigating this subject:

Karine Beauchard, Jean-Michel Coron, Claude Le Bris, Mazyar Mirrahimi, Jean-Pierre Puel, Gabriel Turinici, . . .

ACI Simulation Moléculaire 2003-2006 ANR Project CQUID 2007-2009

Outline

I Non controllability of the quantum harmonic oscillator (spring). (Classical sources generate only quasi-classical light)

I Two-states atom (spin) coupled with a resonant electro-magnetic cavity mode: the controlled Jaynes-Cummings Hamiltonian.

I Similar systems: several trapped ions controlled via lasers;

qubit and quantum gate.

I Quantum Monte-Carlo trajectories, stochastic control and feedback.

Le système atome-cavité

Spin 1/2 couplé à un

oscillateur harmonique

f(r) ≈ e⁻

x²+y² w² cos(2π^z

λ ⁾

Source: S. Haroche, cours au collège de France.

9

a⁺= mω 2x− i

2mω^p a= mω

2x+ i 2mωp

a,a⁺

[ ]

⎠ ⎟

x0 = mω π

⎛

⎝ ⎜ ⎞

⎠ ⎟

1/ 4

e⁻

mω 2x²

Δx=

2mω;Δp= mω

2 → ΔxΔp= 2

n =(a⁺)ⁿ n! 0

Paquet d ’onde gaussien « minimal »

Rappels sur l ’oscillateur harmonique

Opérateurs d ’annihilation et de création de quanta H = p ²/ 2m + mωωωω²x²/ 2

Etat fondamental |||| 0000 >>>>de l ’oscillateur (énergie ω/2)ω/2)ω/2)ω/2)

Etat excité à n quanta (énergie : (n+1/2) ωωωω

Source: S. Haroche, cours au collège de France.

Quantification of the controlled harmonic oscillator

Classical dynamics: _dt^d2₂x=−x+u

d

dtx =p= ∂H

∂p, d

dtp=−x+u=−∂H

∂x withH(x,p,t) = ¹₂(p²+x²)−u(t)x.

Quantification: x 7→X,p7→P=−ı_∂x^∂ , the Hamiltonian becomes an operator

H= 1

2(P²+X²)−u(t)X =−1 2

∂²

∂x² +1

2x²−u(t)x (~=1 here) and the Shrödinger equation

ıd

dtψ=Hψ.

describes the time evolution ofψ, the probability amplitude.

Quantification of ... (end)

ıd

dtψ=Hψ Probability amplitudeψ(t,x)∈C,R+∞

−∞ |ψ(t,x)|²dx =1 obeys the Shrödinger equation:

ı∂ψ

∂t(x,t) =Hψ=−1 2

∂²ψ

∂x²(x,t)+x²ψ(x,t)−u(t)xψ(x,t), x ∈R The averaged position:

X¯(t) =hψ|X|ψi= Z +∞

−∞

x|ψ|²dx,

The averaged impulsion:

P(t) =¯ hψ|P|ψi=−ı Z +∞

−∞

ψ^∗∂ψ

∂xdx

The language of operators

ı∂ψ

∂t = (1

2(P²+X²)−uX)ψ=−1 2

∂²ψ

∂x² +x²ψ−uxψ With theannihilationandcreationoperators

a= X+ıP

√

2 = 1

√ 2

x+ ∂

∂x

, a^†= X−ıP

√

2 = 1

√ 2

x− ∂

∂x

we get (u/√ 27→u)

[a,a^†] =1, H =a^†a+1

2 −u(a+a^†).

The spectral decomposition of a

a

The Hermitian operatora^†aadmitsNas non-degenerate spectrum. The eigen-vector associated ton∈Nis denoted|ni (Fock state,nbeing the number of vibration quanta ):

a|ni=√

n|n−1i, a^†|ni=√

n+1|n+1i and|0i(a|0i=0) corresponds to a Gaussian distribution:

ψ0(x) = 1

π^1/4exp(−x²/2) Remember that

a= X +ıP

√

2 = 1

√ 2

x+ ∂

∂x

, a^†= X −ıP

√

2 = 1

√ 2

x − ∂

∂x

Modal approximation is controllable

ıd

dtψ= (a^†a+1/2)−u(a+a^†)ψ Truncation ofψ=P+∞

n=0c_n(t)|nito orderN leads to

ıd dt

0 B B B B B B

@ c₀ c₁ .. . c_N−1

c_N 1 C C C C C C A

=

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 B B B B B B B B B B

@ 1

2 0 . . . 0

0 ³₂ 0

0 ⁵₂ 0

.. . ..

.

... 0

0 . . . 0 ^2N+1₂

1 C C C C C C C C C C A

−u 0 B B B B B B B B B B B B B B B B

@

0 1 0 . . . 0

1 0 √

2 ...

.. .

0 √

2 0 √

3

.. . ..

. √

3

... 0

.. .

...

... √

N+1

0 . . . 0 √

N+1 0

1 C C C C C C C C C C C C C C C C A 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 0 B B B B B B

@ c₀ c₁ .. . c_N−1

c_N 1 C C C C C C A

Infinite dimensional system not controllable

ıd

dtψ=(a^†a+1/2)−u(a+a^†)ψ This system reads formallyı_dt^dψ= (H₀+uH₁)ψ. Its

controllability is given formally by the Lie algebra spanned by the skew Hermitian operatorsıH₀andıH₁. Using[a,a^†] =1, [a^†a,a+a^†] =a^†−a, [a^†a,a^†−a] =a+a^†, [a^†+a,a^†−a] =2.

we get a Lie algebra of dimension 4 containing ıa^†a, ı(a+a^†), a−a^†, ıI_d

Controllable and un-controllable part

ıd

dtψ= (a^†a+1/2)−u(a+a^†)ψ Setα=hψ|a|ψi ∈Cthen

ιd

dtα =hψ|[a,H]|ψi=α−u and consider the unitary operator

T(t) =exp h

α^∗(t)a−α(t)a^†i .

IfAandBcommute with[A,B](Glauber):

exp(A+B) =exp(A)exp(B)exp(−[A,B]/2), we have T(t)aT^†(t) =a+α(t), T(t)a^†T^†(t) =a^†+α^∗(t) Takeφ=T(t)ψ, then

ιd dtφ=

a^†a+1 2

φ+h

|α|²−u(α+α^∗)i φ

Decomposition .... (end)

ιd dtφ=

a^†a+1 2

φ+h

|α|²−u(α+α^∗)i φ

With a global phase changeχ=e(^ıR0^t[|α|²−u(α+α^∗])φwe obtain the following control free Schrödinger equation:

ι d dtχ=

a^†a+1 2

χ.

Two parts, a controllable oneα∈C,ı_dt^dα =α−u, an uncontrollable oneχ, the quantum fluctuations aroundα.

These computations might be extended to an arbitrary number of harmonic oscillators admitting the same controlubut with different frequencies.

Classical EM fields generated by classical sources

Bounded smooth domainΩ⊂R³, scalar controluassociated to time-varying currents insideΩ. The Maxwell equation reads (perfect mirrors on the boundary,c=1):

∂²E~

∂t² (r,t) = ∆E~(r,t)+u(t)~J(r)forr ∈Ω, E~(r,t) =0 forr ∈∂Ω.

Modal decompositionE(r~ ,t) =P+∞

j=0 x_j(t)E~_j(r)leads to an infinite collection of controlled harmonic oscillators:

d²

dt²x_j =−ω_j²x_j+b_ju, b_j = Z

Ω

E~_j(r)·~J(r)dr.

Quantum EM fields generated by classical sources

d²

dt²x_j =−ω²_jx_j+b_ju Quantification based ona_j =

√ωjXj+√^ı

ωjPj

√

2 we get the following Hamiltonian (~=1):

H=

∞

X

j=0

[ω_j(a^†_ja_j+1

2)−b_ju(a_j+a^†_j)].

an operator on the "state space"N

jL²(R,C).

Controllability decomposition

H =

∞

X

j=0

[ωj(a^†_ja_j+1/2)−b_ju[a_j+a^†_j].

Setαj =

ψ|a_j|ψ then

ιd dtα_j =

ψ|[a_j,H]|ψ

=ω_jα_j −b_ju

TakeT =N

jT_j withT_j =exph

α^∗_jan−α_ja^†_ji . We haveTa_jT^†=a_j+α_j. Taking

φ=e^ı

Rt 0

hP

jω_j|α_j|²−b_ju(α_j+α^∗_j)i

T(t)ψ we get:

ıd dtφ=



 X

j

ω_j (a^†_ja_j+1 2)



φ.

Controllability decomposition (end)

The electro-magnetic field operator

E(r~ ,t) =X

j

a_j+a^†_j p2ω_j

−

→E_j(r)+−→ E(r,t)

is the sum of two terms: the vacuum fluctuation operatorsa_j that obey the autonomous dynamicsı_dt^da_j =ωja_j ; the classical field (scalar operators)E(r~ ,t) =P

jα_j(t)−→

E_j(r)solution of the classical wave equation

∂²E~

∂t² (r,t) = ∆E~(r,t)+u(t)~J(r)for r ∈Ω, E~(r,t) =0for r ∈∂Ω.

Control interpretation of a classical result (MR CDC/ECC 05):

classical motions of charges generate only quasi-classical (coherent) light (see Glauber, CDG2, ...).

Mode gaussien avec un diamètre de 6mm Grand champ par photon (1,5mV/m) Grande durée de vie du champ (1ms) allongée par l ’anneau autour des miroirs

Accord en fréquence facile Faible champ thermique ( < 0,1 photon )

La Cavité Fabry-Perot supraconductrice

Source: S. Haroche, cours au collège de France.

Le système atome-cavité

Spin 1/2 couplé à un

oscillateur harmonique

f(r) ≈ e⁻

x²+y² w² cos(2π^z

λ ⁾

Source: S. Haroche, cours au collège de France.

Two-states quantum systems (

-spin)

Ground state|giand excited state|ei:ψ= (ψ_g, ψ_e)∈C²à linear superposition of|giand|ei(ω₀the Bohr frequency of the transition).

ıd dt

ψ_e ψg

= ω0

2

1 0 0 −1

ψ_e ψg

=H₀ψ

withH₀= ^ω₂⁰(|ei he| − |gi hg|).

Interaction with classical electrical fieldu(t)∈R(µ∈Rdipole coefficient ....):

ıd dt

ψe

ψg

= ω₀

2

1 0 0 −1

+µu

0 1 1 0

ψe

ψg

= (H₀+u(t)H₁)ψ

withH₁=µ(|ei hg|+|gi he|).

Two-states system coupled to a resonant cavity mode

State space:L²(R,C)N C².

Cavity Hamiltonian (an isolated resonant modeω≈ω₀driven by a classical controlu):

Hc=ωa^†a−u(a+a^†) Two-states Hamiltonian

H_a= ω0

2 (|ei he| − |gi hg|) = ω0

2 σz

Interaction Hamiltonian (ΩRabi vacuum frequencyΩω, ω₀) H_int= Ω

2(a+a^†)(|ei hg|+|gi he|) = Ω

2(a+a^†)σx

System Hamiltonian

H_c+H_a+H_int

The rotating wave approximation (ω = ω

)

ıd

dtψ= (Hc+Ha+H_int)ψ Setψ=exp −ıωta^†a

exp(−ıωtσ_z)φ(interaction frame). The Hamiltionian becomes

−u(e^−ıωta+e^ıωta^†)+Ω

2(e^−ıωta+e^ıωta^†)(e^−ıωt|gi he|+e^ıωt|ei hg|) because

exp

ıωta^†a

aexp

−ıωta^†a

=e^−ıωta

exp(ıωtσz)σxexp(−ıωtσ_z) =e^−ıωt|gi he|+e^ıωt|ei hg|

The rotating wave approximation (end)

H=−u(e^−ıωta+e^ıωta^†)+Ω

2(e^−ıωta+e^ıωta^†)(e^−ıωt|gi he|+e^ıωt|ei hg|) Setu=ve^−ıωt +v^∗e^ıωt withva slowly varying complex

amplitude (new controlv∈C) we get neglecting oscillating termse^±2ıωt, the controlled Jaynes-Cummings Hamiltonian (in the interaction representation)

H_JC= Ω

2(a|ei hg|+a^†|gi he|)−(va^†+v^∗a)

PDE behind the Jaynes-Cummings controlled model

ıd

dtψ= Ω

2(a|ei hg|+a^†|gi he|)ψ−(va^†+v^∗a)ψ Sinceψ∈L²(R,C)N

C²andL²(R,C)N

C²∼(L²(R,C))²we representψby two componentsψgandψeelements of L²(R,C). Up-to some scaling:

ı∂ψg

∂t =

v₁x +ıv₂ ∂

∂x

ψg+

x+ ∂

∂x

ψe

ı∂ψ_e

∂t =

x− ∂

∂x

ψ_g+

v₁x +ıv₂ ∂

∂x

ψ_e

wherev=v₁+ıv₂∈Cis the control. Remember that

a= 1

√2

x+ ∂

∂x

, a^†= 1

√2

x− ∂

∂x

Another form of the control Jaynes-Cummings Hamiltonian

H_JC= Ω

2(a|ei hg|+a^†|gi he|)−(va^†+v^∗a) Setw∈Csuch thatı_dt^dw=−vand take the unitary transformationT(t) =exp

w^∗a−wa^†

.ThenH_JC becomes TH_JCT^†−ıTT˙^†and up-to a global phase change we get

H˜_JC = Ω

2(a+w)|ei hg|+ (a^†+w^∗)|gi he|)

wherew∈Cis the new control corresponding to the integral of the physical controlv.

Another PDE formulation

ıd

dtψ= Ω

2[(a+w)|ei hg|+ (a^†+w^∗)|gi he|)]ψ Up-to some scaling, withψ= (ψg,ψe):

ı∂ψg

∂t =

x+w₁+ ∂

∂x +ıw₂

ψe

ı∂ψe

∂t =

x+w₁− ∂

∂x −ıw₂

ψg

wherew =w₁+ıw₂∈Cis the control (time integral of the physical controlv the amplitude and phase modulations).

Elementary facts relative to controllability

The Jaynes-Cummings systems is equivalent to ıd

dtψ= (H₀+w₁H₁+w₂H₂)ψ with

H₀=Xσx −Pσ_y, H₁=σx, H₂=σy

and the commutations are

[X,P] =ı, σ²_x =1, σ_xσ_y =ıσ_z. . .

The Lie algebra spanned byıH₀,ıH₁andıH₂is infinite dimensional now.

But one can prove that thelinear tangent approximation around any eigen-state ofH₀+w₁H₁+w₂H₂for any control valuew₁andw₂isnot controllable.

Innsbruck linear ion trap

1.0m m

5m m

M H z

≈5

radial

ω M H z 2 7 . 0 −

≈

axial

ω

Source: F. Schmidt-Kaler, séminaire au Collège de France en 2004.

C irac & Zoller gate w ith tw o ions

Source: F. Schmidt-Kaler, séminaire au Collège de France en 2004.

2-level-atom harm onic trap

Laser coupling

dressed system

„m olecular Franck C ondon“

picture

...

...

dressed system

S n−1,

D

n−1, n,D n+1,D

S n+1, S

n,

„energy ladder“

picture

S D D

S

Source: F. Schmidt-Kaler, séminaire au Collège de France en 2004.

A single trapped ion controlled via a laser

H=Ω(a^†a+1/2) +ω0

2 (|ei he| − |gi hg|) +

h

ue^ı(ωt−kX)+u^∗e^{−ı(ωt−kX)}i

(|ei hg|+|gi he|) withkX =η(a+a^†),η 1 Lamb-Dicke parameter,ω≈ω₀and the vibration frequencyΩω,u∈Cthe control(amplitude and phase modulations of the laser of frequencyω).

Assumeω=ω₀. Inı_dt^dψ=Hψ, setψ=exp(−ıωtσ_z/2)φ. Then the Hamiltonian becomes

Ω(a^†a+1/2)+

h

ue^{ı(ωt−η(a+a†))}+u^∗e^{−ı(ωt−η(a+a†))} i

e^−ıωt|gi he|+e^ıωt|ei hg|

Rotating Wave Approximation and PDE formulation

We neglect highly oscillating terms withe^±2ıωt and obtain the averaged Hamiltonian:

H˜ = Ω(a^†a+1/2) +ue^{−ıη(a+a†)}|gi he|+u^∗e^ıη(a+a†)|ei hg|

This corresponds to (η7→η√ 2):

ı∂ψg

∂t = Ω 2

x²− ∂²

∂x²

ψg+ue^−ıηxψe

ı∂ψe

∂t =u^∗e^ıηxψ_g+Ω 2

x²− ∂²

∂x²

ψ_e whereu∈Cis the control

_dt^du

ω|u|andη1,Ωω.

Controllability of this system ?

Logique quantique avec une chaîne d’ions

Des impulsions laser appliqués séquentiellement aux

ions de la chaîne réalisent des portes à un

bit et des portes à deux bits. La détection par

fluorescence (éventuellement précédée par une rotation du bit) extrait l’information du système.

Beaucoup de problèmes à résoudre pour réaliser un tel dispositif…..

Source: S. Haroche, CdF 2006.

Quelques chaînes d‘ions (Innsbruck)

Fluorescence spatialement résolue. Détection en quelques millisecondes (voir première leçon) Source: S. Haroche, CdF 2006.

!_z

Mode du CM

Mode «accordéon»

!

3

Mode « ciseau » Z₁(t)=Z₁₀cos

( )

Z₂(t)=Z₂₀cos

(

)

Z₃(t)=Z₃₀cos

(

)

vrai pour les modes plus élevés.

Visualisation des modes ( N=3)

Source: S. Haroche, CdF 2006.

temps

position

Mode du centre de masse

Mode

“accodéon“:

élongation proportionnelle à la distance de l‘ion au centre de la chaîne

Les deux premiers modes de vibration sont indépendants du nombre d’ions dans la

chaîne (ici cas de 7 ions)

!z 3

!

Source: S. Haroche, CdF 2006.

Spectre résolu de l’ion interprété en terme de création/annihilation de phonons (Γ < ω_z)

g,1 g,2 e,0

e,1 e,2

g,0

Porteuse:

ω_L=ω_eg Δn=0 g,1

g,2 e,0

e,1 e,2

g,0

1ère bande latérale rouge:

ωL=ωeg−ω ; Δn=-1

e,0 e,1 e,2

g,1 g,2

g,0

1ère bande latérale bleue:

ω_L=ω_eg+ω ; Δn=+1 Source: S. Haroche, CdF 2006.

Two trapped ions controlled via two lasers ω ( phonons Ω of the center of mass mode only)

The instantaneous Hamiltionian:

H= Ω(a^†a+1/2) +ω₀

2 (|e₁i he₁| − |g₁i hg₁|) +h

u₁e^{ı(ωt−k(a+a†))}+u^∗₁e−ı(ωt−k(a+a^†))i

(|e₁i hg₁|+|g₁i he₁|) +ω0

2 (|e₂i he₂| − |g₂i hg₂|) +

h

u₂e^{ı(ωt−k(a+a†))}+u^∗₂e−ı(ωt−k(a+a^†))i

(|e₂i hg₂|+|g₂i he₂|)

Two trapped ions controlled via two lasers ω

The averaged Interaction Hamiltonian (RWA) H˜ = Ω(a^†a+1/2)

+u₁e^{−ıη(a+a†)}|g₁i he₁|+u^∗₁e^ıη(a+a†)|e₁i hg₁| +u₂e^{−ıη(a+a†)}|g₂i he₂|+u^∗₂e^ıη(a+a†)|e₂i hg₂| with

a= 1

√2

x+ ∂

∂x

, a^†= 1

√2

x− ∂

∂x

and the wave function (probability amplitudeψ_µν ∈L²(R,C), µ, ν=e,g):

|ψi=ψgg(x,t)|g₁g₂i+ψ_ge(x,t)|g₁e₂i+ψ_eg(x,t)|e₁g₂i+ψ_ee(x,t)|e₁e₂i Qbit notations: |1icorresponds to the ground state|giand|0i

to the excited state|ei.

Two trapped ions controlled via two lasers ω

The PDE satisfied byψ(x,t) = (ψ_gg, ψeg, ψge, ψee):

ı∂ψgg

∂t = Ω 2

x²− ∂²

∂x²

ψgg+u₁e^−ıηxψeg+u₂e^−ıηxψge

ı∂ψge

∂t = Ω 2

x²− ∂²

∂x²

ψge+u₁e^−ıηxψee+u^∗₂e^−ıηxψgg

ı∂ψeg

∂t = Ω 2

x²− ∂²

∂x²

ψeg+u^∗₁e^ıηxψgg+u₂e^−ıηxψee

ı∂ψee

∂t = Ω 2

x²− ∂²

∂x²

ψ_ee+u^∗₁e^ıηxψ_ge+u^∗₂e^ıηxψ_eg

whereu₁,u₂∈Care the two controls, laser amplitudes forion no 1andion no 2

_dt^du_1,2

ω|u_1,2|andη 1,Ωω.

Controllability of this system ?

Two-levels atom with spontaneous emission from |ei

Coherent (conservative) evolution:

ı_dt^d |ψi = H|ψi with the controlled Hamilto- nianH:

ω₀

2 (|ei he| − |gi hq|) +u(|ei hg|+|gi he|) Spontaneous emission (decoherence, dissipation): stochastic jump from |ei to |gi associated to the jump operator L=√

Γ|gi he|withΓ⁻¹the life-time of|ei.

Quantum Monte Carlo Trajectories

Γ|gi he|).

Compute the transition from|ψ(t)ito|ψ(t+τ)ivia the following stochastic jump process:

I Compute jump probabilityp=τhψ(t)|L^†L|ψ(t)iand chose σ randomly between[0,1].

I if 0≤σ≤1−pno jump, no photon, no detector click:

|ψ(t+τ)i= 1−ıτH−^τ₂L^†L

p1−p |ψ(τ)i

I ifp−1< σ≤1, jump from|eito|gi, spontaneous

emission of one photon producing one photo-detector click:

|ψ(t+τ)i= L|ψ(t)i pp/τ .

(collapse of the wave packet)

Lindblad master equation

Valid to represent the evolution of theaverage valueof the projector|ψ(t)i hψ(t)|known as thedensity operatorρ, a positive symmetric operator of trace one:

d

dtρ=−ı[H₀+uH₁, ρ] +LρL^†−1 2

L^†Lρ+ρL^†L where

y(t)=trace(ρL^†L)

is theaverage number of detector clicks per time-unit(the measured output).

Notice that the control appears in the Hamiltonian H=H₀+uH₁via the coherent laser fieldu.

Feedback for open quantum systems ...

Probe laser Control law

Filtering equation

Digital control

Magnetic coils Cav

Detector

Such stochastic models are used for feedback on the Caltech experiment of H. Mabuchi (Van Handel et al, IEEE AC 2005) Stochastic Lyapunovtechniques can be used for stabilization:

see the work of M. Mirrahimi and R. Van Handel on Stabilizing feedback controls for quantum systems to appear in SIAM J.

Cont. Opt. (http://arxiv.org/abs/math-ph/0510066).

References

Recent book.S. Haroche, J-M Raimond. Exploring the quantum:

atoms, cavities and photons. Oxford University Press (Graduate texts), 2006.

Basic books:

C. Cohen-Tannoudji, B. Diu, F. Laloë. Mécanique Quantique.Hermann, Paris 1977 (vol I & II).

R. Feynman. Cours de physique: mécanique quantique. InterEdition, Paris,1979.

A. Aspect, C. Fabre, G. Grynberg. Optique quantique 1 et 2.

Cours de majeure de physique. Ecole Polytechnique (2002).

Other books:

CDG1:C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg. Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley, 1989.

(also in french, CNRS editions)

CDG2:C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg.

Atom-Photon interaction: Basic Processes and Applications. Wiley, 1992. (also in French, CNRS editions)

A conference paper:R. W. Brockett, C. Rangan, A.M. Bloch. The Controllability of Infinite Quantum Systems. CDC03.