Modeling and Control of the LKB Photon-Box: 1 Spin Systems

Modeling and Control of the LKB Photon-Box:

Spin Systems

Pierre Rouchon

[email protected]

Würzburg, July 2011

1LKB: Laboratoire Kastler Brossel, ENS, Paris.

Several slides have been used during the IHP course (fall 2010) given with Mazyar Mirrahimi (INRIA) see:

http://cas.ensmp.fr/~rouchon/QuantumSyst/index.html

Outline

1 The 2-level system

2 RWA and averaging

3 Adiabatic control

4 Complements

Multi-frequency averaging: second order STIRAP

Controllability of finite dimensional Schrödinger systems

2-level system (1/2 spin)

The simplest quantum system: a ground state |gi of energy ωg; an excited state |ei of energy ωe. The quantum state |ψi ∈ C² is a linear superposition|ψi=ψ_g|gi+ψ_e|ei and obeys to the Schrödinger equation (ψ_g andψedepend ont).

Schrödinger equationfor the uncontrolled 2-level system (~=1) :

ı_dt^d |ψi=H₀|ψi= ωe|ei he|+ωg|gi hg|

|ψi whereH₀is the Hamiltonian, a Hermitian operatorH₀^†=H₀. Energy is defined up to a constant:H₀andH₀+$(t)1($(t)∈R arbitrary) are attached to the same physical system. If|ψisatisfies i_dt^d |ψi=H0|ψithen|χi=e^−iϑ(t)|ψiwith _dt^dϑ=$obeys to

i_dt^d |χi= (H₀+$I)|χi. Thus for anyϑ,|ψiande^−iϑ|ψirepresent the same physical system: Theglobal phaseof a quantum system|ψi can be chosenarbitrarily at any time.

The controlled 2-level system

Take origin of energy such thatωg(resp. ωe) becomes−^ωe−ω₂ ^g (resp. ^ωe−ω₂ ^g) and setωeg =ωe−ωg

The solution ofi_dt^d |ψi=H₀|ψi= ^ω₂^eg(|ei he| − |gi hg|)|ψiis

|ψi_t =ψ_g0e

iωeg t

2 |gi+ψ_e0e

−iωeg t

2 |ei.

With a classical electromagnetic field described byu(t)∈R, the coherent evolutionthe controlled Hamiltonian

H(t) = ωeg

2 σz+u(t)

2 σx = ωeg

2 (|ei he|−|gi hg|)+u(t)

2 (|ei hg|+|gi he|) The controlled Schrödinger equationi_dt^d |ψi= (H₀+uH₁)|ψi reads:

i_dt^d ψ_e

ψg

= ωeg

2

1 0 0 −1

ψ_e ψg

+u(t) 2

0 1 1 0

ψ_e ψg

.

The 3 Pauli Matrices²

σx =|ei hg|+|gi he|, σy =−i|ei hg|+i|gi he|, σz =|ei he|−|gi hg|

2They correspond, up to multiplication byi, to the 3 imaginary quaternions.

Pauli matrices and some formula

σ_x =|ei hg|+|gi he|, σ_y =−i|ei hg|+i|gi he|, σ_z =|ei he| − |gi hg|

σ²_x =1, σxσy =iσ_z, [σx, σy] =2iσ_z, circular permutation . . . Since for anyθ∈R,e^iθσx =cosθ+isinθσx (idem forσy

andσz), the solution ofi_dt^d |ψi= ^ω₂^egσz|ψiis

|ψi_t =e

−iωeg t

2 σz|ψi₀=

cos

ωegt 2

1−isin ωegt

2

σ_z

|ψi₀

Forα, β=x,y,z,α6=β we have σ_αe^iθσβ =e^−iθσβσ_α,

e^iθσα−1

=

e^iθσα†

=e^−iθσα. and also

e^−iθ2σασβe^iθ2σα =e^−iθσασβ =σβe^iθσα

Bloch sphere representation of a 2-level system

if |ψiobeys _dt^d |ψi= −iH|ψi, then projectorρ=|ψi hψ|obeys:

d

dtρ=−i[H, ρ].

For|ψi=ψ_g|gi+ψ_e|ei:

|ψi hψ|=|ψ_g|²|gi hg|+ψ_gψ^∗_e|gi he|

+ψ^∗_gψ_e|ei hg|+|ψ_e|²|ei he|. Setx =2<(ψ_gψ^∗_e),y =2=(ψ_gψ_e^∗) andz =|ψ_e|²− |ψ_g|²we get

ρ= 1+xσ_x+yσ_y +zσ_z

2 .

The Bloch vectorM~ =x~ı+y~+z~k evolves on the unit sphere ofR³: i_dt^d |ψi= ^ω₂^xσ_x+^ω₂^yσ_y+^ω₂^zσ_z

|ψi v _dt^dM~ = (ω_x~ı+ω_y~+ω_z~k)×M~ Bloch vectorM~ with Euler angles(θ, φ)corresponds to

|ψi=e^iϕsin ^θ₂

|gi+cos ^θ₂

|ei.

Bilinear Schrödinger equation

Un-measured quantum system→Bilinear Schrödinger equation id

dt|ψi= (H₀+u(t)H₁)|ψi,

|ψi ∈ Hthe system’s wavefunction with |ψi

_H=1;

the free Hamiltonian,H₀, is a Hermitian operator defined onH;

the control Hamiltonian,H₁, is a Hermitian operator defined onH;

the controlu(t) :R⁺7→Ris a scalar control.

Here we consider the case of finite dimensionalHfor mathematical proof.

Almost periodic control

We consider the controls of the form u(t) =





r

X

j=1

u_je^iωjt +u^∗_je^−iωjt





>0 is a small parameter;

u_j is the constant complex amplitude associated to the pulsationω_j ≥0;

r stands for the number of independent pulsations (ω_j 6=ω_k forj 6=k).

We are interested in approximations, fortending to 0⁺, of trajectoriest7→ |ψi_t ont ∈[0,1/]of

d

dt |ψi=



A₀+





r

X

j=1

u_je^iωjt +u^∗_je^−iωjt



A₁



|ψi

whereA₀=−iH₀andA₁=−iH₁are skew-Hermitian.

Rotating frame

Consider the following change of variables

|ψi_t =e^A0t|φi_t.

The resulting system is said to be in the“interaction frame”

d

dt|φi=B(t)|φi whereB(t)is a skew-Hermitian operator whose time-dependence isalmost periodic:

B(t) =

r

X

j=1

u_je^iωjte^−A0tA₁e^A0t+u^∗_je^−iωjte^−A0tA₁e^A0t.

Main idea We can write

B(t) = ¯B+_dt^dB(t),e

whereB¯ is a constant skew-Hermitian matrix andB(t)e is a bounded almost periodic skew-Hermitian matrix.

Multi-frequency averaging: first order

Consider the two systems

d

dt |φi=

B¯+ _dt^dB(t)e

|φi, and

d dt

φ^1st

E

=B¯ φ^1st

E , initialized at the same state

φ^1st

E

0=|φi₀.

Theorem: first order approximation (Rotating Wave Approximation)

Consider the functions|φiand φ^1stE

initialized at the same state and following the above dynamics. Then, there exist M>0 andη >0 such that for all∈]0, η[we have

max

t∈

0,1

|φi_t −

φ^1stE

t

≤M

Multi-frequency averaging: first order

Proof’s idea

Almost periodic change of variables:

|χi= (1−B(t))e |φi

well-defined for >0 sufficiently small.

The dynamics can be written as d

dt |χi= (B¯ +²F(,t))|χi whereF(,t)is uniformly bounded in time.

Approximation recipes

We consider the Hamiltonian

H =H₀+

m

X

k=1

u_kH_k, u_k(t) =

r

X

j=1

u_k,je^ωjt +u^∗_k,je^−ωjt.

The Hamiltonian in interaction frame Hint(t) =X

k,j

u_k,je^ωjt +u^∗_k,je^−ωjt

e^iH0tH_ke^−iH0t

We define thefirst order Hamiltonian H_rwa^1st =Hint= lim

T→∞

1 T

Z T 0

Hint(t)dt,

Slowly varying amplitudes

Remark

In the above analysis we have assumed the complex amplitudesu_k,j to be constant. However, the whole analysis holds for the case where each oneu_k,j’s is of a small magnitude, admits a finite number of discontinuities and,

between two successive discontinuities, is a slowly time varying function that is continuously differentiable.

RWA and resonant control Ini_dt^d |ψi= ^ω₂^egσz+^u₂σx

|ψi, take a resonant control

u=ue^iωegt +u^∗e^−iωegt withuslowly varying complex amplitude

_dt^du

ω_eg|u|. SetH₀= ^ω₂^egσ_z andH₁= ^u₂σ_x and consider

|ψi=e⁻

iωeg t

2 σz|φito eliminate thedriftH₀and to get the Hamiltonian in the interaction frame:

i_dt^d |φi= u 2e

iωeg t

2 σzσxe⁻

iωeg t

2 σz|φi=Hint|φi

withHint = ^u₂e^iωegt

σ⁺=|eihg|

z }| { σ_x+iσ_y

2 +^u₂e^−iωegt

σ⁻=|gihe|

z }| { σ_x−iσ_y

The RWA consists in neglecting the oscillating terms at2 frequency 2ω_eg when|u| Ω:

H_int =

ue^2iωegt+u^∗ 2

σ⁺+

u+u^∗e^−2iωegt 2

σ⁻. Thus

H_int = u^∗σ⁺+uσ⁻

2 .

First order average system and Rabi oscillation

i_dt^d |φi= (u^∗σ⁺+uσ⁻)

2 |φi= (u^∗|ei hg|+u|gi he|)

2 |φi

We setu= Ωre^iθ withΩr >0 andθreal.

u^∗σ⁺+uσ⁻

2 = Ω_r

2 (cosθσx +sinθσy) The system oscillates between|eiand|giwith theRabi pulsation ^Ω₂^r. Since(cosθσx+sinθσy)²=1and

e^{−iΩ2r t(cosθσx+sinθσy)} =cos Ωrt

2

−isin Ωrt

2

(cosθσx +sinθσy), the solution of _dt^d |φi= ^−iΩ₂^r (cosθσx+sinθσy)|φireads

|φi_t =cos Ωrt

2

|gi−isin Ωrt

2

e^−iθ|ei, when |φi₀=|gi,

|φi_t =cos Ωrt

2

|ei −isin Ωrt

2

e^iθ|gi, when |φi₀=|ei,

π/2 andπ Pulses

We start always from|φi₀=|giwe light on the resonant control with the constant amplitudeu=−iΩ_r during[0,T](pulse lengthT). Since

|φi_T =cos ΩrT

2

|gi+sin ΩrT

2

|ei,

we see that

ifΩrT =π(π-pulse ) then|φi_T =|ei: stimulate absorption of one photon. If we measure the system energy

(measurement operator ^ω₂^eg |ei he| −^ω₂^eg |gi hg|), then we will find deterministically ^ω₂^eg).

ifΩrT =π/2 (π/2-pulse ) when|φi_T = (|gi+|ei)/√ 2, a coherent superpositionof|giand|ei. If we measure the energy, the result is stochastic and the probability to get

ωeg

2 is ¹₂ and to get−^ω₂^eg is also ¹₂.

Exercise: controllability of the 2-level systems and Rabi oscillation Take the first order approximation

(Σ) i_dt^d |φi= (u^∗σ⁺+uσ⁻)

2 |φi=(u^∗|ei hg|+u|gi he|)

2 |φi

with controlu∈C.

1 Take constant controlu(t) = Ω_re^iθfort ∈[0,T],T >0. Show thati_dt^d |φi=^Ωr(cosθσ₂^x+sinθσy)|φi.

2 SetΘ_r = ^Ω₂^rT. Show that the solution atT of the propagator U_t ∈SU(2),i_dt^dU=^Ωr(cosθσ₂^x+sinθσy)U,U₀=1is given by

U_T =cosΘ_r1−isinΘ_r(cosθσ_x+sinθσ_y),

3 Take a wave function φ¯

. Show that existΩ_r andθsuch that U_T|gi=e^iα

φ¯

, whereαis some global phase.

4 Prove that for any given two wave functions|φ_aiand|φ_biexists a piece-wise constant control[0,2T]3t 7→u(t)∈Csuch that the solution of(Σ)with|φi₀=|φ_aisatisfies|φi_T =e^iβ|φ_bifor some global phaseβ.

Time-adiabatic approximation without gap conditions³

Take[0,1]3s7→H(s)aC²family of Hermitian matricesn×n:

sets=t ∈[0,1]anda small positive parameter. Consider a solution

0,¹

3t7→ |ψi_t of

i_dt^d |ψi_t =H(t)|ψi_t .

Take[0,s]3s7→P(s)afamily of orthogonal projectorssuch that for eachs∈[0,1],H(s)P(s) =ω(s)P(s)whereω(s)is an eigenvalue ofH(s). Assume that[0,s]3s7→P(s)isC²and that,for almost alls∈[0,1],P(s)is theorthogonal projector on the eigen-spaceassociated to the eigen-valueω(s). Then

7→0lim⁺



 sup

t∈[0,1 ]

|

|



=0.

3Theorem 6.2, page 175 ofAdiabatic Perturbation Theory in Quantum Dynamics, by S. Teufel, Lecture notes in Mathematics, Springer, 2003.

Chirped control of a 2-level system (1)

i_dt^d |ψi = ^ω₂^egσ_z+^u₂σ_x

|ψi with quasi- resonant control (|ω_r − ω_eg| ω_eg)

u(t) =v(t) e^{i(ωrt+θ(t))}+e^{−i(ωrt+θ(t))} where v, θ ∈ R, |v| and |^dθ_dt| are small and slowly varying:

|v|, ^dθ_dt

ω_eg, ^dv_dt

ω_eg|v|,

d²θ dt²

ω_eg ^dθ_dt

.

Passage to the interaction frame|ψi=e⁻ⁱ

ωrt+θ(t) 2 σz|φi:

i_dt^d |φi=

ωeg−ωr−d dtθ

2 σz+^{ve2i(ωr t}₂^+θ)+vσ++^ve−2i(ω₂^{r t+θ)+v}σ−

|φi.

Set∆r =ωeg −ωr andw(t) =−_dt^dθ, RWA yields following averaged Hamiltonian

Hchirp= ^∆r+w(t)₂ σz +^v(t)₂ σx

where(v,w)are two real control inputs.

Chirped control of a 2-level system (2)

InHchirp=^∆r₂^+wσ_z+^v₂σ_x set, fors=tvarying in[0, π],w=acos(t) andv =bsin²(t).Spectral decompositionofHchirpfors∈]0, π[:

Ω₋=−

√

(∆_r+w)²+v²

2 with |−i= cosα|gi −(1−sinα)|ei p2(1−sinα) Ω+=

√

(∆r+w)²+v²

2 with |+i= (1−sinα)|gi+cosα|ei p2(1−sinα)

whereα∈]^−π₂ ,^π₂[is defined by tanα=^∆r_v^+w. Witha>|∆_r|andb>0

s7→0lim⁺α= ^π₂ implies lim

s7→0⁺|−i_s=|gi, lim

s7→0⁺|+i_s=|ei

s7→πlim⁻α=−^π₂ implies lim

s7→π⁻|−i_s =− |ei, lim

s7→π⁻|+i_s =|gi. Adiabatic approximation: the solution ofi_dt^d |φi=Hchirp(t)|φistarting from|φi₀=|gireads

|φi_t≈e^iϑt|−i_s=t, t ∈[0,^π], withϑ_t time-varying global phase.

Att= ^π,|ψicoincides with|eiup to a global phase:robustness versus∆_r,aandb(ensemble controllability).

Chirped control on the Bloch sphere.

The chirped dynamicsi_dt^dφ= ^∆r₂^+wσz+^v₂σx

|φiwith w =acos(t)andv =bsin²(t)reads

d

dtM~ = (bsin²(t)~ı+ (∆r +acos(t))~k)

| {z }

=Ω~t

×M~

The initial condition|φi₀=|gimeans thatM~₀=−~k and Ω~₀= (∆r +a)~k with∆r+a>0.

Since~Ωnever vanishes fort∈[0,^π], adiabatic theorem implies thatM~ follows the direction of−Ω, i.e. that~ M~ ≈ − ^~Ω

k~Ωk (see matlab simulationsAdiabaticBloch.m).

Att = ^π,~Ω = (∆_r −a)~k with∆_r −a<0: M~π

=~k and thus

|φiπ

=e^ϑ|ei.

Adiabatic propagatorU(t)forH(t) = ^∆₂^rσ_z +^v(t)₂ σ_y (1)

Consider the propagatorU∈SU(2), solution of

d

dtU(t) =−iH(t)U(t) =−i

∆r

2 σz+^v(t)₂ σy

U(t), U(0) =I assuming 0< 1,∆r >0 andv =f(s)(s=t) where [0,1]3s 7→f(s)is smooth andf(0) =f(1) =0.

We have

U(1/) =e^{−iϑσ¯ z} +O() whereϑ¯is given by the integral:

ϑ¯= ¹₂ Z 1/

0

q

∆²_r +f²(t)dt.

The phaseϑ¯is only due to the time integral of theH(t) eigenvalues (dynamic phaseonly, no Berry phase for such adiabatic evolution).

Adiabatic propagatorU(t)forH(t) = ^∆₂^rσ_z +^v(t)₂ σ_y (2) The frame(|−i_s,|+i_s)that diagonalizeH(s)(s=t) H(s)|±i_s =±

√

∆²r+f²(s)

2 |±i_s,reads

|−i_s =cosξs|gi+isinξs|ei, |+i_s =isinξs|gi+cosξs|ei whereµs =p

1+ (f(s)/∆r)²gives cosξ_s=p

(µ_s+1)/(2µ_s), sinξ_s =p

(µ_s−1)/(2µ_s) The passage from the(|gi,|ei)to(|−i_s,|+i_s)corresponds, in the Bloch sphere representation, to a rotation around the X-axis of angle−2ξ_s:

|−i_s =e^iξsσx|gi, |+i_s =e^iξsσx|ei

Thus we have

∆r

2 σz+ ^f(s)₂ σy =

√

∆²_r+f²(s)

2 e^−iξsσxσze^iξsσx.

Adiabatic propagatorU(t)forH(t) = ^∆₂^rσ_z +^v(t)₂ σ_y (3) Consider _dt^d |ψi=−iH(t)|ψi, set

ϑ(t) = ¹₂ Z t

0

q

∆²_r +f²(τ)dτ

set|ψi=e^iξtσxe^−iϑ(t)σz|φi. Then, withξ_s⁰ = _ds^dξs,

d

dt |φi=−iξ_t⁰ e^iϑ(t)σz σx e^−iϑ(t)σz|φi=−iξ_t⁰ σx e^{−2iϑ(t)σz}|φi. In averageξ⁰_t σx e^{−2iϑ(t)σz} gives zero up to first order terms in (useRt

0e^{−2iϑ(τ)σz}dτ =A(t)withA(t)bounded on[0,1/]). Then

|φi_t ≈ |φi₀=e^−iξ0σx|ψi₀is almost constant and thus

|ψi_t =e^iξtσxe^−iϑ(t)σze^−iξ0σx |ψi₀+O().

The propagator reads then fort ∈[0,1/], U(t) =e^iξtσxe^−iϑ(t)σz +O() sinceξ0=0 results fromf(0) =0.

Multi-frequency averaging: second order

Consider the two systems

d

dt |φi=

B¯+ _dt^dB(t)e

|φi, and

d dt

φ^2nd

E

= (B¯−²D)¯ φ^2nd

E , initialized at the same state

φ^2nd

E

0=|φi₀. Theorem: second order approximation Consider the functions|φiand

φ^2ndE

initialized at the same state and following the above dynamics. Then, there exist M>0 andη >0 such that for all∈]0, η[we have

max

t∈

0,1 ²

|φi_t− φ^2nd

E

t

≤M

Multi-frequency averaging: second order

Proof’s idea

Another almost periodic change of variables

|ξi=

1−²

[ ¯B,C(t)]e −D(t)e

|χi.

The dynamics can be written as

d dt |ξi=

B¯ −²D¯ +³G(,t)

|ξi

whereGis almost periodic and therefore uniformly bounded in time.

Approximation recipes

We consider the Hamiltonian

H =H₀+

m

X

k=1

u_kH_k, u_k(t) =

r

X

j=1

u_k,je^ωjt +u^∗_k,je^−ωjt.

The Hamiltonian in interaction frame Hint(t) =X

k,j

u_k,je^ωjt +u^∗_k,je^−ωjt

e^iH0tH_ke^−iH0t

We define thefirst order Hamiltonian H_rwa^1st =Hint= lim

T→∞

1 T

Z T 0

Hint(t)dt,

and thesecond order Hamiltonian H_rwa^2nd =H_rwa^1st−i Hint−Hint

Z

t

(Hint−Hint)

Application to a 2-level system Ini_dt^d |ψi= ^ω₂^egσz+^u₂σx

|ψi, take a resonant control

u=ue^iωegt +u^∗e^−iωegt withuslowly varying complex amplitude

_dt^du

ω_eg|u|. SetH₀= ^ω₂^egσ_z andH₁= ^u₂σ_x and consider

|ψi=e⁻

iωeg t

2 σz|φito eliminate thedriftH₀and to get the Hamiltonian in the interaction frame:

i_dt^d |φi= u 2e

iωeg t

2 σzσxe⁻

iωeg t

2 σz|φi=Hint|φi

withHint = ^u₂e^iωegt

σ⁺=|eihg|

z }| { σ_x+iσ_y

2 +^u₂e^−iωegt

σ⁻=|gihe|

z }| { σ_x−iσ_y

The RWA consists in neglecting the oscillating terms at2 frequency 2ω_eg when|u| Ω:

H_int =

ue^2iωegt+u^∗ 2

σ⁺+

u+u^∗e^−2iωegt 2

σ⁻. Thus

H_int = u^∗σ⁺+uσ⁻

2 .

Second order approximation and Bloch-Siegert shift

The decomposition ofHint, Hint = ^u₂^∗σ₊+^u₂σ−

| {z }

Hint

+^ue2iω₂^{eg t}σ₊+^u∗e−2iω₂ ^{eg t}σ−

| {z }

Hint−H_int

,

provides thefirst order approximation(RWA) H_rwa^1st =Hint =lim_{T→∞ T}¹RT

0 Hint(t)dt,and also the second order approximation H_rwa^2nd =H_rwa^1st−i Hint−Hint

R

t(Hint−Hint)

. Since R

tHint−Hint = ^ue_4iω^{2iωeg t}

eg σ₊− ^u∗e_4iω^{−2iωeg t}

eg σ−, we have Hint−Hint

Z

t

(Hint−Hint)

=−_8iω^|u|2

egσz

(useσ₊² =σ₋² =0 andσz =σ₊σ−−σ−σ₊).

Thesecond order approximationreads:

H_rwa^2nd =H_rwa^1st +_|u|2

8ωeg

σz = ^u₂^∗σ₊+^u₂σ−+_|u|2

8ωeg

σz. The 2nd order correction ^|u|_4ω²

rσz is called theBloch-Siegert shift.

Stimulated Raman Adiabatic Passage (STIRAP) (1)

H=ω_g|gi hg|+ω_e|ei he|+ω_f|fi hf| +uµ_gf |gi hf|+|fi hg|

+uµ_ef |ei hf|+|fi he|

. Setω_gf =ω_f−ω_g,ω_ef =ω_f−ω_eand u = u_gfcos(ω_gft) + u_efcos(ω_eft) with slowly varying small real am- plitudesu_gf andu_ef.

Puti_dt^d |ψi=H|ψiin the interaction frame:

|ψi=e^{−it(ωg|gihg|+ωe|eihe|+ωf|fihf|)}|φi.

Rotation Wave Approximation yieldsi_dt^d |φi=Hrwa|φiwith Hrwa= ^Ω₂^gf(|gi hf|+|fi hg|) +^Ω₂^ef(|ei hf|+|fi he|) with slowly varying Rabi pulsationsΩ_gf =µ_gfu_gf and Ω_ef =µ_efu_ef.

Stimulated Raman Adiabatic Passage (STIRAP) (2) Spectral decomposition: as soon asΩ²_gf+ Ω²_ef >0,

Ω_gf(|gihf|+|fihg|)

2 +^{Ωef(|eihf|+|f}₂ ^ihe|) admits 3 distinct eigen-values, Ω₋=−

q Ω²_gf+Ω²_ef

2 , Ω₀=0, Ω₊=

q Ω²_gf+Ω²_ef

2 .

They correspond to the following 3 eigen-vectors,

|−i=q ^Ωgf

2(Ω²_gf+Ω²_ef)|gi+q ^Ωef

2(Ω²_gf+Ω²_ef)|ei −^√1

2|fi

|0i=q^−Ωef

Ω²_gf+Ω²_ef |gi+q ^Ωgf Ω²_gf+Ω²_ef |ei

|+i=q ^Ωgf

2(Ω²_gf+Ω²_ef)|gi+q ^Ωef

2(Ω²_gf+Ω²_ef)|ei+^√1

2|fi. Fort=s∈[0,^3π₂]andΩ¯_g,Ω¯_e>0, the adiabatic control

Ωgf(s) =

Ω¯gcos²s, fors∈[^π₂,^3π₂];

0, elsewhere. , Ωef(s) =

Ω¯esin²s, fors∈[0, π];

0, elsewhere.

provides the passage from|giatt=0 to|eiatt = ^3π₂ . (see matlab simulationsstirap.m).

Stimulated Raman Adiabatic Passage (STIRAP) (3)

Exercice

Design an adiabatic passage s7→(Ω_gf(s),Ω_ef(s))from|gito

−|gi+|ei√

2 , up to a global phase.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

s/π

Ω_gf Ω_ef

Take, e.g., s =t ∈ [0, π]

andΩ¯ >0, and set

Ω_gf(s) = ^Ω¯₂ sins−^Ω¯₄sin 2s Ω_ef(s) = Ω¯sins

Results from|0i= q^−Ωef

Ω²_gf+Ω²_ef |gi+ q ^Ωgf Ω²_gf+Ω²_ef |ei

Controllability of bilinear Schrödinger equations

Schrödinger equation

i_dt^d |ψi= H₀+

m

X

k=1

u_kH_k

!

|ψi

State controllability

For any|ψ_aiand|ψ_bion the unit sphere ofH, there exist a time T >0, a global phaseθ∈[0,2π[and a piecewise continuous control[0,T]3t 7→u(t)such that the solution with initial condition|ψi₀=|ψ_aisatisfies|ψi_T =e^iθ|ψ_bi.

4See, e.g.,Introduction to Quantum Control and Dynamicsby D. D’Alessandro. Chapman & Hall/CRC, 2008.

Controllability of bilinear Schrödinger equations

Propagator equation:

i_dt^dU= H₀+

m

X

k=1

u_kH_k

!

U, U(0) =1

We have|ψi_t =U(t)|ψi₀. Operator controllability

For any unitary operatorV onH, there exist a timeT >0, a global phaseθand a piecewise continuous control

[0,T]3t7→u(t)such that the solution of propagator equation satisfiesU_T =e^iθV.

Operator controllability implies state controllability

Lie-algebra rank condition

d

dtU= A0+

m

X

k=1

ukAk

! U

withAk =Hk/i are skew-Hermitian. We define L₀=span{A₀,A₁, . . . ,Am} L₁=span(L₀,[L₀,L₀]) L₂=span(L₁,[L₁,L₁])

...

L=Lν =span(Lν−1,[Lν−1,Lν−1]) Lie Algebra Rank Condition

Operator controllableif, and only if, the Lie algebra generated by them+1 skew-Hermitian matrices{−iH₀,−iH₁, . . . ,−iH_m}is either su(n)oru(n).

Exercice

Show that i_dt^d |ψi= ^ω₂^egσ_z+^u₂σ_x

|ψi,|ψi ∈C²is controllable.

A simple sufficient condition

We considerH=H₀+uH₁,(|ji)_j=1,...,nthe eigenbasis ofH₀. We assumeH₀|ji=ω_j|jiwhereω_j ∈R, we consider a graphG:

V ={|1i, . . . ,|ni}, E ={(|j₁i,|j₂i)|1≤j₁<j₂≤n, hj₁|H₁|j₂i 6=0}. Gamits a degenerate transition if there exist(|j₁i,|j₂i)∈E and (|l₁i,|l₂i)∈E, admitting the same transition frequencies,

|ω_j1−ω_j2|=|ω_l1−ω_l2|.

A sufficient controllability condition

Remove fromE, all the edges with identical transition frequencies.

Denote byE¯ ⊂Ethe reduced set of edges without degenerate transitions and byG¯ = (V,E). If¯ G¯ is connected, then the system is operator controllable.