Quantum Systems: Dynamics and Control1

Mazyar Mirrahimi2 and Pierre Rouchon3

Feb 6, 2018

1See the web page:http://cas.ensmp.fr/~rouchon/MasterUPMC/index.html

2INRIA Paris, QUANTIC research team3Mines ParisTech, QUANTIC research team

Outline

1 Spin/spring systems

2 Quantum systems and almost periodic control

3 Single-frequency averaging and Kapitza’s pendulum

4 Averaging and control : the recipe

Outline

1 Spin/spring systems

2 Quantum systems and almost periodic control

3 Single-frequency averaging and Kapitza’s pendulum

4 Averaging and control : the recipe

Composite system: 2-level and harmonic oscillator

|g〉

|e〉

1

2-level system lives on C2 with Hq =ωeg2 σz

oscillator lives on L2(R,C) ∼ l2(C) with

Hc = −ωc

2∂2

∂x2 +ωc

2x2 ∼ ωc

(N + I

2

)N = a†a and a = X + iP ∼ 1√

2

(x + ∂

∂x

)The composite system lives on the tensor productC2 ⊗ L2(R,C) ∼ C2 ⊗ l2(C) with spin-spring Hamiltonian

H =ωeg2 σz ⊗ Ic + ωc Iq ⊗

(N + I

2

)+ i Ω

2 σx ⊗ (a† − a)

with the typical scales Ω ωc , ωeg and |ωc − ωeg| ωc , ωeg.Shortcut notations:

H =ωeg2 σz︸ ︷︷ ︸Hq

+ωc(N + I

2

)︸ ︷︷ ︸Hc

+ i Ω2 σx (a† − a)︸ ︷︷ ︸

H int

The spin-spring PDE

The Schrödinger system

iddt|ψ〉 =

(ωeg2 σz + ωc

(N +

I2

)+ i Ω

2 σx (a† − a)

)|ψ〉

corresponds to two coupled scalar PDE’s:

i∂ψe

∂t= +

ωeg2 ψe + ωc

2

(x2 − ∂2

∂x2

)ψe − i

Ω√2∂

∂xψg

i∂ψg

∂t= −ωeg

2 ψg + ωc2

(x2 − ∂2

∂x2

)ψg − i

Ω√2∂

∂xψe

since N = a†a, a = 1√2

(x + ∂

∂x

)and |ψ〉 = (ψe(x , t), ψg(x , t)),

ψg(., t), ψe(., t) ∈ L2(R,C) and ‖ψg‖2 + ‖ψe‖2 = 1.

Exercise: write the PDE for the controlled Hamiltonianωeg2 σz + ωc

(N + I

2

)+ i Ω

2 σx (a† − a) + uc(a + a†) + uqσxwhere uc ,uq ∈ R are local control inputs associated to the oscillatorand qubit, respectively.

The spin-spring ODE’s

The Schrödinger system

iddt|ψ〉 =

(ωeg2 σz + ωc

(N + I

2

)+ i Ω

2 σx (a† − a))|ψ〉

corresponds also to an infinite set of ODE’s

iddtψe,n = ((n + 1/2)ωc + ωeg/2)ψe,n + i Ω

2

(√nψg,n−1 −

√n + 1ψg,n+1

)i

ddtψg,n = ((n + 1/2)ωc − ωeg/2)ψg,n + i Ω

2

(√nψe,n−1 −

√n + 1ψe,n+1

)where |ψ〉 =

∑+∞n=0 ψg,n|g,n〉+ ψe,n|e,n〉, ψg,n, ψe,n ∈ C.

Exercise: write the infinite set of ODE’s forωeg2 σz + ωc

(N + I

2

)+ i Ω

2 σx (a† − a) + uc(a + a†) + uqσxwhere uc ,uq ∈ R are local control inputs associated to the oscillatorand qubit, respectively.

Dispersive case: approximate Hamiltonian for Ω |ωc − ωeg|.

H ≈ Hdisp =ωeg2 σz + ωc

(N + I

2

)− χ

2σz(N + I

2

)with χ = Ω2

2(ωc−ωeg)

The corresponding PDE is :

i∂ψe

∂t= +

ωeg

2ψe +

12

(ωc −χ

2)(x2 − ∂2

∂x2 )ψe

i∂ψg

∂t= −ωeg

2ψg +

12

(ωc +χ

2)(x2 − ∂2

∂x2 )ψg

The propagator, the t-dependant unitary operator U solution ofi d

dt U = HU with U(0) = I , reads:

U(t) = eiωegt/2 exp(−i(ωc + χ/2)t(N + I

2 ))⊗ |g〉〈g|

+ e−iωegt/2 exp(−i(ωc − χ/2)t(N + I

2 ))⊗ |e〉〈e|

Exercise: write the infinite set of ODE’s attached to the dispersiveHamiltonian Hdisp.

Resonant case: approximate Hamiltonian for ωc = ωeg = ω.

The Hamiltonian becomes (Jaynes-Cummings Hamiltonian):

H ≈ HJC = ω2 σz + ω

(N +

I2

)+ i Ω

2 (σ-a† − σ+a).

The corresponding PDE is :

i∂ψe

∂t= +

ω

2ψe +

ω

2(x2 − ∂2

∂x2 )ψe − i Ω2√

2

(x +

∂

∂x

)ψg

i∂ψg

∂t= −ω

2ψg +

ω

2(x2 − ∂2

∂x2 )ψg + i Ω2√

2

(x − ∂

∂x

)ψe

Exercise: Write the infinite set of ODE’s attached to theJaynes-Cummings Hamiltonian H.

Jaynes-Cummings propagator

Exercise: For HJC = ω2 σz + ω

(N + I

2

)+ i Ω

2 (σ-a† −σ+a) show that thepropagator, the t-dependant unitary operator U solution ofi d

dt U = HJCU with U(0) = I , reads

U(t) = e−iωt

(σz2 +N+

12

)e

Ωt2 (σ-a†−σ+a) where for any angle θ,

eθ(σ-a†−σ+a) = |g〉〈g| ⊗ cos(θ√

N) + |e〉〈e| ⊗ cos(θ√

N + I)

− σ+ ⊗ asin(θ

√N)√

N+ σ- ⊗

sin(θ√

N)√N

a†

Hint: show that[σz2 + N , σ-a† − σ+a

]= 0(

σ-a† − σ+a)2k

= (−1)k(|g〉〈g| ⊗ Nk + |e〉〈e| ⊗ (N + I)k

)(σ-a† − σ+a

)2k+1= (−1)k

(σ- ⊗ Nk a† − σ+ ⊗ aNk

)and compute de series defining the exponential of an operator.

Outline

1 Spin/spring systems

2 Quantum systems and almost periodic control

3 Single-frequency averaging and Kapitza’s pendulum

4 Averaging and control : the recipe

Controlled Schrödinger equation

iddt|ψ〉 = (H0 + u(t)H1)|ψ〉,

|ψ〉 ∈ H the system’s wavefunction with∥∥∥|ψ〉∥∥∥

H= 1;

the free Hamiltonian, H0, and the control Hamiltonian, H1, areHermitian operators on H;

the control u(t) : R+ 7→ R is a scalar control.

Two key examples:

Qubit: H0 + u(t)H1 = ωeg/2σz + u(t)/2σx .

Quantum harmonic oscillator:H0 + u(t)H1 = ωc(a†a + I

2 ) + u(t)(a + a†).

Almost periodic control of small amplitudes

We consider the controls of the form

u(t) = ε

r∑j=1

ujeiωj t + u∗j e−iωj t

ε > 0 is a small parameter;

εuj is the constant complex amplitude associated to thefrequency ωj ≥ 0;

r stands for the number of independent frequencies (ωj 6= ωk forj 6= k ).

We are interested in approximations, for ε tending to 0+, oftrajectories t 7→ |ψε〉t of

ddt|ψε〉 =

A0 + ε

r∑j=1

ujeiωj t + u∗j e−iωj t

A1

|ψε〉where A0 = −iH0 and A1 = −iH1 are skew-Hermitian.

Outline

1 Spin/spring systems

2 Quantum systems and almost periodic control

3 Single-frequency averaging and Kapitza’s pendulum

4 Averaging and control : the recipe

Time-periodic non-linear systems

We consider a non-linear ODE of the form:

ddt

x = εf (x , t), x ∈ Rn, ε 1,

where f is T -periodic in t .

We will see how its solution is well-approximated by thesolution of the time-independent system:

ddt

z = εf (z)

where f (z) = 1T

∫ T0 f (z, t)dt .

The Averaging Theorem

Consider ddt x = εf (x , t) with x ∈ U ⊂ Rn, 0 ≤ ε 1, and

f : Rn × R→ Rn smooth and period T > 0 in t . Also assume U to bebounded.

If z is the solution of ddt z = εf (z) with the initial condition z0, and

assuming |x0 − z0| = O(ε), we have |x(t)− z(t)| = O(ε) on atime-scale t ∼ 1/ε.

If z is a hyperbolic fixed point of the averaged system then thereexists ε0 > 0 such that, for all 0 < ε ≤ ε0, the main systempossesses a unique hyperbolic periodic orbit γε(t) = z +O(ε) ofthe same stability type as z.

J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical systemsand Bifurcation of Vector Fields, Springer, 1983.

Theory of Kapitza’s pendulum

Fixed suspension point:

d2

dt2 θ =gl

sin θ

g: free fall acceleration, l : pendulum’s length, θ: angle to the vertical;θ = π stable and θ = 0 unstable equilibrium.

Suspension point in vertical oscillation:

Dynamics of the suspension point: z = vΩ cos(Ωt) (a = v/Ω > 0

amplitude and Ω frequency).

Pendulum’s dynamics: replace acceleration g byg + z = g − vΩ cos(Ωt),

ddtθ = ω,

ddtω =

g − vΩ cos(Ωt)l

sin θ.

Replacing the velocity ω by the momentum pθ = ω + v sin(Ωt)l sin θ:

ddtθ = pθ − v sin(Ωt)

l sin θ,

ddt

pθ =(

gl −

v2 sin2(Ωt)l2 cos θ

)sin θ + v sin(Ωt)

l pθ cos θ.

For large enough Ω, we can average these time-periodic dynamicsover [t − π/Ω, t + π/Ω]:

ddtθ = pθ,

ddt

pθ =(

gl − v2

2l2 cos θ)

sin θ.

Around θ = 0 the approximation of small angles gives d2

dt2 θ = g−v2/2ll θ.

If v2/2l > g then the system becomes stable around θ = 0.

Outline

1 Spin/spring systems

2 Quantum systems and almost periodic control

3 Single-frequency averaging and Kapitza’s pendulum

4 Averaging and control : the recipe

Bilinear Schrödinger equation

Un-measured quantum system→ Bilinear Schrödinger equation

i~ddt|ψ〉 = (H0 + u(t)H1)|ψ〉,

|ψ〉 ∈ H the system’s wavefunction with∥∥∥|ψ〉∥∥∥

H= 1;

the free Hamiltonian, H0, is a Hermitian operator definedon H;the control Hamiltonian, H1, is a Hermitian operatordefined on H;the control u(t) : R+ 7→ R is a scalar control.

Here we consider the case of finite dimensional H

Almost periodic control

We consider the controls of the form

u(t) = ε

r∑j=1

ujeiωj t + u∗j e−iωj t

ε > 0 is a small parameter;εuj is the constant complex amplitude associated to thepulsation ωj ≥ 0;r stands for the number of independent frequencies(ωj 6= ωk for j 6= k ).

We are interested in approximations, for ε tending to 0+, oftrajectories t 7→ |ψε〉t of

ddt|ψε〉 =

A0 + ε

r∑j=1

ujeiωj t + u∗j e−iωj t

A1

|ψε〉where A0 = −iH0/~ and A1 = −iH1/~ are skew-Hermitian.

Rotating frame

Consider the following change of variables

|ψε〉t = eA0t |φε〉t .The resulting system is said to be in the “interaction frame”

ddt|φε〉 = εB(t)|φε〉

where B(t) is a skew-Hermitian operator whosetime-dependence is almost periodic:

B(t) =r∑

j=1

ujeiωj te−A0tA1eA0t + u∗j e−iωj te−A0tA1eA0t .

Main idea

We can writeB(t) = B +

ddt

B(t),

where B is a constant skew-Hermitian matrix and B(t) is abounded almost periodic skew-Hermitian matrix.

Multi-frequency averaging: first order

Consider the two systemsddt|φε〉 = ε

(B +

ddt

B(t))|φε〉,

andddt|φ1stε 〉 = εB|φ1st

ε 〉,

initialized at the same state |φ1stε 〉0 = |φε〉0.

Theorem: first order approximation (Rotating WaveApproximation)

Consider the functions |φε〉 and |φ1stε 〉 initialized at the same

state and following the above dynamics. Then, there existM > 0 and η > 0 such that for all ε ∈]0, η[ we have

maxt∈

[0,1ε

]∥∥∥|φε〉t − |φ1stε 〉t

∥∥∥ ≤ Mε

Multi-frequency averaging: first order

Proof’s idea

Almost periodic change of variables:

|χε〉 = (1− εB(t))|φε〉

well-defined for ε > 0 sufficiently small.The dynamics can be written as

ddt|χε〉 = (εB + ε2F (ε, t))|χε〉

where F (ε, t) is uniformly bounded in time.

Multi-frequency averaging: second orderMore precisely, the dynamics of |χε〉 is given by

ddt|χε〉 =

(εB + ε2[B, B(t)]− ε2B(t)

ddt

B(t) + ε3E(ε, t))|χε〉

E(ε, t) is still almost periodic but its entries are no more linearcombinations of time-exponentials;

B(t) ddt B(t) is an almost periodic operator whose entries are

linear combinations of oscillating time-exponentials.

We can write

B(t) =ddt

C(t) and B(t)ddt

B(t) = D +ddt

D(t)

where C(t) and D(t) are almost periodic. We have

ddt|χε〉 =

(εB − ε2D + ε2

ddt

([B, C(t)]− D(t)

)+ ε3E(ε, t)

)|χε〉

where the skew-Hermitian operators B and D are constants and theother ones C, D, and E are almost periodic.

Multi-frequency averaging: second order

Consider the two systems

ddt|φε〉 = ε

(B +

ddt

B(t))|φε〉,

andddt|φ2ndε 〉 = (εB − ε2D)|φ2nd

ε 〉,

initialized at the same state |φ2ndε 〉0 = |φε〉0.

Theorem: second order approximation

Consider the functions |φε〉 and |φ2ndε 〉 initialized at the same

state and following the above dynamics. Then, there existM > 0 and η > 0 such that for all ε ∈]0, η[ we have

maxt∈

[0, 1ε2

] ∥∥∥|φε〉t − |φ2ndε 〉t

∥∥∥ ≤ Mε

Multi-frequency averaging: second order

Proof’s idea

Another almost periodic change of variables

|ξε〉 =(

I − ε2(

[B, C(t)]− D(t)))|χε〉.

The dynamics can be written as

ddt|ξε〉 =

(εB − ε2D + ε3F (ε, t)

)|ξε〉

where εB − ε2D is skew Hermitian and F is almost periodic andtherefore uniformly bounded in time.

The Rotating Wave Approximation (RWA) recipes

Schrödinger dynamics i~ ddt |ψ〉 = H(t)|ψ〉, with

H(t) = H0 +m∑

k=1

uk (t)Hk , uk (t) =r∑

j=1

uk,jeiωj t + u∗k,je−iωj t .

The Hamiltonian in interaction frame

H int(t) =∑k,j

(uk,jeiωj t + u∗k,je

−iωj t)

eiH0tHk e−iH0t

We define the first order Hamiltonian

H1strwa = H int = lim

T→∞

1T

∫ T

0H int(t)dt ,

and the second order Hamiltonian

H2ndrwa = H1st

rwa − i(H int − H int

)(∫t(H int − H int)

)Choose the amplitudes uk,j and the frequencies ωj such that the

propagators of H1strwa or H2nd

rwa admit simple explicit forms that are usedto find t 7→ u(t) steering |ψ〉 from one location to another one.