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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.
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