Bibliography [1] M. MIRRAHIMI, P. ROUCHON, AND G.TURINICI, Lyapunov control of bilinear Schr¨odinger
equations AUTOMATICA, 41 :1987 1994, 2005.
[2] H.K. KHALIL. Nonlinear Systems. MACMILLAN, 1992.
[3] H. RABITZ. AND W. ZHU. Quantum control design via adaptive tracking. J. CHEM. PHYS., 119(7), 2003.
Convergence analysis
Theorem 2. Consider (2) with Ψ ∈ S2n−1C and an eigen-state φ ∈ S2n−1C of H0 associated to the eigenvalue λ. Take the feedback (9)
with c, k > 0. If H0 is not degenerate and either hΦα|H1φi 6= 0 or hΦα|H2φi 6= 0 ∀α, Φα 6= φ, , then the Ω-limit set of S1Ψ0 reduces to a solution of the uncontrolled system, with |I1|, |I2| ≤ Cδ with a constant C depending only on H0.
Proof of Theorem 2
• Observe that neither S1 or S2 define a classical dynamical system, that is the semigroup property is lost. One has instead S1(t + s)Ψ0 = S1(t)S1(s)Ψ0 or S1(t+s)Ψ0 = S2(t)S1(s)Ψ0 according to the control we have when touching S1(s)Ψ0.
• On the limit set Ω(Ψ0) corresponding to S1, V is constant and Ω(Ψ0) is invariant either to S1 or to S2, that is if Ψ1 ∈ Ω(Ψ0) then S1(t)Ψ1 ∈ Ω(Ψ0), t > 0 or S2(t)Ψ1 ∈ Ω(Ψ0), t > 0. But then, since V is constant along the trajectory, u = 0 and we have that Ω(Ψ0) is in-variant under the Schr¨odinger flow for H0.
• On Ω(Ψ0) we have that |I1| ≤ δ and I2 ≥ −δ. Since Ω(Ψ0) is a trajectory to the uncontrolled equation, we have that there exists a constant C > 1 depending only on H0 such that I2 ≤ Cδ.
Remark 5. Now, the stabilizing strategy is obvious since, when δ → 0, the trajectories in Ω(Ψ0) = Ωδ(Ψ0) converge to a trajectory of the uncontrolled system with I1 = I2 = 0.
FIG. 4 – Population |Ψ1|2(solid line) and control u (dashed line) ;
ini-tial condition : Ψ(t = 0) = (0, 1/√2, 1/√2) ; system defined by (8) with feedback (9). We take k = c = 0.8 and δ = 10−3.
FIG. 5 – Time evolution of I1,I2 ; initial condition : Ψ(t = 0) =
(0, 1/√2, 1/√2) ; system defined by (8) with feedback (9). We take k = c = 0.8 and δ = 10−3 ; I1, I2 oscilates and its absolute value converges to zero.
Examples and simulations
We use the previous Lyapunov control in order to reach the first eigen-state
φ = Φ1 = (1, 0, 0) of energy λ = 0, at the final time T .
Remark 2. For the system defined by (8) ,we note that hΦ3|H1φi = 0.
FIG. 2 – Population |Ψ1|2 (blue line) and control u (green line) ; initial
condi-tion Ψ(t = 0) = (0, 1/√2, 1/√2) ; system defined by (8) with feedback (6). We take k = c = 0.8 The feedback (6) fails to reach the target (quality is only 30%).
In order to overcome the lack of convergence for cases similar to that pre-sented above, we consider the feedback :
u(I1, I2) = kI2, in A = {|I1| < δ and I2 < −δ} 0 , in B = { |I1| < δ and I2 > δ} −kI1/(1 + kI2), in C = {|I1| > δ/2 or |I2| < 2δ}. ω = −λ − cIm(hΨ(t) | φi). (9)
Remark 3. With c > 0, k > 0 and δ > 0 we ensure : dV /dt ≤ 0, i.e V is decreasing.
Remark 4. The regions A, C, respectively B, C are overlapping such that ∂A ⊂ intB, ∂B ⊂ intA and, respectively, ∂B ⊂ intC, ∂C ⊂ intB.
• We define the evolutor S1Ψ0 by solving the feedback equation such that :
• if the initial state Ψ0 ∈ A∩C, we initiate with the feedback corresponding to C
• and if the initial state Ψ0 ∈ B ∩ C, we initiate also with the feedback corresponding to C.
• We define also the evolutor S2Ψ0 by solving the feedback equation such that :
• if the initial state Ψ0 ∈ A∩C, we initiate with the feedback corresponding to A
• and if the initial state Ψ0 ∈ B ∩ C, we initiate also with the feedback corresponding to B.
FIG. 3 – u(I1, I2), u defined by (9), for δ = 0.5.
Examples and simulations
Numerical simulations have been performed for a three-dimensional test sys-tem with H0, H1 and H2 given by :
H0 = 0 0 0 0 1 0 0 0 32 , H1 = 0 1 1 1 0 0 1 0 0 , H2 = 0 0 1 0 0 0 1 0 0 . (7)
In this case the wave function is Ψ = (Ψ1, Ψ2, Ψ3)T . We use the previous Lyapunov control in order to reach the first eigen-state φ = Φ1 = (1, 0, 0) of
energy λ = 0, at the final time T .
Remark 1. We note that the conditions of T heorem 1 are fulfilled since : hΦ2|H1φi 6= 0 and hΦ3|H1φi 6= 0.
FIG. 1 – Population |Ψ1|2 (blue line) and control u (green line) ; initial
condi-tion Ψ(t = 0) = (0, 1/√2, 1/√2) ; system defined by (8) with feedback (6). We take k = c = 0.8 .
We consider another three-dimensional test system with H0, H1 and H2 gi-ven by : H0 = 0 0 0 0 1 0 0 0 32 , H1 = 0 1 0 1 0 0 0 0 0 , H2 = 0 0 1 0 0 0 1 0 0 . (8)
Convergence analysis
We denote by λi, with i = 1, ..., N the eigenvalues of the matrix H0. We say that H0 has non degenerate spectrum if for all (i, j), with i 6= j, i, j = 1, ..., N , λi 6= λj.
Theorem 1. Consider (2) with Ψ ∈ S2n−1C and an eigen-state φ ∈ S2n−1
C of
H0 associated to the eigenvalue λ. Take the feedback (6) with c, k > 0. Then the two following propositions are true :
1. If the spectrum of H0 is not degenerate, the Ω-limit set of the closed loop system is the intersection of S2n−1 with the vector space E = Rφ Sα CΦα where Φα is any eigenvector of H0 not co-linear to φ such that hΦα|H1φi = 0.
2. If H0 is not degenerate and hΦα|H1φi 6= 0, ∀α, Φα 6= φ, , the Ω-limit set reduces to {φ, −φ}.
Lyapunov control design
For convenience we denote : I1 = Im(hH1Ψ(t)|φi) and I2 = Im(hH2Ψ(t)|φi. Then (5) becomes the feedback :
u = −kI1/(1 + kI2)
ω = −λ − cIm(hΨ(t) | φi). (6)
Lyapunov control design
Take the following time varying function V (Ψ, t) :
V (Ψ, t) = hΨ − Ψr|Ψ − Ψri (3)
where
• h.|.i denotes the Hermitian product,
• t 7→ (Ψr(t), ur(t), ωr(t)) a reference trajectory i.e., a smooth solution of (2). The fonction V
• is positive for all t > 0 and all Ψ ∈ S2n−1
C ,
• vanishes when Ψ = Ψr.
The derivative of V is given by : dV
dt = 2(u − ur)
Im(hH1Ψ(t)|Ψri) + (u + ur)Im(hH2Ψ(t)|Ψri)+
2(ω − ωr)Im(hΨ(t) | Ψri), (4)
where Im denotes the imaginary part. When for instance we take :
u = ur(t) − kIm(hH1Ψ(t)|Ψri)(1 + kIm(hH2Ψ(t)|Ψri))
ω = ωr(t) − cIm(hΨ(t) | Ψri), (5)
with k and c strictly positive parameters, we ensure dV /dt ≤ 0, i.e. V is de-creasing.
In the following we will focus on the important case when the reference trajectory corresponds to an equilibrium :
ur = 0, ωr = −λ and Ψr = φ
where φ is an eigen-vector of H0 associated to the eigenvalue λ ∈ R.
General settings
We consider a n-level quantum system (~ = 1) evolving under the equation : i d
dtΨ(t) = (H0 + u(t)H1 + u
2(t)H
2)Ψ(t), (1)
where
• Ψ is the wave function, Ψ ∈ S2n−1
C = {Ψ ∈ C n
kΨk
Cn = 1};
• H0, H1 and H2 are n × n Hermitian matrices with complex coefficients ;
• u(t) ∈ R is the control (for instant the intensity of the laser field) ; • H0 is the internal Hamiltonian ;
• H1, H2 are operators that couple the system with the control u.
Since Ψ and eıθ(t)Ψ describe the same physical state for any global phase t 7→ θ(t) ∈ R, i.e Ψ1 and Ψ2 are identified when exists θ ∈ R such that
Ψ1 = eıθΨ2.
• we add a second control ω corresponding to ˙θ (see also [1]) ;
• we consider the following control system i d
dtΨ(t) = (H0 + u(t)H1 + u
2(t)H
2 + ω(t))Ψ(t), (2)
where ω ∈ R is a new virtual control .
Introduction
• We analyse the Lyapunov trajectory tracking of the Schr¨odinger equation for a second order coupling operator ;
• We present a theoretical convergence result ;
• For situations not covered by the first theorem we propose a numerical approach and complement it with a second theoretical result.
Lyapunov control of Schr¨odinger equations : beyond the dipole approximation
Andreea Grigoriu, CEREMADE - Universit´e Paris Dauphine, France ; Catalin Lefter, Faculty of Mathematics, University ”Al.I.Cuza”, Iasi, Romania ;Gabriel Turinici, CEREMADE - Universit´e Paris Dauphine, France.
