Control of Ultracold Atoms Trapped in an Optical Lattice
by Jieqiu Shao B.S., University of Iowa, 2020
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment
of the requirement for the degree of Master of Science Department of Mechanical Engineering
2020
Committee Members:
Marco Nicotra
Shalom Ruben
Kaushik Jayaram ii
Shao, Jieqiu (M.S., Mechanical Engineering) Control of Ultracold Atoms Trapped in an Optical Lattice Thesis directed by Assistant Professor Marco M. Nicotra
This thesis is a first step towards the integration of quantum physics and control engineering. The goal of the work is to model and control ultracold atoms trapped in an optical lattice. First, after introducing the fundamental concept of quantum mechanics, we model the quantum system and use Bloch basis decomposition to obtain an ordinary differential equation. We then discuss the geometric properties of the system and show that the system is fully controllable. After that, we design and apply two different control strategies to the system, the
Lyapunov control method and optimal control method. Finally, we discuss the simulation results from different control methods and detail the next step of this work. iii
Contents
1 Introduction 1 1.1 Introduction to Quantum Mechanics ...... 2 1.2 Modeling of Trapped Ultracold Atoms ...... 3 1.2.1 Potential Energy Operator ...... 3 1.2.2 Bloch Basis Decomposition ...... 4 1.3 Stationary Solutions ...... 5
2 Geometric Properties 9 2.1 General Concepts ...... 9 2.2 Geometric Properties of Quantum Systems ...... 12 2.3 Controllability ...... 12
3 Control Design 16 3.1 Lyapunov Function ...... 17 3.1.1 Bilinear Case ...... 18 3.1.2 Nonlinear Case ...... 22 3.2 Optimal Control ...... 24 3.3 Shooting Method ...... 26 3.4 Newton Step Method ...... 26 3.4.1 Finite Element Method ...... 28 3.4.2 Continuation Method ...... 29 3.4.3 Simulation Results ...... 29 iv
4 Conclusions 33
Bibliography 34 v
List of Figures
1.1 Optical lattice, the blue ball represents the ultracold atom and the red wave
represents the laser beam; the lattice wave number is kL = 2π/λL for a lattice
wavelength λL; The lattice depth is V0; φ(t) is a time varying phase...... 3 1.2 Eigenstates of the Bloch Hamiltonian; this figure shows the eigenstates asso-
ciated to the four lowest eigenvalues λ1, λ2, λ3, λ4...... 7
2 3 2 2 2 2.1 Top Left: the unit sphere S = {(x1, x2, x3) ∈ R |x1 + x2 + x3 = 1} is a 2 manifold; Top Right: the tangent space TxS is a plane that contains x and
∂F is normal to the vector ∂x = [2x1 2x2 2x3]; Bottom Left: the two vector T T fields f(x) = [−x2 x1 0] and g(x) = [−x3 0 x1] belong to the tangent space; Bottom Right: the systemx ˙ = f(x) + ug(x) is controllable. Given u = 0,
2 x˙ = f(x) spans the entirety of the level-set {x ∈ S | x3 = c}. Given u 6= 0, g(x) allows the system to move from one level-set to another...... 15
3.1 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0.2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0.2...... 20 3.2 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0.5; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0.5...... 20 vi
3.3 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 1; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 1...... 21 3.4 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 2...... 21 ˙ ∗ dV˙ 3.5 V vs u, given α = −1 and β = −0.1; u is the solution to du = 0 that minimizes V˙ ; the green region is the control law u = sin−1(K sin u?), where K ∈ (0, 1]...... 22 3.6 Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 1...... 23 3.7 Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 0.1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 0.1...... 23 3.8 Left: the control input u vs time using optimal control law (Newton method) on the bilinear model; Right: the population vs time using using optimal control law (Newton method) on the bilinear model...... 30 3.9 Left: the control input u vs time using optimal control law (Newton method) on the nonlinear model; Right: the population vs time using using optimal control law (Newton method) on the nonlinear model...... 30 3.10 The optimal control input u vs time obtained from bilinear and nonlinear model. 31 3.11 Left: the population vs time using optimal control input from bilinear model applied to nonlinear model; Right: the population vs time using using opti- mal control law (Newton method) on nonlinear model...... 32 1
Chapter 1
Introduction
This thesis investigates the control of ultracold atoms trapped in optical lattice. Ultra- cold atoms are atoms that are maintained at temperatures close to 0 Kelvin (absolute zero). Ultracold atoms offer remarkable opportunities for investigating quantum problems that are relevant to fields as diverse as condensed matter physics, statistical physics, quantum chem- istry, and high-energy physics [1]. The high degree of controllability and new observation tools, which enable the detection of each individual atom, make ultracold atoms ideal as analog quantum simulators [2, 3, 4, 5] high-accuracy measurements such as quantum clocks [6] and interferometers [7]. The optical lattices used in the experiments are typically formed by interfering several laser beams in order to realize a fully controllable periodic light struc- ture meant to mimic the crystal lattice of a solid. Quantum technology is currently at the tipping point and is aiming to transition from experimental physics laboratories to real-world engineering applications. This transition requires a strong degree of collaboration between quantum physicists and specialists from other disciplines, such as control engineering. This thesis brings together quantum mechanics and control engineering. I am a mechanical engi- neering student with little previous knowledge in quantum theory, and my role is to adapt classic tools from systems and control theory to the new and exciting setting of quantum systems. Before we delve deeper into the specifics of our system, we provide a brief overview of quantum mechanics. The objective is to provide just enough background information to 2
enable someone with a traditional engineering background to gain insight into the intricacies of quantum theory.
1.1 Introduction to Quantum Mechanics
Quantum mechanics describes the physical properties of small objects (atomic scale) that are well isolated from other objects. The purpose of this section is to introduce the fundamental concept of quantum mechanics. Quantum mechanics follows from four primary postulates: Wave Function: The state of a quantum system is fully described by a wave function Ψ(x, t), which is a complex valued function of two real variables, where x is position and t is time. The modulus square of the wave function, |Ψ(x, t)|2, is the probability density, which represents the probability of finding the atom in a given position at a given time. If there are many atoms, then the wave function represents the density of atoms at given position and time.
Observables: An observable is a physical quantity that can be measured and is captured by a Hermitian operator applied to the wave function Ψ(x, t). For example, the kinetic energy of a particle is Kˆ (x)Ψ(x, t), where
2 Kˆ (x) = − ~ ∇2 (1.1) 2m x
is the kinetic energy operator, m is the mass of particle and ~ is the reduced Planck constant. Measurements: For a given observable A, the only possible result of the measurement is
one of the eigenvalues ai of the operator, where ai is associated to an eigenvector νi satis-
fying A(x)νi(x) = aiνi(x). This means that the more closely the wave function matches an
eigenvector of the observable νi, the more likely the measurement is to reveal the associated eigenvalue. This property is fundamentally different from classical mechanics, where a mea- surement can yield a continuum of results. Indeed, the whole premise of quantum mechanics is that measurements are “quantized” and any measurement can only yield a discrete set of values. 3
Schr¨odingerequation: The dynamics of the wave function is described by the Schr¨odinger equation
i~Ψ(˙ x, t) = Hˆ (x)Ψ(x, t), (1.2) where Hˆ (x) = Kˆ (x) + Uˆ(x) is the Hamiltonian, Kˆ (x) is the kinetic energy operator and Uˆ(x) is the potential energy operator. This can be paralleled with classical mechanics where the total energy of the system is equal to the sum of kinetic and potential energy.
1.2 Modeling of Trapped Ultracold Atoms
1.2.1 Potential Energy Operator
For our optical lattice, the laser beam defines the potential energy of the wave function referring to Figure 1.1, it is possible to define the potential energy operator as
V Uˆ(x) = − 0 cos(2k x + φ(t)), (1.3) 2 L where V0 is the depth of the lattice, λL is the wavelength, kL = 2π/λL is the wave number, and φ(t) is the phase. Figure 1 below shows these parameters for the optical lattice.
Figure 1.1: Optical lattice, the blue ball represents the ultracold atom and the red wave represents the laser beam; the lattice wave number is kL = 2π/λL for a lattice wavelength
λL; The lattice depth is V0; φ(t) is a time varying phase.
Given the presence of an optical lattice, the time dependent Schr¨odingerequation (1.2) 4
can be specialized to
d 2 V i Ψ(x, t) = − ~ ∇2Ψ(x, t) − 0 cos(2k x + φ(t))Ψ(x, t). (1.4) ~dt 2m x 2 L
1.2.2 Bloch Basis Decomposition
The objective of this section is to reduce the infinite dimensional PDE (1.4) to a finite
dimensional ODE. Since the potential energy operator (1.3) is periodic with periodicity 2kL, we can apply the Bloch Theorem to write (1.4) as a linear combination of periodic functions
∞ X i2nkLx Ψ(x, t) = cn(t)e , (1.5) n=−∞
where cn(t) are time-dependent complex valued scalars. The Schr¨odingerequation (1.4) can then be written as
∞ ∞ 2 X i2nk x X ~ 2 i2nk x V0 i2nk x i c˙ (t)e L = − c (t)∇ e L − c (t) cos(2k x + φ)e L . (1.6) ~ n 2m n x 2 n L n=−∞ n=−∞
Noting that
e−i2kLxe−iφ + ei2kLxeiφ ∇2ei2nkLx = −(2nk )2ei2nkLx, cos(2k x + φ) = , (1.7) x L L 2
we can then rewrite equation (1.6) as
∞ ∞ 2 X i2nk x X ~ 2 2 V0 −iφ iφ i2nk x i c˙ (t)e L = ( (2nk ) c (t) − (c (t)e + c (t)e )e L . ~ n 2m L n 4 n−1 n+1 n=−∞ n=−∞
After eliminating the common terms ei2nkLx, we get an infinite set of ODEs. For each n, we have cn−1(t)
h 2 i i c˙ (t) = V0 iφ ~ 2 V0 −iφ , (1.8) ~ n − 4 e 2m (2nkL) 4 e cn(t) cn+1(t) where eiφ = cos φ + i sin φ = 1 + i sin φ + (cos φ − 1).
2 2 To simplify notation, we define V0 = 4αEr, where Er = ~ kL/2m is the recoil energy. T To obtain a finite ODE, we define the truncated vector of coefficients ψ = [c−N , . . . , cN ] . 5
Moreover, we apply a finite approximation that cn = 0, ∀|n| > N. The Schr¨odingerequation can then be written as
˙ iψ = H0ψ + sin φH1ψ + (cos φ − 1)H2ψ, (1.9)
where (2N)2 −α 2 −α [2(N − 1)] α . . . ...... H0 = Er −α 0 −α , ...... 2 −α [2(N − 1)] −α −α (2N)2 0 αi 0 −α −αi 0 αi −α 0 −α . . . . . . H1 = Er ...... ,H2 = Er ...... . −αi 0 αi −α 0 −α −αi 0 −α 0
Using small angle approximation, i.e. sin φ ≈ φ and cos φ ≈ 1, we can further reduce the equation to
iψ = H0ψ + φH1ψ, (1.10) which is the equation that is classically found in quantum control literature [8, 9, 10]. This thesis will investigate control solutions for both the bilinear approximation (1.10) and the fully nonlinear model (1.9).
1.3 Stationary Solutions
The typical control objective is to steer the system from an initial state to an equilibrium point, satisfyingx ˙ = 0. Now that we have the equation for our quantum system, we wish to 6 identify the stationary solutions in the absence of a shaking lattice, i.e. φ = 0 the equation for the autonomous system (1.9) and (1.10) then becomes
˙ iψ = H0ψ. (1.11)
In this context, we note that the only equilibrium point of the system is ψ = 0. However, this point does not satisfy the fundamental property of the wave function |ψ|2 = 1 and is therefore inadmissible. Thus, The control objective “design a control law such thatx ¯ is a global asymptotically stable equilibrium point” is not a well-posed problem for the quantum system.
To identify a suitable control objective, let λk be the eigenvalues of H0, and let νk be the
∗ associated eigenvector such that H0νk = λkνk and νk νk = 1. 7
Figure 1.2: Eigenstates of the Bloch Hamiltonian; this figure shows the eigenstates associated to the four lowest eigenvalues λ1, λ2, λ3, λ4.
˙ Given the initial condition ψ(0) = νk, the autonomous system iψ = H0ψ admits the
−iλkt solution ψ(t) = νke , since H0νk = λkνk implies iν˙k = λkνk, which belongs to the periodic
−iλkt orbit Ok = {νke |t ∈ [0, 2π/λk]}. A suitable control object for the quantum system then becomes “design a control law such that the periodic orbit Ok is a globally asymptotically stable limit circle”. To explain why this control objective is physically relevant in addition to being well-posed, we note that the eigenvectors of H0 are either odd or even vectors, as shown in Figure 1.2. Specifically, the eigenvector associated to the smallest eigenvalue λ0 represents the ground state of the system, i.e. the solution with the lowest energy. The eigenvector associated with λ1 is an odd vector that has most of its population in ±2~kL. Physically, this is equivalent to having two clouds of atoms moving in opposite directions with a momentum 8 of 2~kL. Therefore, transitioning from the ground state to the first eigenstate is equivalent to separating the atoms into two clouds moving in opposite directions. Likewise, transitioning to the third eigenstate provides a similar effect with the two clouds of atoms having a higher momentum of 4~kL, i.e. two clouds of atoms that are separating at higher speeds. Finally, it is worth noting that the eigenvectors of our system have the particular property of being either even vectors (i.e. φ0 6= 0, φn = φ−n or odd vectors (i.e. φ0 = 0, φn = −φ−n) 9
Chapter 2
Geometric Properties
In this chapter we are going to study the geometric properties of our system to formally define the manifold in which it evolves and to verify that it is possible to design a control law that can steer the system from any initial state to any target state. There are several studies related to the controllability of quantum systems, where [11] shows the controllability of a system using geometric control theory and [12] analyzes the controllability of the quantum system.
Before analyzing our specific system, however, the following section provides an overview of the key concepts from geometric control that will then be used throughout the chapter.
2.1 General Concepts
Geometric control is a branch of systems and control theory that studies the application of differential geometry. The purpose of this section is to introduce the general definitions used in geometric control and interpret their meaning using intuitive examples. These definitions will then be used to determine the structural properties of our quantum system.
Definition 2.1 (Manifold). A manifold M is a topological space that locally resembles Eu- clidean space near each point [13, 14]. Given a differentiable function F : CN −→ Rk, we 10
define a differentiable manifold as M = {x ∈ CN | F (x) = c} coloneqqF −1(c).
Example 2.1. The unit sphere S2 = {x ∈ R3 | xT x = 1} is a differentiable manifold 2 −1 2 2 2 satisfying S = F (1), with F = x1 + x2 + x3.
Definition 2.2 (Tangent Space). The tangent space TxM is the collection of all the vectors tangent to the manifold M [13, 14]. Given a differentiable manifold M = F −1(c), the tangent
N ∂F ∂F space is TxM = {v ∈ C | ∂x v = 0} := ker ∂x .
2 −1 2 2 2 Example 2.2. Given the unit sphere S = F (1), with F = x1 + x2 + x3, the tangent space 2 TxS = ker([2x1 2x2 2x3]) is a plane that contains x and is normal to the vector [2x1 2x2 2x3].
Definition 2.3 (Vector Field). Given a differentiable manifold M, a vector field f is a map
from the manifold to the tangent space: f : M −→ TxM [13, 14].
2 T T Example 2.3. Given the unit sphere S , f(x) = [−x2 x1 0] and g(x) = [−x3 0 x1] are
2 both vector fields since [2x1 2x2 2x3]f(x) = 0 and [2x1 2x2 2x3]g(x) = 0, ∀x ∈ S .
By taking advantage of these definitions, the following can be proven.
Theorem 2.1. Let the dynamic system x˙ = f(x)+u(t)g(x) be subject to the initial condition x(0) ∈ M. Then, ∀u : R+ → R, the system trajectories satisfy x(t) ∈ M, ∀t ∈ R+, if and only if f(x) and g(x) are both vector fields of M.
Proof. See [15].
Theorem 2.1 states that the state vector x(t) cannot exit the manifold M regardless of the control input u(t). Nevertheless, being unable to exit M is not the same as being able to reach any point within M. To achieve the latter, the following definitions are also needed.
Definition 2.4 (Controllability). A system is controllable if, for any initial state x(0) = x0,
any finial state x1, and time T , there exists a piece wise continuous control law such that the
solution x(t) of the system satisfies x(T ) = x1 [16]. 11
Definition 2.5 (Distribution). Let M be a manifold. A distribution D assigns to each
x ∈ M a subspace Dx ⊆ TxM [13], i.e.
D : M −→ TxM.
Consider the dynamicsx ˙(t) = f(x) + u(t)g(x), the distribution D is obtained as follow:
0 1. Dx := span of f(x) and g(x). 1 0 0 2. Dx := Dx plus span of [f(x), h(x)] plus span of [g(x), h(x)], ∀h(x) ∈ Dx. k k−1 k−1 3. Dx := Dx plus span of [f(x), h(x)] plus span of [g(x), h(x)], ∀h(x) ∈ Dx . k A distribution is said to be full ranked when Dx = TxM.
Theorem 2.2. Let M be a compact set, then for a system whose dynamics equation is
x˙ = f(x) + ug(x),
the system is controllable if the distribution generated by f(x) and g(x) is full rank.
Proof. See [17, Theorem 7.1].
Figure 1.1 provides a geometric interpretation of Definitions 1.1-1.3 and Theorem 2.2. These general principles are specialized to our quantum system in Section 2.2. It should be noted, however, that the ability to steer the system to a target state does not imply that the system will remain in that state. To identify a meaningful control objective, the following definitions are given.
Definition 2.6 (Invariant Subspace). Consider a dynamics x˙ = f(x), where x ∈ M. Ω ⊂ M is an invariant subspace if for any x(0) ∈ Ω, x(t) ∈ Ω ∀t > 0 [18].
Definition 2.7 (Projection Operator). Define as ΠΩ(x). The projection operator minimize the distance between x and y, where y belongs to the subspace Ω:
ΠΩ(x) = arg min ||x − y||2. y∈Ω
Definition 2.8 (Global Asymptotic Stability). An invariant subspace Ω ⊂ M is Globally
Asymptotically Stable (GAS) if, ∀x(0) ∈ M, limt→∞ kx(t) − ΠΩ(x(t))k = 0 [19]. 12
Rather than steering the system to an arbitrary target state, a suitable control objective is to ensure that a given invariant subspace is GAS.
Definition 2.9 (Lie bracket). Let M be a differentiable manifold; let f and g be vector fields on M. [f, g] is called the Lie bracket [13] of f and g where
∂g ∂f [f, g](x) = f(x) − g(x). ∂x ∂x
2.2 Geometric Properties of Quantum Systems
This section specializes the concepts introduced in Section 2.1 to our quantum system. Manifold: In our quantum system, the only constraint for the state vector ψ is ψ∗(t)·ψ(t) =
1, for all t > 0. Since ψ(t) ∈ Cn, the manifold of the system is then the unit sphere S2n−1. Tangent Space: Let P (ψ) = ψ∗ψ = 1, it is reasonable to expect that the system dynamics ˙ ˙ ∗ ∗ ˙ 2n−1 ∗ ∗ satisfy P = ψ ψ + ψ ψ = 0. The tangent space is TψS = {Hψ | (Hψ) ψ + ψ Hψ = 0}, and this holds if and only if ψ∗(H∗ + H)ψ = 0, where H is a skew-Hermitian matrix (H∗ + H = 0). ˙ For the bilinear model, we have the dynamics equation iψ = H0ψ + φH1ψ, which can be ˙ rewritten as ψ = (−iH0)ψ + φ(−iH1)ψ. We define f(ψ) = (−iH0)ψ and g(ψ) = (−iH1)ψ.
Noting that here both −iH0 and −iH1 are skew-hermitian matrices, so f(x) and g(x) are indeed two vector fields on the manifold. If the system is controllable, then for a given
2n−1 2n−1 reference r ∈ S , we can always reach it from any initial condition x0 ∈ S .
Invariant Subspace: As detailed in Section 1.3, the periodic orbits Ok are, indeed, invari- ant subspaces.
2.3 Controllability
We now wish to apply Theorem 2.2 to prove that the system is controllable. To do so,
we begin by specializing our system to Er = 1, α = 2.5 and N = 1. The quantum system 13
˙ then becomes finite dimensional with the dynamics equation ψ = A1ψ + φA2ψ, where −2i 1.25i 0 0 1.25 0 A1 = 1.25i 0 1.25i ,A2 = −1.25 0 1.25 0 1.25i −2i 0 −1.25 0
and φ is the control input. The system takes the form ofx ˙ = f(x) + ug(x), so we can analyze the controllability of the quantum system. We have to prove the distribution is full ranked. The dimension of n × n skew-hermitian matrices is n2, since the real part of the skew-hermitian matrices is skew-symmetric (dimension n(n−1)/2) while the imaginary part is symmetric (dimension n(n + 1)/2). Thus, the dimension of n × n skew-hermitian matrices is n(n − 1)/2 + n(n + 1)/2 = n2. We now compute the distribution and check whether it is
0 full ranked. The first distribution Dψ is trivial, where
0 Dψ = [A1ψ, A2ψ].
In this case, the Lie bracket [A1ψ, A2ψ] = [A1,A2]com · ψ, where [, ]com represents the matrix
k commutator ([A, B]com = AB − BA). To compute Dψ, we have to check whether the new matrices generated by matrix commutator are linearly independent. We used MATLAB to check: first, convert a 3x3 matrix into a column vector, where a11 a12 a13 h iT A = −→ a = . a21 a22 a23 a11 a12 a13 a21 a22 a23 a31 a32 a33 a31 a32 a33
Then we used a MATLAB built-in function rank() to check whether these column vectors are linearly independent; if there are n vectors and the function returns a value equals to n, then these matrices are linearly independent; if the value is less than n, they are linearly 14 dependent. By using this method, we get the distribution computed as follow:
0 Dψ = {A1,A2}ψ
1 Dψ = {A1,A2,A3}ψ, A3 = [A1,A2]
2 Dψ = {A1,A2,A3,A4,A5}ψ, A4 = [A1,A3],A5 = [A2,A3]
3 Dψ = {A1,A2,A3,A4,A5,A6,A7}ψ, A6 = [A1,A4],A7 = [A1,A5]
4 Dψ = {A1,A2,A3,A4,A5,A6,A7,A8,A9}ψ, A8 = [A1,A7],A9 = [A2,A6]
Note that the matrix generated by matrix commutator of two skew-hermitian matrices are still skew-hermitian. As a result, these 9(32) linearly independent skew-hermitian matrices span the whole vector space of 3x3 skew-hermitian matrices, thus proving that the distri- bution generated by f(ψ) = A1ψ and g(ψ) = A2ψ is full rank. Therefore, the quantum system in bilinear form is controllable when we apply finite approximation N = 1. We then increase N and use the same method to check the controllability of the system. This works up to N = 3, but after that the numerical conditioning becomes very poor and the test becomes inconclusive. This is because when N = 4, we have to find 81 linear independent skew-hermitian matrices via the matrix commutator. However, the first row, first column
2 entry of A1 is (2N) = 64, which gets multiplied by itself each time there is a commutation and is equal to 1.8 × 1016 after just 9 commutations. Considering the fact that MATLAB uses a floating point precision of 10−15, the result of the rank() function becomes unreliable after 9 commutations. 15
2 3 2 2 2 Figure 2.1: Top Left: the unit sphere S = {(x1, x2, x3) ∈ R |x1 + x2 + x3 = 1} is a 2 manifold; Top Right: the tangent space TxS is a plane that contains x and is normal to
∂F T the vector ∂x = [2x1 2x2 2x3]; Bottom Left: the two vector fields f(x) = [−x2 x1 0] and T g(x) = [−x3 0 x1] belong to the tangent space; Bottom Right: the systemx ˙ = f(x)+ug(x)
2 is controllable. Given u = 0,x ˙ = f(x) spans the entirety of the level-set {x ∈ S | x3 = c}. Given u 6= 0, g(x) allows the system to move from one level-set to another. 16
Chapter 3
Control Design
Having shown that the system is controllable and having identified a suitable control objective, our goal is to design and develop a control law that achieves the required control objective. In this thesis, the control objective is to steer the quantum system from the ground state to an eigenstate. The design of controllers for quantum systems is an active field of research and notable results have been surveyed in [8, 9, 10]. One successful control strategy on the quantum systems is the Lyapunov control method [20, 21], where the control input is designed to ensure that a given distance between the current state and the target state is time-decreasing. Another approach is optimal control, where the control input is obtained by minimizing the desired cost function. In fact, this method has been successfully applied to real world quantum applications. Rigorous optimal control theory is used to achieve given control objectives in physical chemistry [22, 23, 24]. For spin systems, time-optimal problems have been solved to achieve control objectives in minimum time [25, 26, 27]. Optimal control methods have also been applied to multi-dimensional nuclear magnetic resonance (NMR) experiments to improve the sensitivity of these systems in presence of relaxation [28, 29, 30]. 17 3.1 Lyapunov Function
We will introduce and compare different kinds of control schemes, and the first is Lya- punov control method. For control systems, the Lyapunov function V (x) can be considered
as the distance from state x(t) to target orbit Of . To be considered as a distance, V (x) needs to satisfy the following properties:
1 V (x) is zero if and only if the state x(t) belongs to the periodic orbit Of ; otherwise, V (x) must be strictly positive. 2 The time derivative of V (x) is negative, meaning that the distance between the state x(t) and target state xf is always decreasing over time. To explain the above idea in equations, we can say that V (x) is a Lyapunov function if the followings hold:
1 V (x) > 0 if x(t) ∈/ Of and V (x) = 0 if and only if x(t) ∈ Of ˙ 2 V (x) < 0 if x∈ / Of For closed quantum systems, we select the Hilbert-Schmidt distance [31] between the con- trolled state ψ and the target eigenvector νf as a Lyapunov function [20]
1 V = (1 − |ν∗ψ|2). (3.1) 2 f
2n−1 To verify that (2.1) is a Lyapunov function, given νf ∈ S and H0νf = λf νf , we note that the following holds:
∗ 2 2n−1 1 |νf ψ| ∈ [0, 1], ∀ψ ∈ S . ∗ 2 2 |νf ψ| = 1 ⇐⇒ ψ ∈ Of . ∗ 2 ∗ ∗ ˙ Noting that |νf ψ| = (νf ψ)(ψ νf ), the time derivative of the Lyapunov function V is com- puted as follows:
1 d 1 d V˙ = − ( (ν∗ψ)(ψ∗ν )) − (ν∗ψ)( (ψ∗ν )). 2 dt f f 2 f dt f
The Schr¨odingerequation of the quantum system is
d i ψ = (H + H(u))ψ, dt 0 18
where H(u) = uH1 in the bilinear model and H(u) = sin uH1 +(1−cos u)H2 in the nonlinear model. We compute
d (ν∗ψ) = ν∗ψ˙ = ν∗(−i(H + H(u)))ψ = −iλ ν∗ψ − i(ν∗H(u)ψ) dt f f f 0 f f f d (ψ∗ν ) = ψν˙ = ψ∗(−i(H + H(u)))∗ν = iλ ψ∗ν + i(ψ∗H(u)∗ν ), dt f f 0 f f f f then V˙ can be written as
1 V˙ = ((iλ (ν∗ψ) + i(ν∗H(u)ψ))(ψ∗ν ) − (ν∗ψ)(iλ ψ∗ν + i(ψ∗ν + i(ψ∗H(u)ν )))) 2 f f f f f f f f f i i i i = λ |ν∗ψ|2 + (ν∗H(u)ψ)(ψ∗ν ) − λ |ν∗ψ|2 − (ψ∗H(u)∗ν )(ν∗ψ). 2 f f 2 f f 2 f f 2 f f
Using the property i(a − a∗) = −2=(a) and after eliminating and combining like terms, the time derivative of Lyapunov function is
˙ ∗ ∗ V = −=((νf H(u)ψ)(ψ νf )). (3.2)
3.1.1 Bilinear Case
By using small angle approximation, H(u) = uH1 in our bilinear model, equation then becomes ˙ ∗ ∗ V = −u=((νf H1ψ)(ψ νf )). (3.3)
Here, we can choose a control law such that V˙ is negative semi-definite
∗ (ψ ψf ) ∗ u = K=( ∗ (ψf H1ψ)), (3.4) ||ψ ψf || ˙ where K > 0. This control law ensures that V 6 0, which is sufficient to prove stability but does not guarantee asymptotic stability. In fact, it has been proven in [32] that, if
∗ ψf H1ψ 6= 0 ∀ψ∈ / Of , then the system is asymptotically stable. However, this property doesn’t hold in our case. Indeed, given an odd eigenvector νf , it follows that any odd vector
ψ satisfies vf H1ψ = 0. The analogous is also true if νf is an even eigenvector. To prove asymptotic stability, we will apply the KLS principle and modify our control law to ensure ˙ that the smallest invariant set contained in S = {ψ|V (ψ) = 0} is the target state νf . Given 19
an odd eigenvector νf , the set S is a linear combination of all the odd eigenvectors ν2k+1 of
T H0, i.e. ψ = [α β . . . 0 ... − β − α] . To destabilize the undesired stationary solutions and ensure that only Of is an invariant set we can then choose our control law u = 1 when
ψ ∈ S \ {Of }. From the system dynamics, we get
˙ iψ = H0ψ + H1ψ
Since every odd vector is a linear combination of odd eigenvectors, H0ψ will result in an odd vector for any odd vector ψ. As for H1ψ, we note α β β −α + γ γ −β + δ 0 i δ −γ + −i 0 i −δ .. .. H1ψ = −i . . 0 = 0 , .. . 0 i − −δ −i 0 −δ − γ −γ δ − β −β γ − α −α β which is an even vector. As a result, H0ψ + H1ψ is not an odd vector and the set S \ {Of } is not invariant. As a result, the only invariant subset of S is Of , and the control law in equation guarantee the convergence from the initial state ν0 to the target Of . The complete Lyapunov control law is ∗ ∗ 2n−1 u = K Im (ψf H1ψ)(ψ ψf ) , if ψ ∈ S \ S, (3.5) ∗ u = K(1 − ψ ψf ), if ψ ∈ S.
Simulation Results
We apply the control law from equation to the bilinear model, picking time horizon T = 40. In this thesis, we are using different K values, starting from a small positive 20 number.
Figure 3.1: Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0.2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0.2.
Figure 3.2: Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0.5; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0.5.
Comparing figure 3.1 and 3.2, we can say that if K increases (from 0.2 to 0.5), the control input u also increases and the state of the system converges faster to the target state. This would lead us to believe that increasing K makes the closed-loop system evolve faster. However, since our bilinear model uses the small angle approximation, sin u = u, our 21
control input cannot exceed ±1. As we keep increasing the value of K, we have to saturate the control input: if u(t) > 1 at time t, then u(t) = 1; if u(t) < −1 at time t, then u(t) = −1.
Figure 3.3: Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 1; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 1.
Figure 3.4: Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 2.
We now note that increasing K does not necessarily lead to a better convergence rate, meaning that some tuning is required in order to achieve the fastest response. 22
3.1.2 Nonlinear Case
In our nonlinear model, H(u) = sin uH1 + (1 − cos u)H2, equation then becomes
˙ ∗ ∗ ∗ ∗ V = −=((νf H1ψ)(ψ νf )) sin u − =((νf H2ψ)(ψ νf ))(1 − cos u). (3.6)
∗ ∗ ∗ ∗ We define α = =((νf H1ψ)(ψ νf )) and β = =((νf H2ψ)(ψ νf )), the time derivative of Lya- punov function V can then be written as V˙ = −α sin u − β(1 − cos u).
˙ ∗ dV˙ Figure 3.5: V vs u, given α = −1 and β = −0.1; u is the solution to du = 0 that minimizes V˙ ; the green region is the control law u = sin−1(K sin u?), where K ∈ (0, 1].
˙ dV˙ The control input that minimizes V can then be computed by solving du = 0 ! −α(ψ) −β(ψ) u? = −atan2 , . (3.7) pα(ψ)2 + β(ψ)2 pα(ψ)2 + β(ψ)2 However, we note that any control such that V˙ is negative semi-definite is a suitable solution. Therefore, we will consider u = arcsin(K sin u?), (3.8)
where K ∈ (0, 1]. 23
Simulation Results
We apply the control law from equation to the nonlinear model, picking time horizon T = 100. First we apply the control law that minimizes V˙ , i.e. K = 1.
Figure 3.6: Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 1.
However, we note from Figure 3.5, the performance of the system is not well behaved. To get better performance, we can reduce K = 0.1,
Figure 3.7: Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 0.1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 0.1. 24
Comparing Figure 3.5 and 3.6, we find an interesting fact that the control input u? (K = 1) which minimizes V˙ does not necessarily provide the best performance. In our case, when K = 0.1, the state of the system not only converges faster to the target space, but also has less wiggling when it becomes stable. This result is consistent with the observations made on the linear system, where we noted that increasing K as much as possible was actually counter-productive. It is also worth noting that the nonlinear control law displays a natural saturation for the control input u.
3.2 Optimal Control
Although the Lyapunov control method provides a control law that steers the quantum system from the initial state to the target state, we wish to find a faster control law. Thus, we turn to optimal control. Before we can start, however, it is important to note that the numerical solver we use for optimal control requires real vector space whereas the state vector ψ is a complex vector. As a result, we have to re-define the state as real vector. Given
ψ ∈ Cn, consider the bilinear mapping Re(ψ) h i h i x = ψ = In 0 x + i 0 In x, (3.9) Im(ψ)
2n where x ∈ R is the new real state vector and In is the identity matrix of size n. Without ˙ loss of generality, it can be shown that the complex ODE iψ = H0ψ + φH1ψ, which is ˙ equivalent to ψ = (−iH0)ψ + φ(−iH1)ψ, can be transformed into the real ODE
x˙ = Ax + uBx, (3.10) Im(H0) Re(H0) Im(H1) Re(H1) where A = , B = , and u = φ. −Re(H0) Im(H0) −Re(H1) Im(H1)
In general, an optimal control law can be obtained by solving the following optimization 25
problem, Z T min J(x(t), u(t))dt 0 s.t. x(0) = 0
x˙(t) = f(x(t), u(t))
where J(x(t), u(t)) is the cost function and f(x(t), u(t)) is the system dynamics. For quantum
1 T 1 T systems, we select the cost function J(x(t), u(t)) = 2 x Qx + 2 u Ru, where Q is a positive definite matrix and R is a positive scalar. We want the cost be strictly greater than 0 when x∈ / Of , and equal to 0 if and only if x ∈ Of . To satisfy this condition , since the periodicorbit Of is entirely defined by the eigenvector νf , we can design matrix Q as follows: T I − νf νf 0 Q = , (3.11) T 0 I − νf νf where νf is the desired eigenvector of H0. Then, the optimal control problem becomes Z T 1 1 min [ xT (t)Qx(t) + u(t)T Ru(t)]dt 0 2 2
s.t. x(0) = [ν0; 0]
x˙(t) = Ax(t) + u(t)B1x(t)
Here, we introduce the Hamiltonian based on Pontryagin’s maximum principle to solve the problem H(x, u, λ) = J(x, u) + λT f(x, u) (3.12)
where λ(t) ∈ Rn is the co-state vector. The necessary conditions for the optimal solution are:
Hu(x, u, λ) = 0
Hλ(x, u, λ) =x ˙ ˙ Hx(x, u, λ) = −λ (3.13)
x(0) = x0
λ(T ) = 0 26
The optimization control problem then become a two-point boundary value problem; the
boundary values are x(0) = x0 and λ(T ) = 0. We now address possible methods for solving the optimal control problem.
3.3 Shooting Method
To solve the two-point boundary value problem, we first implement the shooting method
1 T 1 T T proposed in [33]. The Hamiltonian is H = 2 x Qx + 2 u Ru + λ (Ax + uBx), and the co- state vector λ can be written as λ = P x, and its time derivative is λ˙ = P˙ x + P x˙. From the
necessary condition Hu = 0, we can obtain the optimal solution
u∗ = R−1xT BT P x. (3.14)
˙ ˙ Based on the necessary condition Hx = −λ = −P x−P x˙ and system dynamicsx ˙ = Ax+uBx,
P˙ = −Q − (AT P + PA) − u(BT P + PB). (3.15)
To solve for u and P , we first use a initial guess uguess(t) to solve for P backward in time t; then we use P to solve for a new u(t) forward in time t. This u(t) can then be used to solve for a new P , and thus the two-point boundary value problem is solved by such an iteration, which is also known as the shooting method. Although the method works for simple (e.g. up to 4 states) quantum systems, we found it to be numerically unstable for our purposes. This is likely due to the fact that by separately solving the problem backward and forward in time, the shooting method accumulates too much error at each iteration.
3.4 Newton Step Method
We consider implementing the Newton step method to compute x and lambda simul- taneously as opposed to sequentially. The necessary conditions in equation (3.13) can be 27 re-written as F (z) = 0 described in [34], with z(t) = (x(t), u(t), λ(t)) and Hu(z) x˙ − f(z) F (z) = λ˙ + H (z) . (3.16) x x(0) − x0 λ(T )
Given a guess zk we seek an update law in the form zk+1 = zk + dk, where dk is obtained by
0 solving the Newton iteration F (zk, dk) + F (zk) = 0, with Huu(z)du + Hux(z)dx + Huλ(z)dλ ˙ dx − fx(z)dx − fu(z)du F 0(z, d) = d˙ + H (z)d + H (z)d + H (z)d . λ xx x xu u xλ λ dx(0) dλ(T ) This leads to
Huu(z)du + Hux(z)dx + Huλ(z)dλ + Hu(z) = 0 ˙ dx − fx(z)dx − fu(z)du +x ˙ − f(z) = 0 ˙ ˙ dλ + Hxx(z)dx + Hxu(z)du + Hxλ(z)dλ + λ + Hx(z) = 0
dx(0) + x(0) − x0 = 0
dλ(T ) + λ(T ) = 0.
Assuming Huu(x(t), u(t), λ(t)) is invertible for all t ∈ [0,T ], we can write the first equation as
−1 du = −Huu (z)(Hux(z)dx + Huλ(z)dλ + Hu(z)), (3.17) then the remaining equations above can be rewritten as a two point boundary value problem, ˙ −1 −1 −1 dx fx − fuHuu Hux −fuHuu Huλ dx f − fuHuu Hu − x˙ = + , ˙ −1 −1 −1 ˙ dλ −Hxx + HxuHuu Hux HxuHuu Huλ − Hxλ dλ HxuHuu Hu − Hx − λ (3.18) with the boundary conditions x(0) = x0 and λ(T ) = 0. 28
3.4.1 Finite Element Method
To solve the two-point boundary value problem (3.18), we first discretize the system by
T sampling for t ∈ [0,T ] into N intervals, with ∆T = N . As for the time derivative terms, these can be obtained by the central difference approximation x − x x˙ = i+1 i−1 , i 2∆T except for the very first and last terms, which can be obtained by x − x x − x x˙ = 1 0 , x˙ = N N−1 . 0 ∆T N ∆T The TPBVP (3.18) can then be written as a big linear equation as follows: −II 0 ... 0 Γ0 0 ... 0 . d0 . d0 b0 I I . . − 0 . 0 Γ1 . 2 2 1 . . d1 . . d1 b1 0 .. .. 0 = . .. + , (3.19) ∆T . . . . . . . . − I 0 I Γ 0 2 2 N−1 dN dN bN 0 ... 0 −II 0 ... 0 ΓN
−1 −1 dxi fxi − fuiHuuiHuxi −fuiHuuiHuλi where di = ,Γi = , −1 −1 dλi −Hxxi + HxuiHuuiHuxi HxuiHuuiHuλi − Hxλi
−1 fi − fuiHuuiHui − x˙i and bi = . To account for the boundary conditions we also modify −1 ˙ HxuiHuuiHui − Hxi − λi the equations: h i I 0 d0 = x0 − x(0) 1 h i d0 h i h i 0 − 0 = 0 Γ0d0 + 0 b0 ∆T I I I I d1 1 h i dN−1 h i h i − 0 0 = 0 ΓN dN + 0 bN ∆T I I I I dN h i 0 I dN = −λ(T ). 29
Equation (3.19) can thus be viewed as a linear system of equations ∆d = Γd + b. As a result, the two-point boundary value problem can be solved by computing d = (∆ − Γ)−1b and using the update law
xk+1 = xk + dxk
λk+1 = λk + dλk.
Here, both x and λ are solved simultaneously. The control input can then be obtained as
1 u = − xT BT λ . (3.20) k+1 R k+1 k+1
3.4.2 Continuation Method
At first, we applied the optimal control law directly to the quantum system and tried
to get converged from the initial state ν0 (the ground state) all the way to the target state
ν1, which is the eigenstate corresponding to the second smallest eigenvalue of H0. However, the simulation result behave poorly and convergence was not achieved likely due to the fact that the Newton method only guarantees local convergence. To account for this, we decided to apply the continuation method, which means instead of starting from the ground state, we picked an initial state that was close to our target state and computed the solution. We could then update the initial state by using the optimal solution obtained from the previous initial state as an initial guess for the new otimal control problem. In this thesis, the update law for the initial state is
0 p p x0 = β · ν0 + 1 − β · ν1, (3.21)
where β = 0.1 : 0.1 : 1.
3.4.3 Simulation Results T I − ν1ν1 0 We pick time horizon T = 20, R = 10, and Q = . T 0 I − ν1ν1 30
Figure 3.8: Left: the control input u vs time using optimal control law (Newton method) on the bilinear model; Right: the population vs time using using optimal control law (Newton method) on the bilinear model.
Figure 3.9: Left: the control input u vs time using optimal control law (Newton method) on the nonlinear model; Right: the population vs time using using optimal control law (Newton method) on the nonlinear model.
First, we compare the optimal control input u(t) from two different models. 31
Figure 3.10: The optimal control input u vs time obtained from bilinear and nonlinear model.
According to the figure above, it is hard to tell the difference between two models. We then use the optimal control input u(t) from the bilinear model to solve the nonlinear ODE
x˙ = H0x+sin uH1x+(1−cos u)H2x and compare the control law obtained using the optimal population from the nonlinear model. 32
Figure 3.11: Left: the population vs time using optimal control input from bilinear model applied to nonlinear model; Right: the population vs time using using optimal control law (Newton method) on nonlinear model.
As we can see from the figures above, this difference between the two models is clear: although the control input obtained from the reduced (bilinear) model is relatively similar to the one obtained using the full (nonlinear) model, the difference between the two becomes significant when applied to the nonlinear system dynamics. While using the optimal control input from the reduced (bilinear) model, the full (nonlinear) system does not converge to the target state. By comparing Figure 3.7 and 3.9, we can see that the optimal controller achieves faster convergence than the Lyapunov controller. 33
Chapter 4
Conclusions
In this thesis, we have successfully designed and developed control laws for a cloud of ultracold atoms trapped in an optical lattice. Starting from the general Schrodinger Equation, we showed how to use Block Basis decomposition to obtain a finite dimensional ODE. We then proposed two different models: a nonlinear model and a bilinear model by using small angle approximation. We then studied the geometric properties of the quantum system, characterizing the manifold in which it evolves, identifying the periodic solutions, and proving that it is fully controllable. After that, we used two different methods, Lyapunov control and optimal control, to steer the system from one periodic orbit to another. The Lyapunov control method guarantees the convergence from the initial state to the target state, although it may take longer time horizon to converge; optimal control minimizes the desired cost function and provides a control law that converges faster. The difference between the bilinear and nonlinear models are also compared and discussed. This thesis represents a first step in the development of a quantum interferometer based on trapped ultracold atoms. Future steps will include an experimental validation of the proposed control methods. 34
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