Open Quantum Systems Under Coherent Control with Continuous Monitoring and Unitary Feedback Master Thesis to Obtain the Academic Degree Master of Science

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Albert-Ludwigs University of Freiburg Faculty of Mathematics and Physics Institute of Physics Quantum Optics and Statistics

Open Quantum Systems under Coherent Control with Continuous Monitoring and Unitary Feedback Master Thesis to obtain the academic degree Master of Science

Tobias Brünner

Supervising Professor: Prof. Dr. Andreas Buchleitner, Albert-Ludwigs University of Freiburg Faculty of Mathematics and Physics Institute of Physics Quantum Optics and Statistics

Advisor: Dr. Clemens Gneiting, Albert-Ludwigs University of Freiburg Faculty of Mathematics and Physics Institute of Physics Quantum Optics and Statistics

Freiburg, April 02, 2014

Danksagung

Diese Arbeit ist ein Zufallsprodukt im weiteren Sinne, denn sie begründet sich aus einer Verkettung unterschiedlichster Gegebenheiten und wurde während ihrer Entste- hung von den verschiedensten Einflussfaktoren geprägt. Unter anderen Umständen hätte ich mein Physik Masterstudium wohl mit einem völlig anderen Werk beendet. Macht dies die hier vorliegende Wort-, Formel- und Bildansammlung "wertlos"? In meinen Augen ganz und garnicht! Die obige Feststellung verdeutlicht doch nur die Einzigartigkeit der Prägung und Unterstützung, die ich in meinem bisherigen Leben und vor allem während der vergangen zwölf Monate erfahren habe. Ohne den Boss: Andreas Buchleitner wäre ich vielleicht niemals im wunderschönen Freiburg geblieben und es gäbe in erster Instanz garkeine so anspornende und gleichzeitig warmherzige Arbeitsgruppe "Quantenoptik und -statistik", derer ich Teil sein durfte. Zu dieser gehörig ist auch Clemens, mein direkter Betreuer, dem ich zu tiefstem Dank für unzählige Diskussionsstuden, erhellende Momente, herausfordernde Fragen, Re- hearsalsitzungen, Korrekturlesungen, Kritik und Verbesserungsvorschläge verpflichtet bin, vielen herzlichen Dank! Auch der Rest der Gruppe, vorallem Edoardo, Chahan, Juliane und Slava hatten immer ein offenes Ohr (und Auge), um bei anfallenden Fragen ihre Expertise mit mir zu teilen. Abseits vom Forschungsalltag bescherte mir die Mittagscrew stets vergnügliche und erholsame Pausen auf dem Dach, in der Küche oder im Garten. Die dabei verzehrten Lebensmittel, erworben auf dem Freiburger Münstermarkt, hatten sicher auch einen endlichen Einfluss auf meine Produktivität. Wesentlich bedeutsamer ist da aber natürlich die moralische Unterstützung meiner lieben Mitbewohner in Kappel und meiner Freunde in- und ausserhalb Freiburgs. More than a friend, Michelle is an integral part of my overall well-being and gives me the necessary strength to keep fighting, Having met her is another perfect example of a random concatenation of circumstances that is not ‘worthless’ because of this very nature. Und schliesslich bin ich meinen Eltern und Familienangehörigen in ganz besonderem Maße für ihre immerwährende Fürsorge dankbar, zumal ich ihnen nicht immer ein mustergültiger Sohn, Enkel, Urenkel, Bruder, Cousin und Neffe bin. All diese mir entgegengebrachte Hilfe ermöglichte erst das Nachfolge und stimmt mich zutiefst glücklich und zufrieden.

Abstract

The screening of quantum properties, such as for example coherence and entanglement, against detrimental environmental noise is a key issue in the advancement of quantum technologies, like quantum computation, cryptography, or teleportation. While coherent control techniques are well-established, it has been shown that in case of Markovian open quantum systems, only a limited degree of coherence can be uphold against the detrimental environmental noise. We investigate to what extent the stabilization potential can be improved with continuous monitoring and unitary feedback. To this end, we monitor single two-level atom quantum trajectories in the presence of spontaneous decay. Already without feedback, we ﬁnd that - in contrast to unmonitored systems - arbitrary pure states can be stabilized by post-selecting on the measurement outcomes. Employing unitary feedback, any desired qubit state can be rendered stationary. As we show, this still remains mainly valid when realistic assumptions, such as perturbed control Hamiltonians, inefﬁcient monitoring, and time-delayed feedback are taken into account.

Zusammenfassung

Zur Realisierung von Quantentechnologien ist es zwingend notwendig, die hierzu notwendigen Quanteneigenschaften, wie zum Beispiel Kohärenz und Verschränkung, gegen schädliche Umgebungseinflüsse zu schützen. Eine mögliche Strategie dies zu erreichen, beruht in kohärenter Kontrolle, die experimentell vergleichsweise einfach realisierbar ist, die gewünschten Eigen- schaften allerdings nur begrenzt aufrecht erhalten kann. Wir untersuchen, inwiefern die Kontrollmöglichkeiten durch eine Erweiterung mittels kon- tinuierlicher Messung und Rückkopplungsmechanismen verbessert werden können. Dabei beschränken wir uns in dieser Arbeit auf das grundlegende Beispiel eines einzelnen Zwei-Niveau-Atoms, welches durch Emission eines Photons in die Umgebung spontan in seinen Grundzustand relaxieren kann. Wir finden, dass man allein schon durch Post-Selektion der Messereignisse jeden beliebigen reinen Qubitzustand stabilisieren kann, was durch kohärente Kontrolle allein nicht möglich scheint. Unter Anwendung von unitären Oper- atoren als Rückkopplungsoperationen ist schliesslich jeder gewünschte Zwei- Niveau-Zustand vollständig gegen den Umwelteinfluss stabilisierbar. Auch unter realistischen Annahmen, die beispielsweise fehlerbehaftete kohärente Kontrolle, ineffizientes Messen und zeitverzögerte Rückkopplungsmechanis- men berücksichtigen müssen, bleiben die verbesserten Kontrollmöglichkeiten erhalten.

Contents

1 Introduction 11

2 Representation of Quantum States 15 2.1 Mathematical Representation ...... 16 2.2 Pure versus Mixed Quantum States ...... 16 2.3 Classical versus Quantum Correlated Systems ...... 17 2.4 Representation of Two- & Three-Dimensional Quantum States ..... 19

3 Open Quantum Systems 23 3.1 Introduction to Open Systems ...... 24 3.2 Classical Open Systems ...... 28 3.3 Quantum Mechanical Piecewise Deterministic Processes ...... 32 3.4 Detrimental Inﬂuence of the Environment - Decoherence ...... 34

4 Control of Open Quantum Systems 39 4.1 Different Control Approaches ...... 40 4.2 Optimal Coherent Control ...... 43 4.3 Optimal Stationary Qutrit State for Different Dissipations ...... 45

5 Continuous Monitoring to Enhance Control 59 5.1 Introduction to Jump Unraveling as a Control Approach ...... 60 5.2 Jump Unraveling for a Stabilized Single Qubit ...... 63 5.3 Stationary Point of the Conditional Dynamics ...... 70 5.4 Jump Unraveling for a Stabilized Single Qubit with Unitary Feedback 75 5.5 Towards an Experimental Realization ...... 80

6 Conclusions & Outlook 87

1 Introduction

Up to today, there still exists no proof that quantum computers are more powerful than classical ones, independent of whether the latter are based on a deterministic or a probabilistic model [1]. In the literature, this open problem is coined in the question whether the strong Church-Turing thesis [2]:

‘A classical, probabilistic Turing machine can efficiently simulate any realistic model of computation.’ can be falsified with quantum Turing machines or not. In other words, the open question is, whether a computation unit relying on quantum resources can solve a problem in a number of computation steps scaling polynomially with the input size, i.e. efficiently, while the same problem can only be solved inefficiently with classical methods. A strong hint for the incorrectness of the strong Church Turing thesis was given by Shors’ algorithm of efficiently finding the prime factors of any integer number [3], using quantum resources, since no classical, probabilistic algorithm has yet been found for efficiently performing this task. Even though a final proof for the latter is still lacking, and thus the initial problem remains unresolved, quantum mechanics provides promising method to efficiently solve computation tasks that appear to be computationally hard for classical computers. The essential quantum resource enabling this is coherence, i.e. coherent superpositions of states which possibly involve different degrees of freedom. For instance, coherence is speculated to play a key role in the transport efficiency of excitations in photosynthetic complexes [4]. The strongest manifestation of coherence for multi-partite systems, reflected in quantum correlations, is entanglement. It is a necessary ingredient for many quantum information protocols, such as quantum teleportation [5], the aforementioned Shors’ algorithm [3] for quantum cryptography, or super dense coding [6]. In experimental implementations, the physical carriers of quantum information have to be considered concurrently with their ambient environment. In other words, the underlying quantum systems must be treated as open [7–9]. As a matter of fact, the environmental influence usually leads to detrimental decoherence effects: Entangled subsystems de-correlate when coupled to individual environments [10] - for example initially entangled photons propagating through a turbulent atmosphere [11] lose entanglement rapidly,

11 and in general the coupling to thermal baths renders any quantum system classical over time by destroying coherences [12]. In order to uphold the necessary quantum resources for technological utilization, it is necessary to protect quantum superpositions against the detrimental effect of the environment. Depending on the desired application, like quantum computation, quantum memory or quantum chemistry, different degrees of coherence are required and need to be stabilized for various time scales. Consequently, different control techniques have to be utilized. For instance, isolating a system can never be achieved completely and will therefore never uphold coherence in the asymptotic time limit. This is usually not an issue for quantum computation tasks, where the operations are performed under strong coupling between the system and the external control fields, such that the environment influence can be neglected. However, to e.g. realize quantum memories, long term stability of coherence is essential. A natural approach to enable such long term stability is to manipulate the Hamiltonian of the quantum system under consideration. This manipulation is one variant of coherent control and is experimentally achieved by coupling the quantum system to external classical potentials, like electro-magnetic fields or shaped laser fields. Such coherent control is already implemented in systems ranging from trapped atoms [13] and ions [14] in a cavity, to semi- conducting quantum dots [15], or solid state systems [16], to mention only a few. The essential question is, to what extent can the restriction to coherent control, i.e. to only manipulating the system Hamiltonian, stabilize desired degrees of coherence. This question was answered in [17] for the case of open quantum systems whose evolution can be described with a Markovian master equation [7]. The authors characterize all quantum states that can be stabilized against environmental influence for a given open system under any possible form of stationary and periodic manipulation of the system Hamiltonian. It turns out that the resulting sets of stabilizable states, depending on the form of environmental influence, indicate a substantial, but limited improvement as compared to uncontrolled systems. This is however often not sufficient for the realization of quantum protocols, and more advanced methods are required to stabilize maximally coherent and/or entangled states. At the same time, it is desirable to keep the required control protocol as simple as possible, such that an experimental realization is straight-forward. This is exactly the aim of the present thesis. Beyond coherent control, we stabilize open quantum systems by continuously monitoring the system and applying appropriate unitary feedback operations, as suggested for trapped atoms in a cavity in [18]. The monitoring can be achieved with standard measurement apparatuses that provide continuous information on the state of the quantum system. For example, in the case of a trapped atom, a suitable measurement on the surrounding light field provides information about when the atom de-excites by the emission of a photon, which goes along with a state

12 Chapter 1 Introduction change in the atom. This continuous information can then be used to apply the proper feedback operations onto the monitored system. In our example, electro-magnetic field pulses could induce well-defined rotations of the state of the atom, to guide it back into the desired stationary state. Since the approach only requires classical measurement apparatuses and state-of-the-art feedback operations, not too much additional experimental overhead is to be expected, due to additional continuous monitoring and feedback operations. We show in the course of this thesis that indeed states with optimal coherence properties can be stabilized in the case of a single qubit under spontaneous decay. It is possible, even without any feedback operations, to stabilize pure quantum states by employing coherent control combined with post-selection on the measurement outcomes. With unitary feedback it is furthermore possible to render any arbitrary qubit state stable against the environmental influence. This is very surprising, since it means that the non-unitary influence of the environment can be fully compensated by unitary control fields and proper unitary feedback operations at every point in state space. With unitary coherent control fields alone, this is impossible [17]. Consequently, any desired degree of coherence in a single qubit state can be uphold for all times. We additionally investigate the impact of realistic conditions on our control approach, such as perturbed stabilizing Hamiltonians, inefficient monitoring, and time-delayed feedback. The results suggest that even under such realistic assumptions, the stabilization of quantum properties is still improved, as compared to coherent control alone.1

Structure of this thesis To embed the obtained results into the right context, the ﬁrst chapter of this thesis introduces the mathematical and physical representation of the two- and three-dimensional quantum states we are dealing with, and, in particular, the notion of a (generalized) Bloch sphere is established. The latter provides a geometric interpretation of pure and mixed quantum states. Furthermore, we explain the difference between classical and quantum correlation, as well as the relation of the latter to the fundamental notion of entanglement. The second chapter describes the theoretical treatment of open quantum systems by effective master equations, leading to the widely used Lindblad form. With the help of classical analogies, we introduce the unraveling approach to describe continuously monitored, and potentially feedback-stabilized open systems, leading to a stochastic Schrödinger equation. The chapter is closed with exemplary demonstrations of the detrimental effect of the environment on quantum properties of open systems. Different control approaches to shield the system against these detrimental effects are brieﬂy introduced in the third chapter, along with an in-depth

1Similar, but less general, results were found in related control schemes [19,20].

13 presentation of the method to characterize the stabilization capabilities of coherent control for fixed open system setups. This method is applied to three- dimensional quantum systems under various environmental influences, to obtain a deeper understanding of coherent control in these specific scenarios. In the fourth chapter, the unraveling of the master equations of two- dimensional quantum systems is presented, and resulting quantum trajectories of individual open quantum systems are considered. Furthermore, the potential of the feedback stabilization is investigated on the effective master equation level, as well as for unraveled evolutions. In the end, realistic assumptions about the employed control methods are considered, to pave the way for future experimental realizations. Finally, we close with some conclusions and an outlook.

14 2 Representation of Quantum States

In this section we show how coherences and non-classical correlations are expressed in the mathematical framework of quantum theory. Quan- tum systems with such properties potentially come with big advantages compared to classical systems for the development of new technologies, as already explained in the introduction.

To mathematically describe these speciﬁc quantum properties, density matrices and their deﬁning properties are introduced, and the difference between pure and mixed quantum states and their respective potential to exhibit quantum coherences is explained. The concept of bi-partite systems is presented, which permits us to formulate the emergence of non- classical correlations between quantum systems. Finally, we present a geometrical representation of the state space of two-level systems (qubits) - the Bloch sphere - and show how it can also be generalized to three-level systems (qutrits). These geometrical interpretations allow us to visualize our later results and by that will render them easier more tangible. 2.2 Pure versus Mixed Quantum States

2.1 Mathematical Representation

In quantum theory, the state of a quantum system in continuous position space is described by a wavefunction Ψ(~x) in an infinite dimensional Hilbertspace H which obeys the fundamental postulates of quantum mechanics [21]. How- ever, in this thesis we consider discrete quantum systems restricted to a finite dimensional Hilberspace. In that case an N-dimensional quantum system is most generally described by a bounded operator ρ, the so-called density matrix, in a (N × N) dimensional Hilbertspace B(H(N×N)) of bounded operators [22]. A matrix represents a physical quantum state if and only if it is normalized, hermitean, positive-semidefinite, and if it has a bounded purity [1, 23]. Mathematically these condition are expressed respectively via

tr{ρ} = 1 (2.1a) ρ† = ρ (2.1b)

ρii ≥ 0 ∀i (2.1c) tr{ρ2} ≤ 1 , (2.1d) where ρii is the ith diagonal element in the diagonal representation of ρ. A unit norm (Cond. (2.1a)) guarantees that the quantum system is cer- tainly in one of the states in the Hilbertspace and hermiticity (Cond. (2.1b)) ensures that the eigenvalues of ρ are real, which together with the positive- semideﬁniteness (Cond. (2.1c)) and bounded purity (Cond. (2.1d)) enables a probabilistic interpretation of the eigenvalues of ρ. The following section is devoted to explaining this interpretation.

2.2 Pure versus Mixed Quantum States

If equality holds in Cond. (2.1d), one speaks of a pure quantum state. In that case it is always possible to ﬁnd a representation of the state |Ψi ∈ H(N), such that

ρpure = |Ψi hΨ| ,(2.2) where we employ bra-ket notation [24]. In this case the density matrix has exactly one eigenvalue equal to one, corresponding to the eigenstate |Ψi, and all other eigenvalues are zero. Otherwise, if tr{ρ2} < 1, the state is said to be mixed, which means that it consists of more than one pure state ρi = |Ψii hΨi|, i.e.

ρmixed = ∑ piρi ,(2.3) i

16 Chapter 2 Representation of Quantum States

with pi being the eigenvalues corresponding to the eigenstates |Ψii. Due to conditions 2.1a to 2.1c the eigenvalues fulﬁll the condition ∑i pi = 1 and can be understood as the probabilities of the system being in the associated eigenstates. Mixed quantum states are often also called classical superposition states, since the superpositions of the corresponding pure states behave according to classical probability theory: the system is always in exactly one of the pure states with the corresponding probability - like a classical dice always gives exactly one result. In a double-slit experiment, no interferences between the pure states would be visible. In contrast to that, pure states can exhibit quantum coherences. A state can be in a coherent superposition of several basis states, √ 1 e.g. the state |ΨCohi = ( / 2)(|0i + |1i) is in a maximal coherent superposition of the states |0i and |1i.1 Such states can show interference patterns in a double-slit experiments and coherence appears to be the fundamental quantum property that is responsible for non-classical correlations between multiple quantum systems, which will be introduced in the next section. Note that mixed states can also contain quantum coherences, due to coherent superpositions in the pure states they are mixed off, but they can’t be coherent superpositions of pure states.

2.3 Classical versus Quantum Correlated Systems

A general quantum system must not necessarily be a single unity, as in the description above, but it might as well be composed of M smaller pieces, i.e. subsystems. In that case, the dimension of the systems state space is (N1 · N2 · ... · NM) × (N1 · N2 · ... · NM), where Nm is the Hilbertspace dimension of the m-th subsystem. A special case is a bipartite system (M = 2), which consists of two subsystems. In the following we restrict our description to this case, since it is sufﬁcient to describe all properties of interest to us. The state of a bipartite system is said to be separable if it can be written as a sum of tensor product states of the subsystems’ states ρ(1) ∈ B (H(N1×N1)) and ρ(2) ∈ B (H(N2×N2)) as follows

(1) (2) ρSep = ∑ pi ρi ⊗ ρi .(2.4) i

Here, ρ(1) and ρ(2) are called reduced density matrices, where, e.g, ρ(1) is deﬁned as N1 N2 (1) (1) (1) (1) (2) (2) ρ = ∑ pi |χi i hχi | = ∑ hχj |ρ|χj i = tr2 ρ ,(2.5) i=1 j=1

1Note that coherence is basis dependent, in the sense that the state |Ψ i is maximally Coh √ coherent in the {|0i , |1i} basis, however it displays no coherence in the {(1/ 2)(|0i + √ |1i), (1/ 2(|0i − |1i)} basis.

17 2.3 Classical versus Quantum Correlated Systems

(1) (2) with {χi } and {χi } the orthonormal basis vectors of the two subsystems. tr2 is the partial trace over subsystem (2), which extracts the density matrix of subsystem (1) by tracing out all the degrees of freedom of subsystem (2). (2) (2) Analogously, the reduced density matrix ρ can be obtained via ρ = tr1 ρ. For a pure state |Ψi ∈ H of a bipartite system, the separability-condition, deﬁned through Eq. (2.4), reduces to

|Ψi = |Ψ(1)i ⊗ |Ψ(2)i ,(2.6) where |Ψ(1)i ∈ H(N1) and |Ψ(2)i ∈ H(N2) are pure subsystem states. Physically speaking, the two subsystems are separable when the outcome of measuring an observable A(1) in subsystem (1) via A = A(1) ⊗ 1 does not depend on the outcome of measuring an observable with the same physical meaning A(2) in subsystem (2) via A0 = 1 ⊗ A(2), i.e. the probabilities of two measurements in different subsystems factorize. This occurs either when one exactly knows the pure state of one of the subsystems, e.g. because it was measured, or when one has two well separated quantum systems that do not (and never did) interact with each other. Such separable states are either classical or quantum correlated. On the one hand, they are classically correlated, if a projective measurement gives full information on the reduced state of the associated subsystem, i.e. when the separable state can be written as

(1) (1) (2) ρCCorr = ∑ pi |Ψi i hΨi | ⊗ ρi .(2.7) i

Such classically correlated states can be simulated by classical systems alone. For example connecting a certain event to the outcome of throwing a dice: Throwing the dice corresponds to the projective measurement on subsystem (1), in our case, and the resulting event corresponds to the state of subsystem (2). On the other hand, the more general case, as given via Eq. (2.4), represents a quantum correlated separable state, since in that case no classical analog can be found. The amount of quantum correlation that are present in a bipartite system can be quantiﬁed via quantum discord [25] or more generally by the rank of the correlation matrix [26]. Quantum correlations are the building blocks for several applications, like e.g. in quantum metrology [27] or for information encoding [28]. There is an even stronger manifestation of non-classical correlations, namely when the state of a multi partite system is not even separable. In that case one speaks of entanglement [29, 30] between the subsystems. For example the well-known maximally entangled Bell state

1 |ΨBelli = √ (|0i |1i − |1i |0i) (2.8) 2

18 Chapter 2 Representation of Quantum States cannot be written in a separable form of Eq. (2.4). In that case, the system is in a coherent superposition of the two separable states |0i |1i and |1i |0i. Applying the physical interpretation for separable state from above also provides an understanding of entanglement here: Measuring locally the state of one of the subsystems leads to a collapse of the total superposition state into one of the separable ones and thereby inﬂuencing the corresponding second subsystem’s state (and an associated measurement outcome). This even holds true if the two subsystems were spatially well separated and didn’t share any classically obvious means of exchanging information. This encouraged Albert Einstein to question the existance of such a ‘spooky action over a distance’[29], which shattered the believes in the foundations of quantum mechanics. In 1964, John S. Bell published [31] a famous inequality which carries the potential to experimentally test the existance of entanglement. And indeed the existence of such non-classical correlations could be successfully proven in the past [32,33]. Ultimately, entanglement is now widely used as a resource for quantum teleportation [5], quantum cryptography [34, 35] or quantum computing [1,36, 37].

2.4 Representation of Two- & Three-Dimensional Quantum States

A general two-dimensional quantum state ρqubit - a so-called qubit - may be representated as

1 ρ = (1 +~n(t) ·~σ) ,(2.9) qubit 2

3 where the σi are the Pauli matrices [24], and ~n ∈ R is the so-called Bloch vector. A pure quantum state is represented by a Bloch vector with unit length (|~n| = 1), i.e. pure states deﬁne the unit sphere S2, which is the surface of a unit ball in R3. This unit ball is also called the Bloch sphere, and its interior contains all the mixed quantum states described by a Bloch vector with |~n| < 1. One often employs the parameterization

ρ = 1 + r cos φ sin θ σx + r sin φ sin θ σy + r cos θ σz ,(2.10) such that a qubit state ρ is uniquely deﬁned by the two Bloch angles φ and θ and the distance from the center r of the Bloch sphere. Then the poles of the Bloch ball correspond to the states |0i and |1i, and the equator marks all maximally coherent states. The Bloch sphere is schematically shown in Fig. 2.1. Later we will use the Bloch representation to visualize the set of quantum

19 2.4 Representation of Two- & Three-Dimensional Quantum States

Figure 2.1: Geometrix representation of a qubit state ρ with the Bloch parameters φ, θ and r.

states, which can be stabilized against the environmental inﬂuence. Fur- thermore, we use it to illustrate the evolution of single realizations of qubit systems under investigation. This visualization enables an easy identiﬁca- tion of interesting properties and behaviors. Especially the straight-forward interpretation of the Bloch parameters φ, θ and r will come very handy. In analogy, a three-dimensional quantum state ρqutrit - a so-called qutrit - 2 may be expanded with respect to the Gell-Mann matrices Λi [39] as 1 √ ρ = 1 + 3~n · ~Λ ,(2.11) qutrit 3 where ~n ∈ R8 is then the so-called generalized Bloch vector. In the computational basis the Gell-Mann matrices read 0 1 0 0 −i 0 1 0 0 Λ1 = 1 0 0 , Λ2 = i 0 0 , Λ3 = 0 −1 0 , 0 0 0 0 0 0 0 0 0 0 0 1 0 0 −i 0 0 0 Λ4 = 0 0 0 , Λ5 = 0 0 0  , Λ6 = 0 0 1 , 1 0 0 i 0 0 0 1 0 0 0 0  1 0 0  1 Λ7 = 0 0 −i , Λ8 = √ 0 1 0  .(2.12) 0 i 0 3 0 0 −2

2Alternative bases would be the polarization operator basis or the Weyl operator basis, for both see [38].

20 Chapter 2 Representation of Quantum States

Like the Pauli matrices, they are traceless and satisfy

2 Λ Λ = δ + d Λ + i f Λ ,(2.13) i j 3 ij ijk k ijk k where the structure constants of the SU(3) algebra fijk are completely anti- symmetric and the dijk are completely symmetric, see [40] for explicit values of these constants. A qutrit state is pure, if the corresponding density matrix tr{ρ2} = 1 (remember Cond. (2.1d)). This condition implies [41, 42] for the elements of the Bloch vector

~n2 = 1 (2.14a) √ ~ej 3dijknjnk = ~n .(2.14b)

According to the ﬁrst condition, ~n is restricted to the unit sphere S7 of the R8. However, due to the second condition, which places three3 additional constraints on the Bloch vector, not all points on the 7 dimensional hyperplane represent valid pure states. Hence, there is no bijective map between pure quantum states and the points on the unit sphere S7. Mixed states are to be found in the inside of the unit hyperball in R8, but again there is no bijective map, since the matrix representation of a mixed quantum state has to satisfy the Conditions (2.1). They translate to the Bloch vector [42] as

~n2 ≤ 1 , (2.15a) √ 2 and ~n − 2 3~ejdijknjnk ·~n ≤ 1 . (2.15b)

Many further studies were devoted to understanding the structure and geometry of the qutrit state space, see for example [43–45]. Despite of these complications, the parameterization of qutrit states in terms of Gell-Mann matrices is still useful for a consistent representation of the to be encountered states and state spaces in this thesis.

3This leaves us with four real parameters for pure states, which is already clear from the general parameterization of a pure state via |Ψi = (A, Beiφ, Ceiψ) with the normalization constraint.

3 Open Quantum Systems

Often, quantum systems are considered to be closed, and are described by purely coherent dynamics. However, this is an idealization, since a system is bound to be in contact with other quantum systems, like neighboring atoms, or surrounding electro-magnetic ﬁelds, even if these are in the vacuum state. Thus, the quantum system of interest needs to be treated as an open system which interacts with a surrounding environment, the latter often modifying the reduced system dynamics signif- icantly. This has usually a detrimental effect on the systems’ quantum properties introduced in the previous section. Since we want to preserve these properties in realistic settings, i.e. taking the presence of an environment into account, it is necessary to study the concept of open quantum systems and to understand the inﬂuence of the environment on coherences and quantum correlations.

This section will introduce open quantum systems, their mathematical treatment, and the physical consequences. We start with the Lindblad master equation describing the effective evolution of a system interacting with an environment, explain the classical analog, and introduce an approach to describe the non-averaged evolution of classical systems under environmental inﬂuence. This is used to also derive an analogous stochastic evolution equation for quantum systems, which describes the dynamics of pure states over time, similar to the Schrödinger equation for closed systems. By averaging over many evolutions of such single realizations, we obtain the mean evolution, which is shown to follow a Lindblad master equation. From the effective description it becomes apparent that the environment usually has a detrimental effect on the desired system properties. We show this via selected examples. In the course of this thesis, the stochastic evolution equation introduced here, as well as the Lindblad master equation, will be often utilized to advance ways of shielding the properties of open quantum systems against the detrimental effects of the environment. 3.1 Introduction to Open Systems

3.1 Introduction to Open Systems

One speaks of a closed quantum system when the system, with a fixed number of degrees of freedom (DoFs), is well separated from any other quantum system und thus isolated from any source of uncontrollable influence. In contrast to that, an open quantum system S is influenced by surrounding quantum systems and classical fields, the environment E. In principle, the environment could contain the whole universe, excluding S itself, but it is usually sufficient to consider only the immediate surroundings. The union of S and E forms a bipartite system, which is assumed to be closed. For example, take a two-level atom as the system S, with its level populations as the DoFs of interest. If the atom was in vacuum and free of any external source of influence, it could be considered a closed system, as shown in Figure (3.1a). However, in a realistic situation, this atom would at least be coupled to the surrounding (vacuum) modes of the electro-magnetic field~k, such that it can emit and absorb photons, which change the level population of the atom. Additionally, the motional DoFs M of the atom could influence the level population, too. In the theory of open quantum systems, these two influencing quantum DoFs are subsumed into the environment E and its interactions with the system S are treated effectively with the operators Li, see Figure (3.1b) for an illustration. In the following, we explain the emergence of the effective interaction operators Li from the microscopic description, which determine the evolution of the quantum system S.

(a) (b)

Figure 3.1: Examples of a (a) closed and an (b) open quantum system. The exemplary closed system (a) here is an atom, which has one DoF, the occupations of the levels |0i and |1i, and which is completely isolated from any external source of influence. An open system (b) could then be the same atom whose level population is, however, influenced by a coupling to surrounding electro-magnetic field modes~k and motional DoFs M. The latter two form the environment E and its interaction with the system is treated with Lindblad operators Li, according to the theory of open quantum systems.

The most general Hamiltonian of an open quantum system can be consid-

24 Chapter 3 Open Quantum Systems ered as

HT = HS + HE + HI ,(3.1) where HS is the internal Hamiltonian of the system S, HE the Hamiltonian of the environment E and HI describes the interaction between the system and environment. The time evolution of this closed bipartite system’s state ρ(t) = ρS(t) ⊗ ρE(t) is governed by the von Neumann equation [21]:

dρ(t) i = − [H , ρ(t)] .(3.2) dt h¯ T In principle, for a given total Hamiltonian, the above equation can be solved, at least numerically, and the according system’s state evolution deduced. However, in many cases the evolution of the environment is actually not of interest, which renders it unnecessary to solve Eq. (3.2) for the total bipartite system state ρ(t). Instead, a lower-dimensional description only of the system’s evolution is desirable. In fact, by employing a couple of often well-justiﬁed approximations, one can derive such a master equation, which describes the evolution of the system state ρS(t) by treating the environment and its interactions with the system effectively. By transforming the von Neumann equation into the interaction picture † † [46], via ρ˜(t)) = U (t)ρ(t)U(t) and H˜ (t) = U (t)HIU(t), where U(t) = exp(−i(HS + HE)t/¯h), we get rid of the less interesting coherent evolution of the system (environment) due to its internal Hamiltonian HS (HE). It can be reintroduced later on, but for now we use the transformed von Neumann equation as follows

dρ˜(t) i = − [H˜ (t), ρ˜(t)] .(3.3) dt h¯ The formal solution of this equation is given in terms of an inﬁnite series

n ∞ Z t Z tn−1 i ˜ ˜ ρ˜(t) = ρ˜(0) − ∑ dt1 ... dtn[H(t1),..., [H(tn), ρ˜(tn)] ... ] . n=1 h¯ 0 0 (3.4)

By assuming that the interaction between system and environment is weak, we can in the following resort to considering the above solution perturba- tively only to second order. This is in the literature also known as the Born approximation [46]. Plugging the solution (3.4) into Eq. (3.3) gives

dρ˜(t) i 1 Z t = − [ ˜ ( ) ( )] − [ ˜ ( ) [ ˜ ( 0) ( 0)]] 0 H t , ρ˜ 0 2 H t , H t , ρ˜ t dt .(3.5) dt h¯ h¯ 0 In the following we drop the ﬁrst term on the right hand side, since such a

25 3.1 Introduction to Open Systems coherent part can in principle always be included in the system Hamiltonian and need not occur in the interaction picture. Because we are interested in the evolution of the system alone, we trace out the environmental degrees of freedom, as given in Eq. (2.5), and ﬁnd

Z t dρ˜S(t) i 0 0 0 = − trE{[H˜ (t), [H˜ (t ), ρ˜(t )]]}dt .(3.6) dt h¯ 0 The environment can be assumed to be virtually unaffected by its interaction with the system, since it is very large. This implies that the timescale over which the environment is correlated to past system states is short, compared to timescales of the system evolution. As a consequence, we can assume the state of the environment to be constant over time ρ˜E(t) = ρ˜E(0) = const and thus the total system state is separable at any time ρ˜(t) = ρ˜S(t) ⊗ ρ˜E(0). This also implies that interactions between system and environment can be considered local in time so that we only need to consider the local density matrix by replacing ρ˜(t0) → ρ˜(t) in the integral. With the same argument we can get rid of the particular initial time t = 0 and instead integrate over all previous times. This procedure is known as the Markov approximation [7] and leads to the following Born-Markov master equation in the interaction picture

∞ dρ˜S(t) i Z = − trE{[H˜ (t), [H˜ (t − τ), ρ˜S(t) ⊗ ρ˜E]]}dτ .(3.7) dt h¯ 0 To transform the equation back into the Schrödinger picture, one has to reverse the interaction picture transformation and reintroduce the coherent dynamics of the system ( ) ( ) dρS t i −iHSt/¯h dρ˜S t iHSt/¯h = − [HS, ρS(t)] + e e (3.8) dt h¯ dt The beauty of the master equation (3.7) lies in the fact that the evolution of the system state is described without needing to treat a single DoF of the environment explicitly, only the average environmental state ρ˜E is needed. By considering the interaction Hamiltonian explicitly, the equation can be cast into a very intelligible form. To derive this, we write the interaction Hamiltonian in its most general (time-independent) form [7]

HI = ∑ Ai(ω) ⊗ Bi ,(3.9) i,ω where the Bi are bounded operators in the environment Hilbertspace B(HE) and Ai(ω) ∈ B(HS) are eigenoperators of the system Hamiltonian HS. I.e.

26 Chapter 3 Open Quantum Systems they fulﬁll the commutation relations

[Ai(ω), HS] = ωAi(ω) (3.10) † † [Ai (ω), HS] = −ωAi (ω) .(3.11)

With that, the Hamiltonian in the interaction picture becomes

˜ −iωt ˜ H(t) = ∑ e Ai(ω) ⊗ Bi(t) ,(3.12) i,ω where

iHEt/¯h −iHEt/¯h B˜i(t) = e Bie .(3.13)

The interaction picture Hamiltonian (3.12) can now be plugged into Eq. (3.7) and one obtains [7]

dρ˜S(t) −i(ω0−ω)t † 0 † 0 = ∑ e Γi,j(ω) Aj(ω)ρ˜S(t)Ai (ω ) − Ai (ω )Aj(ω)ρ˜S(t) dt ω,ω0,i,j + h.c. , (3.14)

Here, we introduced the Fourier-transform of the environment correlation † function trE{Bi (τ)Bj(0)ρ˜E} as

Z ∞ iωτ † Γi,j(ω) = e trE{Bi (τ)Bj(0)ρ˜E}dτ 0 1 = (γ (ω) + iS (ω)) .(3.15) 2 i,j i,j and split it into its real and imaginary parts γi,j(ω) and Si,j(ω), respectively. Note, that the previously introduced Markov approximation appears here in the assumption that the correlation function must decay quickly in τ. In the typical example, where the environment is simulated with harmonic oscillators, this is the case when the number and density of harmonic oscillator frequencies ωE is high [7], i.e. when the environment is large. In a next step, we assume that all terms with ω 6= ω0 in Equation (3.14) can be neglected, because they usually oscillate very quickly, as compared to the typical time scale over which the state of the system ρ˜S(t) changes appreciably [7], and as a result don’t give a contribution to the integral. This is in quantum optics usually termed a rotating wave approximation and leads to the following form of the master equation dρ˜S(t) † 0 1 † = ∑ γi,j(ω) Aj(ω)ρ˜S(t)Ai (ω ) − {Ai (ω)Aj(ω), ρ˜S(t)} ,(3.16) dt ω,i,j 2

27 3.2 Classical Open Systems

where we omitted the imaginary part of Γi,j(ω) (Eq. (3.15)), since it only introduces an additional Hamiltonian contribution (a so called Lamb shift [46]) to the dynamics of the system S and can thus be included in the system Hamiltonian HS. Finally, by diagonalizing the matrix of the decay rates γi,j, and back-transforming into the Schrödinger picture, we arrive at the desired Lindblad form [47] of the master equation

dρS(t) i = − [HS, ρS(t)] dt h¯ † 1 n † o + ∑ γi(ω) Li(ω)ρS(t)Li (ω) − Li (ω)Li(ω), ρS(t) i 2 i = − [HS, ρS(t)] + D({L (ω)}; ρS(t)) .(3.17) h¯ i

The so-called Lindblad operators Li are then linear combinations of the interaction operators acting on the system Li(ω) = ∑i,k βk,i Ak(ω), determined by the diagonalization of γi,j. A Lindblad master equation, as given in Eq. (3.17), is a positive semideﬁ- nite map, which preserves all properties of a density matrix representing a quantum state (Conds. (2.1a)-(2.1d)). It is widely used to describe Markovian open quantum system, since the Lindblad operators represent the effective action of the environment on the system and can often be guesses directly from the interaction between system and environment. For example, in case of a two-level atom coupled to a surrounding electro-magnetic ﬁeld effectively the environment induces excitations and de-excitations of the atom via the Lindblad operators σ+ and σ−, for other environments it could be a dephasing between the levels |0i and |1i via the Lindblad operator L = σz or for higher dimensional systems more involved operators are considered. In Section (4.3), we investigate three-dimensional systems with different Lindblad operators.

3.2 Classical Open Systems

Originally, master equations were introduced to describe the evolution of classical systems under environmental inﬂuence. The most prominent examples are the damped harmonic oscillator and the diffusive motion of a particle in a medium, where in both cases the behavior of the actual system, and not so much the environment is of interest. In this section we introduce a classical master equation, analogous to the Born-Markov master equation from the previous section. Presuming this master equation, we can then straight- forwardly introduce an description of the evolution of single classical systems, in contrast to the average evolution provided by the master equation. This description can later on (in the following section) be transfered to quantum systems, where we can then directly identify the unique quantum features of

28 Chapter 3 Open Quantum Systems the approach. Let us start with a master equation for an open classical system. We have a system in mind that in principle undergoes a deterministic evolution, according to its internal DoFs and is at random times exposed to an interaction with the DoFs of the environment. For example, consider the diffusive motion of a particle in a medium: While the particle is left alone, it moves deterministically according to its Hamiltonian dynamics, but at random times it collides with other particles from the medium, inducing a random change of its momentum. The Markovian assumption that the correlation timescale between system and environment is negligible compared to the system’s dynamics, translates in our example to the assumption that the surrounding medium is large and sparse. The situation above can in general be described with a Liouville master equation [48]

∂ 0 0 ∂ 0 0 T(x, t|x , t ) = − [gi(x)T(x, t|x , t )] ∂t ∂xi Z + dx00[W(x|x00)T(x00, t00|x0, t0) − W(x00|x)T(x, t|x0, t0)] (3.18) for the conditional transition probability T(x, t|x0, t0) for the system to take value x at time t when having the value x0 at time t0. It fulfills the properties Z dxT(x, t|x0, t0) = 1 lim T(x, t|x0, t0) = δ(x − x0) , t→t0 i.e. that with unit probability the system takes a value at time t when having the value x0 at time t0 and when the time t approaches time t0 the values x and x0 must be equal. The deterministic part of the system’s evolution is governed in the Liouville master equation (3.18) by the first term on the right hand site via the autonomous vector field g(x). It is defined as [48]

d x(t) = g(x) .(3.19) dt In our example, where x(t) represents the canonical coordinates position q and momentum p of the particle at time t, the ﬁeld g(x) represents the Hamiltonian ﬁelds ∂H/∂p and ∂H/ − ∂q, where H = H(q, p, t) is the classical Hamiltonian of the system. The interaction with the environment is contained in the second part of the master equation with the time-independent probability W(x|x0) for an instantaneous change (or jump) of the value x0 to x. The rate of jumps

29 3.2 Classical Open Systems from x to any other value at a ﬁxed time t is given by Z Γ(x, t) = dx0W(x0|x, t) (3.20)

The interpretation of the jump probability in our example is straight-forward: A collision of the system particle with other particles induces an instantaneous change of the momentum p → p0 of both involved particles, while the change in the environmental particle is neglected. In classical probability theory, the process described above is also called a piecewise deterministic process (PDP). It is well-known that as an alternative to the effective description of probability evolutions via a master equation, one can also directly derive a differential equation for the random evolution of the value x(t), so-called trajectories, of a single realization of the environmental conﬁguration. Thus, instead of asking what is the probability for a system to take the value x at time t when having the value x0 at time t0, one could ask what is an exemplary evolution of the value x(t) for a time in the interval t0 ≤ t ≤ tn. By averaging over many such exemplary evolutions the underlying master equation evolution is recovered. For example, consider again the diffusion process: A particle in the environment realization α may have the initial value xα(t0) = x0 at the initial time t0, then it moves deterministically until it hits a particle from the surrounding medium at time tα,1, which changes the value xα(tα,1) spontaneously to the 0 new value xα(tα,1). Now it again undergoes a period of deterministic evolution until it next hits a particle at t2 and undergoes another instantaneous change, and so forth. The same particle in another environment realization β, also starting at x0 at t0 also moves deterministically until it hits a particle at tβ,1, however in different realizations α and β not the same value tβ,1 and tα,1 0 need be realized, also the induced value after the xβ(tβ,1) need not coincide 0 with xα(tα,1). The considered scenario is depicted in Figure (3.2). The dynamics of a random trajectory x(t) of a PDP can be described by means of its increment dx(t) = x(t + dt) − x(t) by the following stochastic differential equation [7]

dx(t) = g(x)dt + dJ(x, t) .(3.21)

The deterministic part of the evolution is treated in the same manner as in the Liouville master equation (3.18). In order for the statistical term dJ(x, t) to act in accordance with the master equation, it must fulﬁll the following statistical properties:

• dJ(x, t) = 0 when no jump occurs in the interval t ≤ t0 ≤ t + dt.

• dJ(x, t) = x0(t) − x(t) when a jump occurs in the interval t ≤ t0 ≤ t + dt from x to x0.

30 Chapter 3 Open Quantum Systems

Figure 3.2: Two realizations of α and β of the environment lead to two different evolutions xα(tα) and xβ(tβ) of a PDP, which both start at the same time with the same value. Exempliﬁed via a particle moving diffusively in a surrounding medium. An average over many such evolutions reproduces the evolution given by a corresponding Liouville master equation (3.18).

• The jump probability from x into x0 is given by the already introduced probability density W(x0|x).

• For inﬁnitesimally small time intervals dt at most one jump must happen.

To make the following investigations more transparent, we assume without loss of generality that only a ﬁnite number of discrete jumps is possible, such 0 that the value x after a jump has to be one of the discrete values xα. With that, the properties from above lead to the ansatz Z dJ(x, t) = {xi(t) − x(t)}dNi(t) ,(3.22) where dNα(t) are independent Poisson increments [48], which can either all be zero or one of them is equal to one and the rest is zero at each time t. When all dNit) are zero then dJ(X(t)) = 0, i.e. no jump occurred, and when one dNi(t) = 1 then a jump from x(t) into xi(t) took place. The expectation value of dNi(t) is proportional to the probability density W(xi(t)|x(t)):

E[dNi(t)] = W(xi(t)|x(t))dt .(3.23)

Combining Eq. (3.22) with Eq. (3.21) ﬁnally yields the stochastic differential equation for a PDP

dx(t) = g(x(t))dt + ∑{xi(t) − x(t)}dNi(t) .(3.24) i

It can be shown [7] that transition probabilities, which are obtainable by averaging over many realizations x(t), perfectly coincide with those obtained

31 3.3 Quantum Mechanical Piecewise Deterministic Processes from directly solving the Liouville master equation (3.18). In the following section we introduce an analogous stochastic differential equation for open quantum systems, and there we show explicitly that the average dynamics over many realizations coincides with the corresponding Lindblad master equation description.

3.3 Quantum Mechanical Piecewise Deterministic Processes

By virtue of equation (3.24), we can deﬁne an analogous stochastic differential equation for the evolution of an open quantum mechanical system as

L Ψ(x, t) dΨ(x, t) = −iG(Ψ(x, t))dt + i − Ψ(x, t) dN (t) ,(3.25) ∑ || ( )|| i i LiΨ x, t where the wave function Ψ(x, t) represents the pure state of the system and the operators Li describe the effective interactions between the system S and environment E. They induce instantaneous jumps from the state Ψ(x, t) into L Ψ(x,t) the new state i /||LiΨ(x,t)||. We show in the following that the operators Li are in fact identical to the Lindblad operators of a corresponding Lindblad master equation, by performing an ensemble average over many evolution realizations Ψ(x, t). The second term on the right hand side can be identiﬁed with the statistical part of the dynamics, where dNi are again Poisson increments with the property

dNi(t)dNj(t) = δi,jdNi(t) .(3.26) deﬁning whether jumps corresponding to the operators Li occur, in analogy to the classical dynamics. In order to obtain the right jump statistics, such that the average evolution of Ψ(x, t) corresponds to a master equation description, the mean value of dNi(t) has to be

Mtraj dNi,k(t) † M[dNi](t) = ∑ = dtγihLi Lii(t) ,(3.27) k=1 Mtraj where Mtraj is the number of considered environment realizations k (remember the previous section for an explanation of environment realizations) in the average. The rates γi are given by the corresponding master equations description. The ﬁrst term describes the deterministic evolution. In order to preserve the

32 Chapter 3 Open Quantum Systems normalization of the wave function it is deﬁned via the non-linear operator

HΨ(x, t) i † 2 G(Ψ(x, t)) = − ∑ γi Ai Ai − ||AiΨ(x, t)|| Ψ(x, t) .(3.28) h¯ 2 i

An intuitive understanding of this operator can be gained by splitting it into ˜ i¯h † a non-hermitian Hamiltonian part H = H − ( /2) ∑i γi Ai Ai, which contains the regular Hamiltonian H and the coherent part of the dynamics induced by the environment, and the remaining non-linear part. The formal solution to the Schrödinger-type equation corresponding to H˜ with the normalized initial state Ψ(x, 0) is then given by

exp(−iHt˜ /¯h)Ψ(x, 0) Ψˆ (x, t) = (3.29) ||exp(−iHt˜ /¯h)Ψ(x, 0)||

When we differentiate the norm with respect to time we ﬁnd

∞ d ˜ Z ˜ † ˜ ||e−iHt/¯hΨ(x, 0)||2 = i Ψ∗(x, 0)eiH t/¯h H˜ † − H˜ e−iHt/¯hΨ(x, 0)dx dt −∞ (3.30) −iHt˜ /¯h 2 = − ∑ γi||Aie Ψ(x, t)|| .(3.31) i

Hence we see that the norm decreases in time, which is exactly compensated by the non-linear term in G(Ψ(x, t)). Finally, combining the explicit form for G(Ψ(x, t)) as given in Eq. (3.28) with the general evolution equation (3.25) gives the non-linear stochastic Schrödinger equation (SSE) for jump unraveling. In bra-ket notation it reads

h √ A † 1 d |Ψ(t)i = ∑ dNi(t) i/ hAi Aii(t) − i γ i i + dt i (hA † A i(t) − A † A ) − dt H |Ψ(t)i .(3.32) 2 i i i i h¯

Further details on the jump unraveling approach and the resuling trajectories can be found in [49–52]. Equation (3.32) is said to unravel the underlying Lindblad master equation in the sense that an ensemble average over many realizations of the system’s evolution coincides with the effective evolution of the density matrix given by

33 3.4 Detrimental Inﬂuence of the Environment - Decoherence a master equation. Performing the average explicitely yields

N dρ(t) dρ˜(t) traj dρ˜ (t) = M = ∑ k (3.33a) dt dt k Ntrajdt |dΨ(t)i hΨ(t)| + |Ψ(t)i hdΨ(t)| + |dΨ(t)i hdΨ(t)| = M (3.33b) dt |dΨ(t)i hΨ(t)| + |Ψ(t)i hdΨ(t)| = M + dt       A † Li 1 j 1 + dNi(t)  q −  |Ψ(t)i hΨ(t)| dNj(t)  q −  + † † hLi Lii(t) hAj Aji(t) o + O(dN(t)dt) + O(dt2) (3.33c)      L γ i ≈  ( )  i −1 + i (h † i( ) − † ) −  ( )+ M dNi t q :: dt Li Li t Li Li H ρ˜ t   †  2 ::::::: h¯   hLi Lii(t)       †     Li 1 γi † † i  + ρ˜(t) dNi(t) q − + dt (hLi Lii(t) − Li Li) + H +   †  2 ::::::: h¯    hLi Lii(t)       L ρ˜(t)L † ρ˜(t)L † L ρ˜(t)  + dN (t)  i i − i − i +ρ˜(t) (3.33d) i  h † i( ) q q   Li Li t hL † L i(t) hL † L i(t)  i i i i  i 1 n o = − [H, ρ(t)] + γ L ρ(t)L† − L† L , ρ(t) ,(3.33e) h¯ i i i 2 i i where we sum over the indices i and j, employ the notations d |Ψ(t)i = |dΨ(t)i and |Ψ(t)i hΨ(t)| = ρ˜(t), from (3.33c) to (3.33d) we neglected all higher orders in dt and employ condition (3.26), and from (3.33d) to (3.33e) the mean is performed, while corresponding terms (indicated by matching underlines) cancel. Alternatively to the jump unraveling, as presented above, single realizations of open quantum system can also be described via diffusive unraveling [53–55], where not random finite jumps simulate the coupling to the environment but infinitesimal fluctuations overlay the deterministic evolution. This approach leads to a slightly different stochastic Schrödinger equation, which is however not employed by us.

3.4 Detrimental Inﬂuence of the Environment - Decoherence

In essence, decoherence is the destruction of coherences (in a speciﬁc basis) and quantum correlations between multiple quantum systems. In that sense

34 Chapter 3 Open Quantum Systems one also often refers to decoherence as destroying the ‘quantumness’ of the system, rendering it classical. Even though usually not termed decoherence, the described effect occurs whenever a projective measurement is performed on a quantum system. In that case any superpositions in the measurement basis and also correlations with other systems are destroyed in favor of gaining precise knowledge about the projected state. In contrast, weak measurements only lead to a partial decoherence at the cost of only providing inprecise information about the system state. In most cases when speaking about decoherence, one actually refers to the destruction of quantum coherences and correlations due to environmental inﬂuences. This is, decoherence as a result of the interaction of an environment driving the quantum system into a classically uncertain state ρclassical = ∑p pp |pi hp| by suppressing all coherences between the states |pi and at the same time correlating it with the environment1. This increase of correlation with the local environment will automatically suppressing all correlations with other systems, that are not effected by the same local environment, since a three-point correlation can only exist when everything is correlated with everything. This process is often termed environmentally induced superselection, or short einselection [12, 56]. The resulting states |pi are then called pointer states, forming the preferred basis of the environment. The term ‘pointer state’ originates from measurement theory [57], where these states can be understood as those a classical measurement device (pointer) would be in after measuring the corresponding quantum system with the outcome corresponding to the pointer state. And the ‘preferred basis’ is formed by the states corresponding to all possible measurement results. As an example for a decoherence process, consider a two-level atom with the two possible states |0i and |1i, which can spontaneously emit or absorb a photon by interacting with the surrounding light ﬁeld. On the one hand, the emission of a photon happens with the rate γ− is modeled by the Lindblad operator σ− = |0i h1|, such that the atom decays to the ground state |0i after it emitted the photon. On the other hand, γ+ is the rate for absorbing a photon and jumping into the excited state |1i by means of the effective interaction operator σ+ = |1i h0|. The corresponding Lindblad master equation, with

1This can be understood by the classical interpretation of correlations as presented in Sec. (2.3): Initially, information about the environment would tell nothing about the state of the system, assuming they are initially uncorrelated), but after the environment interacted with the system for a while, full information about the environment’s evolution would sufﬁce to ﬁgure out the state of the system.

35 3.4 Detrimental Inﬂuence of the Environment - Decoherence arbitrary Hamiltonian H, is then given by

dρ(t) i 1 = − [H, ρ(t)] + γ− σ−ρ(t)σ+ − {σ+σ−, ρ(t)} dt h¯ 2 1 + γ+ σ+ρ(t)σ− − {σ−σ+, ρ(t)} .(3.34) 2

In order to show the detrimental effect of the environment in this example, we shall solve the above master equation (3.34) for an initial state which exhibits the desired property. Since we only have a single system, no quantum correlations are possible. Let us consider the maximally coherent state ρ(0) = (1/2)(|0i + |1i)(h0| + h1|) as initial state. We are then interested in how much of this initial coherence is lost over time, where the coherence in the {|0i , |1i} basis is quantiﬁed via

Coh(ρ) = 2| h0| ρ |1i | ,(3.35) such that a maximally coherent state gives 1 and a fully incoherent state gives 0. Since we are solely interested in the effect of the environmental coupling, we neglect the Hamiltonian, i.e. set H = 0, without loss of generality. Equation (3.34) can be solved analytically and the resulting evolution of the coherence Coh(ρ(t)) is then given via

− t (γ +γ )t Coh(ρ(t)) = e 2 − + (3.36) and exemplarily shown in Fig. (3.3) for γ− = (2/3)γ+.

Figure 3.3: Evolution of coherence between the ground |0i and excited state |1i of a qubit coupled to a photon bath with decay rates ratio γ− = (2/3)γ+, starting in an initially maximally coherent state.

One can see that the state ρ(t) loses its coherence exponentially over time. The asymptotic state into which the system evolves for large times t → ∞ can be obtained by solving Eq. (3.34) with the condition dρ(t)/dt = 0. In our case this state is ρ(t → ∞) = γ−/(γ−+γ+) |0i h0| + γ+/(γ−+γ+) |1i h1| - a classically mixed

36 Chapter 3 Open Quantum Systems state with its pure components having no coherence between the ground and excited state. Hence, a two-level atom coupled to a photon bath as described by Eq. (3.34) completely decoheres with time. Let us remark, that one would observe similar coherence destruction if the atom was coupled to other forms of incoherent inﬂuence, like a dephasing environment. Another example for the detrimental effect of the environment is a bipartite system consisting of two qubits which are both individually coupled to a dephasing environment. The associated Lindblad operators are L1 = σz ⊗ 1 1 and L2 = ⊗ σz with the corresponding decay rates γd,1 and γd,2. The Lindblad master equation is then given by

dρ(t) i 1 n o = − [H, ρ(t)] + γ L ρ(t)L† − L† L , ρ(t) dt h¯ d,1 1 1 2 1 1 1 n o + γ L ρ(t)L† − L† L , ρ(t) .(3.37) d,2 2 2 2 2 2

Figure 3.4: Evolution of entanglement, quantiﬁed with concurrence [58,59], of a bipartite system coupled to individually dephasing environments with a decay rate ratio γ1 = (1/2γ2 and starting initially in a maximally entangled Bell state.

The figure of merit is now entanglement as a specific kind of quantum correlations. To show that the coupling to the environment reduces the amount of entanglement in the bipartite system over time, we initialize our system in a maximally entangled Bell state ρ(t = 0) = |Ψ i hΨ |, where √ Bell Bell 1 |ΨBelli = ( / 2)(|0i |1i + |1i |0i), and quantify the amount of entanglement with concurrence as introduced by Wooters [58, 59]. Concurrence is an entanglement monotone [30] quantifying the exact amount of entanglement of for- mation [60]. The evolution of the concurrence for γ1 = (2/3)γ2 is shown in Fig. (3.4). An analysis of the asymptotic state yields the fully separable state ρ(t → ∞) = 1/2(|01i h01| + |10i h10|), according to definition (2.4). This result is not only true for the dephasing environment, but in fact any environment

37 3.4 Detrimental Inﬂuence of the Environment - Decoherence inducing incoherent dynamics locally will inevitably couple the subsystems to their individual environments while decoupling them from each other, which renders their overall state separable.

38 4 Control of Open Quantum Systems

The two examples at the end of the previous chapter, spontaneous decay of a single qubit and dephasing of two entangled qubits, shows how environmental influences can have detrimental effects. Especially, decoherence happens in experimental realizations on relatively short time scales, which is tolerable as long as the system state only needs to be stable for a short time period, e.g. throughout the course of a few calculation steps of a quantum computer. However, in order to keep quantum properties up for time periods required for quantum memories or for long calculations, it is necessary to isolate the quantum system, shield it against the influence of the environment, but still enable external ma- nipulations. This requires control of open quantum systems. Several different approaches have been developed in the past, some of them easier and others harder to implement experimentally. One approach - coherent control - is in principle always possible and experimentally feasible by employing external control Hamiltonians in the form of classical fields. However, as we show this approach is limited in its capabilities to shield the system against the detrimental effect of the environment. In the next (final) chapter of this thesis, we therefor introduce an extension of this coherent control approach to overcome these limitations.

It follows an introduction into various control approaches, like decoherence free subspaces, environmental engineering and coherent control, together with their specific pros and cons. Thereafter, we describe how to find the optimal coherent control, i.e. the optimal stabilizing Hamil- tonian, with respect to properties of interest. The applicability of this method is exemplified with qutrits under different environmental influences in the final section. 4.1 Different Control Approaches

4.1 Different Control Approaches

Since decoherence builds up during the interaction time between system and environment, it is often possible to neglect the environment, i.e. treat the system as if it was closed, when the interaction time is very short, such that the environment had no time to noticeably effect the system. This is for instance the case when a quantum processing units [1] only performs a few sequences of quantum operations on a quantum system and the result is read out right after. To further minimize the influence of the environment it is then sufficient to just isolate the quantum system from the environment to an extent, which leads to a time scale separation between computation time and decoherence time. However, the influence of the environment can never be fully avoided and thus errors occurring during a computation are unavoidable. An isolation of the system, as suggested above, will only reduce the amount of errors occurring, but it can never ensure that such errors won’t accumulate during computation. In order to prevent the latter, further reaching methods are required. One proposal for error-robust quantum computation is given by the topological quantum computation approach [61]. It employs quasi-particles, so-called non-Abelian anyons, which have a topological degeneracy in their state space. An encoding of information in these states is stable against local perturbations of the state and thus can tolerate local environmental influences. Not only to make quantum computations stable against errors, but also to provide quantum properties for longer time periods, it becomes very inefficient1 (and in some experimental realizations even impossible) to further isolate the system from the environment. Thus, apart from the already introduced topological quantum computation, other forms of long term, i.e. asymptotic, quantum control, where the quantum states are meant to be stabilized for long (potentially infinite) time scales, despite the coupling to an environment, are needed. Up to date there are three promising approaches that shall be shortly introduced in the subsequent sections.

Decoherence Free Subspaces

M For a given set of Lindblad operators {Li}i=1, a decoherence free subspace N (DFS) [62] is characterized by a set of pure states {|ki}k=1 that form a basis in an N-dimensional sub-Hilbertspace H˜ ∈ H and for which the resulting density matrix

N ρ = ∑ pk |ki hk| (4.1) k=1

1Achieving very high isolation usually demands lots of experimental resources, which is okay in a lab but not practical for realistic applications

40 Chapter 4 Control of Open Quantum Systems is decoherence free, i.e.

M † 1 † D(ρ) = ∑ γi LiρLi − {ρ, Li Li} = 0 . (4.2) i=1 2

It was proven [63] that a necessary and sufﬁcient condition for the subspace ˜ N H = Span[{|ki}k=1] to be decoherence-free is that all basis states |ki are degenerate eigenstates of all Lindblad operators {Li}. This condition reads formally

Li |ki = ci |ki ∀i, k (4.3) and means that a DFS can be found in a system as soon as there is more than one pure state on which the Lindblad operators have no impact. To not confuse DFSs with pointer states, it should be highlighted that, in contrast to the basis states |ki of the DFS, linear combinations of pointer states do not necessarily lead to states that are unaffected by the dissipation. Furthermore, even though states within the decoherence free subspace are not influenced by the environment, it is not guaranteed that the coupling to the environment drives an initial quantum state into a state within the DFS, while a specific linear combination of pointer states is always the asymptotic state for a system coupled to an environment. The great quality of DFSs is that they allow for linear combinations of pure quantum states being stable against environmental influence for any duration of time. In particular, also coherent superposition states, which are also the basic ingredient for entanglement, can be shielded from the environment. This all can be done without any additional effort, since states within the subspace will forever stay in it, as long as there is no additional coherent dynamics. A first experimental demonstration was given by Kwiat et al., who protected 2-qubit photon states against collective dephasing [64], followed by a realization in ion trap experiments [65] and an application of DFSs in 3-qubit experiments [66]. The major drawback of the DFS concept is that readily usable DFSs are very hard to find in reality. In most cases the DFSs of an open quantum system are merely the one-dimensional state-spaces of the environmentally superselected mixed states. We will later see in the examples of qutrits under environmental influence, that DFSs only occur when multiple environmental interactions happen coherently. To actually make the above concept applicable on larger scales and for a wide range of dissipative effects, additional experimental effort is usually needed to combine the concept with other control approaches, for example to engineer an environment that has the desired DFSs.

41 4.1 Different Control Approaches

Environmental Engineering As mentioned above, if decoherence free subspaces are not readily available for the actual environment in the experimental setup, one could try to man- ufacture it. In this sense, another strategy for long term control consists in artificially manipulating the coupling between system and environment such that the asymptotic state of the system is a quantum state with the desired properties, or such that the man-made coupling exhibit a desired decoherence free subspace. It has been shown [67] that any desired (multipartite) quantum state can be turned into the steady state of a corresponding environment. The authors provide a general scheme to find the (in general non-local) effective interaction operators leading to the desired unique stationary state into which any initial state relaxes within times scale independent on the number of qubits. This scheme basically [68] employs an unitary transformation of the original Lind- † blad operators {Li} into new ones {L˜ i} = U{Li}U , which yield the desired state ρ stationary under the transformed dissipator D({L˜ }; ρ) = 0. This is however only a ‘proof of principle’ and as such hardly experimentally useful, since one still needs to engineer the according environment. In [69] the authors discuss theoretically a possible realization of environmental engineering. They suggest to couple the quantum system to another auxiliary system with a specific interaction Hamiltonian, such that properties of the actual system are adiabatically suppressed and another effective Lind- blad operator comes into action. Another group [70] considers a realization for many-body systems, where the jump operators are chosen such that they fix the phases between adjacent systems, and thus the whole system is driven into a well-specified pure state. To engineer such operators they, similarly to the previous suggestion, couple each lattice site of the many-body system to an extra auxiliary site and drive the extra sites leading to an adiabatic elimination of unwanted interactions between the actual system and the environment. An alternative approach [71] employs continuous monitoring of the quantum system to effectively reshape the interaction operators. The problem with such environmental engineering is, that they usually demand high experimental overhead, such as introducing auxiliary quantum systems, which themselves require to be controlled and suffer from uncontrolled coupling to the surrounding environment, i.e. well-controlled means of constantly pumping the system and/or having to continuously monitor the quantum system in an artificial manner must be applied.

Coherent Control The general drawback of the previous approach - requiring a great deal of experimental effort - is the advantage of the coherent control approach. Instead of introducing nouveau features to the quantum system, external control,

42 Chapter 4 Control of Open Quantum Systems usually in the form of classical fields, is employed, which manifests itself mathematically as an additional Hamiltonian acting on the system. In the theory of open quantum system, they can be readily included as extra terms in the coherent part of the Lindblad master equation. Experimentally, coherent control is usually realized via classical electro- magnetic fields. For example, trapped atomic ions can be influenced coherently by laser fields [14] or lately also via the near field of microwave currents [72]. Semi-conducting quantum dots are controlled optically in a multitude of different approaches [15], such as optical detection of spin reso- nances in NV centers [73] or electric manipulation of confined electrons [74], and nuclear spins of molecules or solid state systems are manipulated by highly developed nuclear magnetic resonance methods [16,75]. Many of the aforementioned examples have led to a realization of a quantum computer, which again highlights the importance of the coherent control approach.

4.2 Optimal Coherent Control

Until very recently, it was unclear to what extent coherent control can be employed to protect quantum systems against the detrimental effect of an environment. Is it possible to uphold coherences and entanglement even in the asymptotic state by coherent control alone? An answer to this question was given by Simeon Sauer during his dissertation [76]. He considered open quantum systems that can be described by Lindblad master equations and developed a formalism to determine the optimal coherent control, such that with his methods it is now possible to precisely quantify what coherent control can achieve for a speciﬁc environment under consideration. In the following we introduce and explain his ﬁndings. A quantum state ρ(t) is said to be a stationary state if ρ˙(t) = 0, i.e. ρ(t) = ρss = const, for all times t in a considered time interval {ti, t f }. Then the Lindblad master equation becomes

0 = −i[H, ρss] + D({Li}; ρss) ∀t ∈ {ti, t f } .(4.4)

When the quantum system does not initially start in the stationary state, i.e. ρ(ti) 6= ρss, then the state ρ(t) is only going to approach the stationary state asymptotically, which means the state is only stable in the asymptotic limit of t → ∞, i.e. ρ(t → ∞) = ρss. Usually the speciﬁc quantum system and environment under consideration ﬁx the set of Lindblad operators {Li}. By assuming these cannot be changed or engineered, which is the underlying principle of the coherent control approach, one can only manipulate the external Hamiltonian H in order to stabilize a desired quantum state ρss. Here, two kinds of questions directly arise: a) what state(s) ρss is(are) stabilized for a given Hamiltonian H, and b)

43 4.2 Optimal Coherent Control

which Hamiltonian H stabilizes a particular state ρss. Both questions yield systems of linear equations, which can be solved with standard methods of linear algebra to either give zero, one unique or infinitely many solutions. As we will see below, it turns out that a given Hamiltonian (question (a)) 2 usually stabilizes a unique stationary state ρss. Whereas, given a desired stationary state ρss one cannot find a stabilizing Hamiltonian (questions (b)) in most cases. Thus it seems worthwhile to ask the more general questions: ‘Which states ρss can be stabilized by a suitable Hamiltonian?’. The answer to this, mathematically termed the set of stabilizable states S, is defined as

S = {ρ ∈ Q|∃H : 0 = −i[H, ρ] + D({Li}; ρ)} .(4.5)

It was proven [76] that S, which is a bounded subset of the set of all physical quantum states Q given by the Conditions (2.1), is well speciﬁed by the conditions

d tr{ρssD({Li}; ρss)} = 0, ∀d ≤ dim(H) .(4.6)

Hence, the above conditions are sufficient and necessary3 conditions for the existence of coherently stabilizable state ρss - a solution of Eq. (4.4) for a given set of Lindblad operators {Li}. There exists an intuitive physical interpretation of the above conditions. They ensure that the dissipator has no incoherent effect at the specific point d ρss in state space, i.e. it does not change the dth moment tr{ρ } of the state ρss. Then it is given for granted that the ‘leftover’ coherent dynamics from the dissipator can be counteracted by an accordingly chosen stabilizing Hamiltonian Hss. In the case of two-dimensional systems, the stabilizing condition ensures that the dissipator does not change the purity of the state, since any non-purity modifying dynamics can be compensated by an appropriate Hamiltonian. These conditions are very ‘handy’ because they can provide answers to optimality questions about coherent control. If one would try to protect a specific property of an open quantum system one can figure out the optimal stabilizable state with respect to this quantity by performing an optimization over the set of stabilizable states S, as specified by the Conditions (4.6), with respect to this property of interest. Note in particular that this optimization is performed over the set of states solely - no specific form or parameterization of the control Hamiltonian is assumed - such that the optimization result gives the global optimum for what can be done with any possible control Hamiltonian for the specific quantum setup. For a specific steady state ρss, the corresponding stabilizing Hamiltonian Hss can be found easily by inverting the stationary Lindblad equation (4.4)

2Depending on the dissipator. 3 As long as ρss is not degenerate, otherwise they are only necessary conditions.

44 Chapter 4 Control of Open Quantum Systems and solving for the Hamiltonian [76]. One obtains

Hss = H0 + HFree

i hα| D({Li}; ρss) |βi = ∑ |αi hβ| + ∑ Hαβ |αi hβ| ,(4.7) pα − pβ pα6=pβ pα=pβ where |αi and pα are the eigenstates and -values of the steady state ρss = ∑α pα |αi hα|, respectively, and Hαβ can be chosen freely. The latter is true because the additional free Hamiltonian HFree is constructed of eigenstates of ρss, which means mathematically that it commutes with the stationary state and it means in the geometric picture of Hamiltonians inducing rotations in the state space that the axis of HFree points exactly through the stationary state. The stabilizing Hamiltonian Hss is well defined, except for when ρss is degenerate, in which case not all eigenvalues are mutually different, i.e. 0 0 pα0 = pβ0 for some α and β , and consequently the corresponding stabilizing Hamiltonian Hss would become singular. However, this is not a serious issue, 0 since it is always possible to find a non-degenerate state ρss in the vicinity of 0 the degenerate state ρss, which is stabilized by a bounded Hamiltonian Hss. 0 By bringing ρss arbitrarily close to ρss one can asymptotically stabilize the 0 state ρss with increasingly strong Hamiltonians Hss. In the original work [76], the above theory is exemplified via single and two qubits under various environmental influences. Here we extent the number of investigated examples by applying the theory to three-level quantum systems, so-called qutrits, again subject to different environments. The qualitative result is the same in both cases: Coherent control potentially brings substantial improvement, but it is nevertheless quite limited in its capability of counteracting the effect of incoherent dissipation.

4.3 Optimal Stationary Qutrit State for Different Dissipations

The most general density matrix representation for qutrits is usually given by an expansion in terms of the eight Gell-Mann matrices {Λi}, as introduced in Section (2.4). It reads in the computational basis {|1i , |2i , |3i}

1  n3− √ n8  1 + √ 3 n1√−n2i n4√−n5i 3 3 3 3 1 √  1  1 ~  −n3− √ n8  ρ = ( + 3~nΛ) =  n1√+n2i 1 + √ 3 n6√−n7i  ,(4.8) 3  3 3 3 3  n4√+n5i n6√+n7i 1 − 2n8 3 3 3 3

45 4.3 Optimal Stationary Qutrit State for Different Dissipations where ~n is the generalized Bloch vector. According to the theory (Sec. (4.2), the necessary and sufﬁcient (in case of non-degenerate states) conditions for a qutrit state ρss to be stationary are

tr{ρssD(ρss)} ≡ 0 (4.9a) 2 and tr{ρssD(ρss)} ≡ 0 . (4.9b)

In the following, we investigate systems with different level structures and environmental influences. First, the case when two excited states |1i, |2i decay independently to the ground state |3i, modeled by the Lindblad operators L = |3i h1| and L2 = |3i h2|, coherently, modeled by the single 1 √ Lindblad operator L = 1/ 2 (|3i h1| + |3i h2|), and in form of a ladder via L1 = |2i h1| and L2 = |3i h2|. Afterwards we briefly discuss what happens for a ring decay process where the dissipation is described via L1 = |2i h1|, L2 = |3i h2| and L3 = |1i h3|, and we explore the coherent ring decay for √ L = 1/ 3(L1 + L2 + L3). For these given dissipations, the conditions above define the corresponding sets of stabilizable states. In these sets one can identify the optimal state with respect to a certain property of interest. We chosse, in order to find out to what extent can coherent dynamics counteract the incoherent driving of the environment, to minimize the occupation of the ground state |3i, quantified by P3 = tr{ρ |3i h3|}. This is the state to which all other levels would decay in absence of coherent control in all cases except of the ring decay. For the latter, where no distinct ground state exists, we optimize the occupation of one arbitrarily picked level, to see how far away from equilibrium we can stabilize the system with coherent control. Optimizations with respect to coherences between levels |mi and |ni could be another property of interest, however we found that no new insights can be gained from this, compared to optimizing with respect to specific level occupations. In summary, we will show that different environments can be counteracted to very differing extents. Let us make one technical remark before the examples are addressed: The optimizations in the sets of stabilizable states are analytically very difficult, since, in contrast to the two-dimensional case, we also need to explicitly take the second order constraint (4.9b) into account. As a consequence, we resort to numerical methods in the following. In principle, we would also always have to additionally take the positivity constraint 2.1c into account, since this constraint is not automatically fulfilled for all generalized Bloch vectors ~n. However, it turns out that it is satisfied for all optimization results, thus we will not mention their verification in the following. Actually, we suspect that that the stationarity constraints (4.9) include the positivity constraint, since solutions of the stationary Lindblad master equation are always valid density matrices and hence the stationarity constraints by default yields positive semidefinite matrices.

46 Chapter 4 Control of Open Quantum Systems

Independent Decay to a single Ground State

We start with the case of a three-system, where two states, |1i, |2i, decay independently into |3i. The system’s level structure with the Lindblad operators L1 = |3i h1| and L2 = |3i h2| corresponding to the decay rates γ1 and γ2 respectively, is illustrated in Fig. (4.1). In the following we usually speak about ground (|3i) and excited states (|1i and |2i), since this is for instance the usual situation for a doubly excited ion. However, the energies of the levels must not necessarily correspond to the level positions in the illustration, in particular, the two excited levels do not have to be energetically degenerate as it might come across, nor must level |3i correspond to the ground state, it could just as well be the state favored by the environment.

Figure 4.1: Level structure of a qutrit subject to independent spontaneous decay, into the ground state.

In this case, the environment drives the system into the unique ground state |3i. In case of no driving, i.e. vanishing Hamiltonian, the ground state is the unique asymptotic stationary state. By means of optimal coherent control we would like to know now which stabilizable state minimizes the ground state population P3 = tr{ρss |3i h3|}. To ﬁnd this optimal state, we optimize the generalized Bloch parameters ~n under the stationarity constraints (4.9) with respect to the quantity P3. In the case of equal decay rates γ1 ≡ γ2, which is for instance the case for degenerate excited levels |1i and |2i that couple uniformly to the ground level |3i of a quantum system, we ﬁnd that the optimal stationary state is the maximally mixed state

ρopt = 0.33(|1i h1| + |2i h2| + |3i h3|) ,(4.10) yielding a ground state occupation of P3 = 0.33. However, this state has a three-fold degeneracy and is therefore not guaranteed to be stabilizable, since Eqs. (4.9) are then only necessary stabilization conditions. In fact, it is easy to see that it is not stabilizable for non-vanishing dissipation, since this diagonal state commutes with any Hamiltonian, reducing the homogeneous master equation to the unfulﬁllable equation D(ρs) ≡ 0. However, one ﬁnds that a

47 4.3 Optimal Stationary Qutrit State for Different Dissipations nearly stabilizable state is given in the computational basis {|1i , |2i , |3i} by √  3/2 2  1 − 2e + 2e √e √−e 3 31/4 3 √ 3 3  e 1 2e3/2 2e2 e  ργ ≡γ ,ss ≈  √ + + √  ,(4.11) 1 2 3 3 31/4 3 3  2  √−e √e 1 + 4e 3 3 3 3 which is asymptotically close to the maximally mixed in the limit e → 0. The corresponding Hamiltonian that stabilizes the above state can be identified to be  0 i i  γ1 Hγ ≡γ ≈ √ −i 0 i ,(4.12) 1 2 3/4 3/2   2 23 e −i −i 0 where γ1 = γ2. Note that there are infinitely many other asymptotically close stabilizable states and corresponding stabilizing Hamiltonians, depending on from which direction in the state space the maximally mixed state is approached. However, we suspect that all Hamiltonians have in common that they couple the three states |1i , |2i , |3i of the system very strongly to counteract the effect of the dissipator. This is in strong analogy to the qubit case [17], and we assume that these results still hold for qudits (d-dimensional quantum systems), where d − 1 states decay separately to a single ground state with equal decay rates. Next, we investigate the more involved case when the decay rates are not equal, which occurs when the two-levels |1i and |2i are not degenerate and/or couple differently to |3i. The numerical analysis of the stabilizable minimal ground level occupation P3(γ1, γ2) for γ1, γ2 ∈ (0, 1) is plotted in Figure (4.2a). In case of equal rates (γ1 = γ2), we find the already mentioned minimal occupation of P3(γ1 = γ2) = 0.33 and for γ1 6= γ2 the value decreases the more the decay rates differ, down to in principle zero when one of the rates vanishes. In that extreme case, it is clear that the non-decaying state can be used to store the population. however, due to numerical issues we cannot see this dropping to zero, since for small decay rates the optimization becomes very fragile. This is also the reason for the random spikes in the outer region of the plot (4.2a), which correspond to finding local minima instead of global minima in the numerical minimization. To verify this stabilizable drop to to zero of the ground state population by analytical expressions for the minimal ground state population, we exploit the fact that the population depends only on the ratio of the two 4 decay rates γ1/γ2 = b > 1 , which can be seen from Figure (4.2a), since regions with equal minimal ground state occupation (equal color) are straight lines. Furthermore, we find that the subset {n1 = n2 = n4 = n5 = 0}, where states

4 Without loss of generality we assume that γ1 > γ2, hence b > 1.

48 Chapter 4 Control of Open Quantum Systems have neither coherences between |1i and |2i nor between |1i and |3i, still contains states with the globally minimal ground state population for any combination of γ1 and γ2. A restriction to this reduced set of parameters enables an analytic expression for the minimal ground state population P3 as a function of b, given by √ √ −6 − 7b + 2 3 + b + 2b 3 + b P (b) = .(4.13) 3 −8 + (−8 + b)b

The corresponding Bloch vector components for the optimal stabilizable states are given by

n1 = n2 = n4 = n5 = 0 √ √ √ 3(1 + b) 6 − 2 3 + b + b 5 + b − 4 3 + b n = − 3 2(3 + b)(−8 + (−8 + b)b)

120 + 420b + 501b2 + 228b3 + 27b4 n6 = ± − (3 + b) (−8 − 8b + b2)2 √ 2 3 4 !1/2 3 3 + b 24 + 76b + 83b + 32b + b 2 − + n7 (3 + b) (−8 − 8b + b2)2 10 − √18 + b 13 − √24 + b2 1 − √ 6 +b +b +b n = 3 3 3 .(4.14) 8 2 (−8 − 8b + b2)

Note that the optimal stabilizable states are not unique for speciﬁc b since n7 can be chosen freely. The value of P3(b) is shown in Figure (4.2b). For b → 1, corresponding to the case γ1 = γ2, we approach the ground state population for a maximally mixed state of 1/3 and when b → ∞ the achievable occupation drops to zero √ as 1/ b. A diagonalization of the density matrix obtained from the Bloch vector components (4.14) reveals that the optimal states are degenerate, which means they are also only asymptotically stabilizable. To ﬁnd the Hamiltonian that stabilizes the above states asymptotically one can exploit the fact that there are states of the form

n2 = n5 = 0

n1 = n4 = e

n3 = fn3 (b, e)

n6 = fn6 (n7, b, e)

n8 = fn8 (b, e) ,(4.15)

49 4.3 Optimal Stationary Qutrit State for Different Dissipations

(a) (b)

Figure 4.2: Minimal ground state population, (a) numerically obtained as a function of the decay rates γ1 and γ2 and (b) analytically obtained

as a function of the ratio of the decay rates γ1/γ2 = b, both for the case of independent decay of two excited states into one ground state.

which are non-degenerate, stabilizable for an appropriate choice of fn3 , fn6 and fn8 , and approach the optimal states given by the components in Eq. (4.14) for e → 0. With the help of these non-degenerate states, a Hamiltonian can be constructed according to Eq. (4.7), which will then be of the form   0 h1(e, b)i h2(e, b)i

Hγ16=γ2 = −h1(e, b)i 0 h3(e, b)i ,(4.16) −h2(e, b)i −h3(e, b)i 0 similar to the case for equal rates (see Eq. (4.12)) but now h1, h2 and h3 are of differing order and depend explicitly on the ratio b. For example the Hamiltonian  0 101, 5935i 158, 2131i Hb=2 = −101, 5935i 0 −3, 3294i ,(4.17) −158, 2131i 3, 3294i 0 stabilizes the state 0, 3000 0, 0006 0, 0006  ρb=2,ss = 0, 0006 0, 3708 −0, 0455 ,(4.18) 0, 0006 −0, 0455 0, 3292 which is optimal for γ1 = 2γ2. One can see that the stabilizing Hamiltonian strongly couples the two excited states |1i and |2i, as well as the faster

50 Chapter 4 Control of Open Quantum Systems decaying excited state |1i and the ground state |3i. By that, the population of the more stable excited state can be increased above the threshold value of 1/3 and so the ground state population is stabilizable below this threshold. Note that there are also Hamiltonians with h3(b) = 0 that can asymptotically stabilize the optimal states, but then the symmetry between n1 and n4, as stated in Eqs. (4.15), is lost. For larger values of b the qualitative behavior does not change. To verify this we show h1(e, b), h2(e, b) and h3(e, b) as functions of b ∈ {2, 3, . . . , 20} in Figures (4.3a), (4.3b) and (4.3c) for e = 1/1000.

(a) (b) (c)

Figure 4.3: Numerical values for components h1, h2 and h3 of the Hamiltonian that stabilizes the minimal ground state occupation as a function of the decay rates ratio b for ﬁxed e = 1/1000.

Coherent Decay Alternatively, it could also happen that the environment de-excites of both levels |1i and |2i coherently. In that case one speaks of a coherent decay to the ground state |3i, which is characterized by the Lindblad operator L = (L1 + L2) and corresponding rate γ. The corresponding system is illustrated in Figure (4.4).

Figure 4.4: Level structure for a qutrit subject to spontaneous coherent decay to the ground state.

Here we ﬁnd a parameter regime of the density matrix which is unaffected by the dissipation. This regime is characterized by the Bloch vector components √ n2 = 0, n3 = 0, n4 = −n6, n5 = −n7, n8 = −1 − 3n1 ,(4.19)

51 4.3 Optimal Stationary Qutrit State for Different Dissipations corresponding to the subspace spanned by the pure states

1 |φ−i = √ (|1i − |2i) (4.20) 2 |φ3i = |3i .(4.21)

Since these are eigenstates of the Lindblad operator, they deﬁne a decoherence free subspace, in accordance with Eq. (4.3). The corresponding eigenvalues are zero for both eigenstates

L |φ−i = 0 |φ−i (4.22)

L |φ3i = 0 |φ3i .(4.23) √ This subset already contains the state |φ−i (for n8 = 1/2, n1 = − 3/2, and the rest zero), which is globally optimal with respect to the minimal ground state population. Hence, no further optimizations with respect to the minimal ground state population is required. However, we would still like to know whether coherent control could extent the decoherence free subspace. In order to verify this, we can optimize over the set of stabilizable state with respect to the deﬁning constraints of the decoherence free subspace, as given via Eq. (4.19). √To do so most transparently, we only consider the condition n8 = −1 − 1 √3n1, since it describes the boundaries of the DFS: for n8 = /2 and n1 = − 3/2 the state |φ−i is realized, and for n8 = −1 and n1 = 0 it is the ground state |3i. To see,√ whether we can break this condition, we maximize the quantity n8 + 1 + 3n1, such that a non-zero result would tell us that states outside the DFS can be stabilized. And indeed, the numerical maximization yields the value 1.5 with the corresponding Bloch vector components √ n1 = 0.433 ≈ 3/2

n8 ≈ 1/4 (4.24)

n2 = n3 = n4 = n5 = n6 = n7 ≈ 0 , which correspond to the mixed state

1 ρ = (|φ+i hφ+| + |3i h3|) ,(4.25) ss 2 √ where |φ+i = (1/ 2) (|1i + |2i). It is clear that this stationary state is not part of the DFS, because it cannot be formed by any probabilistic mixture of the basis states of the DFS |φ−i and |3i. Hence, it seems that via coherent control, additional superpositions5 of the excited states |1i and |2i, apart from the

5An optimization with respect to n gives similar results, where |φ i is replaced by √ √ 2 + (1/ 2) (|1i + i |2i) or (1/ 2) (|1i − i |2i)

52 Chapter 4 Control of Open Quantum Systems

decoherence free state |φ−i, can be stabilized, by mixing it with the ground state. This sounds very similar to the already discovered pattern that coherent control is always able to stabilize highly mixed states. Note that also here the maximally mixed state can be stabilized, which is not part of the DFS.

Ladder Decay One speaks of a ladder decay scenario when the ﬁrst state |1i decays to the second state |2i via L12 = |2i h1| and the second state decays to the ground state |3i via L23 = |3i h2| (and corresponding decay rates), as depicted in Fig. (4.5). This is often the case when the two excited levels are energetically non-degenerate and there is no direct coupling between the highest energetic state |1i and the lowest |3i, e.g. because a transition is spin forbidden or the necessary energy transfer to or from the environment is not possible.

Figure 4.5: Level structure for a qutrit subject to ladder decay into the ground state.

We are again interested in minimizing the ground state population P3 = tr{ρss |3i h3|}, since we want to coherently counteract the environment, which tends to drive the system into the asymptotically stationary state |3i. As for the previous case of independent decay to a single ground state, also here only γ the ratio between the decay rates b = 12/γ23 is relevant.√ We ﬁnd two regimes for the minimal ground state population: a) for 0 ≤ b ≤ 2 the minimal value is √ −6 − 3b + b2 + 2 3 + 2b P (b) = (4.26) 3, case a) −8 + (−4 + b)b √ and b) for b > 2 it is

1 P (b) = .(4.27) 3, case b) 2 + b We refrain from explicitly showing the Bloch parameters for the corresponding stationary states and the expression of the associated stabilizing Hamiltonian for the sake of conciseness, but the reader can be assured that they can be

53 4.3 Optimal Stationary Qutrit State for Different Dissipations found in a similar manner as for the previous case. The occurrence of the two regimes is assumed to be of purely mathematical nature, nice physically the resulting function for the minimal stabilizable ground state population ( √ P3, case a)(b) , 0 ≤ b ≤ 2 P3(b) = √ (4.28) P3, case b)(b) , b > 2 is perfectly smooth. Probably the parameterization of the density matrix in terms of Gell-Mann matrices is not optimal for the considered situation. The corresponding curves for the minimal ground state populations as functions of the ratio b are√ shown in Figure (4.6). One can see that the two curves exactly meet at b = 2 and even though the ground state population√ for the states of case a) is in general lower than that for case b) for b > 2 they do not describe the actual minimal ground state population since the corresponding√ density matrix is not hermitian (n6 becomes imaginary for b > 2). In contrast to the previous results, the maximally mixed state is now optimal not for equal decay rates (b = 1) but in the case when b = 1/2, i.e when γ23 = 2γ12. Hence, this situation seems to be the most detrimental for the ground state population.

Figure 4.6: Minimal ground state occupation as√ a function of the√ ratio of the decay rates b = γ12/γ23 for a) 0 ≤ b ≤ 2 and b) b > 2.

Ring Decay

Now we consider the situation of a one-way ring decay described by the Lind- blad operators L1 = |2i h1|, L2 = |3i h2| and L3 = |1i h3| with corresponding decay rates γ1, γ2 and γ3. This situation occurs in periodic systems where the environment induces a transport of population from one state to another. For

54 Chapter 4 Control of Open Quantum Systems vanishing Hamiltonian the system is then driven into the stationary state

 γ2γ3  Γ 0 0 γ1γ3 ρss =  0 Γ 0  ,(4.29) γ1γ2 0 0 Γ where Γ = γ1γ2 + γ1γ3 + γ2γ3, and once more we want to see to what extent coherent control can modify this.

Figure 4.7: Level structure for a qutrit subject to ring decay.

We start with the case of equal decay rates γ1 = γ2 = γ3 = γ. Interestingly, there exists no Hamiltonian that can render any other state than the maximally mixed one stationary. In other words, the inﬂuence of the Hamiltonian is always orthogonal to that of the dissipator. To show that, we expand the density operator around the maximally mixed state and then verify that the stability condition tr{ρD(ρ)} 6 has a local extremum at the maximally mixed state. To investigate the behavior of the stability condition around the maximally mixed state, we expand ρ, as expressed in terms of the Gell-Mann matrices, as given in Equation (2.11), about this extreme point:

1 0 ρ(~e) = 1 +~e∇~ 0 ρ(~e )| 0= ,(4.30) 3 e e 0 The first order partial derivatives of tr{ρ(~e)D(ρ(~e))} with respect to each component ei are  0 if i ∈ {1, 2, 4, 5, 6, 7}  ∂tr{ρ(~e)D(ρ(~e))} −2γ1+γ2+γ3 | = = √ if i ≡ 3 .(4.31) e 0 3 3 ∂ei  1 3 (−γ2 + γ3) if i ≡ 8. It is easy to see that for equal decay rates all first order derivatives are zero. When calculating the second order partial derivatives in the form of the Jacobian matrix, the latter turns out to be negative definite with eigenvalues {−2γ, −2γ, −4γ/3, −4γ/3, −4γ/3, −4γ/3, −4γ/3, −4γ/3 }. Hence we can conclude

6We need not to also investigate the second condition tr{ρ2D(ρ)}, since a non-zero value for tr{ρD(ρ)} is already sufﬁcient for the state to be not stabilizable.

55 4.3 Optimal Stationary Qutrit State for Different Dissipations

γ1 = 2, γ2 = 1, γ3 = 1 γ1 = 1, γ2 = 2, γ3 = 3 1 0 0 6 0 0 1 1 ρss for H = 0 5 0 2 0 11 0 3 0 0 0 2 0 0 2 Min tr{ρ |1i h1|} 1/5 1/3 Max tr{ρ |1i h1|} 1/3 3/5 Min tr{ρ |2i h2|} 1/3 1/5 Max tr{ρ |2i h2|} 1/2 3/8 Min tr{ρ |3i h3|} 1/4 2/11 Max tr{ρ |3i h3|} 2/5 2/5

Table 4.1: Minimal and maximal stabilizable level populations of a dissipator describing a ring decay for two different decay rate configurations. that the maximally mixed state corresponds to a local minimum for the stability condition and consequently it is locally the only stabilizable state for a ring decay with independent and equally strong decay between the levels. Further numerical investigations suggest that this holds also globally, i.e. there is no additionally stabilizable state. The reason is that additional mixing of two-levels’ population due to coherent coupling between them won’t change the fact that all level are equally populated. However, for differing decay rates the level populations can be manipulated, which can already be seen via the non-vanishing first order derivatives via Eq. (4.31). To give an idea about what can be achieved with optimal Hamiltonians, we exemplarily show in Table (4.1) the stationary state ρss for vanishing Hamiltonian, and the maximal and minimal stabilizable populations of all three levels for two arbitrarily chosen decay rate configurations. We see that almost all level populations can be enhanced in comparison to the stationary state, only the global minimal single level population from the stationary state cannot be further decreased, and also the globally maximal single level population cannot be (or only slightly) increased. Furthermore, in nearly all cases (only not for Min tr{ρ h3| |3i} of the second example when the optimization could not do any better) cases the optimal states have at least a two-fold degeneracy and are therefore only asymptotically stabilizable.

Coherent Ring Decay

Instead of having independent decay rates from level to level, the levels could also be coherently coupled, such that there is only one Lindblad operator L = L1 + L2 + L3, where L1 = |2i h1|, L2 = |3i h2| and L3 = |1i h3|. This situation is illustrated in Fig. (4.8). In that case we ﬁnd, similar to the case of coherent decay to a single ground state, a subspace of states which are unaffected by the dissipator. This regime

56 Chapter 4 Control of Open Quantum Systems

Figure 4.8: Level structure for a qutrit subject to coherent ring decay. is characterized by the Bloch vector components

n3 = n8 = 0

n1 = n4 = n6

−n2 = −n7 = n5 .(4.32)

The most general density matrix in this subset can be expressed with two parameters n1 and n2 as

 1 n1√−n2i n1√+n2i  3 3 3 + −  n1√n2i 1 n1√n2i  .(4.33)  3 3 3  n1√−n2i n1√+n2i 1 3 3 3 The pure states,

±2πi/3 ±4πi/3 ±6πi/3 |Ψ±i = e |1i + e |2i + e |3i (4.34) √ |Ψ0i = 1/ 3(|1i + |2i + |3i) ,(4.35) representing discrete ‘plane waves’ with maximal positive (negative) and zero probability current are the basis states of this decoherence free subspace, since they are eigenstates of the Lindblad operator

∓2πi/3 L |φ±i = e |φ±i

L |φ0i = |φ0i .

Therefore, the asymptotic state of such a system for vanishing Hamiltonian is a linear combination of basis states of the DFS. Introducing an additional stabilizing Hamiltonian does not lead to an augmented set of stabilizable states, neither different phases between the levels, nor different level populations can be stabilized coherently, since maximizing n3 and n8 yields 0, as well as maximizing n1 − n4, n2 − n7, n5 + n2, and so forth, does not give anything but zero. The explanation for this is presumably similar to the one for the case of independent ring decay with equal decay rates: An additional mixing of the states’ populations due to extra coherent couplings would only support driving the state more towards the maximally

57 4.3 Optimal Stationary Qutrit State for Different Dissipations mixed one, which is already part of the DFS.

58 5 Continuous Monitoring to Enhance Control

From the qutrit examples from above, as well as from previous research results [17], one can infer that coherent control alone is limited in its possibilities to uphold desired quantum properties in open systems. For above considered three-level systems under different environmental influences, it was mostly the maximally mixed state that optimizes the occupation of levels evacuated by the environment. To stabilize such maximally mixed states, universally an infinitely strong driving Hamil- tonian needs to be employed. To stabilize states with improved properties, the coherent control approach has therefore to be extended. For this, a continuous monitoring approach combined with feedback loops poses a suitable option. One then still performs coherent control, but on top of that also continuously monitors the system. This measurement information is used to re-stabilize the system with specifically designed coherent control pulses (feedback), whenever the environment acts detri- mentally. The extension retains the universality of the control approach, since only classical measurements and fields are needed, but we find a drastic improvement over the pure coherent control approach.

To see how this can be achieved, we first give a short introduction into the continuous monitoring approach, together with an explanation of how to numerically solve the associated stochastic differential equations. As a result, we obtain the evolution of single quantum trajectories, i.e. pure states conditioned on the stream of obtained measurement results in a single realization of the experiment. Thereafter we elaborate how the sets of stabilizable states can be modified by including the feedback protocol. In the last part, more realistic assumptions about the control approach, such as perturbed coherent control, inefficient monitoring and time-delayed feedback, are considered and their consequences elaborated. 5.1 Introduction to Jump Unraveling as a Control Approach

5.1 Introduction to Jump Unraveling as a Control Approach

When speaking about continuous monitoring, we have an open quantum system in mind, whose environment is continuously measured, such that any quantum jump of the system, induced by the interaction between system and environment, is detected. For example, in the case of a two-level atom under spontaneous decay, the corresponding experimental setup is schematically depicted in Figure (5.1). As soon as the atom decays to the ground state, it emits a photon. We assume that this emitted photon can be detected, updating our knowledge about the quantum state of the atom. Whenever no photon is detected, we know that the atom evolves coherently according to the applied classical ﬁelds. Thus, the monitoring provides continuous information about the evolution of a single quantum system, described by a quantum trajectory - the evolution of a pure state. Since an emission of a photon induces a jump in the state of the monitored system, we expect the state evolution to describe a quantum mechanical piecewise deterministic process, as introduced in Sec. (3.3). This process can be simulated with a stochastic Schrödinger equation given by Eq. (3.32) and the whole procedure is then termed jump unraveling of the corresponding Lindblad master equation. In contrast, a diffusive unraveling, which underlies a continuous state change, due to the environmental inﬂuence, results in a diffusive state evolution.

Figure 5.1: Schematic setup for continuously monitoring a single two-level atom coupled to the vacuum of the light ﬁeld. We assume that any emitted photon γ can be detected by the measurement apparatus M, which idealistically completely encloses the atom.

To obtain the evolution of these continuously monitored quantum systems, like the two-level atom from Fig. (5.1), one has to solve the corresponding stochastic differential equation (3.32). Since these coupled differential equations are stochastic and non-linear in nature, we have little hope to find analytic solutions, even in the most simplest cases. Hence, we resort to numerical means, which we would like to present in the following paragraphs, before we finally show the resulting trajectories. As a first step for a numerical treatment, all quantities have to be turned

60 Chapter 5 Continuous Monitoring to Enhance Control dimensionless. One possibility is given by introducing the dimensionless time

t˜ = γ1t (5.1) and ﬁxing the energy scale by setting Planck’s constant h¯ = 1. Then, the dimensionless Lindblad equation becomes

dρ(t˜) i γ 1 n o = − [H˜ , ρ(t˜)] + i L ρ(t˜)L† − L† L , ρ(t˜) ,(5.2) ˜ ∑ i i i i dt h¯ i γ1 2 and the dimensionless SSE reads      L ˜ ˜ i 1 d |Ψ(t)i = ∑ dNi(t)  q −  †  i hLi Lii(t˜) γi † † + dt˜ hLi Lii(t˜) − Li Li − idt˜H˜ |Ψ(t˜)i ,(5.3) 2γ1 where H H˜ = (5.4) γ1 γi † M[dNi(t˜)] = dt˜ hLi Lii(t˜) .(5.5) γ1 This dimensionless SSE is numerically solvable in two ways [52]: Either by following a ‘direct’ procedure, which evaluates the equation at every discrete time step ∆t˜ ≈ dt˜, or by making use of a unnormalized evolution equation. Both methods yield physically valid solutions, but have different fortes, which will get clearer after we introduced each method in a bit more detail. Let us start with the ‘direct’ method, which is comprised of the following steps

1. Choose an initial state |Ψ(t˜ = 0)i = |Ψ0i and set dNi(t˜ = 0) = 0 ∀i.

2. Evaluate the right-hand side of Eq. (5.3) for t˜ = 0 to obtain ∆ |Ψ(t˜ = 0)i.

3. Determine the state at the later time t˜ + ∆t˜ according to |Ψ(t˜ = ∆t˜)i = |Ψ(t˜ = 0)i + ∆ |Ψ(t˜ = 0)i.

† 4. Pick a random number r ∈ {0, . . . , 1}, and if r > ∑i ∆t˜(γi/γ1)hLi Lii(t˜) then no jump happens and dNi(t˜ = ∆t˜) = 0 ∀i, otherwise a jump happens and exactly one dNi(t˜ = ∆t˜) = 1, which is randomly chosen † according to the weights (γi/γ1)hLi Lii(t˜).

5. Repeat steps 5 to 7 for t˜ ∈ {∆t˜, 2∆t˜, 3∆t˜,..., NT∆t˜} and obtain the discrete solution {|Ψ(t˜ = 0)i , |Ψ(t˜ = ∆t˜)i ,... |Ψ(t˜ = NT∆t˜)i}.

61 5.1 Introduction to Jump Unraveling as a Control Approach

The alternative procedure, relying on an unnormalized evolution equation, can be described as follows

1. Solve the unnormalized evolution equation ! d |Ψ(t˜)i L† L = − i i + iH˜ |Ψ(t)i (5.6) ˜ ∑ dt i 2

˜ ˜ for the initial condition |Ψ(t = 0)i = |Ψ0i to obtain |Ψ(t)iinitial with standard numerical tools.

2. Draw a random number r1 ∈ {0, . . . , 1} and determine t˜1 such that hΨ(t˜ = t˜1)|Ψ(t = t˜1)i = r1. At t˜1 a jump happens and one obtains the ‘ﬁrst part’ of the trajectory

˜ ˜ ˜ ˜ |Ψ(0 ≤ t ≤ t1)i = |Ψ(0 ≤ t ≤ t1)iinitial .(5.7)

3. Decide on which jump L1 ∈ {Li} happens by randomly choosing one † according to the weights (γi/γ1)hLi Lii(t1) as in the ﬁrst method.

4. Solve Eq. (5.6) for the new initial condition |Ψ(t˜ = 0)i = L1 |Ψ(t˜1)i, which gives |Ψ(t˜)i . L1

5. Draw a random number r2 ∈ {0, . . . , 1} and determine t˜2 such that hΨ(t˜ = t˜2)|Ψ(t˜ = t˜2)i = r2. At t˜2 another jump happens and the trajectory until the second jump is given by

|Ψ(t˜ ≤ t˜ ≤ t˜ + t˜ )i = |Ψ(0 ≤ t˜ ≤ t˜ )i .(5.8) 1 1 2 2 L1

The latter method is numerically advantageous because the deterministic evolution has to be evaluated only once for every possible initial state realized after every possible jump. However, in contrast to the ‘direct’ ﬁrst method, does it per se not represent the actual physical evolution of a monitored quantum system at every time t˜, since unphysical (because unnormalized) states are generated by solving the unnormalized evolution equation (5.6). Though, when re-normalizing the state at the time t˜0 of interest, one recovers a

62 Chapter 5 Continuous Monitoring to Enhance Control physically valid state, which coincides with the direct solution [52]. Hence, the ﬁrst method is generally employed when the trajectories need to be analyzed at every time step t˜, while the second method is used e.g. to calculate averages over many trajectories at one speciﬁc time t˜0.

5.2 Jump Unraveling for a Stabilized Single Qubit

After getting through the above technical details, we are now able to investigate trajectories with concrete examples. Our system of choice is a single qubit coupled to an environment inducing spontaneous decay, since in that case the systems’ state space is simple enough to allow for an intelligible geometric representation, namely the Bloch sphere (remember Sec. (2.4)). The trajectories can then be nicely represented and understood in this sphere. Furthermore, this setting contains already the physics of interest: the environment drives the system asymptotically into the ground state of the two-level atom, by destroying all coherences, which can be counteracted to some extent by means of coherent control. In this case, there is only one Lindblad operator

{Li} → σ− (5.10) and accordingly only one decay rate γ. The SSE to be solved becomes " ! σ− 1 d |Ψ(t)i = dN(t) p − hσ+σ−i(t) 1 + dt (hσ+σ−i(t) − σ+σ−) − iH |Ψ(t)i .(5.11) 2

In the following, we use the orthonormal basis {|0i , |1i}. The states are then given in Bloch notation by

θ(t) θ(t) |Ψ(t)i = cos |0i + eiφ(t) sin |1i (5.12) 2 2 ρ(t) = 1 + r cos φ(t) cos θ(t)σx + r sin φ(t) cos θ(t)σy + r sin θ(t)σz .(5.13)

Since our aim is to use continuous monitoring to improve the coherent control approach, we explore trajectories of stationary states fulﬁlling the condition

tr{ρssD(σ−; ρss)} = 0 . (5.14)

First, as a proof of principle, we verify that the calculated trajectories indeed unravel the underlying master equation. This is achieved by performing the ensemble average over many trajectories calculated with the second,

63 5.2 Jump Unraveling for a Stabilized Single Qubit numerically less demanding method, and compare the results to the direct solution of the master equation. Thereafter we solve the SSE via the ‘direct’ method for two exemplary stationary states. This allows us to relate the trajectories and the corresponding stabilizing Hamiltonians.

Prooftesting the unraveling Before focusing on single trajectories in detail, we should check that an ensemble average over the generated trajectories indeed recovers the master equation evolution. Since we are ultimately interested in unraveling stationary states, it is instructive to investigate the evolution of an initial state into such a stationary state of interest. The stationary state we pick is deﬁned by the Bloch components

ρss,3 = ρ(r = 3/4, φ = π/3, θ = −2, 04) ,(5.15) where the numerical value for θ stems from the fact that the stationarity condition (4.6) in Bloch notation

1 0 = tr{ρD(σ−; ρ)} = − r(4 cos(θ) + r(cos(2θ) + 3)) (5.16) 8 implicitly ﬁxes θ as a function of r. The above state serves as a good example, since it is neither pure nor completely mixed, and therefore represents the general, most demanding situation. The corresponding stabilizing Hamiltonian is

Hss,3 = h¯ γ(0.4280σx + 0.2471σy) .(5.17)

It is obtained by inverting the stationary Lindblad master equation, as described in Eq. (4.7), and without loss of generality we choose Hα,β = 0 ∀α, β. As an initial state for the calculation of the trajectories we use the ground state |Ψ(t = 0)i = |0i. With that, we expect an initial evolution of the averaged state into the stationary state, the well-known damped Bloch oscillations [46]. In Figure (5.2) we show the density matrix element ! 100 |Ψ (t)i hΨ (t)| ρ (t) = k k ,(5.18) 1,1 ∑ 100 k 1,1 averaged over 100 trajectories |Ψk(t)i with the corresponding standard deviation in blue. The standard deviation for 100 trajectories is deﬁned as v u100 2 u [(|Ψk(t)i hΨk(t)|)1,1 − ρ1,1(t)] σ(ρ1,1(t)]) = t∑ .(5.19) i 9900

64 Chapter 5 Continuous Monitoring to Enhance Control

0 Figure 5.2: Matrix element ρ1,1(t), of a single qubit, as obtained from the master equation (3.33e) with L = σ− (black dashed line), and mean ρ1,1(t) (blue dots) obtained from an average over 100 trajectories (Eq. (5.18)) at discrete time intervals with its standard deviation (Eq. (5.19)). The matrix element ρss,(1,1) of the stationary state is given by the red dotted line. As an initial state we use the ground state ρ(t = 0) = |0i h0|, and we see that the mean over 100 trajectories very precisely (albeit expected statistical ﬂuctuations) recovers the average evolution described with a master equation and the evolution asymptotically approaches the stationary state ρss.

We compare this to the evolution of the matrix element given by the Lindblad master equation with a dashed line and to the value of the stationary state in red. It can be seen that the mean evolution of these 100 trajectories already satisfactorily coincides with the evolution of the master equation within the estimated errors. Naturally, there are a few larger deviations in accordance with theory, which predicts that the true value is included in the standard deviation with ≈ 68% [48]. Thus, the numerical unraveling works. Also, the damped Bloch oscillations can be observed and one can say that at around γt = 6 the initial state ρ(t = 0) = |0i h0| evolved into the stationarity point ρss. Let us remark that, for stationary states, one has the additional, interesting feature that a time average over single trajectories should recover the ensemble average. By comparing the time to the ensemble average, one may thus establish a kind of an ergodic hypothesis for the underlying trajectories. We obtained results that strongly suggest this equivalence between time and ensemble average for trajectories of stationary states, however, since there is no particular motivation to investigate this ergodicity of the trajectories, we refrain from showing these results here. Instead, we investigate, in the following, single trajectories of stationary states.

65 5.2 Jump Unraveling for a Stabilized Single Qubit

Exemplary trajectories

To gain an intuition about the trajectories of coherently stabilized states, we investigate in the following two different exemplary stationary states. As a ﬁrst example, we consider the stationary state deﬁned by the Bloch components

ρss,1 = ρ(r = 1/5, φ = π/3, θ = −1, 67) .(5.20)

As in the case above, the numerical value for θ stems from the stationarity condition (4.6), which has to be fulﬁlled. This state is stabilized by the Hamiltonian

Hss,1 = H0,1 = h¯ γ(2.1322σx + 1.2310σy) ,(5.21)

, as given by Eq. (4.7), where we again chose Hα,β = 0 ∀α, β (remember the freedom in the stabilizing Hamiltonian, introduced by Eq. (4.7)). Since r is close to zero, we know that this state is strongly mixed and thus lies close to the center of the Bloch ball. As an initial state for solving the SSE, without loss of generality, the ground state |Ψ(t = 0)i = |Ψ0i = |0i is chosen. In the ﬁrst frame in Figure (5.3) we depict one realization of the pure states’ evolution on the surface of the Bloch ball, captured by the Bloch angles θ(t) in blue and φ(t) in red, while the second frame shows the absolute value of the angular velocity

∆θ(t) ∆φ(t) |ω(t)| = | | + | sin(θ(t)) | ,(5.22) ∆t ∆t multiplied by the time step size ∆t. This provides us with an intuition about the velocity with which the pure state moves on the surface of the Bloch sphere. The time interval 0 < γt < 10 is chosen to contain several orbits. This is in order to gain an intuition about the jump statistics and at the same time still to be able to resolve single orbits. In the left, where the evolution of the Bloch angles is depicted, we can observe that the polar angle θ(t) takes the value −π at t = 0, corresponding to the ground state |0i, from where it increases monotonically towards the value 0, i.e. the excited state |1i, before falling down to −π again, closing one orbit. In between one such orbit, stochastic jumps to the ground state’ value θ = 0 happen occasionally. The angle φ(t) = φconst ± π jumps between two values that are separated by π, corresponding to the half of the Bloch sphere the state lies in. From this observation we can conclude that the state moves on a great circle around the Bloch sphere, where the plane the circle lies in is deﬁned by φconst ± π, or equivalently by the three points in the plane: the two poles and ρss. The absolute value of the angular velocity |ω(t)| changes periodically. It has its minima and maxima at times when θ = −π/2, i.e. when the state is

66 Chapter 5 Continuous Monitoring to Enhance Control

Figure 5.3: Single realization of a quantum trajectory of a single qubit under spontaneous decay with rate γ. Its mean stationary state ρss,1 = ρ(r = 1/5, φ = π/3, θ = −1, 67) is stabilized with the Hamil- tonian Hss,1, see Eq. (5.21). Depicted is the evolution of the Bloch angles θ(t) and φ(t) (left panel) and the angular velocity |ω(t)| (right panel) for an evolution governed by the stochastic evolution equation (5.11). The initial state |Ψ(0)i of the states’ evolution is the ground state |0i. at the equator. After coming from the ground state (θ = 0), first a minimum is reached and only after crossing through the excited state (θ = π), the maximum is potentially reached. This tells us that the angular velocity through the course of an orbit can be described as: First decreasing towards the equator, then increasing towards the north-pole, followed by further increasing towards the other side of the equator and finally decreasing again. Thus, the second half of the orbit, where the state propagates from the north pole to the south pole, is traversed faster than the first half. In Figure (5.4) we schematically depict the corresponding evolution of the state on the Bloch sphere. The reason for the non-constant angular velocity is that the deterministic evolution induced by the dissipator, as given via (1/2)(hσ+σ−i(t) − σ+σ−) |Ψ(t)i depends non-linearly on the state |Ψ(t)i. Its value is largest at the equators where it counteracts the Hamiltonian dynamics in the first half of the Bloch sphere, and it boosts the latter in the second half. The second exemplary state under investigation is purer and defined by {r = 4/5, φ = 3π/4, θ = 2π/3}. It is numerically given by

ρss,2 = ρ(r = 4/5, φ = 3π/4, θ = 2π/3) (5.23) and stabilized by the Hamiltonian

Hss,2 = H0,2 = h¯ γ(−0.3062σx + 0.3062σy) (5.24)

In the same way as for the ﬁrst example, one realization of the unraveling is shown in Figure (5.5). By comparing the trajectories of the two exemplary stationary states we can

67 5.2 Jump Unraveling for a Stabilized Single Qubit

Figure 5.4: A schematic illustration of the quantum trajectory for a stationary state ρss of a single qubit under spontaneous decay. The pure state propagates on a great circle on the surface of the Bloch ball. The plane of this circle is spanned by the two poles and the stationary state ρss. The angular velocity varies as indicated by the red arrows, and occasionally a jump to the ground state occurs, shown by the dashed red arrows. The axis of the stabilizing Hamiltonian is perpendicular to the plane of the great circle, and it’s direction is given by the arrow. infer several properties about the stabilizing Hamiltonian: First, the purer the stationary state, the weaker the corresponding stabilizing Hamiltonian. We see that from the ﬁrst (less pure) state, which moves much faster and is even able to perform several full orbits on the Bloch sphere without jumping in between, as compared to the second (more pure) state, which never performs a full orbit because it moves much slower, due to a weaker Hamiltonian. Comparing the values for the angular velocity manifests the observation. In more general scenarios it will still hold that stronger Hamiltonians induce faster trajectories. This leads (a) to a reduction of the probability of a jump to occur during one orbit and (b) to the ensemble-averaged state lying closer to the axis of the stabilizing Hamiltonian.1 Second, in case of spontaneous decay, the Hamiltonian can always be chosen as a linear combination of σˆx and σˆy, such that the dynamics between jumps always lies on a great circle on the Bloch sphere. This fact is investigated in more detail in the following subsection. And third, the direction of the induced rotation around the Bloch sphere, i.e. the sign of the unitary induced by the Hamiltonian, is such that the evolution is slowed down at that half of the Bloch sphere, where the stationary state ρss lies in. It equals the half that

1The latter gets clearer, when keeping in mind that a stationary state of pure coherent dynamics ρ(˙t) = −(i/h¯ )[H, ρ(t)] must commute with the Hamiltonian, i.e. lie on the axis of the Hamiltonian. Corresponding quantum trajectories will simply rotate around this stationary state, depending on the initial state. Additional dissipative dynamics will now transform the trajectories such that the ensemble average does not lie on the axis of the Hamiltonian anymore. The distance of the stationary state from the axis of the Hamiltonian is consequently given by the ratio of the strength of the dissipator and the Hamiltonian.

68 Chapter 5 Continuous Monitoring to Enhance Control

Figure 5.5: Single realization of a quantum trajectory for the state of a single qubit under spontaneous decay. Its mean stationary state ρss,2 = ρ(r = 4/5, φ = 3π/4, θ = 2π/3) is stabilized with the Hamil- tonian Hss,2, see Eq. (5.24). Depicted is the evolution of the Bloch angles θ(t) and φ(t) (left panel) and the angular velocity |ω(t)| (right panel) for an evolution governed by the stochastic evolution equation (5.11). The initial state |Ψ(0)i of the states’ evolution is the ground state |0i. is entered ﬁrst after a jump to the ground state happens.

Reduction of the parameter space of stabilizing Hamiltonians

We shall now show that in case of single qubits under spontaneous decay, the stabilizing Hamiltonian can always be chosen with vanishing σz component. The homogeneous Lindblad equation

0 = −i[H, ρ] + D(σ−; ρ) (5.25) gives four coupled equations, where two of them are not linearly independent, thus leaving us with three equations. Combining these by eliminating ρ1,2 † and ρ2,1 = ρ1,2, yields

ρ + 4(H − H )2ρ + 4H H (ρ − ρ ) 2,2 1,1 2,2 2,2 1,2 2,1 2,2 1,1 = 0 . (5.26) 2H1,2(−i + 2(H1,1 − H2,2))

By choosing H1,2 H2,1 appropriately one can always achieve H1,1 − H2,2 = 0, hence one is free to choose the stabilizing Hamiltonian such that it does not contain a σz component. We emphasize, however, that this is not required. In the next section, we will explicitly exploit this freedom to generate stationary post-selected quantum trajectories. Interestingly, it is the invariable part of the Hamiltonian H0, as introduced in Eq. (4.7), which lies in the x-y plane. To show this, we directly investigate

69 5.3 Stationary Point of the Conditional Dynamics

the diagonal elements of H0 in the {|0i , |1i} basis,   i hα| D(σ−; ρss) |βi h0/1| H0 |0/1i = h0/1|  ∑ |αi hβ| |0/1i .(5.27) pα − pβ pα6=pβ

For an arbitrary potentially stabilizable state ρ = pα |αi hα| + pβ |βi hβ|, where |αi = α0 |0i + α1 |1i and |βi = β0 |0i + β1 |1i, we ﬁnd

2 2 ∗ ∗ ∗ ∗ (pα|α0| + pβ|β0| − 2)(α0α1 β0 β1 − α0α1β0β1) h0| H0 |0i = ,(5.28) 2(pα − pβ) 2 2 ∗ ∗ ∗ ∗ (pα|α1| + pβ|β1| )(α0α1 β0 β1 − α0α1β0β1) h1| H0 |1i = − .(5.29) 2(pα − pβ)

For both diagonal elements we see that the first factors are never vanishing, due to the normalization of the state, but the second ones always are, since the spectral decomposition of ρ implies the orthogonality condition hα|βi = hβ|αi = 0. Hence, the fixed part H0 of the stabilizing Hamiltonian has zero diagonal elements in the {|0i , |1i} basis, and thus no σz component. These investigations manifest our initial assumption that the fixed part of the stabilizing Hamiltonian stands perpendicular to the plane spanned by the two poles and the stationary state ρss as shown in Fig. (5.4).

5.3 Stationary Point of the Conditional Dynamics

Even though the effective dynamics of ρss = ∑α pα |αi hα| is not changed by a diagonal component Hα,β (in the {|αi , |βi} basis), of the stabilizing Hamiltonian2, it is clear that such a Hamiltonian will have an impact on the underlying quantum trajectories. We ﬁnd that the evolving state now asymptotically approaches a stationarity point for periods when it does not decay to the ground state. In the following we investigate the emergence of such a stationary point of the conditional dynamics in a more general framework and characterize its stability properties, i.e. whether it is a stable or unstable stationary point. To show an example for such a modiﬁed trajectory, we pick up the second exemplary stabilizable state ρss,2 from the previous section, whose trajectories we already investigated when stabilized with H0,2. Now we stabilize the state with the Hamiltonian

H˜ ss,2 = Hss,2 + |αi hα|

= H0,2 + h¯ (0.51 − 0.306(σx + σy) + 0.25σz) ,(5.30)

2 Because ρss and Hα,β commute.

70 Chapter 5 Continuous Monitoring to Enhance Control

Figure 5.6: Conditional dynamics of a single realization of a quantum trajectory for the stationary state of a single qubit under spontaneous decay. The stationary state ρss,2 = ρ(r = 1/5, φ = 3π/4, θ = 2π/3) is stabilized with the Hamiltonian H˜ ss,2 = Hss,2 + h¯ (0.51 − 0.306(σx + σy) + 0.25σz), that has an additional σz component, as compared to the second example in the previous section (5.2). Depicted is the evolution of the Bloch angles θ(t) and φ(t) (left panel) and the angular velocity |ω(t)| (right panel) for an evolution governed by the stochastic evolution equation (5.11) with the ground state as the initial state |Ψ(0)i = |0i. In Bloch representation, this describes a spiral in the Bloch sphere towards a stationary point.

where |αi is the eigenvector of ρss,2 corresponding to the eigenvalue 9/10. Since the additional Hamiltonian only changes the deterministic part of the evolution, and not the jump statistics, we focus in the following on the former. This is done by conditioning the evolution on the absence of jumps, such that the evolution is completely governed by the deterministic part of the stochastic evolution equation (5.11) when dN(t) ≡ 0∀t. The resulting conditional dynamics is shown in Figure (5.6). We can see that the introduction of the additional component to the Hamiltonian leads to a more involved evolution of the state: Instead of staying constant over time, the azimuthal angle φ(t) roughly oscillates with decreasing amplitude for consecutive orbits; the amplitude of the polar angle θ(t) also loses its amplitude non-linearly, and the velocity performs a non-trivial evolution while decreasing overall. The corresponding conditional trajectory for such an evolution spirals on the surface of the Bloch ball from the ground state to a stationary point. However, as we see in the following, the additional Hamiltonian Hα,β is not responsible for the emergence of a stationary point, but can be reason for a trajectory to approach this point asymptotically. These stationary, or ﬁxed, points of the conditional dynamics emerge, where the deterministic part of the stochastic master equation is stationary. This equation was already indirectly introduced in Eq. (3.33d) and its deterministic part is obtained by conditioning the dynamics on periods of no detection of quantum jumps, i.e. for dNi(t) ≡ 0 ∀t. In terms of a single qubit under

71 5.3 Stationary Point of the Conditional Dynamics spontaneous decay, it reads

dρ(t) i γ = − [H, ρ(t)] + {hσ+σ−i(t) − σ+σ−, ρ(t)} ≡ 0 . (5.31) dt h¯ 2 Now, two kinds of questions can be asked: (a) ‘Given a Hamiltonian H, what are the corresponding fixed points ρ(t) = ρ = const ∀t?’, and (b) ‘Given a fixed point ρ(t) = ρ = const ∀t, what are the corresponding stabilizing Hamiltonians H?’. To answer question (a), it is instructive to restrict3 the Hamiltonian to only induce rotations in the x-z plane of the Bloch sphere, i.e. H = hh¯ yσy. Then, the resulting fixed points, parameterized by the Bloch vector ~n, as given in Eq. (2.9), are for |hy| ≤ γ/4 given by

4hy n = x γ

ny = 0 (5.32) s 2 16hy n = ± 1 − , z γ2 and for |hy| > γ/4 by γ nx = 4hy s γ2 ≤ | | ≤ − 0 ny 1 2 (5.33) 16hy

nz = 0 .

Or in other words, for weak Hamiltonians, i.e. |hy| ≤ γ/4, the fixed points are pure states in the x-z plane, and for stronger Hamiltonians, any state in the x-y plane fulfilling n = γ is stationary due to a compensation of the x 4hy deterministic part of the dissipator and the Hamiltonian. For unrestricted arbitrary Hamiltonians H, the solution of Equation (5.31) yields analogously the corresponding fixed points of the conditional dynamics. Interestingly, for all Hamiltonians with |hy| < γ/4 two fixed point emerge: r 2 = ± − 16hy nz 1 γ2 . It turns out, that the corresponding state in the lower half of the Bloch sphere, i.e. for nz < 0, is a stable fixed point, into which all initial states in the x-z plane asymptotically propagate, while the fixed point in the upper half of the Bloch sphere is unstable. This stability can be

3This restriction enables a simple geometric interpretation of the resulting ﬁxed point and can straight-forwardly be loosened for more general cases.

72 Chapter 5 Continuous Monitoring to Enhance Control

Figure 5.7: Temporal change of the polar angle dθ/dt describing the evolution of a pure state ρ = sin(θ)σx + cos(θ)σz in the x-z plane of the Bloch sphere, governed by the deterministic part of the stochastic master equation (5.31). The dependence on the angle θ is shown for different Hamiltonians H = hyσy, analytically it can be described ( ) with the function dθ/dt = −2hy + γ sin θ /2. One can see that for hy = γ/4 there exists exactly one fixed point of the dynamics, characterized by dθ/dt = 0. For smaller values of Hy two fixed point emerge, where the first one, i.e. θ < π/2, is unstable, because a slight deviation from the fixed point towards smaller θ leads to a further reduction of θ over time, since dθ/dt < 0 then. A slight increase of θ leads to increasing θ over time, since dθ/dt > 0. Exactly the opposite holds true for the second fixed point at larger θ, such that any state ρ = sin(θ)σx + cos(θ)σz is driven into it over time, i.e. this fixed point is stable. understood by investigating the change of the state with time as a function of the position of the state in state space. For a more tangible understanding we keep the restriction of the Hamiltonian H = hyσy. Then, a pure state is characterizable solely by the polar angle θ as ρ = sin(θ)σx + cos(θ)σz, since the whole dynamics induced by any H = hyσy is restricted to the x-z plane. The temporal change of the polar angle dθ/dt, obtained from the deterministic part of the stochastic master Equation (5.31) is given by

( ) dθ/dt = −2hy + γ sin θ /2 .(5.34)

In Figure (5.7) we plot dθ/dt as a function of the angle θ for three different Hamiltonian strengths H = hyσy = γ/10σy, γ/6σy, γ/4σy. As predicted by the Conditions (5.32), exactly one stationarity point occurs when hy = γ/4 at θ = π/2, i.e. at nz = 0. In the other two cases, each Hamiltonian creates two ﬁxed points. The one in the upper half of the Bloch sphere with θ < π/2, i.e. nz > 1, is unstable, since variations away from this ﬁxed point lead to an increased distance from it over time. For example, when the angle θ is

73 5.3 Stationary Point of the Conditional Dynamics

smaller than the value where dθ/dt = 0, then dθ/dt < 0, which means that the angle is further decreased over time. Analogously, a higher value of θ leads to further increase of θ over time. For the fixed point in the lower half of the Bloch sphere, the situation is exactly the opposite, such that it is a stable fixed point. This stability analysis makes clear that only the fixed points in the lower half of the Bloch sphere are asymptotically approached during the deterministic evolution. Pure states in the upper half of the Bloch sphere can only be stabilized when the initial state coincides with the corresponding fixed point. To draw a connection back to the exemplary trajectory from the beginning of this section. There, the fixed part of the stabilizing Hamiltonian H0,2 = h¯ γ(−0.3062σx + 0.3062σy), which induces rotations around an axis in the x- y plane, is such that the fixed point of the conditional dynamics is to be described with Conditions (5.33). This means that the two pure state fixed points lie not in the plane of the state evolution, induced by H0,2. This is the reason why the trajectory, corresponding to the Hamiltonian H0,2, as illustrated in Fig. (5.5), approaches no fixed point during its deterministic evolution, or in other words dθ/dt is never zero. However, as soon as one introduces an additional Hamiltonian Hα,α with a σz component, see Eq. (5.30), the deterministic evolution is not restricted to a single plane in the Bloch sphere anymore, and consequently the stable fixed point, which is the pure state in the lower half of the Bloch sphere, is approached asymptotically. Question (b) can be answered in two different regimes. First, pure states with nz 6= 0 can be rendered fixed points with the Hamiltonian components

h0 ∈ R 4hznx ± γnzny hx = 4nz γnxnz ∓ 4hzny hy = (5.35) 4nz hz ∈ R , ~ where H = h01 + h~σ. This means that every pure state can be rendered a fixed point of the conditional dynamics of a continuously monitored quantum system by a set of Hamiltonians, as defined via Equations (5.35). Second, states in the x-y plane, i.e. nz = 0, are fixed points for

h0 ∈ R γ = ±(h n − h n ) (5.36) 4 x y y x hz = 0 .

Any other mixed state cannot be a stationarity point of the coherent dynamics.

74 Chapter 5 Continuous Monitoring to Enhance Control

This is, because in the here considered spontaneous decay case the coherent part of the dissipator drives any mixed states into the ground state |0i (nz = 1) over time. As a consequence the deterministic part of the dissipator changes the purity of all mixed states other than for nz = 0, which cannot be counteracted with purity preserving coherent dynamics, induced by a Hamiltonian H. In summary, we showed in this section that with the help of continuous monitoring and post-selection, any pure state of a qubit under spontaneous can be rendered stationary with properly manipulated system Hamiltonian. Thus, combining post-selection with coherent control can already improve the stabilization possibilities, as compared to coherent control alone, where the ground state is the only stabilize pure state. We furthermore found that all stationary points in the lower half of the Bloch sphere, i.e. states with a higher ground state than excited state population, are stable in the sense that a state in the vicinity of the ﬁxed points is driven back into stationarity, while ﬁxed points in the upper half of the Bloch sphere are unstable.

5.4 Jump Unraveling for a Stabilized Single Qubit with Unitary Feedback

Let us now introduce another extension of the coherent control, which does not rely on post-selection, as in the previous case. This approach exploits the information from the continuous monitoring to stabilize the system with the help of feedback operations, similar as introduced in [18]. There, an atom in a cavity is continuously monitored to detect emitted photons, which then indicate a jump to the ground state of the state of the atom. The feedback is now implemented in the sense that, whenever the detector registers a photon, a unitary operation is applied onto the atom. With that method, the authors could generate and stabilize entangled states. We use a similar scheme, as shown in Figure (5.8), to investigate in the following the impact of unitary feedback on the set of stabilizable states, and in this sense generalize [18] to the stabilization of arbitrary quantum properties, not only entanglement. We ﬁnd that, with the help of unitary feedback, any desired qubit state can be rendered stationary and the corresponding trajectories of feedback stabilized states are investigated subsequently.

Effective feedback master equation Even though the feedback method is based on the observation of single quantum trajectories, it gives also rise to a modiﬁed Lindblad master equation. This allows us to use the methods described in Sec. (4.3), to ﬁnd the optimal stationary states with respect to any desired quantum property. In other

75 5.4 Jump Unraveling for a Stabilized Single Qubit with Unitary Feedback

Figure 5.8: Extension of the stabilization setup for continuously monitoring a single atom coupled to the vacuum of a light ﬁeld (Fig. (5.1)) with a feedback unitary operation U. When the atom emits a photon γ, which induces a jump to the ground state with σ−, the surrounding measurement apparatus registers this and instantaneously initiates a feedback operation with the unitary U onto the state of the atom, to re-stabilize it.

words, we can identify modified sets of stabilizable states, as implied by fixed unitary feedback. Per se, the introduction of a unitary feedback, as shown in Fig. (5.8), only modifies the jump operators in the statistical part of the SSE (5.11) by σ− → Uσ−, since right after a photon is emitted, the feedback unitary U is applied to the atom. However, by employing the defining property of a unitary U†U = 1, one can then also transform the other jump operators in the † SSE, since σ+σ− = σ+U Uσ−. Hence, all jump operators can be transformed and with that it becomes clear that also the master equation contains the transformed jump operators

ρ˙(t) = −i [H, ρ(t)] + D(Uσ−; ρ(t)) ,(5.37) where † 1 † D(L˜ = Uσ−; ρ(t)) = γ L˜ ρ(t)L˜ − {ρ(t), L˜ L˜ } (5.38) 2 † 1 = γ Uσ−ρ(t)σ+U − {ρ(t), σ+σ−} .(5.39) 2

Given the above Lindblad master equation, we can ﬁnd the corresponding set of stabilizable states S(U) as a function of the feedback unitary U by solving the stationarity condition

S(U) = {ρ : [0 = tr{ρD(Uσ−; ρ)}]} .(5.40)

In order to perform an optimization over this set, the unitary has to be

76 Chapter 5 Continuous Monitoring to Enhance Control parameterized. The most general unitary can be expanded with respect to the Pauli matrices σi as follows,

iαu/2~u·~σ U = e = 1 cos(αu/2) + i~u ·~σ sin(αu/2) .(5.41)

The effect of the unitary on a state ρ can thus be associated with a rotation in the Bloch sphere by the angle αu about the axis ~u. Without loss of generality T we may assume that the rotation axis ~u = (u1, u2, u3) is normalized to one and is hence parameterized with polar coordinates as follows

u1 = cos φu sin θu

u2 = sin φu sin θu (5.42)

u3 = cos θu .

As usual, the density matrix is expanded with respect to the Pauli matrices σi, and spherical coordinates {rn, φn, θn} for the Bloch vector ~n are employed in the following. Plugging the parameterization of U (Eqs. (5.41) and (5.42)) into the stationarity condition (5.40) yields n S(U) = ρ : tr{ρD(Uσ−; ρ)} 1 = r γ − 5r − 2 cos[θ ](3 + cos[α ] + r cos[α ] cos[θ ]) 16 n n n u n u n h α i2 − r cos[2θ ] − 4 cos[θ ](1 + r cos[θ ]) cos[2θ ] sin u n n n n n u 2 h α i2 − 4(1 + r cos[θ ]) cos[φ − φ ] sin u sin[θ ] sin[2θ ] n n n u 2 n u o + 4(1 + rn cos[θn]) sin[αu] sin[θn] sin[θu] sin[φn − φu] = 0 (5.43) which is admittedly not very intelligible. However, one can make a simpli- fication by setting θu = π/2, which restricts the rotation axis of the unitary operator to the x-y plane in Bloch representation. This only eliminates some ambiguities in choosing U, since, as we see later, a fixed stabilizable state does not uniquely define a corresponding unitary. This reduces Equation (5.43) to

n 1 h αu i2 S(U) = ρ : tr{ρD(Uσ−; ρ)} = r γ − 2 cos cos[θ ](1 + r cos[θ ]) 4 n 2 n n n o + sin[θn](−rn sin[θn] + (1 + rn cos[θn]) sin[αu] sin[φn − φu]) = 0 . (5.44)

To understand the above condition physically, we focus on stabilizable pure

77 5.4 Jump Unraveling for a Stabilized Single Qubit with Unitary Feedback

states, i.e. rn = 1, which gives

2 n 1 θn S(U) = ρ : tr{ρD(Uσ−; ρ)} = − γ cos (1 + cos[α ] cos[θ ] 2 2 u n o − sin[αu] sin[θn] sin[φn − φu]) = 0 .(5.45)

This condition is trivially fulﬁlled for θn = 2nπ, which corresponds to the ground state ρ = |0i h0|, thus the ground state can still be a stationary pure state, namely when the stabilizing Hamiltonian vanishes, regardless of the applied unitary feedback. However, there is an additional solution given by

φu = φn + π/2 (5.46)

αu = θn + π .(5.47)

This solution describes the case when the rotation axis is perpendicular to the plane spanned by the states’ Bloch vector and the z-axis, and the rotation angle αu is such that the ground state |0i is rotated into the state deﬁned by the Bloch vector (remember that θn = 0 describes the excited state |1i). In other words, an arbitrary pure state can be stabilized by choosing an appropriate feedback unitary. Remember that in the case of pure spontaneous decay, without feedback, the ground state was the only pure stabilizable state. The above provides us with a recipe to determine the feedback unitary that stabilizes an arbitrary pure state against the detrimental effect of the environment. If we were for example to stabilize the maximally coherent state √ (1/ 2)(|0i + |1i), which is given in terms of the Bloch angles by θn = π/2 and φn = 0, we could use the unitary operator

3π 3π U = 1 cos + iσ sin ,(5.48) 4 2 4 which corresponds to a rotation about the y-axis by the angle 3π/2. Note that, one can also choose another unitary that rotates the ground state into the maximally coherent state, but we removed this ambiguity in the foregoing derivation, as mentioned in the step from Eq. (5.43) to Eq. (5.44). The set of stabilizable states corresponding to this particular choice (Eq. (5.48)) of unitary feedback operator is plotted in Figure (5.9f). To deepen our intuition, we also show the sets of stabilizable states for other rotation angles αu in Figures (5.9b)-(5.9e). Hence, it is clear that with the help of unitary feedback, any qubit state can be stabilized4 against environmentally induced spontaneous decay. The necessary unitary for stabilizing a desired state ρss can be identiﬁed by solving the

4Only the excited-state |1i cannot be stabilized, for reasons which will become clear in the following subsection, where we investigate the trajectories of feedback stabilized states.

78 Chapter 5 Continuous Monitoring to Enhance Control

(a) (b) (c)

(d) (e) (f)

Figure 5.9: Sets of stabilizable states (purple) in the x-z-plane of the Bloch sphere for a single qubit under spontaneous decay. The qubit is continuously monitored and stabilized with coherent control ﬁelds and unitary feedback. The unitary feedback corresponds to rotations about the y-axis (θu = π/2 and φu = π/2) by the angles (a) αu = 0, (b) αu = 3π/2, (c) αu = π/4, (d) αu = π/2, (e) αu = 3π/4 and (f) αu = 99π/100. stationarity condition (5.40), where, in particular for pure states, the unitary is easily identiﬁed by using the rules given in Eq. (5.46). The corresponding stabilizing Hamiltonian is given by Equation (4.7), where the Lindblad operator with unitary must be used in the dissipator D(Uσ−; ρss).

Trajectories for feedback-stabilized quantum states

Let us start with investigating the trajectories of pure states stabilized with feedback. We pick up the example from above, stabilizing a maximally coherent state given by the Bloch angles θn = π/2 and φn = 0. This is achieved by a unitary corresponding to a rotation about the y-axis by the angle 3π/2.

79 5.5 Towards an Experimental Realization

The corresponding stabilizing Hamiltonian to ensure

ρ˙(t) = −i [H, ρ(t)] + D(Uσ−; ρ(t)) = 0 is given by H = (1/4)σ2, as obtained from solving Eq. (4.7). This Hamilto- nian corresponds to a rotation about the y-axis in such a way that it exactly counteracts the deterministic part of the dissipator at the desired stationary point. This is equivalent to the emergence of a fixed point of the conditional dynamics, as introduced in Section (5.3), and it can be verified by plugging everything into the deterministic part of the SME (Eq. (5.31)). Hence, it is clear that the stabilizing Hamiltonian can alternatively to Eq. (4.7) also be directly identified with the Conditions (5.35) for the emergence of a fixed point of the conditional dynamics. Since we know that the unitary feedback operator is chosen such that it rotates the ground state into the to-be stabilized state, it becomes apparent that the trajectory of this feedback stabilized maximally coherent state is √ constant over time, i.e. |Ψ(t)i = const = (1/ 2)(|0i + |1i). In retrospect, it should also be clear now that actually any stationary pure state must have a time-independent trajectory, because otherwise an ensemble average over many such trajectories would not yield the pure state at all times. While the trajectories of mixed states stabilized by unitary feedback will not be constant in time, since trajectories, per definition, only consist of pure states, the realized trajectories won’t differ much in their characteristics, compared to the previously introduced trajectories in Sec. (5.2), apart from that jumps bring the state not into the ground state but one specified by the feedback unitary. It is clear that, here again, the pure quantum trajectories must evolve such that an ensemble average reproduces the in general mixed stabilizable state. In the following chapter more realistic assumptions about the feedback will be investigated.

5.5 Towards an Experimental Realization

During the foregoing investigations, we made idealizations, such as perfectly realizable stabilizing Hamiltonians, unit monitoring efﬁciency, or instantaneous feedback, that do not fully apply in realistic experimental setups. Since an experimental implementation of the presented ideas is clearly a goal of our work, we examine in the following the consequences of experimental imperfections. To this end, we investigate the impact of perturbed stabilizing Hamiltonians on the desired stabilized states. Then we deal with the consequences of less than perfect measurement efﬁciency and in the last subsection the issue of time-delayed feedback operations is addressed.

80 Chapter 5 Continuous Monitoring to Enhance Control

Stability of the stationary state against variations of the stabilizing Hamiltonian

In real experiments, the desired stabilizing Hamiltonian will never be perfectly realized, but will be subject to noise. Thus, it is of interest for the experimenter to know whether the stationary state ρss is stable against variations of the corresponding stabilizing Hamiltonian Hss. To this end, we would like to know whether (a) there exists for a perturbed Hamiltonian a stationary state, other than the ground state, and (b), in case such a state exists, whether it lies in the vicinity of ρss. These two questions will be answered for the case of a single qubit under spontaneous decay by solving the homogeneous Lindblad master equation for the non-perturbed and perturbed Hamiltonian and comparing the obtained solutions. First, let us ﬁnd the solution of the Lindblad master equation for the unperturbed Hamiltonian, which is expanded in terms of Pauli matrices ~σ as follows, ~ Hss = h01 + h~σ .(5.49)

Without loss of generality we can restrict the stabilizing Hamiltonian to the x-y plane by setting h3 = 0, as it is explained in Sec. (5.2). Then, the solution of the homogeneous Lindblad master equation is given by

4h = 2 n1 2 2 1 + 8h1 + 8h2 4h = − 1 n2 2 2 (5.50) 1 + 8h1 + 8h2 1 = n3 2 2 , 1 + 8h1 + 8h2 where ~n is the Bloch vector of the corresponding state ρss. Note that h0 does not appear in the solution, since this component commutes with any density matrix. Now, we shall compare this solution to the one for a perturbed Hamiltonian 0 Hss, which can in general be written as 0 1 Hss = Hss + e0 +~e~σ,(5.51)

81 5.5 Towards an Experimental Realization

with the ei 1. The corresponding new solution for the stationary state reads

4(h + e + 4(h + e )e ) n0 = 2 2 1 1 3 1 2 2 2 1 + 8(h1 + e1) + 8(h2 + e2) + 16e3 −4(h + e ) + 16(h + e )e n0 = 1 1 2 2 3 (5.52) 2 2 2 2 1 + 8(h1 + e1) + 8(h2 + e2) + 16e3 1 + 16e2 n0 = 3 . 3 2 2 2 1 + 8(h1 + e1) + 8(h2 + e2) + 16e3 As a ﬁrst result, we can answer question (a) - concerning the existence of a stationary state also for the perturbed Hamiltonian - positively. To show that this state is close to the unperturbed solution, we expand it in lowest order in ~e. Note that e0 dropped out of the solution for the same reason as h0 does not appear in the unperturbed solution. The expansion yields

4h + 16h e 4(1 + 8h2 − 8h2)e − 64h h e n0 ≈ 2 1 3 + 1 2 2 1 2 1 1 2 2 2 2 2 1 + 8h1 + 8h2 (1 + 8h1 + 8h2) −4h + 16h e 64h h e − 4(1 − 8h2 + 8h2)e n0 ≈ 1 2 3 + 1 2 2 1 2 1 (5.53) 2 2 2 2 2 2 1 + 8h1 + 8h2 (1 + 8h1 + 8h2) 1 16(h e − h e ) n0 ≈ + 2 2 1 1 . 3 2 2 2 2 2 1 + 8h1 + 8h2 (1 + 8h1 + 8h2) Thus, by comparing this to the original solution given via Eqs. (5.50), we can see that the stationary state ~n0 corresponding to the perturbed Hamiltonian remains close to the unperturbed state ~n for small perturbations ~e, i.e. we can answer question (b) positively too. Regarding experimental implementations, this means that the coherently stabilized states are robust with respect to small perturbations of the Hamiltonian.

Inefficient detection of emitted photons A main assumption underlying our feedback approach so far was that any emitted photon is detected by the adjacent measurement apparatus. It is clear that this can only be achieved approximately in experimental settings, since no photon detector has perfect efficiency, there will always be leaking photons that cannot be detected. To estimate the impact of inefficient detection, we treat the feedback approach effectively, since the inefficient detection can be easily introduced into a Lindblad master equation. In our spontaneous decay case, this is done by introducing an efficiency parameter 0 ≤ η ≤ 1 and splitting the dissipative part into a part corresponding to the situation when the emitted photon is detected and we can apply the unitary feedback (multiplied by η), and another part for when it is not (multiplied by 1 − η). Then the master equation

82 Chapter 5 Continuous Monitoring to Enhance Control becomes † 1 † ρ˙(t) = −i[H, ρ(t)] + η Uσ−ρ(t)σ+U − {ρ(t), σ+U Uσ−} 2 1 + (1 − η) σ−ρ(t)σ+ − {ρ(t), σ+σ−} (5.54) 2 = −i[H, ρ(t)] + D({σ−, Uσ−}; ρ(t)) .

With the help of the above master equation we can now directly infer the corresponding set of stabilizable states as a function of η and U. For example for U = (3π/2)σ− we show the corresponding sets of stabilizable states in the x-z plane of the Bloch sphere for differing detection efﬁciencies η = 1, 3/4, 1/2, 1/4, 0 in Figure (5.10). As we can see, for η → 0 the set of stabilizable states approaches the well-known set for non-feedback stabilization in the lower half of the Bloch sphere, as shown in the last frame of Fig. (5.10).

Figure 5.10: Sets of stabilizable states (purple) in the x-z-plane of the Bloch sphere for a single qubit under spontaneous decay, stabilized with coherent control and using the feedback unitary U = (3π/2)σ−. From left to right, different detection efﬁciencies η = 1, 3/4, 1/2, 1/4, 0 of the continuous monitoring are considered.

To quantify the loss of stabilization potential as a function of the detection efﬁciency, we calculate the maximal stabilizable amount of coherence Coh(ρ) in the {|0i , |1i} basis, as given by Eq. (3.35), for values of η ∈ {0, 1}. The feedback unitary U = (3π/2)σ− is chosen such that for perfect efﬁciency η = 1 a maximally coherent state can be stabilized. The respective curve is shown in Figure (5.11). The decay of the maximally stabilizable coherence is not exponential, as the inset shows, but still the decrease is faster for higher

83 5.5 Towards an Experimental Realization efficiencies than for lower ones. This also implies that the detrimental effect of inefficient detection increases the higher the desired coherence is. In this concrete example, a 10% loss of efficiency creates a loss of around 7% of maximally stabilizable coherence.

Figure 5.11: Evolution of the maximally stabilizable coherence for a single qubit under spontaneous decay, stabilized with coherent control ﬁelds and unitary feedback, as a function of the detection in- efﬁciency 1 − η of the continuous monitoring.

Time-delayed feedback

Another generic issue of the feedback scheme is that realistic implementations of the unitary feedback operation cannot change the state of the system instantaneously, but only in a finite time interval. This interval includes the time delay between the measurement signal and the initiation of the unitary operator, but also the finite interaction time required for the implementation of the unitary. For example, if one would engineer a unitary via a correspondingly applied electro-magnetic field pulse, first the information of a measurement event has to be transformed into the initiation of corresponding field pulse, and then the atom needs to couple to the applied field. This takes time, since a finitely strong pulse induces only a finite rotation velocity for the state of the atom, just like an applied Hamiltonian can only gradually change the state. We assume that the above described finite time interval for initiation and information transfer can approximately be summarized in time-delayed instantaneous unitary. In other words, we assume that, in our case, after the system jumps to the ground state it first undergoes a period of coherent dynamics, and only after a finite time the action of the unitary takes place instantaneously. In general, this situation cannot be treated with a Markovian master equation, since the time delay requires time non-local processes. How- ever, it can be simulated very well in the framework of quantum trajectories,

84 Chapter 5 Continuous Monitoring to Enhance Control where the modiﬁed stochastic Schrödinger equation becomes ! h σ− 1 1 d |Ψ(t)i = dn(t) p − + dt (hσ+σ−i(t) − σ+σ−) − iH hσ+σ−i(t) 2 i 1 + δdn(t−∆t),1(U − ) |Ψ(t)i .(5.55)

We simulate the time delay ∆t via the additional third term on the right: When a jump occurred at the previous time t − ∆t, i.e. dn(t − ∆t) = 1, the Kronecker-Delta δ gives one and induces with that the application of the unitary U |Ψ(t)i. If no jump occurred at the previous time dn(t − ∆) = 0, then the Kronecker-Delta yields zero, such that nothing happens. Corresponding trajectories have the characteristic evolution, as illustrated in Figure (5.12): After a jump to the ground state occurs, the system evolves deterministically for a time interval ∆t and then the unitary feedback kick U takes place.

Figure 5.12: Schematic illustration, in the x-z plane of the Bloch sphere, of the trajectories of a single qubit under spontaneous decay with coherent control and time-delayed unitary feedback. After a spontaneous decay to the ground state, the state evolves deterministically during the delay time ∆t, where after the unitary U sets in.

If the time delay ∆t is very small, i.e. ∆t 1/γ, we can assume that the deterministic evolution in that short interval ∆t can be described by an unitary kick UDet. This unitary UDet has to be such that it rotates the ground state into the state that is deterministically reached after the interval ∆t.5 Under this assumption, we can treat the time-delayed feedback with a Lindblad master equation by transforming the feedback Lindblad operators for perfect feedback L = Uσ− → L˜ = UUDetσ−, employing the same trick as when introducing the feedback unitary U into the Lindblad master equation (remember Sec. (5.4).

5The former is only possible because, in our case, all spontaneous jumps bring us into the ground state and the following evolution during ∆t is hence always the same.

85 5.5 Towards an Experimental Realization

In this approximate case, the impact on the set of stabilizable states due to the time-delayed feedback can be completely compensated, because the new feedback unitary U˜ for the modified Lindblad operator L˜ = UU˜ Det can be chosen such that it exactly recovers the original Lindblad operator L for perfect ˜ † feedback, by choosing U = UUDet. This is a very important observation for an experimental realization. It means that, under the realistic assumption that the time delay is short, compared to the inverse of the spontaneous decay rate, i.e. ∆t 1/γ, the impact of the time-delay can be fully compensated by adapting the feedback unitary accordingly. Hence, we found that time-delayed feedback does not pose a strong restriction on the stabilization capabilities. Note that in principle additional jumps to the ground state can occur during the delay time ∆t, before the application of the feedback. Those would trigger additional time-delayed feedback unitaries, rendering the consequences of the time-delay more severe. However, firstly, in a realistic implementation additional unitaries can be easily prevented by programming the feedback such that it cannot be triggered during the delay time ∆t. And secondly, under the previously employed assumption that the time delay is small, such events happen in general very rarely, such that the tiny effect of an additional jump to the ground state, which changes the actual deterministic evolution during ∆t, can be easily ignored. To get a non-approximate picture of the impact of the time-delayed feedback, one would have to solve the modified SSE, given by Equation (5.55), and average over many of the resulting trajectories to get the evolution of ρ(t) for a fixed time delay ∆t, unitary feedback U˜ and stabilizing Hamiltonian H. However, for sufficiently small ∆t it is clear that the detrimental effect on the set of stabilizable states should be negligible.

86 6 Conclusions &Outlook

In this thesis, we investigated an extension of the coherent control approach for the stabilization of open quantum systems against detrimental environmental influence. The extension, in form of continuously monitoring the quantum system and applying unitary feedback, is treated with a jump unraveling approach, yielding quantum trajectories. We investigated exemplary trajectories for a coherently controlled single qubit under spontaneous decay with and without additional feedback stabilization. Furthermore, an effective treatment with Lindblad master equations was considered. As a consequence, we obtained sets of stabilizable states by employing the method introduced in [17], and could thus make general statements about the stabilization capabilities of our approach. Finally, also realistic assumptions, to pave the way for an experimental implementation, were considered. We found that, already with continuous monitoring and appropriate coherent control alone, pure states can be fixed points of the state evolution of single qubits under spontaneous decay. As we showed, the quantum state can be ensured to asymptotically approach the desired fixed point conditioned on when no spontaneous jump to the ground state occur. In other words, by performing post-selection on the monitoring outcome, pure single qubit states, such as maximally coherent states, can be rendered stationary against the environmental influence. In comparison, with coherent control alone, the only stationary pure state is the ground state. Hence, continuous monitoring alone already substantially increases the control capabilities. This can be understood intuitively, because continuous monitoring provides continuous information about the system state, such that the states’ purity increases constantly, or remains pure for already pure states. Consequently, the environmental influence is locally unitary for any pure state, which can be canceled exactly by properly manipulating the system Hamiltonian. By applying appropriate unitary feedback operations onto the quantum system, we were able to show that any arbitrary qubit state is stabilizable now. A simple recipe for finding the corresponding unitary feedback operator for a desired stationary state is provided alongside. In case of stabilized pure states, it turns out that the unitary has to be chosen such that it rotates the ground state into the desired pure state. The coherent control Hamiltonian can then be chosen such that it exactly counteracts the deterministic influence of the environment at this point, just as in the previous case of emerging sta-

87 tionary points for the conditional dynamics. As a result, the overall quantum trajectories corresponding to feedback-stabilized pure states are constant over all times. Stabilized mixed states have necessarily non-constant trajectories that on average recover the stationarity point. Finally, we also subjected the above very promising results to realistic experimental conditions. Perturbed stabilizing Hamiltonians, inefficient monitoring and time-delayed feedback, as the most apparent sources for experimental imperfections, were considered. Interestingly, one finds that the effect of time-delayed feedback can, under rather mild assumptions, can be fully compensated by suitably adjusting the feedback unitary, hence rendering it unproblematic for realistic implementations. The inefficient monitoring is a bit more subtle, because it cannot be counteracted. But still, in our example, a loss of 10% efficiency only leads to a 7% loss of maximally stabilizable coherence. And last, a perturbed Hamiltonian turns out to lead to a correspondingly perturbed stabilized state close to the original state, such that experimental noise does remain under control. Thus, in conclusion, we can say that the here presented control approach leads to substantial improvements of the control capabilities, as compared to coherent control alone, and these improvements persist even under realistic conditions.

Outlook There are still many open questions, which appear worthwhile to be addressed in the future. For instance, it is interesting to consolidate the ergodic hypothesis for trajectories of stationary states, with the potential consequence that a time average over one single trajectory gives the same result as an ensemble average over many individual trajectories. This can lead to numerically more efficient means of calculating the stationary state of the average evolution for given dissipator and Hamiltonian. The fixed point of the conditioned dynamics of stabilized states is also worthwhile to be investigated under more general conditions, like in higher- dimensional systems and under different environmental influences. The fixed point facilitate in principle a probabilistic (post-selection is needed) method of pure state generation, since any initial states approaches a corresponding fixed point asymptotically, for a proper choice of employed Hamiltonians. In this thesis, we only focused on single qubits. A natural generalization is to consider several qubits, which enables an optimization of entanglement, too. We suspect that non-local unitary feedback is required to stabilize maximally entangled systems under the effect of local decoherence, since local unitaries are by definition unable to generate entanglement. By treating the feedback effectively, and calculating the corresponding sets of stabilizable states, one can find the maximally stabilizable amount of entanglement for local and non-local feedback-stabilized states - so far unknown quantities. In alternative to the here employed jump unraveling approach, also diffusive unraveling can be considered. It arises when instead of discrete observables, like the detection of photons in our case, a homodyne measurement of continuous variables is performed. The resulting quantum trajectories will differ from the onces we observe and also the consequences for realistic implementations potentially change then. Finally, in order to examine the non-approximate effect of time-delayed feedback, a numerical treatment of the latter, with quantum trajectories, as described in Sec. (5.5), is to be considered. Ultimately, we would be very happy about an experimental realization of our control approach, to prove its high stabilization capability.

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Erklärung: Hiermit versichere ich, Tobias Brünner, dass die vorliegende Arbeit ohne Hilfe Dritter und ohne Benutzung anderer, als der angegebenen Quellen und Hilfsmittel angefertigt wurde. Die den Quellen inhaltlich oder wörtlich entnomme- nen Stellen sind als solche kenntlich gemacht. Diese Arbeit wurde in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegt.

Freiburg, April 02, 2014 ...... Tobias Brünner