Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications

Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.

Influence of super-ohmic dissipation on a disordered quantum critical point

We investigate the combined influence of quenched randomness and dissipation on a quantum critical point with O(N) order-parameter symmetry. Utilizing a strong-disorder renormalization group, we determine the critical behavior in one space dimension exactly. For super-ohmic dissipation, we find a Kosterlitz-Thouless type transition with conventional (power-law) dynamical scaling. The dynamical critical exponent depends on the spectral density of the dissipative baths. We also discuss the Griffiths singularities, and we determine observables.

Self-energies for interacting fields with a compactified spatial dimension

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Energy-dependent point interactions in one dimension

We consider a new type of point interaction in one-dimensional quantum mechanics. It is characterized by a boundary condition at the origin that involves the second and/or higher order derivatives of the wavefunction. The interaction is effectively energy dependent. It leads to a unitary S-matrix for the transmission-reflection problem. The energy dependence of the interaction can be chosen such that any given unitary S-matrix (or the transmission and reflection coefficients) can be reproduced at all energies. Generalization of the results to coupled-channel cases is discussed.

Cubic interactions and quantum criticality in dimerized antiferromagnets

In certain Mott-insulating dimerized antiferromagnets, triplet excitations of the paramagnetic phase display both three-particle and four-particle interactions. When such a magnet undergoes a quantum phase transition into a magnetically ordered state, the three-particle interaction becomes part of the critical theory provided that the lattice ordering wave vector is zero. One microscopic example is the staggered-dimer antiferromagnet on the square lattice, for which deviations from O(3) universality have been reported in numerical studies. Using both symmetry arguments and microscopic calculations, we show that a nontrivial cubic term arises in the relevant order-parameter quantum field theory, and we assess its consequences using a combination of analytical and numerical methods. We also present finite-temperature quantum Monte Carlo data for the staggered-dimer antiferromagnet which complement recently published results. The data can be consistently interpreted in terms of critical exponents identical to that of the standard O(3) universality class, but with anomalously large corrections to scaling. We argue that the cubic interaction of critical triplons, although irrelevant in two spatial dimensions, is responsible for the leading corrections to scaling due to its small scaling dimension.

Self-bound droplet of Bose and Fermi atoms in one dimension: Collective properties in mean-field and Tonks-Girardeau regimes

We investigate a dilute mixture of bosons and spin-polarized fermions in one dimension. With an attractive Bose-Fermi scattering length the ground state is a self-bound droplet, i.e., a Bose-Fermi bright soliton where the Bose and Fermi clouds are superimposed. We find that the quantum fluctuations stabilize the Bose-Fermi soliton such that the one-dimensional bright soliton exists for any finite attractive Bose-Fermi scattering length. We study density profile and collective excitations of the atomic bright soliton showing that they depend on the bosonic regime involved: mean-field or Tonks-Girardeau.

Field-theory Two-point Functions via Negative Dimension Integration

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Quantum topology change and large-N gauge theories

We study a model for dynamical localization of topology using ideas from non-commutative geometry and topology in quantum mechanics. We consider a collection X of N one-dimensional manifolds and the corresponding set of boundary conditions (self-adjoint extensions) of the Dirac operator D. The set of boundary conditions encodes the topology and is parameterized by unitary matrices g. A particular geometry is described by a spectral triple x(g) = (A X, script H sign X, D(g)). We define a partition function for the sum over all g. In this model topology fluctuates but the dimension is kept fixed. We use the spectral principle to obtain an action for the set of boundary conditions. Together with invariance principles the procedure fixes the partition function for fluctuating topologies. The model has one free-parameter β and it is equivalent to a one plaquette gauge theory. We argue that topology becomes localized at β = ∞ for any value of N. Moreover, the system undergoes a third-order phase transition at β = 1 for large-N. We give a topological interpretation of the phase transition by looking how it affects the topology. © SISSA/ISAS 2004.

Dimensionality effects in the local density of states of ferromagnetic hosts probed via STM: Spin-polarized quantum beats and spin filtering

We theoretically investigate the local density of states (LDOS) probed by an STM tip of ferromagnetic metals hosting a single adatom and a subsurface impurity. We model the system via the two-impurity Anderson Hamiltonian. By using the equation of motion with the relevant Green's functions, we derive analytical expressions for the LDOS of two host types: a surface and a quantum wire. The LDOS reveals Friedel-like oscillations and Fano interference as a function of the STM tip position. These oscillations strongly depend on the host dimension. Interestingly, we find that the spin-dependent Fermi wave numbers of the hosts give rise to spin-polarized quantum beats in the LDOS. Although the LDOS for the metallic surface shows a damped beating pattern, it exhibits the opposite behavior in the quantum wire. Due to this absence of damping, the wire operates as a spatially resolved spin filter with a high efficiency. © 2013 American Physical Society.

Theoretical and experimental study of the excitonic binding energy in GaAs/AlGaAs single and coupled double quantum wells

This paper discusses the theoretical and experimental results obtained for the excitonic binding energy (Eb) in a set of single and coupled double quantum wells (SQWs and CDQWs) of GaAs/AlGaAs with different Al concentrations (Al%) and inter-well barrier thicknesses. To obtain the theoretical Eb the method proposed by Mathieu, Lefebvre and Christol (MLC) was used, which is based on the idea of fractional-dimension space, together with the approach proposed by Zhao et al., which extends the MLC method for application in CDQWs. Through magnetophotoluminescence (MPL) measurements performed at 4 K with magnetic fields ranging from 0 T to 12 T, the diamagnetic shift curves were plotted and adjusted using two expressions: one appropriate to fit the curve in the range of low intensity fields and another for the range of high intensity fields, providing the experimental Eb values. The effects of increasing the Al% and the inter-well barrier thickness on E b are discussed. The Eb reduction when going from the SQW to the CDQW with 5 Å inter-well barrier is clearly observed experimentally for 35% Al concentration and this trend can be noticed even for concentrations as low as 25% and 15%, although the Eb variations in these latter cases are within the error bars. As the Zhao's approach is unable to describe this effect, the wave functions and the probability densities for electrons and holes were calculated, allowing us to explain this effect as being due to a decrease in the spatial superposition of the wave functions caused by the thin inter-well barrier. © 2013 Elsevier B.V.

Quantum entanglement of bound particles under free center of mass dispersion

On the basis of the full analytical solution of the overall unitary dynamics, the time evolution of entanglement is studied in a simple bipartite model system evolving unitarily from a pure initial state. The system consists of two particles in one spatial dimension bound by harmonic forces and having its free center of mass initially localized in space in a minimum uncertainty wavepacket. The existence of such initial states in which the bound particles are not entangled is discussed. Galilean invariance of the system ensures that the dynamics of entanglement between the two particles is independent of the wavepacket mean momentum. In fact, as shown, it is driven by the dispersive center of mass free dynamics, and evolves in a time scale that depends on the interparticle interaction in an essential way.

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T, a quantum parameter g, and the ratio p = -J(2)/J(1) where J(1) > 0 refers to ferromagnetic interactions between first-neighbour sites along the d directions of a hypercubic lattice, and J(2) < 0 is associated with competing anti ferromagnetic interactions between second neighbours along m <= d directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g = 0 space, with a Lifshitz point at p = 1/4, for d > 2, and closed-form expressions for the decay of the pair correlations in one dimension. In the T = 0 phase diagram, there is a critical border, g(c) = g(c) (p) for d >= 2, with a singularity at the Lifshitz point if d < (m + 4)/2. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p = 1/4. 2012 (C) Elsevier B.V. All rights reserved.

Modeling the gluon propagator in Landau gauge: Lattice estimates of pole masses and dimension-two condensates

We present an analytic description of numerical results for the Landau-gauge SU(2) gluon propagator D(p(2)), obtained from lattice simulations (in the scaling region) for the largest lattice sizes to date, in d = 2, 3 and 4 space-time dimensions. Fits to the gluon data in 3d and in 4d show very good agreement with the tree-level prediction of the refined Gribov-Zwanziger (RGZ) framework, supporting a massive behavior for D(p(2)) in the infrared limit. In particular, we investigate the propagator's pole structure and provide estimates of the dynamical mass scales that can be associated with dimension-two condensates in the theory. In the 2d case, fitting the data requires a noninteger power of the momentum p in the numerator of the expression for D(p(2)). In this case, an infinite-volume-limit extrapolation gives D(0) = 0. Our analysis suggests that this result is related to a particular symmetry in the complex-pole structure of the propagator and not to purely imaginary poles, as would be expected in the original Gribov-Zwanziger scenario.

Theoretical and experimental study of the excitonic binding energy in GaAs/AlGaAs single and coupled double quantum wells

This paper discusses the theoretical and experimental results obtained for the excitonic binding energy (Eb) in a set of single and coupled double quantum wells (SQWs and CDQWs) of GaAs/AlGaAs with different Al concentrations (Al%) and inter-well barrier thicknesses. To obtain the theoretical Eb the method proposed by Mathieu, Lefebvre and Christol (MLC) was used, which is based on the idea of fractional-dimension space, together with the approach proposed by Zhao et al., which extends the MLC method for application in CDQWs. Through magnetophotoluminescence (MPL) measurements performed at 4 K with magnetic fields ranging from 0 T to 12 T, the diamagnetic shift curves were plotted and adjusted using two expressions: one appropriate to fit the curve in the range of low intensity fields and another for the range of high intensity fields, providing the experimental Eb values. The effects of increasing the Al% and the inter-well barrier thickness on Eb are discussed. The Eb reduction when going from the SQW to the CDQW with 5 Å inter-well barrier is clearly observed experimentally for 35% Al concentration and this trend can be noticed even for concentrations as low as 25% and 15%, although the Eb variations in these latter cases are within the error bars. As the Zhao's approach is unable to describe this effect, the wave functions and the probability densities for electrons and holes were calculated, allowing us to explain this effect as being due to a decrease in the spatial superposition of the wave functions caused by the thin inter-well barrier.

Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.