In the present work we develop a formalism to tackle the problem of optimal execution when trading market securities. More precisely, we introduce a utility function that balances market impact and timing risk, with this last being modelled as the very negative transaction costs incurred by our order execution. The framework is built upon existing theory on optimal trading strategies, but incorporates characteristics that enable distinctive execution strategies. The formalism is complemented by an analysis of various impact models and different distributional properties of market returns.

We propose a platform for quantum many body simulations of dipolar spin models using current circuit QED technology. Our basic building blocks are 3D transmon qubits where we use the naturally occurring dipolar interactions to realize interacting spin systems. This opens the way toward the realization of a broad class of tunable spin models in both two- and one-dimensional geometries. We illustrate the potential offered by these systems in the context of dimerized Majumdar-Ghosh-type phases, archetypical examples of quantum magnetism, showing how such phases are robust against disorder and decoherence and could be observed within state-of-the-art experiments.

A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

Generalized Dicke models can be implemented in hybrid quantum systems built from ensembles of nitrogen-vacancy (NV) centers in diamond coupled to superconducting microwave cavities. By engineering cavity assisted Raman transitions between two spin states of the NV defect, a fully tunable model for collective light-matter interactions in the ultrastrong coupling limit can be obtained. Our analysis of the resulting nonequilibrium phases for a single cavity and for coupled cavity arrays shows that different superradiant phase transitions can be observed using existing experimental technologies, even in the presence of large inhomogeneous broadening of the spin ensemble. The phase diagram of the Dicke lattice model displays distinct features induced by dissipation, which can serve as a genuine experimental signature for phase transitions in driven open quantum systems.

We show how engineered classical noise can be used to generate constrained Hamiltonian dynamics in atomic quantum simulators of many-body systems, taking advantage of the continuous Zeno effect. After discussing the general theoretical framework, we focus on applications in the context of lattice gauge theories, where imposing exotic, quasilocal constraints is usually challenging. We demonstrate the effectiveness of the scheme for both Abelian and non-Abelian gauge theories, and discuss how engineering dissipative constraints substitutes complicated, nonlocal interaction patterns by global coupling to laser fields.

We discuss how a lattice Schwinger model can be realized in a linear ion trap, allowing a detailed study of the physics of Abelian lattice gauge theories related to one-dimensional quantum electrodynamics. Relying on the rich quantum-simulation toolbox available in state-of-the-art trapped-ion experiments, we show how one can engineer an effectively gauge-invariant dynamics by imposing energetic constraints, provided by strong Ising-like interactions. Applying exact diagonalization to ground-state and time-dependent properties, we study the underlying microscopic model and discuss undesired interaction terms and other imperfections. As our analysis shows, the proposed scheme allows for the observation in realistic setups of spontaneous parity-and charge-symmetry breaking, as well as false-vacuum decay. Besides an implementation aimed at larger ion chains, we also discuss a minimal setting, consisting of only four ions in a simpler experimental setup, which enables us to probe basic physical phenomena related to the full many-body problem. The proposal opens a new route for analog quantum simulation of high-energy and condensed-matter models where gauge symmetries play a prominent role.

We describe a superconducting-circuit lattice design for the implementation and simulation of dynamical lattice gauge theories. We illustrate our proposal by analyzing a one-dimensional U(1) quantum-link model, where superconducting qubits play the role of matter fields on the lattice sites and the gauge fields are represented by two coupled microwave resonators on each link between neighboring sites. A detailed analysis of a minimal experimental protocol for probing the physics related to string breaking effects shows that despite the presence of decoherence in these systems, distinctive phenomena from condensed-matter and high-energy physics can be visualized with state-of-the-art technology in small superconducting-circuit arrays.

We study photon condensation phenomena in a driven and dissipative array of superconducting microwave resonators. Specifically, we show that by using an appropriately designed coupling of microwave photons to superconducting qubits, an effective dissipative mechanism can be engineered, which scatters photons towards low-momentum states while conserving their number. This mimics a tunable coupling of bosons to a low-temperature bath, and leads to the formation of a stationary photon condensate in the presence of losses and under continuous-driving conditions. In this paper, we propose a realistic experimental setup to observe this effect in two or multiple coupled cavities, and study the characteristics of such an out-of-equilibrium condensate, which arise from the competition between pumping and dissipation processes.

This thesis is devoted to the study of quantum mechanical effects that arise in systems of reduced dimensionality. Specifically, we investigate coherence and correlation effects in quantum transport models. In the first part, we present a theory of Markovian and non- Markovian current correlations in nanoscopic conductors. The theory is applied to obtain the spectrum of quantum noise and high-order current correlations at finite frequencies in quantum-dot systems. One of the main conclusions is that only the non-Markovian approach contains the physics of vacuum fluctuations. In the second part, we study the coupling of superconducting qubits to optical atomic systems and to cavity resonators. We propose a hybrid quantum system consisting of a flux qubit coupled to NV centers in diamond. We also demonstrate the existence of the so-called Bloch-Siegert shift in the ultra-strong coupling regime between a flux qubit and a LC resonator. Throughout the thesis, we make special emphasis on the study of decoherence effects produced by the distinct dissipative baths to which the various types of qubits presented in this thesis are inevitably coupled.

We present a theory of finite-frequency noise in nonequilibrium conductors. It is shown that Non-Markovian correlations are essential to describe the physics of quantum noise. In particular, we show the importance of a correct treatment of the initial system-bath correlations, and how these can be calculated using the formalism of quantum master equations. Our method is particularly important in interacting systems, and when the measured frequencies are larger than the temperature and applied voltage. In this regime, quantum-noise steps are expected in the power spectrum due to vacuum fluctuations. This is illustrated in the current noise spectrum of a single resonant level model and of a double quantum dot-charge qubit-attached to electronic reservoirs. Furthermore, the method allows for the calculation of the single-time counting statistics in quantum dots, measured in recent experiments.

We measure the dispersive energy-level shift of an LC resonator magnetically coupled to a superconducting qubit, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of ≈ 0.82 GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50 MHz Bloch-Siegert shift when the qubit is in its symmetry point, fully consistent with our analytical model.

We present a theory of frequency-dependent counting statistics of electron transport through nanostructures within the framework of Markovian quantum master equations. Our method allows the calculation of finite-frequency current cumulants of arbitrary order, as we explicitly show for the second- and third-order cumulants. Our formulae generalize previous zero-frequency expressions in the literature and can be viewed as an extension of MacDonald's formula beyond shot noise. When combined with an appropriate treatment of tunneling using, e.g., the Liouvillian perturbation theory in Laplace space, our method can deal with arbitrary bias voltages and frequencies, as we illustrate with the paradigmatic example of transport through a single resonant level model. We discuss various interesting limits, including the recovery of the fluctuation-dissipation theorem near linear response, as well as some drawbacks inherent to the Markovian description arising from the neglect of quantum fluctuations.

We propose a method to achieve coherent coupling between nitrogen-vacancy (NV) centers in diamond and superconducting (SC) flux qubits. The resulting coupling can be used to create a coherent interaction between the spin states of distant NV centers mediated by the flux qubit. Furthermore, the magnetic coupling can be used to achieve a coherent transfer of quantum information between the flux qubit and an ensemble of NV centers. This enables a long-term memory for a SC quantum processor and possibly an interface between SC qubits and light.

Hyperfine coupling to nuclei can be detrimental to the coherence of electron spins, but properly harnessed it can provide a mechanism for manipulation and storage of quantum information. Spin-blockade measurements in 13C carbon nanotubes now show surprisingly strong effects of electron–nuclear interaction, with a hyperfine coupling two orders of magnitude larger than previously anticipated. For coherent electron spins, hyperfine coupling to nuclei in the host material can either be a dominant source of unwanted spin decoherence1,2,3 or, if controlled effectively, a resource enabling storage and retrieval of quantum information4,5,6,7. To investigate the effect of a controllable nuclear environment on the evolution of confined electron spins, we have fabricated and measured gate-defined double quantum dots with integrated charge sensors made from single-walled carbon nanotubes with a variable concentration of 13C (nuclear spin I=1/2) among the majority zero-nuclear-spin 12C atoms. We observe strong isotope effects in spin-blockaded transport, and from the magnetic field dependence estimate the hyperfine coupling in 13C nanotubes to be of the order of 100 μeV, two orders of magnitude larger than anticipated8,9. 13C-enhanced nanotubes are an interesting system for spin-based quantum information processing and memory: the 13C nuclei differ from those in the substrate, are naturally confined to one dimension, lack quadrupolar coupling and have a readily controllable concentration from less than one to 105 per electron.

We present a formalism to calculate finite-frequency current correlations in interacting nanoscopic conductors. We work within the n-resolved density matrix approach and obtain a multitime cumulant generating function that provides the fluctuation statistics solely from the spectral decomposition of the Liouvillian. We apply the method to the frequency-dependent third cumulant of the current through a single resonant level and through a double quantum dot. Our results, which show that deviations from Poissonian behavior strongly depend on frequency, demonstrate the importance of finite-frequency higher-order cumulants in fully characterizing transport.