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International Journal of Innovative Technology and Exploring Engineering

Bosons-Bosons Oscillator Dynamics in a Quantum Control System

Roberto P. L. Caporali

Published Feb 20, 2025• 0 citations
Publisher

Abstract

In this work, we developed a study of a set of Cooper pairs in a superconducting system and the control dynamics of the corresponding Quantum Control System. This paper deals specifically with the interaction in a crystal between a bosonic field given by the Cooper pairs and another bosonic field given by the System of vibrational modes of the phonons in the crystal, regarding applications related to Quantum Controllers. We consider an interaction of the bosons-bosons type, which specifically refers to the possible coupling between the phonon field and the bosonic field constituted by Cooper pairs. We develop the Hamiltonian of the whole system, giving the Hamiltonian of the Cooper Pairs, the Hamiltonian of the interacting phonon system, and the interaction Hamiltonian. In our study, we define, for the Quantum Control System relative to the bosons-bosons interaction, a Quantum Kalman Filter. We obtain, as a first result of this work, the expression that makes explicit the Quantum Kushner-Stratonovich equation in our case of interaction between Cooper pairs and phonons of the crystal lattice of a superconductor. It expresses the density of states ρ of the Cooper pairs as a function of time in the non-Markovian statistical system described by us. Furthermore, we obtain a second fundamental result of this work, expressing the Hamiltonian of the overall open system as a function of the Pauli matrix operator. This operator allows us to represent the same Hamiltonian of the open system described by us through the logic gates of a quantum controller. So, in the end, we obtain the control law of the temporal evolution of the bosonic system and the possibility of representing it through the logic gates of a quantum control system.

DOI
10.35940/ijitee.b1049.14030225

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In this work, we developed a study of a set of Cooper pairs in a superconducting system and the control dynamics of the corresponding Quantum Control System. This paper deals specifically with the interaction in a crystal between a bosonic field given by the Cooper pairs and another bosonic field given by the System of vibrational modes of the phonons in the crystal, regarding applications related to Quantum Controllers. We consider an interaction of the bosons-bosons type, which specifically refers to the possible coupling between the phonon field and the bosonic field constituted by Cooper pairs. We develop the Hamiltonian of the whole system, giving the Hamiltonian of the Cooper Pairs, the Hamiltonian of the interacting phonon system, and the interaction Hamiltonian. In our study, we define, for the Quantum Control System relative to the bosons-bosons interaction, a Quantum Kalman Filter. We obtain, as a first result of this work, the expression that makes explicit the Quantum Kushner-Stratonovich equation in our case of interaction between Cooper pairs and phonons of the crystal lattice of a superconductor. It expresses the density of states ρ of the Cooper pairs as a function of time in the non-Markovian statistical system described by us. Furthermore, we obtain a second fundamental result of this work, expressing the Hamiltonian of the overall open system as a function of the Pauli matrix operator. This operator allows us to represent the same Hamiltonian of the open system described by us through the logic gates of a quantum controller. So, in the end, we obtain the control law of the temporal evolution of the bosonic system and the possibility of representing it through the logic gates of a quantum control system.
We develop a general approach for monitoring and controlling evolution of open quantum systems. In contrast to the master equations describing time evolution of density operators, here, we formulate a dynamical equation for the evolution of the process matrix acting on a system. This equation is applicable to non-Markovian and/or strong-coupling regimes. We propose two distinct applications for this dynamical equation. We first demonstrate identification of quantum Hamiltonians generating dynamics of closed or open systems via performing process tomography. In particular, we argue how one can efficiently estimate certain classes of sparse Hamiltonians by performing partial tomography schemes. In addition, we introduce an optimal control theoretic setting for manipulating quantum dynamics of Hamiltonian systems, specifically for the task of decoherence suppression.
This work establishes the algebraic structure of the Kohn-Sham equations to be solved in a density formulation of electron and phonon dynamics, including the superconducting order parameter. A Bogoliubov transform is required to diagonalize both the fermionic and bosonic Kohn-Sham Hamiltonians since they both represent a non-interacting quantum field theory. The Bogoliubov transform for phonons is non-Hermitian in the general case, and the corresponding time-evolution is non-unitary. Several sufficient conditions for ensuring that the bosonic eigenvalues are real are provided and a practical method for solving the system is described. Finally, we produce a set of approximate mean-field potentials which are functionals of the electronic and phononic density matrices and depend on the electron-phonon vertex.
We investigate the controllability of quantum systems in two different settings: the standard "closed" setting, in which a quantum system is seen as isolated and the control problem is formulated on the Schrödinger equation; the open setting that describes a quantum system in interaction with a larger one, of which just qualitative parameters are known, by means of the Lindblad equation on states. In the context of closed systems we focus our attention to an interesting class of models, namely the spin-boson models. The latter describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. We discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (C-QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S. Haroche. We investigate the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding -in the application to cavity QED- to an external electric and magnetic field, respectively. We review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. This result is based on a spectral analysis exploiting the non-resonances of the spectrum. As far as the relation between the Rabi and the Jaynes-Cummings Hamiltonians concerns, we treat the so called rotating wave approximation in a rigorous framework. We formulate the problem as an adiabatic limit in which the detuning frequency and the interaction strength parameter goes to zero, known as the weak-coupling regime. We prove that, under certain hypothesis on the ratio between the detuning and the coupling, the Jaynes-Cumming and the Rabi dynamics exhibit the same behaviour, more precisely the evolution operators they generate are close in norm. In the framework of open quantum systems we investigate the controllability of the Lindblad equation. We consider a control acting adiabatically on the internal part of the system, which we see as a degree of freedom that can be used to contrast the action of the environment. The adiabatic action of the control is chosen to produce a robust transition. We prove, in the prototype case of a two-level system, that the system approach a set of equilibrium points determined by the environment, i.e. the parameters that specify the Lindblad operator. On that set the system can be adiabatically steered choosing a suitable control. The analysis is based on the application of geometrical singular perturbation methods.
A Markovian master equation describing the evolution of open quantum systems in the presence of a time-dependent external field is derived within the Bloch-Redfield formalism. It leads to a system-bath interaction which depends on the control field. Optimal control theory is used to select control fields which allow accelerated or decelerated system relaxation, or suppression of relaxation (dissipation) altogether, depending on the dynamics we impose on the quantum system. The control-dissipation correlation and the nonperturbative treatment of the control field are essential for reaching this goal. The optimal control problem is formulated within Pontryagin's minimum principle and the resulting optimal differential system is solved numerically. As an application, we study the dynamics of a spin-boson model in the strong coupling regime under the influence of an external control field. We show how trapping the system in unstable quantum states and transfer of population can be achieved by optimized control of the dissipative quantum system. We also used optimal control theory to find the driving field that generates the quantum ZZZ gate. In several cases studied, we find that the selected optimal field which reduces the purity loss significantly is a multicomponent low-frequency field including higher harmonics, all of which lie below the phonon cutoff frequency. Finally, in the undriven case we present an analytic result for the Lamb shift at zero temperature.
Accurate control of a quantum system is complicated to achieve partly due to the system being coupled to its surrounding environment. The coupling induces dissipation and decoherence not only destroying the coherent quantum state but making the application of control unpredictable. When the control results from manipulation of external fields, it is usually referred to as driving and its joint effect with decoherence constitutes an active field of study in reduced-density-operator theory. Recently, this field has been pushed forward by its necessity in simulating Cooper pair pumping where the geometric nature of quantum evolution allows for controlled transport of charge carriers in superconducting circuits. Such circuits themselves are under constant investigation as they grant access to fundamental quantum phenomena and facilitate promising applications such as those in quantum information processing.\nIn this dissertation, driven quantum systems under decoherence are investigated. Extensions and improvements to a recent master equation for nearly adiabatic driving are presented and analyzed. The emergent properties in quantum evolution are studied both analytically and numerically. Related to the findings, a general conservation law of operator current is derived and used to explain observed nonconservation in previous theoretical studies. The derived theory is applied to Cooper-pair pumping and shown to lead to important properties such as superadiabatic ground-state pumping. A recent pumping experiment is simulated and the breakdown point for ideal pumping is found with feasible physical parameters. A study of charge transport in the presence of flux noise is presented leading to detectable dissipative currents and the typical description of the device used for pumping is extended to include a nonvanishing loop inductance.\nA novel approach to implement control of quantum systems is proposed based on constructing a tunable coupling to an artificial environment using either a coplanar waveguide cavity or coupled quantum LC resonators. Tunability allowing for both efficient initialization and protected evolution is theoretically demonstrated. Finally, a general framework for quantum driving is constructed without the typical assumption of a classical driving force leading to peculiar results.\nThis dissertation presents original research on both modeling the control of open quantum systems as well as the realization of such control. The work simulates physical phenomena in superconducting circuits and makes predictions for future experiments. In addition, it introduces novel theoretical tools and approaches that advance the state of the art.
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.