# Time evolution in integrable systems

# Time evolution in integrable systems

Promotor(en):**17FUND03**/ Many-particle physics**D. Van Neck**/One of the most intriguing phenomena in quantum many-particle physics is the appearance of strong quantum correlations and complexity. In these systems it's not just the interaction between single particles, but the synergy of all interactions together that gives rise to many different phases of matter. The whole system is greater than the sum of its parts, or in Nobel laureate P. W. Anderson's words: “More is different!”.

Theoretical quantum many-particle physics is tasked with explaining, understanding, and even predicting the appearance and evolution of these complex phenomena. For instance, it is known that many quantum systems undergo a quantum phase transition when the density (or number of particles) reaches a certain threshold. However, we are confronted by the fact that the Hilbert space of the fully-interacting quantum system increases exponentially with increasing system size. This means that the exact theoretical investigation of these interesting systems is simply impossible, and one must usually resort to approximation techniques. Interestingly, there exists a special class of so-called integrable systems, where the system can be solved exactly, despite the strong quantum correlations and exponential size-explosion of the Hilbert space. For these systems, the wave function is given by a factorisable Bethe Ansatz and the correlations are encoded explicitly by means of a set of generalized momenta, or rapidities. Apart from the nice physical interpretation, these rapidities can also be efficiently computed by solving a set of non-linear algebraic (Bethe) equations. The major importance of this result is that the diagonalization of a Hamiltonian matrix in an exponentially scaling Hilbert space has been reduced to solving a set of non-linear equations scaling linearly with system size. This allows for an exact study of systems far out of the reach of conventional many-body techniques.

Due to the algebraic formulation of integrable systems, they appear in many branches of physics, such as

• Superconductivity in crystals and nanoparticles,

• Magnetism in spin chains,

• Atoms in a cavity interacting with an electromagnetic field,

• Pairing interactions in atomic nuclei,

• Hyperfine interactions in condensed matter systems.One of the open problems in (integrable) quantum many-body systems is the incorporation of time. This touches at the heart of fundamental quantum many-body physics, i.e. controlling and manipulating the Schroedinger equation. Indeed, one needs to have a sufficiently good control over the time-independent Schroedinger equation as inaccuracies and approximations can easily accumulate over time, leading to quantum states that evolve to wrong corners in the Hilbert space. Furthermore, in view of the exponential size explosion and complexity of the Hilbert space, it is a formidable task of quantum engineering to know which interactions to tune in the system in order to drive the quantum state into a desired position in Hilbert space. It is clear that integrable systems offer a unique platform to investigate these aspects.

**Goal**The goal of this thesis is to investigate the role of time in integrable quantum systems. This can be done on multiple levels, ranging from fundamental to applied. On a fundamental level, it is currently an open question how to generalize quantum integrable systems to incorporate time dependence. From a classical point of view, the situation is clear via the concept of Liouville integrability, but it becomes ambiguous when quantum degrees of freedom enter. In this thesis, the student can scrutinize the link between classical and quantum integrable systems, and put the concept of integrability in time-dependent quantum systems on a firm fundamental footing. On the applied level, the on-line control of a quantum state is one of the holy grails of quantum computing. For instance, decoherence effects in which a system under study leaks (quantum) information to the environment, hampers the usefulness of these systems, and remains a fundamental problem to date. It is known that integrability inhibits these decoherence effects due to a hidden symmetry in the system, facilitating the investigation of these processes in the system. Using the present mathematical machinery available in our research group, the student can devise schemes to tune and control quantum states in, e.g., Nitrogen-Vacancy defects in diamond, either using conventional or newly proposed protocols.

**Aspects**The engineering aspect of this thesis concerns the modeling and subsequent computational investigation of systems from different branches of physics, where the quantum effects have become of experimental importance, for example, in the Nitrogen-Vacancy defects in diamond.

This research topic will be conducted in the framework of a strong international network and if possible the student will be actively involved in work discussions with collaborative partners.