Research projects for master students are divided into two parts: during the first 6 months, the student must scientifically explore the master's thesis topic (module Physics910), plan the project and develop the required research methods (module Physics 920). A short report (2-4 pages) on the exploration and the planning must be handed in. The last 6 months of the research project are reserved to the master’s thesis work itself. The master’s thesis has in general 30 to 60 pages. The results of the master’s thesis are presented in a talk near the end of the research phase. The formal aspects are summarized in the Module-Handbook Master in Physics.

The research topic of the master's thesis will be discussed together with the candidate. It is possible to bring in your own ideas. Examples of research topics are listed below. Other opportunities are also available and can be discussed in person in confidential terms. Please come and talk to us. We look forward to your enthusiastic participation.

Those who are interested to join our group, please contact:

**Topological insulators are quantum materials behaving like an ordinary insulator in the bulk and yet allowing, in two dimensions and above, matter waves to propagate along their boundaries through a discrete number of edge modes free of any scattering ****or back-reflection. We want to use quantum walks of ultracold atoms to realize novel topological phases that cannot be implemented straightforwardly in solid-state materials.**

We plan to realize spatial domains with different topology, and to study the topologically protected transport of ultracold matter waves along the edges separating these domains [1,2]. To realize sharp edges, we will use structured laser intensity patterns, which can be generated with a liquid crystal on silicon spatial light modulator (LCoS-SLM) either by holographic projection or by directly modulating the light intensity. In both cases, the structured intensity pattern will be projected onto the atoms trapped in a two-dimensional (2D) spin-dependent optical lattice. In a previous work [3], we have used pixel-wise phase-shifting interferometry to realize gamma correction and phase-calibration for each pixel, achieving an overall RMS wavefront flatness of ~ λ/80. Subsequently, we have demonstrated the generation of intensity patterns with holographic techniques, extending the popular Gerchberg-Saxton algorithm to avoid optical vortices and to suppress speckles. In the first part of this master project, you will compare the holographic technique with direct intensity modulation. Depending on the outcome of this comparative study, one of the two techniques will be adopted to generate structured intensity patterns for the 2D quantum walk experiment. The second part of this master project will thus be focused on integrating the SLM setup into the main experimental apparatus, and testing it by projecting structured light patterns onto the atoms.

[1] M. Sajid, J.K. Asbóth, D. Meschede, R. Werner, A. Alberti, “Creating Floquet Chern insulators with magnetic quantum walks,” 2018, arXiv:1808.08923v1.

[2] T. Groh, S. Brakhane, W. Alt, D. Meschede, J. Asbóth and A. Alberti, “Robustness of topologically protected edge states in quantum walk experiments with neutral atoms,” Phys. Rev. A

[3] W. Zhou-Hanf, “Robust Holographic Generation of Arbitrary Light Patterns: Method and Implementation,” Master's thesis (2018, IAP, University Bonn).

**Ultracold atoms walking on a two-dimensional optical lattice move very differently compared to their classical counterpart. Interference among different quantum paths makes this quantum transport very intriguing. To a first approximation, atoms move like Dirac particles in two dimensions. We plan several experiments to unravel the rich physics of 2D discrete-time quantum walks.**

Neutral atoms confined in optical lattice potentials are ideal candidates to perform digital quantum simulations and novel quantum computational schemes. A one-dimensional optical lattice has already been used in our group to perform discrete-time quantum walks of single atoms. With this system, we have simulated the physics of charged particles in a crystal subject to an external electric field.

For many physical problems it is important to go beyond one dimension like for quantum transport experiments (see, e.g., graphene), simulation of artificial magnetic fields (see, e.g., quantum Hall effect), disordered materials (see, e.g., Anderson localization), topological insulators (see, e.g., geometric phase), and novel paradigms of quantum information science (see, e.g., one-way quantum computer). Our original approach consists in employing two-dimensional spin-dependent transport, i.e. the ability to deterministically transport atoms depending on their internal state, in order to experimentally investigate these physical models.

Within this project, a high-power (>10W) phase-locked Ti:sapphire laser source shall be used to generate the optical lattice at the magic wavelength (866 nm), which is necessary for state-dependent transport. The 2D lattice will be placed at exactly 150 μm from the first surface of a large numerical-aperture (NA~0.9) objective lens, which is situated in a twelve-sided ultrahigh vacuum cell (see image). Two-dimensional spin-dependent transport will be performed by means of a newly developed technology, which allows to digitally synthesize the light polarization of the lattice laser beams. The final goal of the project consists in using the spin-dependent lattice to implement a two-dimensional discrete-time quantum walk with neutral atoms.

[1] M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, A. Widera:

[2] M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede and A. Alberti:

**Topological insulators behave like an ordinary band insulator in the bulk, but exhibit frictionless quantized transport along the edges. Such a topologically protected transport can be explained in terms of topological integer numbers that are associated with the different bands and that depend on the topological structure of the Bloch wavefunctions forming those bands. Because of their topological nature, these numbers are invariant under continuous deformations of the system's parameters, provided that the band gaps and the relevant symmetries of the system are preserved.**

Although topological invariants are well-defined physical quantities, these are generally difficult to measure because they are properties of the whole system, rather than of a specific quantum state. Thus, new type of measurement protocols are necessary to extract such a topological information from a quantum system. In this master's thesis, we plan to measure the topological invariants characterizing a one-dimensional (1D) quantum walk [1]. A 1D quantum walk represents the space- and time-discrete analogue of a spin-1/2 quantum particle constrained to move on a line. In a theoretical paper [2], we have shown that controlled losses in a 1D one-quantum walk can be used to reconstruct its topological invariants. More specifically, if we extract atoms in a given spin state at the end of each time step of the quantum walk, the average position where atoms are extracted turns out to be an integer, which remarkably is equal to the sum (or difference, depending on the details) of the two relevant topological invariants. In our quantum walk setup, we realize 1D quantum walks by coherently transporting atoms at discrete time intervals one site to the right or to the left depending on their spin state. To implement the lossy protocol, we plan to use fast, long-distance spin-dependent shift operations, allowing us to selectively remove atoms in a particular spin state, while allowing atoms in the other spin state to continue their quantum walk. Single-site-resolved fluorescence imaging [3] will allow us to detect with high efficiency the precise site where the atom is extracted and, thereby, to compute the average extraction position.

[1] T. Groh, S. Brakhane, W. Alt, D. Meschede, J. Asbóth and A. Alberti, “Robustness of topologically protected edge states in quantum walk experiments with neutral atoms,” Phys. Rev. A

[2] T. Rakovszky, J. K. Asbóth, and A. Alberti, “Detecting topological invariants in chiral symmetric insulators via losses,” Phys. Rev. B

[3] A. Alberti, W. Alt, R. Werner, and D. Meschede, “Decoherence models for discrete-time quantum walks and their application to neutral atom experiments,” New J. Phys.

**Cooling trapped atoms into the lowest motional state of an optical dipole trap is a crucial requisite to exploit coherent controlled interactions among ultracold atoms. When two atoms are prepared in the motional ground state, their collisional interaction yields a well-defined quantum phase, which is acquired by their two-atom wavefunction, and which only depend on the internal state of the atoms, but not on unknown (thermal) motional states.**

We plan to employ resolved sideband cooling techniques to prepare ^{133}Cs atoms into the motional ground state of a three-dimensional, deep optical lattice. In essence, resolved sideband cooling uses transitions between two atomic states that change the motional state (i.e., address a motional sideband), so as to reduce the number of motional excitations. In our present apparatus, Cs atoms are initially cooled by an optical molasses, resulting in about 3 motional states occupied on average for each of the three dimensions. The goal of this master's thesis is to employ sideband cooling to bring atoms into the motional ground state (see figure). While sideband cooling is a well established technique to cool trapped ions [1], it is much less spread among ultracold atom experiments because it requires steep optical potentials in order to enter the recoil-free Lamb-Dicke regime. In our setup, we operate in a deep lattice with motional sidebands separated by 30 up to 100 kHz from the carrier frequency, satisfying the Lamb-Dicke condition. Such a large frequency separation should also enable high cooling rates and, thus, high cooling efficiencies. The first part of this master's thesis concerns the precise characterization of our system, carrying out sideband spectroscopy experiments, inducing Rabi oscillations on different sidebands, and measuring Franck-Condon factors for the microwave-induced sideband transitions [2]. This investigation will be supported theoretically by an ab-initio calculation of the energy band structure of the three-dimensional optical lattice, which is carried out using Quantum Espresso software. The goal of the second part of the master's thesis is to prepare atoms into the motional ground state with a success probability of 99% per dimension. This requires well-designed cooling sequences, which operate on different sidebands to circumvent population trapping [3] when atoms are relatively hot, and on the first sideband in a pulsed fashion when atoms are close to the ground state. As an outlook of this master's thesis, ultracold coherent collisions among two atoms prepared in the motional ground state shall allow us to realize a CPHASE gate entangling the two atoms.

[1] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys.

[2] N. Belmechri, L. Förster, W. Alt, A. Widera, D. Meschede, and A. Alberti, “Microwave control of atomic motional states in a spin-dependent optical lattice,” J. Phys. B:. At. Mol. Phys.

[3] M. K. Joshi, P. Hrmo, V. Jarlaud, F. Oehl, and R. C. Thompson, “Population dynamics in sideband cooling of trapped ions outside the Lamb-Dicke regime,” Phys. Rev. A

**We have all learned from quantum mechanics textbooks: If we swap two particles that differ only in their position but are otherwise identical in the other degrees of freedom, the quantum state acquires a phase 0 for bosons and π for fermions. We take up the challenge to validate this fundamental law of nature in a two-atom interferometer experiment.**

In nature elementary particles are either bosons or fermions depending on whether their angular momentum is an integer number or a half-integer number of *ħ*. This classification of particles into two large families has deep physical consequences in relation to identical particles. The spin-statistics theorem states that when we exchange two particles – namely, when we transport one particle into the position of the other and vice versa – we obtain the same two-particle quantum mechanical state except for a quantum phase [1]. This phase is 0 for bosons and *π* for fermions. The different exchange phase between bosons and fermions can be revealed with a new type of two-particle interferometry experiment, which probes the spin-spin correlations between the two particles. The scheme illustrated in the figure is robust against decoherence mechanisms and can be implemented with bosons and fermions.

[1] A. Jabs, Connecting Spin and Statistics in Quantum Mechanics, Foundations of Physics

[2] M. Berry and J. Robbins, Quantum indistinguishability: Spin-statistics without relativity or field theory? AIP Conf. Proc.

[3] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett.

[4] C. F. Roos, A. Alberti, D. Meschede, P. Hauke, and H. Häffner, “Revealing Quantum Statistics with a Pair of Distant Atoms,” Phys. Rev. Lett.

**How far can we delocalize the wave function of a single particle preserving quantum coherences? We intend to use spin-dependent optical lattices to transport the spin-up and the spin-down components of a single atom by 1 mm apart, and recombine them to observe interference fringes.**

State-dependent optical lattices are periodic potentials created by laser fields, which we use to transport atoms along a direction determined by their qubit state, which can either be spin up or spin down. With this system we could, for instance, demonstrate the splitting of the wave function of a single atom into two spatial locations separated by 10 μm.

In order to achieve even larger distances, we have recently developed a new technology for transporting atoms over arbitrary long distances. The new system is based on a digital synthesis of light polarization, which is used to control the state-dependent optical lattice. The state-dependent transport of single atoms is controlled by digitally programming the phase of a RF signal through a direct digital synthesizer (DDS). The new technology allows us to shape arbitrary transport shape.

The project consists in applying the concept of quantum optimal control theory to optimize the atom transport over large distances while preserving the very fragile quantum "coherences". We aim at demonstrating a macroscopic single-atom interferometer, where atoms are coherently split over hundreds of lattice sites (1 lattice site = 433 nm). We will tailor the interferometer geometry with ultrahigh spatiotemporal control, ultimately achieving a spatial and temporal resolution of 100 pm and 10 ns, respectively.

[1] Analog Devices Direct Digital Synthesizer AD9910.

[2] A. Steffen, A. Alberti, W. Alt, N. Belmechri, S. Hild, M. Karski, A. Widera and D. Meschede: A digital atom interferometer with single particle control on a discretized spacetime geometry, PNAS

[3] J. Werschnik and E. K. U. Gross: Tutorial on Quantum optimal control theory, J. Phys. B: At. Mol. Opt. Phys.

[4] G. de Chiara, T. Calarco, M. Anderlini, S. Montangero, P. J. Lee, B. L. Brown, W. D. Phillips, and J. V. Porto: Optimal control of atom transport for quantum gates in optical lattices, Phys. Rev. A

**There is significant interest in reducing size and cost of Faraday Isolators, key components in high performance laser diode laser systems, and a basic question is: what permanent magnet configuration optimizes ∫Bdz? This project comprises an innovation project with industrial relevance and addresses the theoretical solution of the basic optimization task, analysis of cost and technical constraints and realization of a practical device.**

[1] V. Frerichs, W. G, Kaenders, and D. Meschede: Analytic Construction of Magnetic Multipoles from Cylindric Permanent Magnets, Appl. Phys. A 55, 242 (1992).

[2] Gérard Trénec, William Volondat, Orphée Cugat, and Jacques Vigué: Permanent magnets for Faraday rotators inspired by the design of the magic sphere, App. Opt. 50, 4788 (2011).