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:

The exact manipulation of quantum states is crucial for realizing quantum devices for information processing [1]. Quantum tomography is a well established method that allows to fully characterize the action of an arbitrary quantum devices [2].

During the master thesis an experimental setup shall be implemented to perform quantum tomography on the polarization degree of freedom of single photons. The student contributes to work towards an atom-cavity based quantum memory for optical qubits.

[1] “Quantum Computation and Quantum Information”, Michael A. Nielsen and Isaac L. Chuang, Cambridge University Press, Cambridge, UK, 20000.

[2] “Prescription for experimental determination of the dynamics of a quantum black box”, I. L. Chuang and M. A. Nielsen, Journal of Modern Optics,

**Can two photons that were indipendently generated, that will never meet each other, and that will be independently detected by separate detectors, exhibit correlations in their polarisation degree of freedom? We plan to set up a two-photon experiment to answer this question. **

Quantum mechanics predicts that indistinguishable particles exhibit correlations even when the two particles have never interacted with each other. Experiments like the Hanbury Brown and Twiss interferometer [1] and the Hong-Ou-Mandel effect [2] have played a crucial role in the last fifty years to develop a solid understanding of the physics of indistinguishable particles. Traditional single-particle interferometers are the result of interference between single-particle quantum paths. Two-particle interferometers, instead, can be thought of as interference between two-particle quantum paths – with reference to the figure, between the violet and cyan paths. We plan to use two SPDC sources to independently generate two photons. We shall post-select only those events in which the two photons are coincidentally detected at the upper and lower part detector assemblies. If the two photons are indistinguishable (with respect to their transverse mode, frequency, polarization, etc.), there is no possible way to say whether they travelled along the twisted violet paths or rather through the straight cyan paths. The inherent lack of physical information is at the origin of the two-particle correlations. We want to demonstrate correlations in the polarisation of the two photons by observing oscillating fringes as we vary the relative phase between the violet and cyan paths. The optical setup can be constructed in such a way that the two photons never cross. This experiment shall provide a dramatic demonstration of the nature of indistinguishable quantum particles.

[1] R. H. Brown and R. Q. Twiss, A Test of a New Type of Stellar Interferometer on Sirius, Nature

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

[3] R. J. Glauber, M. Kleber, A. K. Patnaik, M. O. Scully, and H. Walther, A simple study of photon correlations from Hanbury-Brown and Twiss to Einstein, Podolsky, Rosen and beyond, J. Phys. B: at. Mol. Phys.

[4] C. F. Roos, A. Alberti, D. Meschede, P. Hauke and H. Häffner, Revealing quantum statistics with a pair of distant atoms, arXiv:1706.04231 [quant-ph] (accepted in Phys. Rev. Lett.), (2017)

**We have learned from quantum mechanics textbooks: If we swap two particles that differ only in their position but are otherwise identical in the other degree 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 40, 776 (2009).

[2] M. Berry and J. Robbins, Quantum indistinguishability: Spin-statistics without relativity or field theory? AIP Conf. Proc. 545, 3 (2000)

[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, arXiv:1706.04231 [quant-ph] (accepted in Phys. Rev. Lett.), (2017)

**Ultracold atoms walking on a two-dimensional optical lattice move very differently than does their classical counterpart. Interference among different quantum paths makes this quantum transport very intriguing. To a first approximation, atoms move like 2D Dirac particles. 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. For instance, a one-dimensional optical lattice has been used in our group to perform discrete-time quantum walks of single atoms, thus demonstrating an elementary quantum cellular automaton. With this system, we have recently 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 graphene), simulation of artificial magnetic fields (see quantum Hall effect), disordered materials (see Anderson localization), topological insulators (see geometric phase), and novel paradigms of quantum information science (see 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 commercial 4W 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:

**We are witnessing a fast transition from analog to digital electronics. The advantages of digital signal processing are really impressive. However, when dealing with precision measurements, it becomes difficult to decide upon the winning technology. We take up the challenge to realize a low-noise large-bandwidth digital servo controller based on state-of-the-art digital electronics.**

The project consists in building a large bandwidth (~5 MHz) low-noise digital servo-loop controller. In our experiments, feedback is a central part of almost every application, which is essential for the precision control and coherent manipulation of single atoms. For the most demanding applications from the point of view of bandwidth and signal-to-noise ratio, we currently use analogue servo-loop controllers, which are based on the widely-known PID control loop mechanism. In the past years, digital electronic devices have enormously improved to the point that digital servo-loop controllers constitute a competitive and affordable alternative to state-of-the-art analogue logic boards. In addition, digital servo-loop controllers make available a whole new range of possibilities (e.g. internal model control, smooth relock), which are not simple to implement with analogue logic devices because of their limited flexibility.

In this project we plan to use a system-on-chip digital device to close the servo loop. A digital control employing the RedPitaya development board has already been demonstrated in our group [1], reaching a bandwidth close 5 MHz. However, the effective 12-bit resolution of the A/D converters on this board limits the range of applicability to applications where high precision (or alternaively, large dynamic range) is not strictly required. For most applications in quantum optics (like intensity stabilization of laser beams forming optical dipole traps and optical phase-lock loops for Raman lasers), high-bandwidth high-precision A/D converters are necessary. We plan to increase the effective bit resolution by either combining the two channels available from the A/D converters on the RedPitaya board, or by employing A/D converters with a higher number of bits, for instance, the 16-Bit 30 MSPS DAC AD768 and the 16-bit 10 MSPS ADC AD7626. An alternative solution is provided by Altera DE2-115 development board together with the AD/DA module.

The final goal of the project consists in stabilizing the intensity of laser beams used to create periodic optical lattice potentials for atoms. A test setup with opto-mechanical components is already available for characterizing the servo-loop and measure its closed-loop response function.

[1] Florian Seidler, Digital high bandwidth control, *Masterarbeit* (2015)

[3] Tutorial based on Matlab Simulink toolbox about PID controller design.

[4] J. Bechhoefer, Feedback for physicists: *A tutorial essay on control*, Rev. Mod. Phys. **77**, 783 (2005)

**How far can we coherently split a single particle preserving quantum coherences? We intend to use spin-dependent optical lattices to transport the electron spin-up and the electron spin-down components of a single atom by 1 mm apart, and recombine them together 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 a single atom at 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 use 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 arbritrary transport shape.

The project consists in applying the concept of quantum optimal control theory to optimize the atom transport over large disances 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 interface a fast FPGA with the DDS chip to program complex transport sequences in real time. This will allow us to 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

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