## Final Report Summary - DQSIM (Discrete Quantum Simulator)

The DQSIM project set out to experimentally master so-called quantum walks by controlling cold atoms trapped in a two-dimensional optical lattice. Quantum walks are the quantum mechanical analogue of random walks. In contrast to their classical counterpart characterized by diffusive motion with random trajectories in space, quantum walks cause individual atoms to split their motion into a superposition of numerous partial trajectories, which give rise to complex interference patterns. One noticeable result is ballistic spreading of the particle wave function in space.

Our quantum walkers (for DQSIM, neutral cesium atoms) behave effectively as spin one-half particles, where the up and a down pseudo-spin components are implemented using long-lived internal states. In the experiment, quantum walkers resemble individual single-atom clocks moving around in an optical lattice and exploring the “quantum world”. Their quantum motion is controlled by a rotation of the pseudo spin of the atoms, which is called “coin” operation in analogy with the random walk of the Galton board. Atoms in spin up move e.g. to the left, those in spin down to the right, and spin superposition states result in split trajectories which depend on the spin rotation. In stark contrast to the random walk situation, however, the quantum dynamics is fully reversible.

The most basic case of a quantum walk consists in splitting the atom into two trajectories once and recombining it again. This scheme is fully equivalent to the realization of a single atom Mach-Zehnder interferometer. In our DQSIM experiments, we could realize precisely this device. The single atom interferometer could detect inertial accelerations at the level of 0.0001 g (g is the gravity acceleration).

Quantum superpositions of trajectories (in coordinate space as well as in parameter space) are at the heart of quantum physics. Leggett and Garg have formulated a criterion how to quantitatively test alternative theories like the Ghirardi-Rimini-Weber theory or Penrose gravitationally induced collapse theory, which extend quantum mechanics to include a macro-realistic assumption: particles of sufficiently large mass must be at all times on a specific trajectory, thus excluding the superposition of macroscopically distinct states. Under the assumption of macro-realism, if an experiment checks whether a particle is absent from a selected partial trajectory (“ideal negative measurement”), the dynamic evolution should continue without alteration. The DQSIM experiment has allowed us to implement this fundamental test of the quantum superposition principle, by means of which we disprove—in the spirit of falsification experiments of the philosopher Karl Popper—macro-realistic theories for massive particles on the scale of the mass of cesium atoms.

Atoms “walking” in an optical lattice resemble electrons moving in the conduction band of a metal. Electric fields can be simulated by applying external inertial forces onto the atoms, which is accomplished by simply accelerating the optical lattice. The acceleration is imprinted by controlling the frequency difference of two counterpropagating optical waves forming the optical lattice with radiofrequency precision. A plethora of phenomena including the band structure of the system, Bloch oscillations and Anderson-like localization could be observed by appropriate tuning of the radiofrequency just using the same experimental set-up! One interesting aspect of the DQSIM scheme is the realization of the extreme strong field regime, which is has no equivalent in conventional (continuous-time) metals. This regime originate from the discrete-time nature of the quantum walk, which evolves through discrete steps repeated periodically in time. In this extreme limit, which corresponds to an external field accelerating a particle from edge to edge of the Brillouin zone within one step of the walk, the quantum walks for zero and extremely high acceleration are identical.

DQSIM has constructed a new apparatus which opens the route to investigate 2D quantum walk phenomena. Interesting foreseen situations comprise quantum Hall physics with artificial magnetic fields, extensions into topological quantum walks, many-particle interacting quantum walks and more. The apparatus features several novel highlights as an experimental device including a novel NA=0.92 microscope objective lens with long working distance, a novel glass cell with ultralow birefringence, and a dynamic polarization synthesizer, which outperforms electro-optical devices for the control of the polarization state of light.

The future of DQSIM looks very bright on its road to the long-term goal – the realization of universal quantum computation processors and quantum cellular automata.

Our quantum walkers (for DQSIM, neutral cesium atoms) behave effectively as spin one-half particles, where the up and a down pseudo-spin components are implemented using long-lived internal states. In the experiment, quantum walkers resemble individual single-atom clocks moving around in an optical lattice and exploring the “quantum world”. Their quantum motion is controlled by a rotation of the pseudo spin of the atoms, which is called “coin” operation in analogy with the random walk of the Galton board. Atoms in spin up move e.g. to the left, those in spin down to the right, and spin superposition states result in split trajectories which depend on the spin rotation. In stark contrast to the random walk situation, however, the quantum dynamics is fully reversible.

The most basic case of a quantum walk consists in splitting the atom into two trajectories once and recombining it again. This scheme is fully equivalent to the realization of a single atom Mach-Zehnder interferometer. In our DQSIM experiments, we could realize precisely this device. The single atom interferometer could detect inertial accelerations at the level of 0.0001 g (g is the gravity acceleration).

Quantum superpositions of trajectories (in coordinate space as well as in parameter space) are at the heart of quantum physics. Leggett and Garg have formulated a criterion how to quantitatively test alternative theories like the Ghirardi-Rimini-Weber theory or Penrose gravitationally induced collapse theory, which extend quantum mechanics to include a macro-realistic assumption: particles of sufficiently large mass must be at all times on a specific trajectory, thus excluding the superposition of macroscopically distinct states. Under the assumption of macro-realism, if an experiment checks whether a particle is absent from a selected partial trajectory (“ideal negative measurement”), the dynamic evolution should continue without alteration. The DQSIM experiment has allowed us to implement this fundamental test of the quantum superposition principle, by means of which we disprove—in the spirit of falsification experiments of the philosopher Karl Popper—macro-realistic theories for massive particles on the scale of the mass of cesium atoms.

Atoms “walking” in an optical lattice resemble electrons moving in the conduction band of a metal. Electric fields can be simulated by applying external inertial forces onto the atoms, which is accomplished by simply accelerating the optical lattice. The acceleration is imprinted by controlling the frequency difference of two counterpropagating optical waves forming the optical lattice with radiofrequency precision. A plethora of phenomena including the band structure of the system, Bloch oscillations and Anderson-like localization could be observed by appropriate tuning of the radiofrequency just using the same experimental set-up! One interesting aspect of the DQSIM scheme is the realization of the extreme strong field regime, which is has no equivalent in conventional (continuous-time) metals. This regime originate from the discrete-time nature of the quantum walk, which evolves through discrete steps repeated periodically in time. In this extreme limit, which corresponds to an external field accelerating a particle from edge to edge of the Brillouin zone within one step of the walk, the quantum walks for zero and extremely high acceleration are identical.

DQSIM has constructed a new apparatus which opens the route to investigate 2D quantum walk phenomena. Interesting foreseen situations comprise quantum Hall physics with artificial magnetic fields, extensions into topological quantum walks, many-particle interacting quantum walks and more. The apparatus features several novel highlights as an experimental device including a novel NA=0.92 microscope objective lens with long working distance, a novel glass cell with ultralow birefringence, and a dynamic polarization synthesizer, which outperforms electro-optical devices for the control of the polarization state of light.

The future of DQSIM looks very bright on its road to the long-term goal – the realization of universal quantum computation processors and quantum cellular automata.