# Quantum Information and Quantum Simulations with Trapped Ions (QuIQS)

**Quantum physics as developed by Bohr, Schrödinger, Heisenberg and many others provides us with a framework for accurately predicting the quantum dynamics of any non-relativistic physical system. In our experiments, we investigate the basic notions of quantum theory and the physical implications of storing information in quantum systems. Moreover, we want to find out whether it is possible to carry out quantum simulations, that is to engineer one quantum system for studying the physics of another quantum system.**

The discovery of quantum theory in the first decades of the twentieth century has provided physicists with a theory that has been immensely successful in explaining physical phenomena in many different areas of physics. The study of very simple quantum systems, like for example the hydrogen atom, the quantum harmonic oscillator, or the process of absorption and emission of single photons, played a crucial role in the development of the mathematical framework of quantum theory. On this basis, a theory was developed that defined in a highly abstract way the quantum state of a system, its dynamical evolution in time, the quantum measurement process and its back-action on the system.

### Testing quantum physics

Interestingly, in the early days of quantum mechanics, the complete body of evidence available for testing the predictions of the new theory was of a completely different nature: all experiments that could be carried out at that time and for decades to come were done on ensembles of quantum systems. In contrast, the direct experimental investigation of basic notions of quantum physics like the preparation of pure states, projective quantum measurements, and quantum entanglement is a fairly recent development that became more and more widespread in the last 30 years. The reason for this is of a technical nature: while it is very easy to solve the equations describing a simple quantum system like for example the quantum harmonic oscillator, it can be a challenge to engineer and measure such a quantum system and to prevent it from interacting with other quantum systems which would spoil its simple character. Here, the development of lasers and of traps for confining atoms and ions has made it possible to indeed engineer quantum systems that come very close to the idealizations studied in theory. With these systems, one can now test quantum physics at the most fundamental level by reliably and repeatedly preparing a quantum system in a pure quantum state, controlling its dynamical evolution and carrying out quantum measurements. With trapped ions, entanglement can be deterministically created and subsequently characterized. Moreover, this kind of quantum engineering has applications in metrology (in atomic clocks, for example) and might also be used in future quantum information processing devices.

### Quantum simulations

Another field of applications is quantum simulations that started in the 1980's when Richard Feynman and others proposed a new method for approaching quantum mechanical problems that are too hard to solve on ordinary computers. The idea is to simulate quantum effects of interest by using a more accessible quantum system as the computing device. This other system which might be easier to control and measure is engineered such that it mimics the system one wants to simulate. To date, only a few quantum systems can be controlled well enough to act as a quantum simulator: neutral atoms held in optical lattices are considered to be very interesting for simulating solid state physics. Trapped ions, on the other hand, offer an excellent control over all quantum degrees of freedom and can be measured extremely well which makes them interesting for building small quantum simulators using a bottom-up approach.

### Experimental setup

In our experiments, we work with linear strings of trapped calcium ions that are stored for hours or even days by a combination of radio-frequency fields and electrostatic potentials. Using ultrastable lasers, we cool the ion motion to the ground state of the potential provided by the ion trap and prepare the ions in particular quantum states. In particular, we can encode a quantum bit of information in a combination of two long-lived states. This information can be measured by detecting the fluorescence emitted by the ions when they are excited with laser light. As the absorption of a photon by an ion is accompanied by a momentum transfer from the light field to the ions, it is also possible to control the quantized motion of the ions using laser light and to entangle the ion motion with the ions’ internal states. Moreover, this kind of coupling makes it possible to create entanglement between the internal states of two or more ions. We have two ions of choice: the isotope ^{40}Ca^{+} provides us with an ion that is easy to handle which, however, is sensitive to fluctuating magnetic field. In contrast, the isotope ^{43}Ca^{+} has field-insensitive 'clock' states and allows for much longer coherence times. In this case, the price to pay is a much more complex level structure that needs to be handled.

Linear Paul trap |
Sketch of the experimental setup |

### Experimental highlights

**Quasiparticle spectroscopy: **We present and demonstrate a technique for performing spectroscopy of an engineered quantum-many body system that is well described by a long-range transverse Ising model. By exciting the system into close approximations of many-body spin waves, and observing the site-resolved dynamical response, we are able to characterise the system’s emergent quasiparticle excitations. The technique enables us to measure the quasiparticle dispersion relation and to detect interactions between a pair of quasiparticles [14].Read more...

**Engineering of quasiparticles: **A string of trapped and laser-cooled ions can be made to behave as an interacting quantum many-body system. We have implemented a long-range Ising model in a string of 7-15 ions and investigated the spreading of quantum correlations by quasi-particles in this artificial quantum system [13].** **Read more...

**Single Photon recoil: **In a new take on quantum logic spectroscopy, a sensitive spectroscopy method employing a Schrödinger cat state of motion is investigated. With sensitivity down to the single photon level, the new technique extends the range of transitions that can be investigate in the spectroscopy of atomic and molecular ions. The results of our work are published in the journal Nature Photonics. Read more...

**Digital quantum simulations:** The time evolution of a quantum system under the action of a Hamiltonian acting for a duration t can be simulated by a gate-based approach. Instead of trying to engineer the Hamiltonian of interest as it is done in the analog quantum simulation approach, the digital method is based on sequences of quantum gates that generate the same propagator as the Hamiltonian to be simulated for a given instance of time. By making use of the Trotter formula, very different Hamiltonians can be simulated with the same basic universal set of quantum gates as we demonstrated in an experiment with two to six ions [7]. Read more ...

**Quantum simulation of the Dirac equation: **The physics of a free relativistic quantum particle can be simulated using a non-relativistic trapped ion. We implemented this proposal that makes use of the laser-ion interactions coupling the internal and motional states of a trapped ^{40}Ca^{+} ion and developed a technique for measuring observables of interest without the need of reconstructing the complete quantum state of the ion. In this proof-of-principle quantum simulation [5] where the dynamics of the system can be easily predicted on a computer, we had the possibility of comparing experimental results with theoretical predictions and thus to test the concept of a quantum simulator. The experimental findings convincingly reproduce theoretical predictions like for example Zitterbewegung, the trembling motion of a Dirac particle predicted by Erwin Schrödinger that arises from interference of wave functions with positive and negative energies.Read more...

**Tests of hidden-variable theories:** This fundamental quantum gate can not only be used for creating entanglement but also for carrying out quantum non-demolition measurements of non-local observables. For example, it is possible to detect whether two ions are in the same quantum state or in different states without getting any information about the particular states of ion 1 or ion 2. For this, the relevant information is first transferred to one of the ions by an entangling gate before this ion is read out and the inverse gate operation is applied. In this way, one bit of information is gained that carries no information at all about the individual states of the ions. This technique makes it possible to carry out consecutive quantum non-demolition measurements on an individual quantum system [4] and to use correlations between the measurement results to test noncontextual hidden-variable theories of quantum physics in a way that is independent of the input state. Read more ...

**High-fidelity quantum gates:** Different techniques exist for entangling ions and there are a handful of laboratories worldwide that have demonstrated ion entanglement. In Innsbruck, we have shown that entanglement can persist in trapped ions on the timescale of seconds [1] and that it can be created very reliably. More specifically, we implemented a quantum gate operation originally proposed by K. Mølmer and A. Sørensen and demonstrated that we could entangle a pair of ions with a fidelity of better than 99% [2] ; that is, in 99 out of 100 cases, the operation succeeded. Also, we showed that the ion motion does not even need to be prepared in the lowest quantum state for this [3]. Read more ...

## Project Members

- Christoph Steinlechner (Master student)
- Tiffany Brydges (PhD student)
- Petar Jurcevic (PhD student)
- Christine Maier (PhD student)
- Ben Lanyon (Senior scientist)
- Christian Roos (Project leader)
- Rainer Blatt (Group leader)

Former members: Heng Shen, Cornelius Hempel, Sebastian Schunke, Florian Zähringer, Markus Rambach, René Gerritsma, Gerhard Kirchmair, Jan Benhelm, Timo Körber, Umakant Rapol.

## References

[1] C. F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)

[2] J. Benhelm et al., Nature Phys. 4, 839 (2008)

[3] G. Kirchmair et al., New J. Phys. 11, 023002 (2009)

[4] G. Kirchmair et al., Nature 460, 494 (2009)

[5] R. Gerritsma et al., Nature 463, 68 (2010)

[6] F. Zähringer et al., Phys. Rev. Lett. 104, 100503 (2010)

[7] B. Lanyon et al., Science 334, 57 (2011)

[8] R. Blatt and C. F. Roos, Nature Phys. 8, 277 (2012)

[9] C. Hempel et al., Nature Photon. 7, 630 (2013)

[10] B. P. Lanyon et al., Phys. Rev. Lett. 111, 100504 (2013)

[11] B. P. Lanyon et al., Phys. Rev. Lett. 111, 210501 (2013)

[12] B. P. Lanyon et al., Phys. Rev. Lett. 112, 100403 (2014)

[13] P. Jurcevic et al., Nature 511, 202 (2014)

[14] P. Jurcevic et al., Phys. Rev. Lett. 115, 100501 (2015)

## Further Reading

PhD theses of Jan Benhelm, Gerhard Kirchmair, Florian Zähringer and Cornelius Hempel