Delft University of Technology, 24 September 2001

**Quantum Superpositions of Persistent Josephson Currents**

This thesis presents experimental research on Josephson junction devices that behave quantum mechanically. The devices are formed by micrometer-sized superconducting islands, that are interconnected by a Josephson tunnel junction: a thin insulating layer between two superconductors. With current microfabrication technology, it is possible to make very clean and well-defined junctions. The behavior of such high-quality junctions is defined by two parameters. One is the junction capacitance, which results from the parallel-plate geometry of the tunnel junctions. The second is the junction's Josephson coupling energy, which is a measure for the tunnel coupling between the two superconducting electrodes, and it determines the supercurrent that can flow through the junctions. The supercurrent through the junction is directly related to a phase coordinate of the junction. Circuits with Josephson junctions have charge and current coordinates that are conjugate variables. This leads to quantum mechanical behavior of these devices if the energy scale for the Josephson effect and charging effects are comparable. Due to the superconductivity in these devices, the dynamics is much better decoupled from a dissipative environment than that of other solid-states devices.

The research that is presented in this thesis aimed at investigating whether Josephson junction circuits can have quantum coherent dynamics with a long decoherence time. This question was inspired by three, partly overlapping research themes that are currently of interest in the scientific community.
Firstly, if quantum coherent dynamics of Josephson junction circuits could be accurately controlled, these systems would be a promising candidate for realizing a quantum computer. An advantage of a quantum computer based on small Josephson junction circuits is that the technology for expanding such a system to a large-scale integrated computer is already available. Secondly, the current degrees of freedom of these circuits are macroscopic, in the sense that they correspond to the center-of-mass motion of a very large number of microscopic charge carriers. Josephson junction circuits are therefore unique systems for testing the validity of quantum mechanics at a macroscopic scale. Thirdly, it is very interesting that these devices are artificially fabricated quantum systems. With the technology that is applied, it is possible to engineer and control them in a wide parameter range, and to couple the systems in a controlled manner to environmental degrees of freedom. This allows for detailed research on the boundary between classical and quantum physics, and decoherence.

We report research on two different devices. The first part of this thesis concentrates on a small loop with three Josephson junctions. The second part reports work on a small two-dimensional Josephson junction array with two coupled loops. The two systems have in common, that the loops carry persistent Josephson currents when a small magnetic field is applied to the loops.

The Josephson energy of the three-junction loop forms a double-well potential when the magnetic flux in the loop is close to half a superconducting flux quantum. The states at the bottoms of the two wells correspond to persistent-current states of opposite polarity.
This system was realized in the regime where the junctions' Josephson energy was about a factor fifty larger than the energy scale for single-charge effects.
For this ratio a subtle balance is struck in this device. The system's charging effects are still significant, and allow for quantum tunneling through the barrier between the two stable persistent-current states. At the same time, this large ratio allows for engineering the system such that it is very insensitive to the influence of background charges in the solid-state environment of the loop.
We performed microwave-spectroscopy experiments which demonstrated that this system is an artificially fabricated quantum two-level system; we observed narrow resonance lines that resulted from microwave-induced quantum transitions between the quantum levels. An anti-crossing of the quantum levels proved that quantum superposition states of the two macroscopic persistent-current states occur in this system. The system was measured by placing it in a DC-SQUID magnetometer. The measurement process and the resulting noise is analyzed in detail.

The small two-dimensional arrays were studied in the regime where the energy scales for Josephson effect and single-charge effects were comparable.
The arrays had a self-dual geometry: They can be described as two coupled islands with a high charging energy, but also as two coupled loops that can each carry a persistent current. The latter description corresponds to that of an array in which the meshes can contain a vortex.
The arrays had a ground state with comparable quantum fluctuations in the charge and Josephson phase due to the Heisenberg uncertainty relation between these coordinates.
In this regime, the devices are very sensitive to the influence of background charges in the environment. However, with capacitively coupled gate electrodes we succeeded in obtaining accurate control over the
single-Cooper-pair effects in this system. The Josephson effect could be controlled by applying a small magnetic field to the loops. This allowed for a controlled study of the trade-off between quantum fluctuations in the Josephson phase coordinates and charge coordinates of the array, and our results show that superpositions of charge and vortex states occur in the array in a dual fashion.
Microwave spectroscopy experiments were used to demonstrate a discrete set of excited quantum levels in these devices.

From the research that is presented here one can conclude that it is possible to realize well-defined quantum systems with Josephson junction circuits.
Quantum superpositions of charge and persistent-current states have been demonstrated, and transitions between quantum levels can be controlled with resonant microwave radiation. These results form a good basis for future experiments that investigate whether quantum coherent dynamics with a long decoherence time can be realized with these devices. This requires quantum-state control with pulsed microwaves.
De technology for such experiments, as well as for experiments on circuits that contain multiple coupled quantum systems, is available.

*Caspar van der Wal*

*Delft, June 2001*