Trapped Ions

Right after the discovery of Shor's factoring algorithm in 1994, trapped ions interacting with laser light were identified as one of the most promising candidates to build a small--scale quantum computer. The reason is that, for many years, the technology to control and manipulate single (or few) ions had been very strongly developed in the fields of ultrahigh precision spectroscopy and atomic clocks. In particular, ions can be trapped and cooled such that they remain practically frozen in a specific region of space; their internal states can be precisely manipulated using lasers, and one can perform measurements with practically 100% efficiency; they also interact with each other very strongly due to the Coulomb repulsion, and they can, at the same time, be decoupled from the environment very efficiently.

Feedback cooling

Motivated by experiments in the group of R. Blatt and collaborators, we have developed a theory of quantum feedback cooling of a single ion trapped in front of a mirror [1] (The schematic setup is given in the first picture on the right). By monitoring the motional sidebands of the light emitted into the mirror mode one infers the position of the ion, and acts back with an appropriate force proportional to the momentum to cool the ion. This is done in analogy to the classical case of a harmonic oscillator which is damped by a viscous (velocity-dependend) force. We have derived a feedback master equation, which provides us with cooling times and final temperatures as a function of feedback gain and various system parameters. We have obtained excellent agreement with the experimental data [3]. To overcome the limitations in this setup posed by spontaneous emission, we propose a method to infer the position of the atom using electromagnetically induced transparency (EIT) [2] (Setup in the second picture on the right side), which yields a way of theoretically cooling to the vibrational ground state.

 

[1] V. Steixner, P. Rabl, P. Zoller, Quantum feedback cooling of a single trapped ion in front of a mirror, Phys. Rev. A 72, 043826 (2005).

[2] P. Rabl, V. Steixner, P. Zoller, Quantum-limited velocity readout and quantum feedback cooling of a trapped ion via electromagnetically induced transparency, Phys. Rev. A 72, 043823 (2005).

[3] P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, P. Zoller, Feedback cooling of a single trapped ion, Phys. Rev. Lett. 96, 043003 (2006).

Fast quantum gates and state-dependent forces

In collaboration with I. Cirac and J. J. García-Ripoll (MPQ-theory group) we have developed an optimal control theory employing time varying laser light which allows not only to overcome existing speed limits for 2 qubit quantum gates, but also provides a systematic tool for generation of interesting entangled states and simulation of spin models [4]. In a further publication [5] we have developed a unified framework to study the coherent control of trapped ions subject to state-dependent forces. Taking different limits in our theory, we can reproduce previous designs of quantum gates and propose a different design of fast gates based on continuous laser beams. We demonstrate how to simulate Ising Hamiltonians in a many ions setup, and how to create highly entangled states and induce squeezing. Finally, in a detailed analysis we identify the physical limits of this technique and study the dependence of errors on the temperature.
We also propose in [6] a scheme for quantum logic with neutral atoms stored in an array of holographic dipole traps where the positions of the atoms can be rearranged by using holographic optical tweezers. In particular, this allows for the transport of two atoms to the same well where an external control field is used to perform gate operations via the molecular interaction between the atoms. We show that optimal control techniques allow for the fast implementation of the gates with high fidelity.

For a review see: J. I. Cirac, P. Zoller: Qubits, Gatter und Register, Physik Journal 11, 31 (2005).

[4] J. J. García-Ripoll, P. Zoller, J. I. Cirac, Coherent control of trapped ions using off-resonant lasers, Phys. Rev. A 71, 062309 (2005).

[5] J. J. García-Ripoll, P. Zoller, J. I. Cirac, Quantum information processing with cold atoms and trapped ions, J. Phys. B: At. Mol. Opt. Phys. 38, 567-578 (2005).

[6] U. Dorner, T. Calarco, P. Zoller, A. Browaeys, P. Grangier, Quantum logic via optimal control in holographic dipole traps, J. Opt. B: Quantum Semiclass. Opt. 7, 341-346 (2005).

Interfacing ion traps and solid state systems

We have studied a hybrid quantum computing scheme where the hybrid qubit is made of an ion trap qubit serving as the information storage (due to long decoherence times) and a solid-state charge qubit serving as the quantum processor (due to the short gate times achievable). The connection is given by a superconducting cavity. We study the decoherence, coupling and scalability of such a hybrid system.

L. Tian, R. Blatt, P. Zoller, Scalable ion trap quantum computing without moving ions, Eur. Phys. J. D 32, 201-208 (2005).