Dynamic Zeno manipulation and synthesis of quantum oscillator states

Abstract

The sixth lesson was devoted to describing the manipulation of radiation states by the dynamic Zeno effect in a cavity quantum electrodynamics device with Rydberg atoms. The Zeno effect and its variants correspond to an inhibition of the coherent evolution of a quantum system subjected to repeated perturbation. These include coupling to a measuring device, which periodically throws the system back to its initial state, or the repeated application of pulses that scramble the quantum phases responsible for the system's evolution and block it (so-called " bang-bang " methods). An analysis of these effects has already been presented in the 2004-2005 lecture (lesson⁴ ). Beyond its paradoxical aspect ("observing a system prevents it from evolving"), the Zeno effect can be used to manipulate a system's Hamiltonian to prepare certain quantum states. It can be used to freeze the evolution of certain states, or to block this evolution in a subspace of the state space, while other states, located outside this subspace, can evolve freely. In the case of a harmonic oscillator, for example, phase space tweezers can be used to manipulate and synthesize largely arbitrary states.

The lesson began by recalling the characteristics of the Zeno effect in its various forms. We then described in detail a proposed experiment aimed at restricting the evolution of a field in a part of its Hilbert space by making particular repeated measurements of its photon number. A measurement performed on a Rydberg atom in the cavity is intended to decide whether or not the field contains a well-defined number of photons. This binary measurement, if repeated at regular intervals, should prevent the field from reaching this limit photon number. If the initially empty cavity is fed by a conventional source, the coherent field that builds up sees the support of its Wigner function reflected by a barrier corresponding to this limit value, and the field remains confined in phase space within an exclusion circle from which it cannot escape. When the support of its Wigner function approaches the limit, a 180-degree out-of-phase component appears, corresponding to the creation of a two-component "Schrödinger cat" state. The amplitude of the field thus oscillates within the exclusion circle, splitting periodically to create cat-like states of transient existence. If the carrier of the coherent field is initially outside the exclusion circle, it cannot enter it, and coupling with a classical source leads the field to "avoid" the exclusion circle, again with the possibility of creating transient "cat" states. If the field is initially in a "cat" state, a superposition of two coherent states, it should be possible to trap one of the components in an exclusion circle and, by coupling the cavity to a classical source, displace the free component while immobilizing the "trapped" component. By adiabatically varying the center of the exclusion circle creating this trap, it should be possible to manufacture genuine "tweezers" in phase space, enabling different coherent components to be moved independently. Properly handled, such tweezers should make it possible to synthesize arbitrary superpositions of non-overlapping coherent states.

We discussed in the lesson the practical possibilities of performing such experiments, which requires a novel experimental set-up. A Rydberg atom must remain in the cavity for a time of the order of milliseconds, so as to enable precise measurements of the spectrum of this atom "dressed" by the cavity field. To carry out these experiments, we plan to couple the superconducting cavity to an atomic fountain of ultra-slow atoms, a device whose principle we described in this lesson.