Controlling isolated quantum particles (5)

The refractive index effect induced by a non-resonant atom on the cavity field was first analyzed. This is the atom's feedback action on the field, complementary to the dispersive effect produced by the field on the atomic dipole. We simply show that a single atom induces on the phase of a coherent field a phase shift equal to that produced by a single photon on the atom's dipole. This phase shift can thus reach a value of 180° on a field containing, on average, several photons. What's more, the sign of the phase shift changes depending on whether the atom passes through the cavity in the e or g state. If the atom is prepared before entering C by a_R1 microwave pulse in a superposition of these two states, a coherent field initially injected into C acquires two phases at once when the atom crosses the cavity, and the states of the atom and the field are entangled. This is a situation analogous to Schrödinger's description of a cat suspended between life and death by its coupling to a radioactive atom undergoing decay from an excited state to a final de-excited state. The field can also be seen as a device measuring the atom's energy, its phase taking on two values associated with the two possible values of this energy. After the atom has exited the cavity, its states are mixed again by a second microwave pulse in_R2. As a result, the final detection of the atom in e or g provides no information about the atom's state when it passed through the cavity. Quantum ambiguity is thus preserved, and the field is then projected by the measurement of the atom in a superposed state with two different phase components. This state is known as a Schrödinger cat.

The coherence of this superimposed state can be tested using a second probe atom passing through C after the first, and subjected to the same microwave pulses at the entrance and exit of the cavity. By analyzing the correlations between the finally detected states of the two atoms, the first preparing the Schrödinger cat and the second probing it, we obtain an exponentially decreasing signal that measures the system's loss of coherence as a function of time. This quantitative study of decoherence confirmed the theoretical predictions of this phenomenon and showed that decoherence is faster the more photons the field contains.

In the last part of the lesson, we briefly recalled how we could completely reconstruct the state of the field in the cavity using quantum tomography. This involves combining QND measurements of the field with translations in its phase space obtained by mixing it (homodyning) with coherent fields of various phases and amplitudes. The results of numerous measurements carried out on a large number of copies can be used to reconstruct the Wigner function of the field, be it coherent fields, fields with defined numbers of photons or Schrödinger cat states. By reconstructing the Wigner functions of cats at different times after their preparation, the ENS-Collège de France group has been able to produce movies of the decoherence, showing how the Wigner function, which initially shows interference bangs signalling the existence of coherence between the two components of the "cat", eventually evolves towards a statistical mixture in which these bangs have disappeared.