A Study on the Applications of Atom Optics to Quantum State Measurement

Abstract

In this thesis we studied the role of atom optics In the reconstntction of the quantum state of the radiation field in high-Q cavities. The atomic diffraction method provides a unique tool for the measurement of field photon statistics. In this case, the momentum states of the atom serve as a tool that probes a quantum state. Raman-Nath and Bragg's interaction regimes yield distinct momentum distributions. Here we are concemed with both types of atomic beam diffraction and utilize them for the reconstruction of single mode and entangled multi-modes field-state. The single mode reconstruction scheme is based upon the Raman-Nath regime interaction. Among the multi-mode field-state reconstruction schemes one deal with the two modes in a single high-Q cavity while the other concems with the two modes in two separate cavities. The first scheme relies on the momentum distribution spectntm in Raman-Nath regime while the other utilizes the Bragg's diffraction of the atomic probe. The single mode reconstntction scheme covers both the resonant and off-resonant atom-field interactions. In the first part of the scheme, we use the resonant atoms and measure the corresponding momentum distribution . Our calculations explicitly suggest that the momentum distribution of the atoms is a sensitive function of the field photon statistics. We note that when vacuum Rabi frequency is increased the peaks in the momentum di stribution spectrum begin to resolve with enhanced momentum change. Each momentum peak cOlTesponds to a unique photon number. We exploit the obtained momentum distribution patterns for the reconstruction of the field photon statistics. As an example, we reconstruct the photon statistics of the coherent state. The same scheme is also executed with the off-resonant atom-field interaction and the Schrodinger-cat state is reconstructed in this case. Here we inject two-level atoms having detuning prepared initially in ground state Ib> to the standing wave cavity field. A narrow slit of width ox allows the atoms to interact with a small pOliion of the cavity field as requirement of the Raman-Nath regime interaction. For the reconstruction of the multi-mode field-state in a single high-Q cavity we utilize three-level atom in V configuration in such a way that one mode is resonant with the one transition and the other mode has resonance with the second transition. In this case we get a complex momentum distribution pertaining to simultaneous contribution from both the field modes because the atom may have x-interactions with mode A, and yinteractions with mode B. We get the peaks whenever set of joint photon numbers, equates to the momentum. The basic job in the current case is to resolve the momentum peaks and identify them corresponding to the different set of joint photon numbers. Here, we need to resolve a large number of peaks in momentum distributions spectrum that is why we need a high atom-field coupling. We investigate that the momentum peaks corresponding to the different set of joint photon numbers becomes separated if we take the large separation between the coupling constants. As the each peak in the momentum distribution spectrum corresponds to the unique set of joint photon numbers, so, by knowing the probabilities of the momentum states, we can determine the state of the cavity field. The Bragg's regime interaction scheme uses a two-level atoms interacting offresonantly with the standing wave field of the cavities in cascade fashion . After interacting with two cavities the atoms are detected in either of the two momentum states. The probability of finding the atom in anyone of the momentum state is the product of joint photon statistics and an oscillatory function exhibiting periodic maxima and minima. The argument of the oscillatory function contains the information of the joint photon numbers in two cavities. The mapping of decimation function over the joint photon statistics decimates it at the points where the product of the two functions is close to zero. This causes a reduced joint photon statistics in the cavities. The position of the minima changes with the interaction time. Each atom in different interaction time eliminates some photon numbers probabilities of the field distribution. After the reduction of the photon distribution to a set of joint photon number, the additional atoms send do not alter it. The reduced single set of joint photon numbers is then measured by sending fixed velocity atoms and detecting them in at a particular momentum state. We repeat the same process on an identical system until we get another set of joint photon number. The frequency of the joint photon number distribution is the reconstructed joint photon statistics. Photon statistics alone gives only the diagonal density matrix elements of the field-state. To get the information about the off-diagonal density matrix elements one needs to reconstruct the Wigner function. We propose to displace the photon statistics of the cavity field by injecting coherent state from a local classical oscillator. The momentum distribution of the displaced field-state gives the direct analogy of the Wigner function. To get the Wigner function of the entangled field-state, we propose to displace each mode by injecting coherent states 1(1'> and I,B> into the cavities. Repeating this procedure with identically prepared systems for many phase space points, one may find the Wigner function of the entangled field-state. The recovered photon statistics and the Wigner functions are in good agreement with the original photon statistics and the Wigner function.