In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.

We start from quantum mechanical laser equations which were derived in a previous paper for an inhomogeneously broadened laser and which contain in particular the noise sources due to cavity losses, vacuum fluctuations, interaction with phonons and nonlasing photons and the pump. For the example of frequency locking caused by the nonlinear polarization we derive a quantum mechanical Langevin equation for the relative phase angle ~,= 2q/2-q/i--N3, where q/l, ~,//2, 1[I3 are the total phases of three axial modes which would be equally spaced in the unloaded cavity. In the resulting equation = 6 --t7 sin ~ +f (t) (1) the fluctuating forcef(t) is Markoffian and Gaussian, the second moment being given by d (t --t') 2 (F 1 + F 3 + 4/'2), where Fj is the linewidth of the individual unlocked mode, j.Eq. (1) is solved by Fokker-Planck techniques and numerical results are represented for the characteristic function of order 1. Furthermore the mean locking time is represented in a graphical plot. The results are also applicable to two modes which are coupled to each other, e.g. by loss modulation. In this case the second moment off(t) is given by 8(t

We present a quantum mechanical nonlinear treatment of the phase and amplitude flucutations of gas lasers, i.e. lasers with moving atoms, and of solid state lasers with an inhomogeneously broadened line. The atoms may possess an arbitrary number of levels. As in our preceding papers the noise due to the pump, incoherent decay, lattice vibrations or atomic collisions, as well as due to the thermal and zero point fluctuations of the cavity is completely taken into account. The linewidth (due to phase diffusion), and the intensity fluctuations (due to amplitude noise) are essentially expressed by the threshold inversion, the unsaturated inversion and the saturated population numbers of the two atomic levels, which support the laser modes. Our results apply to the whole threshold region and above up to essentially the same photon number, to which the previous semiclassical theories of inhomogeneously broadened lasers were applicable. For the example of a two-level system we also demonstrate the application of a new technique which allows us to eliminate rigorously the atomic variables (operators), yielding a set of nonlinear coupled equations for the lightfield operators alone. If the elimination procedure is carried out only partially and additional approximations are made, we find essentially the rate equations of McCtJMB~R, in a form derived by LAx. When we neglect noise, the nonlinear equation may be solved exactly in the case of single mode operation. By a suitable expansion of the exact multimode equations we find a convenient set of equations, which reduce in the noiseless case to those derived and used previously by HAKEN and SAU~RMANN as well as LAMB.

As an application of a preceding paper t we set up a Fokker-Planck equation with quantum mechanically defined dissipation and fluctuation coefficients for a distribution function of the atomic variables (dipole moments and level occupation numbers) as well as of the lasing light amplitude in a laser with a homogeneously broadened line. Since the nonlinear coefficients can be linearized in appropriate coordinates well below and well above threshold, the equation can be solved with the Wang-Uhlenbeck method. Then it is easy to obtain correlation functions, spectral densities and expressions for linewidth.

Entanglement of photons is a very active field of research in many theoretical as well as experimental respects. On the first glance photons seem to be a suitable quantum system rather for quantum communication than for quantum computation. But despite the lack of photonphoton interaction it was shown that scaleable quantum computation with photons is indeed possible by the use of linear optics employing photon detection and ancilla qubits1. This includes, among other things, also the preparation and the analysis of multi-particle entangled states, where photons allow the most detailed analysis so far.We experimentally implemented a probabilistic quantum controlled-phase gate for photons operating on the polarization degree of freedom. This two qubit gate applies a phase shift of or to a target photon, conditioned on the 2 polarization of a control photon. Its principal architecture follows the approach of J. L. O'Brien et al. who uses however rail logic encoding.Our gate is realized by overlapping the input photons on a beam splitter with polarization dependent splitting ratio (TH= 1, TV= 1/3). The phase is introduced by a second order interference in case two vertically polarized photons are passing the gate. In order to ensure polarization independent weighting coefficients for the output states of all possible input combinations, two beam splitters with reversed splitting ratio (TH=1/3, TV= 1) are placed after each output of the overlap BS. (see FIG. 1) The simplicity of our scheme allows the usage of the gate for various applications. In particular it enables us to perform a full Bell-State analysis and by this the accomplishment of a complete teleportation experiment. We teleported horizontal, vertical, +45°, and right circular polarized photon states from which we could deduce a full process tomography of our gate for detection of all four Bell states. Whereas in the Bell-State analysis the gate maps an entangled state onto a product state, it can be used as well in the opposite way for an entangling operation. We exploit this fact to generate a special four qubit entangled state, the socalled four-photon cluster state. For this purpose we apply the gate on two photons of two different EPR pairs originating from spontaneous parametric down conversion processes. The resulting experimental state shows a fidelity of 74.4 % +/-1.2 % to the theoretically expected one. By the experimental violation of a specially tailored Bell Inequality3 we are able to proof its non-locality and delimit it from a GHZ state. We demonstrate its genuine fourphoton entanglement by a witness measurement. Furthermore we characterize the generated state by the study of its remarkable entanglement persistency properties with respect to the measurement and the loss of one and two particles. As the cluster state is the key resource of one way quantum computing our work represents an important step forward for the use of photons in information processing tasks besides pure communication applications. IF (A=2nm) beam spliner IF (A=2nm) A/2...