Joint work with Oleksiy Roslyakk (Chemistry department, University of California Irvine, USA). Nonlinear optical spectroscopy is commonly formulated semi-classically, i.e. letting a quantum material interact with classical fields. The key quantity in this approach is the nonlinear polarization, characterizing the microscopic response of the material to the incoming fields. Its calculation can be based on either the density matrix or the wave function. The former involves forward propagation in real time and is represented by double sided Feynman diagrams in Liouville space, whereas the latter requires forward and backward propagation in Hilbert space which is carried out on the Schwinger-Keldysh closed time path loop (CTPL). Such loops are extensively used in quantum field theory of non-equilibrium states, but double-sided Feynman diagrams have become a practical tool for the design and analysis of time-domain nonlinear optical experiments. Several fundamental ambiguities which arise in the semi-classical formulation regarding the intuitive interpretation of optical signals are resolved by combining the CTPL with a quantum description of the laser fields. In nonlinear spectroscopy of single molecules, for example, the signal cannot be given in terms of a classical response functions as predicted by the semi-classical theory. Heterodyne detection can be viewed as a stimulated process and does not require a classical local oscillator. The quantum nature of the field requires the introduction of superoperator nonequilibrium Green’s functions (SNGF), which represent both response and spontaneous fluctuations of the material. This formalism allows the computation of nonlinear optical processes involving any combination of classical and quantum optical modes. Closed correlation-function expressions are derived for the combined effects of causal response and non-causal spontaneous fluctuations. Coherent three wave mixing (sum frequency generation (SFG) and parametric down conversion (PDC)) involving one and two quantum optical modes respectively, are connected to their incoherent counterparts: two-photon-induced fluorescence (TPIF) and two-photon-emitted fluorescence (TPEF). We show how two-photon absorption and homodyne detected difference frequency generation conducted with entangled photons can be used to manipulate interference effects and select desired Liouville space pathways of matter. Recently several groups have applied entangled photon pairs in nonlinear spectroscopy (near resonance homodyne detected sum-frequency generation (SFG), two photon induced fluorescence (TPIF) and two-photon absorption (TPA). It was demonstrated that the normally quadratic scaling of the signal with the intensity of the incoming field becomes linear when using entangled photons. This indicates that the two photons effectively act as a single particle, interacting with matter within a narrow time window. This opens new ways for manipulating nonlinear optical signals and revealing new matter information otherwise erased by interference. Processes involving an arbitrary number of classical and quantum modes of the radiation field are treated within the same framework. Loop diagrams can be used to describe all incoherent and coherent (cooperative) signals. A unified approach is provided for both resonant and off-resonant measurements. In the latter the material enters as a parameter in an effective Hamiltonian for the field. Nonlinear spectroscopy conducted with resonant classical fields only accesses the causal response function. Quantum fields reveal the broader SNGF's family which carry additional information about fluctuations. Spectroscopy with quantum entangled fields may be described. "Nonlinear Spectroscopy with Entangled Photons Manipulating Quantum Pathways of Matter," O. Rosyak, C. Marx and S. Mukamel, Phys. Rev. A. (In press, 2009). "Photon Entanglement Signatures in Homodyne Detected Difference Frequency Gene," O. Roslyak and S. Mukamel, Optics Express 17, 1093 (2009). "Nonlinear Optical Spectroscopy of Single, Few and Many Molecules; Nonequilibrium Green’s Function QED Approach," C.A. Marx, U. Harbola and S. Mukamel, Phys. Rev. A. 77, 022110, 2008. "A Unified Description of Sum Frequency Generation, Parametric Down Conversion and Two Photon Fluoresence," O. Roslyak, C. Marx and S. Mukamel, Molecular Physics. (In press, 2009).