Manipulating Quantum Pathways of Matter by Coherent Nonlinear
Spectroscopy with Classical Fields and Entangled Photons
Shaul Mukamel and Oleksiy Roslyak
Chemistry department, University of California Irvine, USA
E-mail: smukamel@uci.edu
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 twophoton-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
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(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.
1. “Nonlinear Spectroscopy with Entangled Photons Manipulating Quantum Pathways of Matter ”,
O. Rosyak, C. Marx and S. Mukamel, Phys. Rev. A. (In press, 2009).
2. “Photon Entanglement Signatures in Homodyne Detected Difference Frequency Gene”, O.
Roslyak and S. Mukamel, Optics Express 17, 1093 (2009).
3. “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.
4. “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).
IMA, University of Minnesota, MN, March 1-5, 2009