Manipulating quantum pathways of matter by coherent nonlinear spectroscopy with classical fields and entangled photons

Wednesday, March 4, 2009 - 10:40am - 11:10am
EE/CS 3-180
Shaul Mukamel (University of California)
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

  • 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.

  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,

  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,

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