# 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

interference.

(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).

MSC Code:

49M37

Keywords: