Probing fast dynamics of single molecules: non-linear spectroscopy

approach

Eli Barkai

Department of Physics

Bar-Ilan University

Shikerman, Barkai PRL 99, 208302 (2007)Shikerman, Barkai JCP 129, 244702 (2008)

Outline

Influence of Spectral Diffusion on Photon Statistics

Impulsive and Selective limits

Fast modulation limit

Experiments

Photon statistics via Optical Bloch Equations

Stochastic Frequency Modulation – Spectral Diffusion

Time

–bare absorption frequency

-random function of time

Spectral Trail Investigates Slow Dynamics

E. Barkai , … L. Kador, PRL 91, 075502 (2003)

Single-molecule Pump-Probe experiment

Van Dijk,… van Hulst, PRL 94, 078302 (2005)

time

Indistinguishable pair of photons from single Quantum Dot

time0

2 nano seconds

t1 t3t2

Santori et al Nature 419, 594 (2002)

Spectral Diffusion leads to distinguishable photons

N. Katz et al Science 312, 1498 (2006)

Single Molecule Non-linear Spectroscopy

What are the physical limitations of the investigation of fast dynamics ?

How does the information gained by pulsed experiments differ from CW experiments ?

What are the fingerprint of coherence?

How to design the external laser field?

Merge SMS with NLS

Mukamel, Principles of nonlinear optical spectroscopy

Photon Statistics

Glauber, Mandel, Mollow, Zoller, Mukamel, Brown

E. Barkai, J. Jung, R. Silbey Annu. Rev. Phys. Chem. 55, 457 (2004)

Pump and Probe Setup

time

– delay interval

t1

pump

t3t2

probe

Pulses are short :

no photons are emitted during the pulses state of the molecule does not change during the pulse events

0

pupumpmppupumpmp

probprobee

probprobee

Classical

Taurus

The outcome of the experiment does not depend on the path

Semi-Classical

Scorpion

Coherent

Scorpion

QuantumScorpion

pupumpmppupumpmp

probprobee

probprobee

The outcome of the experiment depends on the path

Optical Bloch Equations

Molecule’s density matrix elements

“Single photon emission” operator

Ω = -E0·d/ħ - Rabi Frequency

-laser field time-dependence

Γ - spontaneous emission rate

`1

Path Interpretation

-Propagation without photon emissions

-Molecule’s state at time t conditioned by n photon emission events

Orthonormal Basis

Populations Coherences

Two Separated Pulses

Ω- Rabi frequency

time

Δ – delay interval

t1 t3t20

Photon-propagators for the delay interval

Photon Statistics for Two Square Pulses

time

– delay interval

t1

pump

t3t2

probe

0

Semi-Classical

Scorpion

Coherent

Scorpion

Semiclassical and Coherent paths

t1 t2 t3 timet0 – delay interval

The phase of the laser is important

Semiclassical Approximation

Laser phase is important

Ramsey experiment: laser’s phase coherence is preserved

Probability Density Function

Probability of emitting n photons

Photon statistics

n = 0, 1, 2

time

– delay interval

t1

pump

t3t2

probe

0

Linear CW Spectroscopy:

Impulsive Limit Ω»ν

For π /2 pulses the influence of the coherent paths is strongest

time

– delay interval

t1

pump

t3t2

probe

0

Two-State Poissonian Process – Exact Solution.

time

For the two-state process exact solution was found

For a stochastic Gaussian process numerical semi-classical approximation was obtained

Two-State Process –Selective Limit ν » Ω

t1t0pumpt3t2 time

In Selective Limit temporal In Selective Limit temporal resolution is foundresolution is found..

In Selective Limit temporal In Selective Limit temporalresolution is foundresolution is found..

Selective limit

Impulsive limit

Intermediate case

With selective pulses we distinguish between different stochastic processes. In the Impulsive Limit the photon statistics are independent of the stochastic

process

P0Cla, P1

Cla and P2Cla versus “bare” detuning

R, , R >> , R/²= const

T 1 – to ensure the excitation of the molecule R >> - hence >>

Fast modulation Impulsive Limit

R T << 1 – in order to provide constant detuning during the pulse events

Fast Modulation Limit

R, , R >> , R/²= constFast Modulation Limit

In the fast modulation limit the Kubo-Anderson correlation function reduces to the exponential factor, renormalizing the

decay rate of the coherent paths.

Summary

Nonlinear single molecule spectroscopy-a new tool.

The photon statistics is sensitive to the phase accumulated by the molecule during the delay.

In the Impulsive Limit the information on the spectral diffusion is contained only in the Kubo-Anderson correlation function.

In the Selective Limit the temporal resolution is found.

To benefit from this new method one must make careful choice of the pulse strength, duration and phase.

Shikerman, Barkai PRL 99, 208302 (2007)Shikerman, Barkai JCP 129, 244702 (2008)

Coherent state evolution in a superconductive Qubit

Two-state process : Exact Solution for π-pulses

Pump and Probe Technique « 1

t1t0pump

t3t2probetime

t1t0pump

t3t2probetime