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
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
Photon Statistics for Two Square Pulses
time
– delay interval
t1
pump
t3t2
probe
0
Semi-Classical
Scorpion
Coherent
Scorpion
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)