Probing fast dynamics of single molecules: non-linear spectroscopy approach Eli Barkai Department of Physics Bar-Ilan University Shikerman, Barkai PRL 99, (2007) Shikerman, Barkai JCP 129, (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, (2003)

Single-molecule Pump-Probe experiment Van Dijk,… van Hulst, PRL 94, (2005) time

Indistinguishable pair of photons from single Quantum Dot time0 2 nano seconds t1t1 t3t3 t2t2 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 t1t1 pump t3t3 t2t2 probe Pulses are short :   no photons are emitted during the pulses  state of the molecule does not change during the pulse events 0

pumppumpprobeprobe Classical Taurus The outcome of the experiment does not depend on the path

Semi-Classical Scorpion Coherent Scorpion Quantum Scorpion pumppumpprobeprobe The outcome of the experiment depends on the path

Optical Bloch Equations Molecule’s density matrix elements “Single photon emission” operator Ω = -E 0 ·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 t1t1 t3t3 t2t2 0

Photon-propagators for the delay interval

Photon Statistics for Two Square Pulses time  – delay interval t1t1 pump t3t3 t2t2 probe 0 Semi-Classical Scorpion Coherent Scorpion

Semiclassical and Coherent paths t1t1 t2t2 t3t3 time t0t0  – 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 t1t1 pump t3t3 t2t2 probe 0

Linear CW Spectroscopy : Impulsive Limit Ω»ν For π /2 pulses the influence of the coherent paths is strongest time  – delay interval t1t1 pump t3t3 t2t2 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 t1t1 t0t0 pump t3t3 t2t2 time  In Selective Limit temporal resolution 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  P 0 Cla ,  P 1 Cla  and  P 2 Cla  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= const Fast 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, (2007) Shikerman, Barkai JCP 129, (2008)

Coherent state evolution in a superconductive Qubit

Quantum Paths of Two Pulses

Two-state process : Exact Solution for π -pulses

Pump and Probe Technique  « 1 t1t1 t0t0 pump t3t3 t2t2 probe time t1t1 t0t0 pump t3t3 t2t2 probe time  