Probing Dynamics of Single Molecules: Non-linear Spectroscopy Approach

Abstract

A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes are obtained. We examine under what circumstances the photon statistics emerging from such pump-probe set up provides new information on the stochastic process parameters, and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit, exhibit general behaviors of this new type of spectroscopy. We show, that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line-shape. Comment: 13 pages, 4 figures, 1 table

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arXiv:0809.1544v1 [physics.chem-ph] 9 Sep 2008
Probing Dynamics of Single Molecules: Non-linear Spectroscopy Approach
F. Shikerman, E. Barkai
Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel
A two level model of a single molecule undergoing spectral diffusion dynamics and interacting
with a sequence of two short laser pulses is investigated. Analytical solution for the probability
of n = 0,1,2 photon emission events for the telegraph and Gaussian processes are obtained. We
examine under what circumstances the photon statistics emerging from such pump-probe set up
provides new information on the stochastic process parameters, and what are the measurement
limitations of this technique. The impulsive and selective limits, the semiclassical approximation,
and the fast modulation limit, exhibit general behaviors of this new type of spectroscopy. We show,
that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful
results, the information on the photon statistics is contained in the molecule’s dipole correlation
function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained
within the selective limit depends on the both spectral shifts and rates and exhibits oscillations,
which are not found in the corresponding line-shape.
PACS numbers: 82.37-j, 82.53-k, 05.10.Gg, 42.50.Ar
I.INTRODUCTION
Recently van Dijk et al [1] reported the first experi-
mental ultra-fast pump-probe study of a single molecule
system. Unlike previous approaches to non-linear spec-
troscopy where only the ensemble average response to
the external fields is resolved [2], the new method yields
direct information on single molecule dynamics, gained
through the analysis of photon statistics. Although the
original experiment [1] was conducted on a molecule
undergoing a relatively simple relaxation process, the
potential of combining non-linear spectroscopy with
single molecule spectroscopy inspires many unanswered
questions: What are the limitations of the investigation
of fast dynamics? How does the information contained in
these experiments differ from the information contained
in simpler continuous wave experiments? How to design
the external control fields, so that needed information
on dynamics of molecules is gained? What is the finger
print of coherence in these types of experiments, and
how its influence on photon statistics is suppressed due
to dephasing processes? The answers to these questions
are important for better understanding of a wide variety
of physical phenomena and have implication in the
investigation of ultra-fast dynamics of molecules in the
condensed phase, of quantum properties of light, and
in the field of quantum information and computation
[1, 3, 4, 5, 6, 7, 8, 9]. Here we present a treatment based
on the stochastic Kubo-Anderson model [2, 10, 11, 12],
which yields general insights on the problem.
We consider a sequence of two identical laser pulses
interacting with a single molecule (or an atom, or a
quantum dot) undergoing a spectral diffusion process,
namely a molecule whose absorption frequency is
randomly modulated in time due to interaction with
a thermal bath.The electronic states of the single
emitter are modeled based on the two level system
approximation.Most single molecules have a triplet
state, however the life time of the triplet is much longer
than the time scales under consideration in this paper,
and it can be neglected. It is assumed that the pulses
are very short compared with the inverse rate R of the
spectral diffusion process, as well as with the inverse of
the radiative life-time of the emitter, Γ. The probability
of photon emissions during the pulse event is then
negligible, and a pair of pulses yields two photons at
most. Repeating the experiment many times one obtains
the probabilities ?P0?,?P1? and ?P2? of emitting 0,1
and 2 photons, where ?···? designates the average over
the stochastic modulation of the two level system’s
absorption frequency ω(t). In what follows we generalize
the results obtained in our earlier publication [12] by:
(i) establishing the general expressions for ?P0?,?P1? and
?P2? in the limit of long measurement times without any
restricting assumptions regarding the laser detuning,
(ii) comparing the photon statistics obtained for the
two state Kubo-Anderson and Gaussian processes.
We show, that under certain conditions this type of
photon statistics reveals important information on single
molecule dynamics, information which might be difficult
to obtain using other theoretical approaches to single
molecule spectroscopy [13, 14, 15, 16, 17, 18, 19, 20, 21].
The general expressions for the photon statistics are
obtained starting with the path interpretation of Mollow
and Zoller, Marte and Walls [22, 23] of the optical Bloch
equations [24]. We show, that depending on the char-
acteristics of the stochastic dynamics and the laser field
parameters, different types of non-linear spectroscopies
emerge. In particular, sensitivity to the phase accumu-
lated by the system in the delay interval between the
pulses is found, and impulsive and selective type of spec-
troscopies are considered in detail. The Kubo-Anderson
spectral diffusion process [10] used in this work is found
in many molecular systems [14, 19, 20, 21] and may be
easily detected, when the process is slow by means of
the spectral trail technique. Our goal is developing gen-

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time
Ω?t?
?
t?0 t1
t2t3
FIG. 1: The two square laser pulses as modeled in Eq. (6) on
the background of a schematic realization of spectral diffusion
process ω(t). T = t1 = t3−t2 is the width of single pulse and
∆ is the delay time between two subsequent pulses.
eral methods suitable for detection of the wider range
of dynamics. Some technical details of the calculations
skipped in the main text appear in Appendixes A, B and
C.
II.PHOTON STATISTICS
In our model two identical square pulses interact with
the two level system (see Fig. 1).
system is described by the density matrix, represented
by the 4-vector σ = (σee,σgg,σge,σeg)T. Here σee and
σggrepresent the populations of the excited and ground
states respectively and σge,σeg describe the coherences,
namely the off diagonal matrix elements of the density
matrix. For the sake of mathematical convenience we use
the following 4-dimensional basis [25]: |e? = (1,0,0,0)T,
which means that the system is in the pure excited state,
|g? = (0,1,0,0)T- the ground state, and |c? = (0,0,1,0)T
and |c∗? = (0,0,0,1)Tdescribe the coherences. The dy-
namical evolution of the density matrix in the presence
of the external laser field is given by the optical Bloch
equation [24]
The state of the
˙ σ = L(t)σ +ˆΓσ.(1)
The operator
L(t) =



−Γ
0
0
0
−iΩf(t)
iΩf(t)
iω(t) − Γ/2
0
iΩf(t)
−iΩf(t)
0
−iω(t) − Γ/2
−iΩf(t)
iΩf(t)
iΩf(t)
−iΩf(t)



(2)
describes the interaction of the system with the driv-
ing electromagnetic field through Ωf(t), where Ω =
−(1/¯ h)dge· E0is the Rabi frequency with E0- the am-
plitude of the electric field and dge= degthe transition
dipole moment of the two level system. The operator
ˆΓ = Γ|g??e| describes the transition from the excited
state into the ground state, due to spontaneous emission
with Γ designating the emission rate. Finally ω(t) is the
stochastic time dependent absorption frequency of the
system. The spectral diffusion process ω(t) is modeled
using the Kubo-Anderson approach:
ω(t) = ω0+ δw(t),(3)
where ω0 is the bare absorption frequency of the single
emitter, and δw(t) is a random function of time [2, 10].
We will assume the process δw(t) is stationary, its mean
is zero, its correlation function is
?δw(t0+ t)δw(t0)? = ν2ψ(t),(4)
where ψ(0) = 1 and ψ(∞) = 0. Later we will demon-
strate our results for a particular choice
ψ(t) = e−2Rt,(5)
obtaining semi-analytical solution for the Gaussian
process and analytical solution for the two-state Kubo-
Anderson process [14, 27], where ω(t) = ω0+ ν or
ω(t) = ω0− ν, with the rate R determining the tran-
sition between the + and − states. The later is used
to model single molecules in low temperature glasses [21].
In the case of two identical square pulses the modulat-
ing function f(t) in Eq. (2) is:
f(t) =





cos(ωLt)
0
cos[ωL(t − t2)] t2< t < t3
0
0 < t < t1
t1< t < t2
t3< t
,(6)
where ωL is the laser frequency, t1 = t3− t2 = T
is the pulses duration, and ∆ = t2− t1 is the delay
between the pulses. In our calculations we assume: (i)
that the system is always found in its ground state at
the beginning of the experiment.
meaningful measurable results one has to use sufficiently
intense laser fields such that ΩT ∼ 1 - weak fields cannot
excite the emitter even once. (iii) The pulses are short
enough, so that the lifetime of the excited state is much
longer than pulse’s duration - ΓT ≪ 1 (hence, Ω ≫ Γ).
(iv) The rate R of changes of the stochastic process ω(t)
satisfies RT ≪ 1 (hence, Ω ≫ R).
leads to the mentioned negligibility of photon emissions
during the pulse events. The last assumption implies,
that the time-dependent absorption frequency ω(t) is
unchanged during the excitations, and will be taken to
be ω(t1) = ω0+ δw(t1) during the first pulse event and
ω(t2) = ω0+ δw(t2) during the second pulse event.
(ii) In order to get
Assumption (iii)
A calculation given in short in Appendix A yields the
following expression for the probabilities of emitting zero,
one and two photons in the limit of long measurement
times t → ∞ for a particular realization of the stochastic
process ω(t):

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Pn[ω (t1),ω (t2),θ(∆)] = PCla
n
[ω (t1),ω (t2)] + 2e−Γ∆
2Re
?
ACoh
n
[ω (t1),ω (t2)]ei(θ(∆)+ω0(T+∆))?
,(7)
where θ(∆) is the random phase accumulated during the delay interval given by
θ(∆) =
?∆
0
δw(t)dt,(8)
and PCla
defined in Eqs (9,10) below. Note, from Eq. (7) it follows that all accumulated random phase effects become negligible
when Γ∆ ≫ 1. The two summands in Eq. (7) describe two kinds of possible quantum trajectories leading to required
number of photon emission events: the term PCla
n
[ω (t1),ω (t2)] we call semiclassical in the sense that it summarizes
the paths where at the beginning of the delay the system is found in one of the pure states |e? or |g? [26]. For example,
consider the first term of PCla
0
: the system starts in the electronic ground state |g?, then it evolves with the propagator
of the first pulse G [ω(t1)] without photon emissions and reaches the excited state |e?, it stays in the excited state
during the delay interval (with probability e−Γ∆), and afterwards the second pulse with the propagator G [ω(t2)]
stimulates the induced emission, bringing the system back to the ground state |g? without emitting a photon. The
second summand in the right-hand side of Eq. (7), ACoh
n
[ω (t1),ω (t2)], summarizes the contribution of the coherence
effects (i.e. all those quantum paths where at the beginning of the delay interval the system is left in a superposition
of the pure states). It can be shown (see Appendix A), that very strong or resonant π-pulses, where ΩT = π, simply
switch the state of the molecule being in |g? to |e? or vice versa. Hence, they do not excite the coherence - in such
cases the coherent term ACoh
n
[ω (t1),ω (t2)] vanishes.
n
[ω (t1),ω(t2)] and ACoh
n
[ω (t1),ω (t2)] are given in Table 1, where G [ω (ti)] (i = 1,2) is the Green function
n
0 ?g|G [ω(t2)]|e??e|G [ω(t1)]|g?e−Γ∆+ ?g|G [ω(t2)]|g??g|G [ω (t1)]|g? ?g|G [ω (t2)]|c??c|G [ω(t1)]|g?
PCla
n
[ω (t1),ω (t2)]ACoh
n
[ω (t1),ω (t2)]
1?g|G [ω (t2)]|g??e|G [ω(t1)]|g? + ?e|G [ω(t2)]|g??g|G [ω (t1)]|g??e|G [ω(t2)]|c??c|G [ω(t1)]|g?
2?e|G [ω(t2)]|g??e|G [ω (t1)]|g??1 − e−Γ∆?
0
Table 1: Photon statistics for two short pulses and arbitrary spectral diffusion process ω(t).
In Table 1 the propagator of the two level system
during the first and the second pulse events G [ω(ti)]
(i = 1,2) is obtained within the rotating wave approxi-
mation (RWA) (see Appendix A):
G [ω (ti)] = eT·LRWA[ω(ti)],(9)
where
LRWA[ω(ti)] =





0
0
0
0
iΩ
2
−iΩ
2
−iΩ
2
iΩ
2
iΩ
2
−iΩ
2
0
iδ(ti)
−iΩ
2
iΩ
2
−iδ(ti)
0




,(10)
and
δ(ti) = δL− δw(ti)(11)
with
δL= ωL− ω0
(12)
is the detuning.
made with the help of Mathematica 5.0. Evidently, since
the spontaneous emissions during the pulse events are
neglected (Ω ≫ Γ), the Green function Eq. (9) describes
well-known Rabi oscillations.
The mathematical calculations were
III.INFLUENCE OF SPECTRAL DIFFUSION
ON PHOTON STATISTICS.
We now take the average of Pn[ω(t1),ω(t2),θ(∆)]
Eq. (7) over the stochastic process ω(t). This procedure
requires the knowledge of the joint probability density
function (PDF) P[ω(t1),ω(t2),θ(∆)] of finding the sys-
tem’s absorption frequency ω(t) in the infinitesimal range
near ω0+ δw(t1) at t1, near ω0+ δw(t2) at t2, with ac-
cumulated random phase θ(∆). Then by definition the

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4
∆L
0.2
0.4
0.6
0.8
1
?e???Ω?t????g?
∆L??Ν???∆L?Ν??
?2?
FIG. 2: The matrix element ?e|G[ω(ti)]|g? Eq. (54) describing
the probability of transition from the ground to excited state
as a result of interaction with a π-pulse for Ω = 4Γ. The
smooth curve represents ?e|G[ω(ti)]|g? for ω(t) = ω0+ ν with
ν = Ω. The dashed curve represents ?e|G[ω(ti)]|g? for ω(t) =
ω0− ν with ν = Ω. The figure illustrates that since the two
curves practically do not overlap for ν ≥ Ω the probabilities
to excite the molecule being in state ω(t1) = ω0 + ν during
the first pulse event and in the state ω(t2) = ω0− ν during
the second pulse event are measurably different.
average of Pnis:
?Pn? =
?∞
−∞
dθ(∆)
?∞
0
dω(t1)
?∞
0
dω(t2)×
× Pn[ω(t1),ω(t2),θ(∆)]P[ω(t1),ω(t2),θ(∆)].
Later we find exact solution for the three-variable PDF
P[ω(t1),ω(t2),θ(∆)] for the case of two-state Kubo-
Anderson process, thus providing all essential tools for
calculation of ?Pn? in the case of the telegraph noise.
But first, we discuss several limiting cases common for
(13)
all stationary processes.
In Fig. 2 we plotted the probability of transition from
the ground to the excited state ?e|G[ω(ti)]|g? Eq. (54)
for two identical π-pulses. The smooth and the dashed
curves represent ?e|G[ω(ti)]|g? as a function of δL for
ω(t) = ω0+ν and ω(t) = ω0−ν respectively with ν = Ω.
The half width of ?e|G[ω(ti)]|g? is of the order of Ω.
When ν ≫ Ω the two curves practically do not overlap.
Hence, assuming the absorption frequency of the system
at the moment of the first excitation is ω(t1) = ω0+ ν
and of the second excitation is ω(t2) = ω0− ν, the
probabilities to excite the molecule during two pulse
events strictly differ. In this selective limit the photon
statistics is therefore very sensitive to the temporal state
of the molecule at the moments of excitations, and the
particular type of the underlying stochastic process has
large importance. Later we consider this limit in detail
for the case of telegraph noise. The opposite situation is
the impulsive limit.
Impulsive limit Ω ≫ ν.
which we call impulsive, the matrix elements of G[ω(t1)]
and G[ω(t2)] Eqs. (54) become independent of the value
of stochastic detuning δw(t) at the moment of the ex-
citation. Thus instead of the multi variable PDF
P[ω(t1),ω(t2),θ(∆)] we now have to deal only with the
one variable PDF of the phase θ(∆). As a result the
photon statistics shows an interesting relation with lin-
ear continuous wave spectroscopy:
-
In the limit Ω ≫ ν,
lim
Ω≫ν?Pn? = lim
Ω≫νPCla
n
+2e−Γ∆
2Re
?
φ(∆)eiω0(T+∆)lim
Ω≫νACoh
(14)
n
?
,
where using Table 1 and Eqs. (54)
lim
Ω≫ν?PCla
1
? =
2Ω2
?
δ2
L+ Ω2cos2
?
ΩT
2
?
(δ2
1 +
δ2
L
Ω2
??
sin2
?
ΩT
2
?
1 +
δ2
L
Ω2
?
L+ Ω2)2
(15)
lim
Ω≫ν?PCla
2
? =(1 − e−Γ∆)
(δ2
L+ Ω2)2Ω4sin4
?
ΩT
2
?
1 +δ2
L
Ω2
?
,(16)
lim
Ω≫ν?PCla
0
? = 1 − lim
Ω≫ν?PCla
1
? − lim
Ω≫ν?PCla
2
?(17)
and
lim
Ω≫νACoh
0
=
Ω2
?
2δLsin2
?
ΩT
2
?
1 +
δ2
L
Ω2
?
4(δ2
+ i?δ2
L+ Ω2)2
L+ Ω2sin
?
ΩT
?
1 +
δ2
L
Ω2
??2
.(18)
In Eq. (14) the function φ(∆) given by
φ(∆) = ?exp[iθ(∆)]? = ?exp[i
?∆
0
δω(t)dt]? (19)
is the well investigated Kubo-Anderson correlation

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function, whose Fourier transform is the line shape of
the two level system according to the Wiener-Khintchine
theorem [10]. In conclusion, we see, that working with
very strong laser fields Ω ≫ ν under assumptions (iii,iv)
we gain the same information as found in the line-shape
in continuous wave experiments.
Near the resonance, where δL∼ 0, using Eqs. (14-18)
we find
lim
Ω≫ν?P0? = e−Γ∆sin4
?ΩT
2
?
+ cos4
?ΩT
2
?
−
−1
2e−Γ∆/2sin2(ΩT)Re
?
φ(∆)eiω0(T+∆)?
,(20)
lim
Ω≫ν?P1? =1
2sin2(ΩT)
?
1 + e−Γ∆
2Re
?
φ(∆)eiω0(T+∆)??
,
(21)
lim
Ω≫ν?P2? =?1 − e−Γ∆?sin4
From Eqs. (20-21) we see, that for the strong π-pulses
the coherent terms vanish, as mentioned. In contrast,
for π/2-pulses with ΩT = π/2 the importance of the
coherent terms, and hence, the correlation function
φ(∆) on the photon statistics is the strongest, since
the π/2 pulse excites the off diagonal terms of the
pulse-propagators G[ω(ti)] Eq. (54) [25].
?ΩT
2
?
.(22)
Semiclassical approximation. - The influence of the
coherence on photon statistics in many experimental
cases is expected to be difficult to detect.
because of the dephasing effects caused by the damping
coefficient e−Γ∆
Eq. (7). Moreover, because of the large value of the bare
optical transition frequency ω0the coherent paths oscil-
late too fast to be detected (see the term eiω0(T+∆)in
Eq. (7)). In such cases a practical approximation is to
keep only the semiclassical terms PCla
stress, that for multilevel systems or in non optical ex-
periments on Josephson junction coherence contribution
is important [8]. Since the semiclassical paths are inde-
pendent of the random phase θ(∆), for the calculation of
It may be
2 multiplying the coherent terms ACoh
n
in
n
. Nevertheless, we
?PCla
P[ω(t1),ω(t2)].
noise [28] we have:
n
? we need only the marginal, two dimensional PDF
For example, in the case of Gaussian
P[δw(t1),δw(t2)] =
1
2πν2?(1 − ψ2)×
× exp
?
−δw(t1)2+ δw(t2)2− 2δw(t1)δw(t2)ψ
2ν2(1 − ψ2)
where ψ = ψ(|t2− t1|) is the time dependent part of the
correlation function in Eq. (4). Once the two-time PDF
P[ω(t1),ω(t2)] is known, ?Pn? within the semiclassical
approximation is:
?
, (23)
?PCla
n
?=
?∞
0
?∞
0
PCla
n
[ω(t1),ω(t2)]P[ω(t1),ω(t2)]dω(t1)dω(t2).
(24)
In Appendix B we give the explicit semiclassical ap-
proximation for the two state Kubo-Anderson model
Eqs. (58-57). Later in Fig. 3 we compare these results
with similar results for the Gaussian noise, which was
solved semi-analytically with the help of Mathematica
5.0. In the both calculations the same correlation
function Eqs. (4,5) was used.
IV.TWO STATE PROCESS: EXACT SOLUTION
Now we obtain the exact solution for the two state
Kubo-Anderson Poissonian process, where the absorp-
tion frequency of the system jumps between the + and
− states, i.e.
tial state, during the first pulse with ω(t1) = + or
ω(t1) = −. Similarly, the final state at the second pulse
is ω(t2) = + or −. Since the random phase is now given
by ∆θ = ν(T+− T−), where T±are occupation times
in states + and − [27] obeying ∆ = T++ T−, the joint
PDF P[ω(t1),ω(t2),θ(∆)] can be found from the joint
PDF h[ω(t1),ω(t2),T+] of finding the system in state
ω(t1) = ± during the first pulse, state ω(t2) = ± during
the second and with the occupation time T+between the
two pulses. In this case, where the random process takes
only discrete values, ?Pn? Eq. (13) takes the form
ω(t) = ω0± ν.We denote the ini-
?Pn? =
ω(t1),ω(t2)=±
?
P[ω(t1),ω(t2)]PCla
n
[ω(t1),ω(t2)]+2Re
?
eiω0(T+∆)−Γ∆
2ACoh
n
[ω(t1),ω(t2)]
?∞
0
eiν(2T+−∆)h?ω(t1),ω(t2),T+?dT+
?
,
(25)
where
P[±,±] =1 + e−2R∆
4
,P[±,∓] =1 − e−2R∆
4
(26)