Statistical properties of blinking kinetics in single molecule

Chemical Physics Letters 516 (2011) 272–276

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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Statistical properties of blinking kinetics in single molecule Yonggang Peng, Yujun Zheng ⇑ School of Physics, Shandong University, Jinan 250100, China

a r t i c l e

i n f o

Article history: Received 6 July 2011 In final form 4 October 2011 Available online 14 October 2011

a b s t r a c t We study the statistical properties of blinking kinetics of single terrylene molecule in p-terphenyl crystal with intersystem crossing process (ISC) via generating function. The analytical expression of the waiting time distribution of the ‘bright’ and ‘dark’ period is obtained, which demonstrates that the behaviors of the ‘dark’ period show bi-exponential decaying because of two kinds of sublevels. The probability of the single terrylene molecule in the ‘bright’ and ‘dark’ states are also investigated. Our theoretical results are in good agreement with experimental results. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Single molecule spectroscopy has become an useful tool to study the properties of single quantum system interacting with external fields and its surrounding environmental molecules [1–3]. The advantage of the single molecule spectroscopy is that it can reveal the fluctuations of the single molecule itself and its local environments. The optical observation of the single molecule experiment was successfully performed by Moerner et al. [4] and Orrit et al. [5]. The single molecule spectroscopy results demonstrate that the single molecules show stochastic properties in solid matrix or condensed matter matrix, such as spectral diffusion [1,3,6], spectral hole burning [1,5,7] and fluorescence blinking [1,3,8–18]. The spectral diffusion and spectral hole burning are caused by the rearrangements of the local environments of the single molecule. The fluorescence blinking is, however, caused by a long-lived metastable triplet state in the molecules. The fluorescence blinking phenomenon reflects the intra-molecule dynamics of the single molecule, which can help us understand the properties of the triplet state [1,8,19]. A review on single molecule kinetics can be found in Ref. [20]. In this Letter, we consider single terrylene molecule in p-terphenyl crystal, which can be described by a three-level model [1,8,19,21]. In this model, there are one singlet ground state S0, one excited singlet state S1 and one triplet excited state T1 (see the schematic diagram in Figure 1). Some experiments demonstrate that the sublevels of the triplet excited state T1 have markedly different relaxation (depopulation) rates [8,19,21–24]. In this respect, the triplet excited state T1 can be split into two classes: one is the long-lived sublevel Tz and the other is the short-lived sublevels Tx and Ty [24]. The external field is tuned in resonance with S0  S1 transition. The molecule can be excited into S1 state by exciting field and would back into ground state S0 via ⇑ Corresponding author. E-mail address: [email protected] (Y. Zheng). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.10.018

emission a fluorescence photon. The molecule undergoes fluorescent cycles between the singlet ground state S0 and singlet excited state S1, which can be regarded as ON state [12,25]. The molecule has, however, few probability from the singlet state S1 transferring into triplet state T1. When the molecule in the triplet state T1, the molecular transition is out of the resonance with driving field and no fluorescence photon is emitted, which can be regarded as OFF state [12,25]. Considering the triplet state T1 can be split into two kind sublevels, we model this process via two OFF states. When the molecule is trapped in the long-lived sublevel Tz, the molecule is assumed in the OFF1 state; when the molecule is, however, trapped in the short-lived sublevels Tx and Ty, it is denoted as in OFF2 state. The molecule can ‘jump’ back into ON state through the relaxation (depopulation) process of the sublevels Tz, Tx and Ty. The generating function approach has, as a powerful and convenient method to study the counting statistics, been used to study the photon counting statistics of the single molecule driven by continuous wave field and pulse fields [6,26–35], and the interaction between single molecule, its surrounding environments [36–40] etc. Boiron et al. investigated the blinking process using the fluorescence correlation function, which occurred in single molecule system [24]. The fluorescence intensity (average emission photon numbers) are also studied in Ref. [24]. They give an expression of the intensity auto-correlation function g(2)(s), which reflect the distribution of the single molecule emission a photon at time 0 and then emission a photon after a delay time s. For a single mode system, the intensity auto-correlation function g(2)(s) can be expressed as a function of the population of the singlet excited state rS1 ðsÞ, it is [24]

g ð2Þ ðsÞ ¼

rS1 ðsÞ : rS1 ð1Þ

ð1Þ

In this work, we assume the population rS1 ðtÞ  1 (see below for detail), which is independent on the time t. That means, in our model, the intensity auto-correlation function g(2) = 1. We investigate the blinking kinetics of the statistical properties of single terrylene

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S1

where ri(t), (i = on, off1, off2) is the probability of the system in the state i at time t. We consider the kinetic blinking process of single terrylene molecule as follows: the single molecule jumps into OFF1 and OFF2 states from the ON state via the transitions of k21 and k23, respectively; then the single molecule jumps back to ON state through the paths k12 and k32 from the OFF1 and OFF2 states, and so on. To extract the information of kinetic blinking, the ri can, similar with Refs. [6,26,27,30], be expanded as follows

ON

T

1

k12

k

k

23

32

k

21

ri ðtÞ ¼

(2)

X

OFF

rðnÞ i ;

ð4Þ

n ðnÞ

(1)

S0

OFF

Figure 1. The schematic diagram of the three-level single molecule system.

molecule embedded in a p-terphenyl crystal by employing the generating function approach in this Letter. In our investigation, we also discuss the influence of the intersystem crossing process and the waiting time distributions of ‘bright’ and ‘dark’ states in blinking process of single terrylene molecule. The rest part of this Letter is organized as follows: in Section 2, we briefly review the generating function approach, and deduce the equation of generating function from the master equation used to describe blinking kinetics of single terrylene molecule. And, we finally show that how to extract the information from the generating function. We present our results and the discussions in Section 3.

ð1Þ

; ð2Þ

kexc þ c þ kISC þ kISC

ð2Þ

where kexc is the population rate of the single molecule driven by external laser, which is proportional to the intensity of the external ðiÞ laser, c is the spontaneous emission rate of the singlet state S1, kISC is the intersystem crossing rates to the ith sub-state in the triplet state (long-lived state and short-lived state), and satisfy ð1Þ ð2Þ k21 ¼ rS1 kISC and k23 ¼ rS1 kISC . Generally, the transition rates k21 and k23 are dependent on the external laser intensity. If ð1Þ ð2Þ kexc  c; kISC ; kISC is satisfied, the population of the singlet state rS1  1. In this condition, the effective transition rates k21 and k23 are the same as the intersystem crossing rates in the real four level ð1Þ ð2Þ system, namely, k21 ¼ kISC and k23 ¼ kISC . The single terrylene molecule we considered in this Letter satisfies this condition. A set coupled master equations are employed to describe the kinetic evolution of the system’s states with time t,

r_ on ¼ k21 ron  k23 ron þ k12 roff1 þ k32 roff2 ; r_ off1 ¼ k12 roff1 þ k21 ron ; r_ off2 ¼ k32 roff2 þ k23 ron ;

ðnÞ ðn1Þ r_ ðnÞ ; off1 ¼ k12 roff1 þ k21 ron ðnÞ ðnÞ r_ off2 ¼ k32 roff2 þ k23 rðn1Þ : on

ð5Þ

rið0Þ , however, satisfy the following equation ð0Þ ð0Þ r_ ð0Þ on ¼ k21 ron  k23 ron ; ð0Þ r_ ð0Þ off1 ¼ k12 roff1 ; ð0Þ r_ ð0Þ off2 ¼ k32 roff2 :

ð6Þ

The generating function is defined as [6]

X

riðnÞ sn ;

ð7Þ

n

In the three-level model, OFF1 is long-lived sublevels in the triplet state, OFF2 is the short-lived sublevels in the triplet state, ON is the single molecule in the singlet states S0 and S1 and driven by external field. k21 and k12 are the transition rates between ON and OFF1, which correspond to the intersystem crossing rate of the singlet state S1 to the sublevel Tz multiplied the population of S1 and the relaxation rate of the sublevel Tz, respectively. k23 and k32 are the transition rates between ON and OFF2, which correspond to the intersystem crossing rate of the singlet state S1 to the sublevels Tx and Ty multiplied the population of S1 and the relaxation rate of the sublevels Tx and Ty, respectively (see Figure 1). The population of the singlet excited state rS1 can be expressed as (assume the intersystem crossing process is very rarely occurred) [41]

kexc

ðn1Þ ðn1Þ ðnÞ ðnÞ r_ ðnÞ þ k32 roff2 ; on ¼ k21 ron  k23 ron þ k12 roff1

Gi ðs; tÞ ¼

2. Theoretical framework

rS1 

where ri is the probability of the system being in the state i at time t after the system undergoes n jumps between ON and OFF ðnÞ states. The ri (n > 0) satisfy, after some algebras, the following relation

ð3Þ

where the s is auxiliary parameter. It can be used to count the times of the system jumping between ON and OFF states in time interval [0, t]. Using Eqs. (5) and (6), the generating functions satisfy

G_ on ¼  k21 Gon  k23 Gon þ k12 sGoff1 þ k32 sGoff2 ; G_ off1 ¼  k12 Goff1 þ k21 sGon ; G_ off ¼  k32 Goff þ k23 sGon : 2

ð8Þ

2

In the term of generating function, the probability r expressed as

rðnÞ i ðtÞ ¼

ðnÞ i ðtÞ

can be

  1 @n  ; G ðs; tÞ i  n n! @s s¼0

ð9Þ

and, for example, the cumulants j1 and j2 of the switching time between ON and OFF states can be expressed as [41]

j1 ¼ hNiðtÞ ¼

  X @  Gi ðs; tÞ  @s i

j2 ¼ hN2 iðtÞ  hNi2 ðtÞ ¼   X @  Gi ðs; tÞ   @s i

; s¼1

    X @2 X @   G G ðs; tÞ þ ðs; tÞ   i i 2   @s @s i i s¼1 s¼1 !2 :

ð10Þ

s¼1

3. Results and discussions In this section we show our results of single terrylene molecule in p-terphenyl crystal with intersystem crossing process (ISC). To calculate the waiting time distribution of the ‘dark’ and ‘bright’ state, we assume the single molecule is initially in OFF and ON states [42], respectively. Considering the probability of transition from ON to OFF1 and ON to OFF2 is proportional to the transition rate k21 and k23, one can obtain the initial condition of OFF state is as follows:

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ron ðt0 Þ ¼ 0; roff1 ðt0 Þ ¼

roff2 ðt0 Þ ¼

k23 : k21 þ k23

rð0Þ off with the OFF initial condition is

0.8

k21 k23 ¼ ek12 t þ ek32 t : k21 þ k23 k21 þ k23

ð0Þ ð0Þ rð0Þ off ¼ roff1 þ roff2

1

ð11Þ

ð12Þ

The initial condition of ON state is ron ðt 0 Þ ¼ 1; roff1 ðt0 Þ ¼ 0; roff2 ðt0 Þ ¼ 0, employing Eqs. (6) and (8), one can obtain the solution ð0Þ of ron as follows ðk21 þk23 Þt rð0Þ : on ¼ e

Probability

The solution of

k21 ; k21 þ k23

0.6 0.4 0.2

ð13Þ

The waiting time distributions are then defined as [3,41]

0

d ð0Þ wi ðtÞ ¼  ri ðtÞ: dt

ð14Þ

won ðtÞ ¼ ðk21 þ k23 Þeðk21 þk23 Þt ; woff ðtÞ ¼

k21 k12 k12 t k23 k32 k32 t e þ e : k21 þ k23 k21 þ k23

ð15Þ

For the convenience to compare our theoretical results with experimental results of single terrylene molecule in p-terphenyl crystal, we use the same parameters with experiment in our numerical calculations. They are taken from Ref. [8], namely, k32 = 1.79  103 s1, k23 = 0.33  103 s1, k12 = 0.04  103 s1 and k21 = 0.07  103 s1. In Figure 2, we plot the waiting time distribution of the OFF state when the system start from OFF state. The squares in the figure is the experimental result coming from Ref. [8], the solid line is our theoretical result getting from Eq. (15). They are in good agreement with each other. From the expression of woff(t), the two exponential have different weight in the bi-exponential expression, the weight coefficients are proportional to k12k21 and k23k32, respectively. For the single terrylene molecules in p-terphenyl crystal, the coefficient k12k21  k23k32, which means one can omit the part of k12k21 in the bi-exponential. In this view respect, the waiting time distribution of the OFF state can be fitted by mono-exponential decaying function as in Ref. [19]. If k12k21  k23k32 is satisfied, one would obtain the waiting time distribution of the single molecule in the OFF state

1

0.8

0.15 Probability

According to this definition and Eqs. (12) and (13), the waiting time distributions of the single molecule in ON and OFF states can be expressed as

0.1

0.05

0 0

20 time [ms]

30

40

Figure 3. The probabilities of the system in the ON state (upper) and OFF state (lower) as a function of time t after the system undergoes a few times between ON and OFF states with the system initial in the ON state, the dashed-dot line ð2Þ rð0Þ on , the dashed line corresponds to the ron and the solid line ð1Þ rð4Þ on (upper). The dashed-dot line corresponds to the roff , the ð3Þ ð5Þ dashed line corresponds to the roff and the solid line corresponds to the roff . The red

corresponds to the

corresponds to the

lines correspond to the OFF1 state’s contribution to the ON distribution, the blue lines correspond to the OFF2 state’s contribution to the ON distribution (lower). The results obtained by Monte Carlo simulation are marked using circles, squares, and diamonds. The parameters are k32 = 1.79  103 s1, k23 = 0.33  103 s1, k12 = 0.04  103 s1 and k21 = 0.07  103 s1 [8]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

demonstrates an significant bi-exponential decaying behavior. On the other way, from the experimental data, we can obtain the transition rates k21 and k23 and the ratio k12/k32 from Eq. (15). ðnÞ The probabilities of the single molecule in ON (ron , upper panel) ðnÞ and OFF (roff , lower panel) states after the single molecule jumps n times between ON and OFF states as a function of time t are shown in Figure 3. The probabilities of the single molecule in ON and OFF states after its jumping n times between ON and OFF states can also be expressed as follows

0.6

ðnÞ

w(t)

rðnÞ iðjÞ ðtÞ ¼ 0.4

NiðjÞ ðtÞ N

ð16Þ

; ðnÞ

0.2

0

10

−1

10

0

10 time [ms]

1

10

Figure 2. The waiting time distribution of the OFF state, the solid line corresponds to the expression (15), the square marks are the experimental result from the Ref. [8].

where i,j = on, off, and N iðjÞ is the number of the single molecule has jumped n times between ON and OFF states in the time interval [0, t] in the case that the single molecule is initial in j state, and final in i state at time t; N is the number of total measure times. The result of Eq. (16) are demonstrated in Figures 3 and 4 (the circles, squares and diamonds) by Monte Carlo simulation. In our Monte Carlo simulation, the total number N = 10000. The results shown in the figure, the single molecule is supposed to be in ON state initially. The dashed-dot, dashed and solid lines in the upper panel correð0Þ ð2Þ ð4Þ spond to ron , ron , and ron , respectively. The dashed-dot, dashed ð1Þ ð3Þ and solid lines in the lower panel correspond to the roff , roff , and ð5Þ roff , respectively. Also, the black, red and blue lines correspond to

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the probabilities of the single molecule in OFF state, the contributions of OFF1 and OFF2 states to OFF state, respectively. As the single molecule is initial in ON state, after the system jumped 2m + 1 (m is integer) times between ON and OFF states, the system will be in OFF state. Correspondingly, after it jumped 2m times between ON and OFF states, the system will be in ON state. The third and more higher order distribution would be hard to be detected in the experiments as the sample has low emission quantum yields. The weak signal is covered by the noises, which makes more difficult to count the switching times between ON and OFF in experiments. If the molecule jumps from i state to the other j state, the molecule cannot ð1Þ ð1Þ be found in i state. roff1 and roff2 , in the case of the single molecule in ON state initially, can be expressed as

r

ð1Þ off1

r

ð1Þ off2

ð1Þ ron ¼

k23 k32 ek32 t ð1  eðk21 þk23 k32 Þt Þ k21 þ k23  k32 k21 þ k23 k21 k12 ek12 t ð1  eðk21 þk23 k12 Þt Þ þ : k21 þ k23  k32 k21 þ k23

The average time of finishing a circle of ON to OFF to ON can be ð2mþ1Þ given by the nearest two maximum values of ron , and the average time of finishing a circle of ON to OFFi to ON is given by the ð2mÞ nearest two maximum values of roffi . The probability of the system switching zero times at time interval (0, t) and switching n times at time interval (t, tf) is described by P(0, t;n, tf). In our model the joint probability can be expressed as

Pð0; t; n; tf Þ ¼ rð0Þ rðnÞ ðtf  tÞ:

k21 ¼ ek12 t ð1  eðk21 þk23 k12 Þt Þ; k21 þ k23  k12 k23 ¼ ek32 t ð1  eðk21 þk23 k32 Þt Þ: k21 þ k23  k32

ð17Þ

The probabilities of the system in ON state (upper panel) and OFF state (lower panel) at time t with the system initial in OFF state are shown in Figure 4. The probability roff (lower panel black line) shows significant bi-exponential decaying behavior, the red line is the contribution of OFF1 state, the blue line is the contribuð1Þ tion of OFF2 state. The ron , in the case of the single molecule initial being in OFF state, can be expressed as

ð19Þ

The condition probability can be expressed as (using Bayes’ rule)

Pð0; tjn; t f Þ ¼

rð0Þ rðnÞ ðtf  tÞ  rð0Þ ðtÞ; ðtf  tÞ: rðnÞ ðtf Þ

ð20Þ

From the condition waiting time distribution definition

wn ðtÞ ¼ 

d Pð0; tjn; t f Þ; dt

ð21Þ

one can obtain

wn ðtÞ ¼ wðtÞ:

ð22Þ

This means that the condition waiting time distribution has the same behavior with the waiting time distribution with respect to evolution time. Generally, the entirely description a stochastic process is very complex, which needs the all order probability distribution of the process [41]. Some parameters are chosen to reflect the main prop-

0.6

Probability

ð18Þ

0.4

2

6000

0.2 4000

1.5

κ

1

0 0.2 Probability

Probability

0.8 0.6 0.4

0 0

10 time [s]

20

0.15

0.5 0.1 0.05 0 0

0.2

1

2000

0 20 time [ms]

40

0

10

20 time [ms]

30

40

Figure 4. The probabilities of the system in the ON state (upper) and OFF state (lower) after the system undergoes serval times between ON and OFF states with the system initial in the OFF states, joff iðt0 Þ ¼ roff1 ðt 0 Þjoff1 i þ roff2 ðt0 Þjoff2 i, the ð1Þ ð3Þ dashed-dot line corresponds to the ron , the dashed line corresponds to the ron and ð5Þ the solid line corresponds to the ron (upper). The dashed-dot line corresponds to ð0Þ ð2Þ the roff , the dashed line corresponds to the roff and the solid line corresponds to the rð4Þ . The red lines correspond to the OFF state’s contribution to the ON distribution, 1 off the blue lines correspond to the OFF2 state’s contribution to the ON distribution (lower). The results obtained by Monte Carlo simulation are marked using circles, squares, and diamonds. The parameters are the same as Figure 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

κ

0

2

1

0.5

0 0

0.5

1 time [ms]

1.5

2

Figure 5. The cumulants j1 and j2 as a function of time t with the system initial in ON state (solid lines) and in the OFF1 state (dashed-dot lines) and in the OFF2 state (dashed lines). The parameters are the same as in Figure 3.

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erties of the process, such as the first order cumulant j1, which reflect the expectation value of the process and can be directly obtained by experiments, and the second order cumulant j2, which reflect the variance of the process. Particularly, if the process is GAUSSIAN, the cumulants j1 and j2 can be used to entirely describe the process [41]. The cumulants j1 and j2 as a function of time t are demonstrated in Figure 5. The solid, dashed-dot and dashed line represent the system initial being in ON, OFF1 and OFF2 states, respectively. The cumulant j1 reflects the average switching time between ON and OFF state in interval time [0, t]. The j1 shows different behavior with respect to the evolution time t as the single molecule being in different initial condition. The slopes of the j1 corresponding to the initial conditions of ON, OFF1 and OFF2 in short time (close to the origin point) reflect the transition rates k21 + k23, k12 and k32, respectively. In the short time, the single molecule just finished ONE time switching between ON and OFF state. In short time period, the j1 are nonlinear, this shows the jumping process in short time is in non-equilibrium. As the time increasing, after the system jumping few times, the system will be in equilibrium of jumping into and out. Or, the cumulant j1 will be shown the linear behaviors for long time, and the slope of the average switching times hNi (j1) satisfy a simple relationship in the long time limit,

d ðssÞ ðssÞ ðssÞ j1 ¼ ðk21 þ k23 Þron þ k12 roff1 þ k32 roff2 dt k12 k21 k32 þ k12 k23 k32 ¼2 ; k21 k32 þ k12 k23 þ k32 k12

ð23Þ

ðssÞ

where ri is the steady state population of the i state. The j2 reflects the fluctuation of the switching times between ON and OFF state (the variance of the switching time between ON and OFF states). The fluctuation of the switching time of the single molecule initial being in ON state is increasing faster than that of the single molecule initial being in OFF state in long time regime, which is the single molecule has two paths relax into OFF state from ON state. Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant Nos. 91021009 and 10874102).

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