One- and two-photon counting methods in single molecule fluorescence

Journal of Luminescence 87}89 (2000) 184}188

Invited Paper

One- and two-photon counting methods in single molecule #uorescence I.S. Osad'ko*, L.B. Yershova Department of Physics, Moscow State Pedagogical University, Moscow 119882, Russia

Abstract One- and two-photon counting methods in single molecule #uorescence are compared. An advantage of the two-photon method is demonstrated. Two-photon autocorrelation functions enables one to measure both exponential and logarithmic relaxation in polymers and glasses. The logarithmic relaxation is due to the interaction of a molecule with a huge number of intrinsic two-level system (TLS) existing in amorphous solids. The theory is compared with an experiment.  2000 Elsevier Science B.V. All rights reserved. Keywords: Single molecule; Correlator

1. Introduction Spectroscopy of single molecule (SMS) is the most promising method to probe a local environment of individual chromophores-doped low-temperature amorphous matrices like polymers and glasses [1,2]. Indeed, optical lines relating to individual tunneling systems were visualized "rst just by the SMS technique [3]. SMS enabled one to discover a very broad variety of optical dephasing rates and temporal changes in optical lines of individual molecules in low-temperature matrices [4,5]. Light absorption by a single molecule is measured with the help of photons emitted by the molecule irradiated by cw-laser light [6]. Transition from a large molecular ensemble to a single molecule forces us to use photon counting methods in which we are facing #uctuations in the signal detected by a photo-multiplier. Fluctuations of quantum nature cannot be removed in principle. Besides quantum #uctuations due to random character of photon emission from a molecule, there are #uctuations due to tunneling transitions in two-level systems (TLSs) which surround the chromophore. These tunneling transitions occur in random time moments and they are visualized as jumps of optical line between various

* Corresponding author. Fax: #7-095-3340886. E-mail address: [email protected] (I.S. Osad'ko)

spectral positions. The question arises as to how we can get information concerning relaxation of TLS from such jumping lines. In this paper we try to "nd an answer to this question.

2. One- and two-photon counting methods We shall consider a situation where a single molecule is irradiated by cw monochromatic laser light. In each elementary event the molecule absorbs and after some time emits just one photon. A train of emitted photons #ies in space with random time delay, as shown in Fig. 1. In the one-photon method of registration the full number of photons, emitted by a molecule is counted. If the laser frequency u equals the resonance frequency X of a spec tral line, the absorption of light is maximum and consequently the mean interval between the emitted photons is the least. The larger the detuning D "u !X the   longer the mean time delay between emitted photons. Therefore, the number of the photons, emitted by a molecule, N(u !X), counted during the time t, is a function  of detuning. This function describes the absorption line shape. At a typical counting rate of 10}10 s\, 10 s or more is required to record the contour of an absorption line with the help of the emitted photons with good accuracy. This is not a serious disadvantage when TLSs are absent

0022-2313/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 2 5 6 - 2

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185

equal to ratio (1), i.e. =k (t )"S(t ).   k Fig. 1. Train of photons emitted by a single molecule irradiated by a cw-laser.

in a sample studied. However in polymers, where TLS transitions exist, we encounter `jumpsa of spectral line. As shown below, they result from the interaction of a molecule with slowly tunneling TLSs existing in polymers and glasses. It is intuitively clear that the inverse time between random jumps characterizes the rate of TLS relaxation. Unfortunately, the one-photon method of measurements based on counting the total number of emitted photons is least suitable for measurement of TLS relaxation rates. In the two-photon method of registration, the number of pairs N of emitted photons with delay t between   photons in a pair is counted. Three pairs of this sort with delay t are shown in Fig. 1. The count rate of such  photon pairs p(t ) is named a two-photon correlator. It is  also a function of the exciting light frequency, i.e. p(t )"  p(u !X, t ). Our task is to "nd a mathematical expres  sion for the function p(t ).  Measurements taking into account only pairs of photons in which photons follow directly one another with delay t is called the `start}stopa regime. For  example, if the delay between photons 2 and 3 in Fig. 1 equals t , the second event will be registered only after  the arrival of photons 12 and 13. If the total time t of counting de"nite pairs is rather short, a number of such events will undergo signi"cant #uctuations. The longer we count, the smaller the ratio of the #uctuations to an average number of events. Therefore, the ratio N (t , t)   "S(t )  N (t) 

(1)

is a function of t and does not depend on t if the latter is  long. Here, the numerator is a number of counted events, measured during t, and the denominator is the full number of the pairs which have arrived on a PMT during the same time. Let us "nd out in what way ratio (1), which is certainly less than unity, corresponds to the values = , = and   = which describe probabilities of "nding a molecule in I the ground state, in the excited state and in the state when one spontaneous photon k is emitted, respectively. The moment of registration of the "rst emitted photon by the detector should be obviously regarded as t"0. At this moment = "1, i.e. the molecule is in the ground  state. The remaining probabilities are equal to zero. If later the PMT detects a photon spontaneously emitted by a molecule, the probability of such an event equals the sum of all probabilities Wk and on the other hand it is

(2)

As usual the total number of interesting events per unit of time, i.e. the count rate, is measured. This count rate is determined by the following expression: d s(t )"SQ (t )" =k (t ).    dt k 

(3)

Equations for probabilities = , = and =k , describing   the `start}stopa regime, were derived in Ref. [7]. It was also shown that d = s(t )" =k (t )"!(= Q #= Q )"  .     dt k ¹  

(4)

The probability of detecting two sequentially emitted photons will vanish at t <¹ . Therefore, a slow TLS   relaxation cannot be studied with the help of the `start}stopa correlator. However, a slow relaxation can be measured with the help of the so-called full two-photon correlator which takes into account all pairs of photons with delay t . If in  the `start}stopa regime we count, for example, only pairs (2, 3) and (12, 13), depicted in Fig. 1, by measuring the full two-photon correlator for the same delay t , we also  include in our count the pair (4, 6) and others like this. In the time interval between photons 4 and 6 the molecule emits photon 5 in addition. Let p(t ) stand for the count  rate of all photon pairs. A quantum mechanical theory of two-photon correlators for a three-level molecule including a triplet state was developed in Ref. [7]. It was shown that the full two-photon correlator is described by the formula o (t ) p(t )"   ,  ¹ 

(5)

where o (t) is the probability of "nding a molecule in the  excited state which can be found from optical Bloch equations. This correlator is related to an autocorrelation function of #uorescence intensities I(t) as follows: p(t ) 1I(t)I(t#t )2  "g(t )"lim  . (6)  p(R) 1I(t)2 R Correlators s(t) and p(t) are connected by the following equation [7]:



p(t)"s(t)#

R

s(t!x)p(x) dx.

(7)



Solving this equation by an iteration procedure we "nd  p(t)"s(t)# p (t), L L

(8)

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I.S. Osad'ko, L.B. Yershova / Journal of Luminescence 87}89 (2000) 184}188

where the nth term

 

p (t)" L

R

R



RL

dt s(t!t )s(t !t ) L     (9) 2s(tL\ !tL )s(tL ), describes the counting rate of pairs with n intermediate photons like the pair (4, 6) in Fig. 1, including one intermediate photon. Unfortunately, the probability o (t) found in Ref. [7]  does not allow for the existence of a huge number of intrinsic TLSs inherent in polymers and glasses. However, the interaction of a chromophore with these TLSs results in the logarithmic spectral di!usion (SD) in holeburning spectroscopy [8}11]. The interaction with a huge number of TLSs can only be taken into consideration beyond the optical Bloch equations used in Ref. [7]. This restriction of the former theory was eliminated in Ref. [12]. General formula for correlator p(t) derived in this paper will be used in next sections. 

dt 



dt 2 

and TLSs, respectively. The third term depends also on time due to nonequilibrated TLSs which emerge in a polymer in due course of irradiation by light. Indeed since a photon is emitted TLS gets to one of two possible quantum states with the probability equal to unity. This value of the probability is not accessible thermodynamically. When time t describing a delay between photons in a pair approaches in"nity, the function k(u , t, ¹)  approaches conventional absorption coe$cient k(u ,  R, ¹) of a molecule interacting with phonons and TLSs. The term c (t, ¹) describing the optical dephasing 2*1 rate due to the interaction with nonequilibrated TLSs plays an important role in the SD problem. All TLSs of a polymer contribute to it. The formula for c can be 2*1 written as follows [12}15]:



c 2*1 " N(E, R)o(1!o)RI(R) dR dE, 2 where o"f#(o(0)!f )e\0R, f"1/[exp(E/k¹)#1],

3. E4ect of intrinsic TLSs on two-photon correlator

(13)

(14)

describes relaxation of TLS with splitting E and In the absence of photon-induced tunneling a formula for the full two-photon correlator derived in Ref. [12] is given by






t p(t)"k(u , t, ¹) 1!exp !  ¹  where



,

(10)

k(u , t, ¹)  1/¹ (t, ¹)  "2s w (t, ¹) . (11) + (u !X )#[1/2¹ (t, ¹)]  +  + Here s"dE/ is a Rabi frequency. Eq. (11) takes into account the idea developed in Refs. [11,13,14]. In accordance with this idea the equation D(r)"1/¹ determines  the radius of the nearest vicinity of the chromophore. Here D(r) is the dipolar interaction between the chromophore and TLS. Few TLSs, n , from this vicinity interact  strongly with the chromophore. These strongly coupled TLSs determine the number of resolved optical lines in the optical band. w (t, ¹) is a probability to "nd n TLSs +  in quantum state M. X is their excitation frequency. + A huge number N of TLSs situated beyond the nearest  vicinity of the chromophore determined by the equation D(r)"1/¹ in#uence mainly the optical dephasing rate  which is given by 1 1 c (¹) c (t, ¹) " #  # 2*1 , (12) ¹ (t, ¹) 2¹ 2 2   where the second and third terms describe temperaturedependent dephasing due to the interaction with phonons



, D N D(r) Q "  I(R)" d< D(r)#R D#R < 4 Q Q

(15)

is an integral which depends on the type of a multipolar interaction D between a chromophore and TLS [14,15]. Here N /< is a TLS density, and N(E, R) is a distribution  function of splittings and relaxation rates of TLSs. Taking the function N(E, R) in the standard form and assuming the dipole}dipole type of the interaction between a chromophore and TLSs as well as low temperature we can rewrite Eq. (13) in the following form [10,14]:





c 0 dR 0 dR 2*1 Jk¹ (1!e\0R) k¹ 2 R R 0 R "k¹ ln(R t). 

(16)

Here R 10c\ determines the value of the greatest  constant of tunneling in intrinsic TLSs. This formula describes the logarithmic SD, resulting from coupling to all intrinsic TLSs.

4. Comparison with experiment Now let us apply general Eqs. (10) and (11) to the two-photon correlator of a chromophore whose optical band consists of four lines. It implies a chromophore strongly coupled to two TLSs from its local environment and more weakly to the remaining TLSs. The latter interaction can only in#uence optical dephasing time, i.e.

I.S. Osad'ko, L.B. Yershova / Journal of Luminescence 87}89 (2000) 184}188

187

line width. In this case k(D , t, ¹)"w ¸ #w ¸ #w ¸ #w ¸          "2s[o o ¸(D )#o o ¸(D !D)       #o o ¸(D !D)#o o ¸(D !D!D)],       (17) where 1/¹ (t, ¹)  ¸(D )" .  D #1/¹ (t, ¹)  

(18)

The interaction with a huge number of intrinsic TLSs results in optical dephasing rate 1/¹ which depends on  time. The probabilities p (t) look as follows: H o (t#t )"f#[o (t )!f ]e\0R,     o (t#t )"1!o (t#t ).    

Fig. 2. Autocorrelation functions for three terrylene molecules in PE at 1.8 K [5] and theoretical curves calculated by means of Eqs. (20) and (21) at the following values of parameters (¹ "10\ s, R "10 s\): f"0.18, R"8;10 s\, f "0.065,   R"0.5 s\, b/a"9;10\ K, p(R)"p(10 s) (1); f"0.09, R" 0.12 s\, f "R"0, b/a"10\ K, p(R)"p(10 s) (2); f"0.062, R"47 s\, f "R"0, b/a"4;10\ K, p(R)"p(10 s) (3).

(19)

At the excitation in the maximum of the line 0}0, i.e. at D "0, the contribution of the other three Lorentzians  can be neglected. Then at the moment t "0 of regis tration of the "rst emitted photon the probabilities o (0)"o (0)"0. Then Eq. (17) for the full two-photon   correlator takes the following form: p(t)"2s¹ (t, ¹)(1!e\R2 )[1!f (1!e\0R)]  ;[1!f (1!e\0YR)].

(20)

Fig. 3. In#uence of temperature on the autocorrelation function 1 from Fig. 3. ¹"1.8 K (1), 3.6 K (2) and 7.2 K (3) (a); ¹"1.8 K (1), 1.2 K (2) and 0.6 K (3) (b).

The correlator is the product of four functions of time. By taking into consideration the phonon-assisted dephasing with quadratic temperature dependence [16], i.e. c (¹)/2"a¹, we can rewrite Eq. (12) in the following  form: 1 1 " #a¹#b¹ ln R t.  ¹ (t, ¹) 2¹  

(21)

The transition from the linear temperature law to the quadratic one with increasing temperature was observed in photon echo experiments [17]. The logarithmic time broadening of a spectral molecule line in a polymer was discovered in experiments on persistent spectral hole burning [10]. In Fig. 2 the temporal dependence of two-photon correlators measured in Ref. [5] for three di!erent molecules of terrylene in polyethylene is shown. Two-photon correlators calculated with the help of Eq. (20) in the case o (0)"o (0)"0 are also plotted. The exponential relax  ation of a TLS from the nearest vicinity of a chromophore looks like a smooth step of one order of magnitude in the logarithmic time scale. The number of the steps is equal to the number of TLSs strongly coupled to a chromophore. The coupling to a huge number of intrinsic TLSs results in the logarithmic time optical line broadening, which looks as linear dependence of the two-photon

Fig. 4. Autocorrelation function of single tetra-tert-butylterrylene molecule embedded in polyisobutylene at ¹"1.4 K [18] and theoretical curve calculated by means of Eqs. (20) and (21) at the following values of parameters: ¹ "10\ s, R "10 s\,   f"0.025, R"5;10\ s\, f "R"0, b/a"2.7;10\ K, p(R)"p(9 s).

correlator in the logarithmic time scale. As the third term in Eq. (21) must be identical for all molecules, the di!erent declination of the curves in Fig. 2 is due to the di!erence in the value of the phonon term in Eq. (21), which strongly depends on the nearest environment of a chromophore. In Figs. 3a and b the temperature e!ect on a two-step correlator 1 taken from Fig. 2 is shown. The calculation

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I.S. Osad'ko, L.B. Yershova / Journal of Luminescence 87}89 (2000) 184}188

was carried out with the help of Eqs. (20) and (21). With rising temperature, the logarithmic SD increases less than the exponential one. Therefore, the relative role of the exponential SD increases. On the contrary, if the temperature decreases the role of the exponential relaxation becomes inconspicuous and the role of logarithmic SD expands. In Fig. 4 the correlator of a molecule of tetra-tertbutylterrylene in a polyisobutene, measured in Ref. [18], is depicted. It shows strong logarithmic SD. In the same Fig. 4 the theoretical curve calculated with the help of Eq. (20) is plotted. It provides a good "t to the experimental data. 5. Conclusion We considered both one- and two-photon counting methods in SMS. Theoretical expressions (10) and (20) for the two-photon correlator, take into account the relaxation of not only TLSs from the nearest environment of a molecule, but also relaxation of all intrinsic TLSs of the polymer. All the nontrivial information on a TLS relaxation can be found from the function k(D , t, ¹) which transforms into the conventional ab sorption light coe$cient when t approaches in"nity. The main physical results of the work are shown in Figs. 2}4 where the "t to experiment is carried out. According to Figs. 2}4 both the exponential relaxation of TLSs from the local environment of a molecule and the logarithmic relaxation resulting from interaction with all intrinsic TLSs of a polymer manifest themselves in the two-photon correlator. The method based on the measurement of two-photon correlators in the single-molecule spectroscopy has obvious advantages as compared to one-photon methods of measurement of absorption, since the latter does not allow to observe logarithmic time dependence of the optical dephasing rate 1/¹ . 

Acknowledgements The authors thank the Russian Foundation for Basic Researches (grant 97-02-17285) for "nancial support of the work.

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