Oct 22, 1999 - tions which are slow compared to the fluorescence decay time, which is not possible using conventional bulk spectros- copy. SMS signals show ...
JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 16
22 OCTOBER 1999
Stochastic-trajectories and nonPoisson kinetics in single-molecule spectroscopy Vladimir Chernyak, Michael Schulz, and Shaul Mukamel Department of Chemistry and Rochester Theory Center for Optical Science and Engineering, University of Rochester, Rochester, New York 14627
共Received 1 June 1999; accepted 29 July 1999兲 Microscopic correlation-function expressions for the stochastic evolution observed in single molecule spectroscopy are derived using Liouville-space projection operators. The kinetics of a multilevel quantum system coupled to a single collective overdamped Brownian-oscillator coordinate is exactly mapped onto a continuous-time-random-walk 共CTRW兲 involving the transition states共curve-crossing points兲. Closed expressions are derived for the stochastic trajectories and the nonPoissonian distribution of number of flips. When the oscillator relaxation is fast compared with the reaction rates, the waiting time distribution becomes exponential and the standard Poisson kinetics is recovered. © 1999 American Institute of Physics. 关S0021-9606共99兲01340-9兴
I. INTRODUCTION
jectories is strictly Poissonian and carries no additional information beyond the macroscopic rate constants.16,17 Microscopically, rate equations hold when all other degrees of freedom are fast compared with the observed kinetic time scales. Once the kinetic variables are coupled to some slow degrees of freedom, the theoretical description must include them explicitly, and the simple kinetic equations no longer hold. Under these circumstances the stochastic trajectories observed in SMS provide an important source of additional microscopic information about the system. A common model is to couple a multilevel system to an overdamped macroscopic slow coordinate described by the Smoluchowski equation.29–31 In general, solving these equations requires computing the evolution of coupled nuclear wavepackets.32 One of the interesting results of this article is that when the reactions occur at well localized nuclear configurations, the stochastic evolution can be rigorously mapped onto the continuous time random walk 共CTRW兲 model.33,34 In ordinary random walks, the walker jumps between different sites at fixed regular time intervals. Instead, the CTRW adds a distribution of waiting times between successive jumps. The model was first proposed by Montroll and Weiss.33 Extensions and further developments were made by Klafter, Zumofen, and Shlesinger.22,34 Hopping among the curvecrossing points associated with each pair of states results in the discrete CTRW. The distribution of waiting times originates from the evolution of the slow coordinate once it is eliminated from the description. Rather than solving a set of coupled partial differential equations for nuclear wavepackets, we obtain a random walk where the slow dynamics is rigorously incorporated into the waiting time distribution. One technical difference from the standard CTRW is that we need several waiting time distributions, not just one. When the nuclear coordinate becomes fast, we recover the simple kinetic equations with Poisson statistics. In Sec. II we start with one of the basic postulates of quantum mechanics due to von Neumann to derive general expressions for the probability to obtain a certain result in a
Recent advances in fluorescence detection and microscopy allow the studies of single molecules in complex condensed phase environments at cryogenic as well as room temperatures.1–7 Single-molecule spectroscopy 共SMS兲 provides much more detailed microscopic information on the molecule under study and its coupling to the environment, than standard bulk measurements. The technique eliminates ensemble averaging and disorder effects from the outset, without involving spectroscopically-selective methods such as photon echo8 or hole burning.9 SMS allows to probe motions which are slow compared to the fluorescence decay time, which is not possible using conventional bulk spectroscopy. SMS signals show stochastic behavior which originates from translational10–12 and rotational13 diffusion, spectral fluctuations,14 conformational motions,15 and chemicalchanges-induced fluorescence blinking.16,17 Detailed information on these motions can be obtained through a statistical analysis of the stochastic trajectories. Stochastic models of various kinds18–20 including random jump models21 as well as continuous-time random walks22 have been used for the interpretation of SMS experiments. In this article we compute the stochastic SMS trajectories for chemical reactions coupled to a slow solvation coordinate. Apart from providing detailed information on molecular dynamics, SMS also has the capacity to clarify some fundamental questions regarding the microscopic quantum description of single objects.23–28 The following issues will be addressed as well: what is the relation between the microscopic models which describe the dynamics of a system in terms of Liouville space evolution of the density matrix? Does the density matrix merely represent ensemble averages or does it also provide a powerful and useful tool for understanding the dynamics of single objects? Kinetic 共rate兲 equations provide a simple framework for the interpretation of chemical dynamics. When this level of theoretical description applies, the statistics of stochastic tra0021-9606/99/111(16)/7416/10/$15.00
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© 1999 American Institute of Physics
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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
Single-molecule spectroscopy
sequence of measurements in terms of a Liouville space multitime correlation function. It is important to note that the stochastic model describes the probability of getting a certain result in a sequence of measurements, rather than the free dynamics of the unperturbed system.28 The result of a sequence of measurements will hereafter be referred to as a stochastic trajectory. The probability of a given trajectory is expressed in terms of a multitime Liouville-space correlation function of the unperturbed system. This establishes a firm connection between the microscopic and stochastic descriptions of SMS, and provides a prescription for computing the stochastic trajectories and their statistical properties. In Sec. III we apply this approach to a kinetic model of L levels coupled to a single overdamped Brownian mode. We derive the stochastic model corresponding to this microscopic model, and show that it corresponds to a continuous-timerandom-walk. Expressions for the distribution of the number of flips are derived in Appendix A. In Sec. IV we consider the simplest case of two level kinetics. Numerical calculations which illustrate the nonPoisson nature of the stochastic trajectories, and a brief summary are given in Sec. V.
II. CORRELATION-FUNCTION EXPRESSIONS FOR THE EFFECTIVE STOCHASTIC MODELS
In this section we derive correlation-function expressions which make it possible to interpret SMS signals starting with a fully microscopic description of the system. Because of the quantum nature of microscopic objects, one can only predict the probability of a certain outcome in a set of measurements rather than the actual result.35 This probability can be expressed in terms of Liouville-space correlation functions using the following postulate due to von Neumann, which reflects the Copenhagen interpretation of quantum mechanics.36 A measurement with L possible outcomes can be described by decomposing the Hilbert space of states V of a quantum system into a direct sum V⫽V 1 丣 V 2 丣 ¯ 丣 V L where the space V j ( j⫽1,2,...,L) corresponds to the jth outcome of a measurement. Let P j be the projection operator associated with V j . According to von Neumann, for a system whose state is described by a wavefunction , the probability to get the result j in a measurement is given by 兩 具 兩 Pˆ j 兩 典 兩 2 and after the measurement the system has the wavefunction Pˆ j 兩 典 兩 具 兩 Pˆ j 兩 典 兩 ⫺1 . If we repeat the same measurement at times 1 , 2 ... N , apply von Neumann’s principle N times, and combine it with the time-dependent Schrodinger equation, we get for the probability P( j N ,..., j 1 ; N ,..., 1 ) to obtain the results j 1 ,..., j N in these measurements:
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P 共 j N ,..., j 1 ; N , . . . , 1 兲 ⫽ 具 P j N 共 n 兲 ...P j 1 共 1 兲 典 ˜
˜
⬅Tr关 P j N e ⫺iLt N⫺1 P j N⫺1 ...e ⫺iLt 1 P j 1˜ 兴 ,
共2.2兲
where ˜ ⫽ 兩 典具 兩 is the equilibrium density matrix, ˜L ⫽ 关 H, 兴 is the Liouville operator, and P j ( )⬅ Pˆ j Pˆ j are projection superoperators. Equation 共2.2兲 expresses the probability P( j N ,..., j 1 ; N ,..., 1 ) as a Liouville-space correlation function of the projection superoperators P j and allows to interpret the SMS signals in terms of Liouville-space dynamics. Even though we have derived Eq. 共2.2兲 for a density matrix ˜ that corresponds to a pure state 兩 典 , it also holds in the more general case of a mixed state. This can be obtained by starting in a pure state in an extended 共system and bath兲 space. By then integrating out the bath coordinates assuming that the projection operators Pˆ j do not depend on the bath, we recover Eq. 共2.2兲, where ˜ now represents the reduced system density matrix. Consider an experiment involving repeated measurements separated by a fixed time interval t j ⬅ j⫹1 ⫺ j ⫽⌬ . ˜ ⌬)P we can reIntroducing the operators Ui j ⬅Pi exp(⫺iL j cast Eq. 共2.2兲 in the form: P 共 j N ,..., j 1 兲 ⫽Tr兵 U j N j N⫺1 ...U j 2 j 1˜ 其 .
共2.3兲
Equation 共2.3兲 can be viewed as the probability of a trajectory determined by a sequence j N , . . . , j 1 for a stochastic model defined on a lattice. The stochastic model described by Eq. 共2.3兲 and formed by the possible outcomes j k provides a continuous limit representation as long as the time interval ⌬ is kept short compared with the Liouville-space dynamics time scale. This implies that the flips are rare and the description of the stochastic dynamics is given by the probability P f ( j N ,..., j 0 ; N ,..., 1 ) that the flips occur at times 1 ,..., N while at time k the system flips from j k⫺1 to j k . Typically nⰆN where n is the number of flips. Expanding U j j ⬇I⫹(⌬ )L j , Ui j ⬇(⌬ ) v i j for i⫽ j the probability P f finally assumes the form: P f 共 j n ,..., j 0 ; n⬘ ,..., ⬘1 兲 ⫽
冕
1
1 ⫺⌬
d 1⬘ ...
冕
n
n ⫺⌬
d n⬘ P s 共 j n ,..., j 0 ; n⬘ ,..., 1⬘ 兲 , 共2.4兲
with the probability density P s 共 j n ,..., j 0 ; n ,..., 1 兲
P 共 j N ,..., j 1 ; N , . . . , 1 兲 ⫽ 具 兩 Pˆ j 1 e iHt 1 . . . e iHt N⫺1 Pˆ j N e ⫺iHt N⫺1 . . . e ⫺iHt 1 Pˆ j 1 兩 典 ,
⫽Tr兵 e ⫺iL j n ( ⫺ n ) v j n j nN⫺1 . . . e ⫺iL j 1 ( 2 ⫺ 1 ) v j 1 j 0 e ⫺iL j 0 1˜ 其 .
共2.1兲
共2.5兲
where 兩 典 is the initial wavefunction and t m ⬅ m⫹1 ⫺ m is the nth time interval between successive measurements. This distribution, normalized as 兺 j 1 ,..., j N P( j N ,..., j 1 ; N ,..., 1 ) ⫽1, can be recast in Liouville-space form
Equation 共2.5兲 relates the stochastic model of SMS signals to microscopic quantities. P s can be interpreted as the probability of a continuous stochastic trajectory whereas P f is its regularized form.
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where ¯ ⫽ ␦ (q), q mn are the crossing points of the potential surfaces of states 兩 n 典 and 兩 m 典
III. CONTINUOUS-TIME-RANDOM-WALKS FOR MULTISTATE KINETICS
In this section we apply Eq. 共2.5兲 to derive a stochastic model for SMS starting with a microscopic model of an L-level quantum system coupled linearly to an overdamped Browninan oscillator with an arbitrary relaxation time scale. We assume that a measurement determines that the system occupies one of the L levels 兩 n 典 , which are not eigenstates of the system Hamiltonian H s : n⫽m
H s⫽
兺n E n兩 n 典具 n 兩 ⫹ 兺 nm
J nm 兩 n 典具 m 兩 .
共3.1兲
We further assume a diagonal coupling to a harmonic bath described by the Hamiltonian: H int⫽⫺
兺n q
(n)
兩 n 典具 n 兩 ,
共3.2兲
where q (n) denote the set of collective bath coordinates. All relevant information about the bath is contained in the matrix of spectral densities37,32,38 C i j共 兲⫽
i 2
冕
⬁
⫺⬁
d exp共 i 兲 具 关 q (i) 共 兲 ,q ( j) 共 0 兲兴 典 .
共3.3兲
The Liouville-space dynamics is described by the evolution of the wavepackets i j (q) where q represent the complete set of bath variables. q can be viewed as a set of collective relevant bath variables and the coordinates q (n) in Eq. 共3.2兲 are linear combinations of the components of q. The number of collective coordinates 共i.e., the number of components of q) needed for a reasonable reduced description is uniquely determined by the matrix of spectral densities C i j ( ) as well as the temperature.32 Below we consider a model which includes a single overdamped Brownian oscillator mode, where the collective coordinates q (n) responsible for the coupling are given by q (n) ⫽2 d n q and the spectral density is: C i j 共 兲 ⫽d i d j
2⌳
2 ⫹⌳ 2
共3.4兲
.
Here ⌳ is the relaxation rate of q whereas denotes its coupling strength with the multilevel system. In the hightemperature (⌳Ⰶk B T) limit, the dynamics can be described by the evolution of wavepackets i j (q) which only depend on a single collective coordinate q. The effective Liouville operator which describes the propagation of the wavepackets in the absence of systembath coupling is represented by the Smoluchowski operator 关Eq. 共3.8兲兴. In the spectral-diffusion (⌳Ⰶ 冑2 k B T) limit, the off-diagonal 共coherence兲 wavepackets 关i.e., i j (q) for i⫽ j] can be eliminated from the equations of motion, resulting in a closed system of equations for the diagonal 共population兲 wavepackets:29 d mm 共 q, 兲 (0) ⫽L m mm 共 q, 兲 ⫹ d
兺n mn¯ 共 q⫺q mn 兲
⫻关 nn 共 q, 兲 ⫺ mm 共 q, 兲兴 ,
共3.5兲
q mn ⫽q nm ⫽
E m ⫺E n , 2 共 d m ⫺d n 兲
共3.6兲
mn are the transition rates at the crossing points: mn ⫽ nm ⫽
2 兩 J mn 兩 2 , 2 兩 d m ⫺d n 兩
共3.7兲
(0) and L m is the Smoluchowski operator for the population wavepacket mm : (0) ⫽⌳ Lm
冉 冊 册
冋
k BT . 共 q⫺d m 兲 ⫹ q 2 q
共3.8兲
To identify the operators L j and v i j required for the evaluation of Eq. 共2.5兲 we need to start with a wavepacket j j (q), propagate it for a short time interval ⌬ , and find these operators from the conditions
j j 共 q, ⫹⌬ 兲 ⫺ j j 共 q, 兲 ⫽ 共 ⌬ 兲 L j j j 共 q, 兲 , ii 共 q, ⫹⌬ 兲 ⫽ 共 ⌬ 兲v i j j j 共 q, 兲 .
共3.9兲
⌬ is taken to be short compared to the time scale of the evolution of the population wavepackets, and only terms linear in ⌬ should be retained in the l.h.s. Combining Eq. 共3.9兲 with Eq. 共3.5兲 immediately yields n⫽ j
L j ⫽L (0) j ⫺
兺n
jn¯ 共 q⫺q jn 兲 ,
v i j ⫽ i j¯ 共 q⫺q i j 兲 .
共3.10兲
Equation 共3.10兲 together with Eq. 共2.5兲 determine the stochastic model for SMS corresponding to the microscopic model defined by Eqs. 共3.1兲–共3.4兲. For practical applications we can describe the wavepackets ¯ (q) in Eq. 共3.10兲 as narrow-Gaussians rather than ␦ -functions. The finite width of ¯ (q) which may result from eliminating fast bath degrees of freedom weakly coupled to the system frees the theory from any divergences. The solution of Eq. 共3.5兲 is carried out in three steps. First, nonadiabatic transitions are neglected. The Smolu(0) is solved analytically in each chowski equation with L m state, resulting in N Gaussian Green functions G. In the second step we add the term representing the leaking out from each state 共but still neglect the return兲. This results in L j 关Eq. 共3.10兲兴. This gives in an (L⫺1)⫻(L⫺1) matrix W ( j) for each state, representing the waiting time distribution between each pair of crossing points involving that state. W ( j) for each state of j is expressed in a closed form using G. Finally we consider the full Smoluchowski equations, express the resulting stochastic trajectories using W ( j) , and show that they represent a continuous-time-random-walk. We start with the first step. According to Eq. 共2.5兲, the evolution of the system between flips in state 兩 j 典 is determined by the effective Liouville operator L j 关Eq. 共3.10兲兴, constitutes which consists of two terms. The first term L (0) j the Smoluchowski operator which describes the diffusion of the oscillator coordinate whereas the second term is related
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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
Single-molecule spectroscopy
to the loss of the population at state 兩 j 典 which occurs at the crossing points q jn . The evolution with the operators L (0) j is described by the Green functions G ( j) (q,q ⬘ ; ) of the Smoluchowski equations, defined as the solutions of dG ( j) 共 q,q ⬘ ; 兲 ⫺L (0j) G ( j) 共 q,q ⬘ ; 兲 ⫽0, d
共3.11兲
with the initial conditions G ( j) 共 q,q ⬘ ;,0兲 ⫽¯ 共 q⫺q ⬘ 兲 .
共3.12兲
In our model the kinetics is fully determined by the values of the Green functions G ( j) (q,q ⬘ ; ) at the crossing points where the populations can change. This yields a set of ( j) (L⫺1)⫻(L⫺1) matrices G mn ( ) with m,n⫽1,2..., j⫺1, j⫹1,...L: ( j) G mn 共 兲 ⬅G ( j) 共 q m j ,q n j ; 兲 ,
which have the form ( j) G mn 共 兲⫽
with
1
冑2 共 兲
共3.13兲
再
exp ⫺
关 q m j ⫺x n j 共 兲兴 2
2 2共 兲
冎
,
冉 冊 冋冉 冊 册
k BT 2共 兲 ⬅ ⫺ 2
k BT ⫺ 20 e ⫺2⌳ , 2
x n j 共 兲 ⬅d j ⫺ 共 d j ⫺q n j 兲 e
⫺⌳
共3.14兲
1
冑2 0
冉 冊
exp ⫺
q2
2 20
,
共3.17兲
which becomes a ␦-function in the limit 0 ⫽0. Turning to the second step, we consider the following sequence of events 共history兲: the system jumps from 兩 j k⫺2 典 to state 兩 j k⫺1 典 , remains in that state for the time interval , and finally jumps to state 兩 j k 典 . It follows from Eq. 共2.5兲 that the probability for this sequence is ˆ ( j k⫺1 ) 共 兲 ⬅Tr关v j j W j k j k⫺2 k k⫺1
exp共 ⫺iL j k⫺1 兲¯ j k⫺1 j k⫺2 兴 ,
共3.18兲
共3.19兲
( j) where ˆ mn ⬅ ␦ mn m j and we adopt the following convention for the Fourier transform:
W ( j) 共 兲 ⫽
冕
d ( j) ˆ 共 兲 exp共 ⫺i 兲 . W ⫺⬁ 2 ⬁
n n⫺1
ˆ ( j n⫺1 ) 共 n ⫺ n⫺1 兲 ... 共 ⫺n兲W j j n n⫺2
( j 1) 共 2 ⫺ 1 兲 D (0) j 1 j 0共 1 兲 . 2 j0
共3.21兲
⫻W j
The first factor from the right D (0) j 1 j 2 ( ) is the probability to stay in state 兩 j 0 典 for time and jump to state 兩 j 1 典 . We denote it the doorway function ˜ ( j 0) 兴 , D (0) j 1 j 0 共 兲 ⬅Tr关v j 1 j 0 exp共 ⫺iL j 0 兲
共3.22兲
and ˜ ( j 0 ) is the j 0 j 0 (q) component of the stationary solution of Eq. 共3.5兲. Following that, Eq. 共3.21兲 contains a series of n⫺1 evolution periods related to the waiting time distributions 关Eq.共3.18兲兴. Finally the last period is represented by the window function W (0) j n j n⫺1 ( ) which gives the probability of state 兩 j n 典 to survive during the time , provided it was created by a flip from state 兩 j n⫺1 典 , W (0) j j
n n⫺1
共 兲 ⬅Tr关 exp共 ⫺iL j n 兲¯ j n j n⫺1 兴 .
A straightforward calculation yields:
冋
1 1⫺ i
W (0) j j
共 兲 ⫽⫺
ˆ (0) D j j ⫽
1 ˜ ⫺ i j1 j0 j1 j0
1 0
冋
兺m Wˆ m( j j ) n
n⫺1
共3.23兲
册
共兲 ,
兺m Wˆ (j j m)共 兲 m j ˜ m j
共3.24兲
0
0
1
0
册
,
共3.25兲
where ˜ m j 0 ⬅˜ ( j 0 ) (q j 0 m ). The distribution of stochastic trajectories Eq. 共3.21兲 can be used to compute any desired statistical property of the system. In Appendix A we use a generating function approach to calculate the probability P ( f ) (n; ) to have n jumps during the time interval . This distribution is given by P ( f ) 共 n; 兲 ⫽
冕
⬁
⫺⬁
d 1⫺e ⫺i mn˜ m ⬘ n ⬘ 2 2 mnm ⬘ n ⬘
兺
⫻ 兵 关 A ⫺1 共 兲 ⫺A ⫺1 共 兲 B 共 兲 A ⫺1 共 兲兴 ⫻关 B 共 兲 A ⫺1 共 兲兴 N⫺1 其 mn,m ⬘ n ⬘ .
共3.26兲
A and B are the following tetradic L 2 ⫻L 2 matrices: A mn,m ⬘ n ⬘ 共 兲 ⬅ ␦ mm ⬘ ␦ nn ⬘ ⫹ ␦ mm ⬘ mn ⬘ G nn ⬘ 共 兲 , (m)
where ¯ j k⫺1 j k⫺2 (q)⫽¯ (q⫺q j k⫺1 j k⫺2 ). The operator in the r.h.s. of Eq. 共3.18兲 can be expressed in terms of the Green function of the Smoluchowski equation 关Eq. 共3.14兲兴 by switching to the frequency representation, which yields: ˆ ( j) 共 兲关 I⫹ ˆ ( j) G ˆ ( j) 共 兲兴 ⫺1 , ˆ ( j) 共 兲 ⬅ ˆ ( j) G W
ˆ (0) ⫽W j j
共3.15兲
In Eq. 共3.15兲 0 is a regularization parameter, representing a finite width at ⫽0, and the wavepacket ¯ in Eq. 共2.5兲 has the form ¯ 共 q 兲 ⫽
P s 共 j n ,..., , j 0 ; n ,..., 1 兲
n n⫺1
共3.16兲
.
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共3.20兲
Using Eq. 共3.10兲, the probability for a stochastic trajectory 关Eq. 共2.5兲兴 becomes:
B mn,m ⬘ n ⬘ 共 兲 ⬅ ␦ mn ⬘ mm ⬘ G nm ⬘ 共 兲 . (m)
共3.27兲
In the next section we shall apply the general expressions derived here to the simplest case of A B chemical reaction. IV. RANDOM WALKS AND TRAJECTORIES FOR TWO-LEVEL KINETICS
In the present continuous-time random walk 共CTRW兲 model, the waiting-time distribution of the state 兩 j k 典 depends on the state 兩 j k⫺1 典 prior to the last flip. This has a simple physical origin. In our model, jumps can only occur at the curve-crossing points q mn , and the jump probability depends on the population at the crossing points. During the waiting time, the Brownian oscillator undergoes diffusion and can
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move among the various crossing points. Since the jump to the state 兩 j k 典 and 兩 j k⫺1 典 occurs at the points q j k j k⫺1 and q j k⫹1 j k , respectively, the time spent in state 兩 j k 典 reflects the flow of population from q j k j k⫺1 . The waiting-time ˆ ( j k) ( ) depends therefore on the three distribution W
P ( f ) 共 0; 兲 ⫽1⫺ ˜ ⫻
j k⫹1 j k⫺1
states 兩 j k⫹1 典 , 兩 j k 典 , and 兩 j k⫺1 典 . Stated differently, the L c ⫽ 21 L(L⫺1) crossing points rather than the L states constitute the sites for stochastic jumps. This is why in Sec. III we expressed the distribution of the number of flips in terms of L 2c ⫻L 2c superoperators rather than ordinary L⫻L operators. The present L⫽2 case is special since it only has a ˆ ( j k) single crossing point q 12⫽q 21 . The distributions W
冋
W ( j) 共 兲 ⫽
冕
⫺⬁
d ⫺i ( j) ˆ 共 兲. e W 2
G (1) 共 兲
G ( j) 共 兲 1⫹ G ( j) 共 兲
共4.2兲
.
The distribution of number of flips P (N, ) 关Eq. 共3.26兲兴 also simplifies considerably in this case. The superoperators A and B 关Eq. 共3.27兲兴 act in a two-dimensional space with the basis 兩 mn 典 represented by 兩 ab 典 and 兩 ba 典 . Denoting these basis elements 兩 1 典 ⬅ 兩 ab 典 , 兩 2 典 ⬅ 兩 ba 典 关Eq. 共3.27兲兴 yields A mn 共 兲 ⫽ ␦ mn 关 1⫹ G (m) 共 兲兴 , 共4.3兲
B mn 共 兲 ⫽ 共 1⫺ ␦ mn 兲 G (m) 共 兲 .
Substituting Eq. 共4.3兲 into Eq. 共3.26兲 and making use of the fact that the matrices A( ) and B( ) have only diagonal and off-diagonal components, respectively, we obtain
⬁
⫺⬁
ˆ (1) 共 兲 W ˆ (2) 共 兲兴 m d 1⫺e ⫺i 关 W , 2 2 2 G (1) 共 兲 G (2) 共 兲
共4.4兲
P ( f ) 共 2m; 兲 ⫽ ˜ ⫻
冕
再
⬁
⫺⬁
G (2) 共 兲
册
共4.6兲
,
G ( j) 共 兲 ⫽
冕
⬁
0
d exp兵 i 其
再
1
冑2 共 兲
关¯x ⫺x j 共 兲兴 2
⫻exp ⫺
2 2共 兲
冎
共4.7兲
,
with x j 共 兲 ⬅d j ⫺ 共 d j ⫺x ¯ 兲 e ⫺⌳ ,
共4.8兲
¯x ⬅
E 1 ⫺E 2 , 2 共 d 1 ⫺d 2 兲
共4.9兲
and ˜ ⫽
1
1
冑2 ˜
exp关共¯x ⫺d 1 兲 /2 兴 ⫹exp关共¯x ⫺d 2 兲 2 /2˜ 2 兴 2
˜2
, 共4.10兲
with ˜ 2 ⬅k B T/2 . Equations 共4.4兲–共4.6兲 together with Eqs. 共4.7兲–共4.10兲 constitute a closed set of expressions for P ( f ) (N; ) in a twolevel kinetics. Let us consider first the limiting case where the bath relaxation rate ⌳ is fast compared with the kinetics. In this case the system can be described by an ordinary rate equation for the level populations rather than by Eq. 共2.5兲 which describes the dynamics of the population wavepackets, and the trajectory has a simple Poisson statistics. To recover this limit we replace G ( j) ( ) by its long-time 共i.e., low-frequency兲 asymptotic form G (0j) ( ): G (0j) 共 兲 ⫽⫺
1 i
冋
1
exp ⫺
冑2 ˜
共¯x ⫺d j 兲 2
2 ˜ 2
册
共4.11兲
,
ˆ (1) 共 兲 W (1)
共 兲兴
for m⫽1,2, . . . and
2
⫹
ˆ (2) 共 兲 W 关G
(2)
共 兲兴
2
冎
,
k nm ⫽ 共 1⫺ ␦ mn 兲
冑2 ˜
冋
exp ⫺
共¯x ⫺d m 兲 2
2 ˜ 2
册
共4.12兲
.
Equation 共4.12兲 represents the Marcus rates.31 Substituting ˜ 2 ⫽k B T/2 and using the expressions for 关Eq. 共3.7兲兴 and ¯x 关Eq. 共4.9兲兴 yields
d 1⫺e ⫺i ˆ (1) 共 兲 W ˆ (2) 共 兲兴 m 关W 2 2
关G
ˆ (2) 共 兲 W
the kinetic rate to go from state m to n now assumes the Marcus form31
P ( f ) 共 2m⫺1; 兲
冕
⫹
where ˜ ⬅˜ 12⫽˜ 21. The Green functions G ( j) ( ) which enter Eqs. 共4.4兲, ˆ ( j) ( ) have the form 共4.5兲, 共4.6兲 both directly and through W
共4.1兲
(f)
⫽2 ˜
⫺⬁
d 1⫺e ⫺i 2 2
the crossing point ⬁
Equation 共3.19兲 which becomes a scalar rather than a matrix expression immediately gives ˆ ( j) 共 兲 ⫽ W
⬁
ˆ (1) 共 兲 W
j k⫹1 j k⫺1
can be considered as depending on 兩 j k 典 only, since given 兩 j k 典 , 兩 j k⫹1 典 , and 兩 j k⫺1 典 are uniquely determined from the ˆ ( j) 关Eq. 共3.19兲兴 and the Green condition j k⫹1 ⫽ j k⫺1 ⫽ j k . W ( j) functions G therefore loose their matrix structure as well, and become scalars. This results in an ordinary two-state CTRW model with the waiting-time distribution.
冕
共4.5兲
k mn ⫽ 共 1⫺ ␦ mn 兲
2J2
冑4 k B T
冋
exp ⫺
册
共 ⌬G mn ⫹ 兲 2 , 共4.13兲 4k B T
where ⬅ (d 2 ⫺d 1 ) 2 is the reorganization energy, and 2 )⫺(E n ⫺ d 2n ) is the reaction free energy. ⌬G mn ⬅(E m ⫺ d m
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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
Single-molecule spectroscopy
7421
Substituting Eq. 共4.11兲 in Eqs. 共4.1兲 and 共4.2兲 yields W (1) ( )⫽⫺k 21(i ⫺k 21) ⫺1 and W (2) ( )⫽⫺k 12(i ⫺1 ⫺k 21) . The waiting time distributions Eq. 共4.1兲 and 共4.2兲 become exponential W (1) 共 兲 ⫽k 21 exp共 ⫺k 21 兲 ;
W (2) 共 兲 ⫽k 12 exp共 ⫺k 12 兲 . 共4.14兲
Substituting these in Eqs. 共4.5兲 and 共4.6兲 yields after some straightforward transformations P ( f ) 共 2m⫺1; 兲 ⫽2e ⫺k 21
k 21k 12 共 k 12 兲 m⫺1 共 k 21 兲 m⫺1 k 12⫹k 21 共 2m⫺1 兲 !
⫻M 共 m,m; 共 k 21⫺k 12兲 兲 , P ( f ) 共 2m; 兲 ⫽
共4.15兲
共 k 12k 21 兲 共 k 12 兲 m⫺1 共 k 21 兲 m⫺1 ) k 12⫹k 21 共 2m 兲 !
⫻ 兵 k 21 e ⫺k 21 M 共 m⫹1,m; 共 k 21⫺k 12兲 兲 ⫹k 12 e ⫺k 21 M 共 m⫹1,m; 共 k 12⫺k 21兲 兲 其 ,
FIG. 1. Waiting time distribution W( ) for relaxation rates ⌳⫽10k 共solid line兲, ⌳⫽k 共dashed line兲, ⌳⫽0.05k 共dotted line兲, and ⌳⫽0.01k 共dasheddotted line兲. The other parameters are given in the text.
共4.16兲 where M denotes the confluent hypergeometric function. When k 12⫽k 21 the distributions given by Eqs. 共4.15兲 and 共4.16兲 are Poissonian. Otherwise the statistics described in Eqs. 共4.4兲–共4.6兲 represents a convolution of two Poissonians. Deviations from Poisson statistics are noticeable once the bath relaxation rate becomes comparable to or slower than the kinetics, and the simple kinetic description no longer holds. These will be demonstrated in the next section.
where g ( j) 共 兲 ⫽C ( j) ⫺i
再
冕
⬁
0
⫻exp ⫺
G ( j) 共 兲 ⫽
冕
⬁
0
d e i
再
⫻exp ⫺
冋
1
冑2 共 兲
关¯x ⫺x j 共 兲兴 2
2 2共 兲
⫺ C ( j) ⫹ C ( j)
i , 共5.1兲
¯⫺d j)2/2˜ 2 兴 / 冑2 ˜ . The integrand now where C ( j) ⫽exp关⫺(x decays to zero, and the numerical computation can be performed by using a standard fast Fourier transform 共FFT兲 routine. We now have G ( j) 共 兲 ⫽⫺g ( j) 共 兲 /i ,
共5.2兲
1
冑2 共 兲
2 2共 兲
冎 册
⫺ C ( j) .
共5.3兲
Substituting this expression for into Eq. 共4.4兲, the 1/ 2 factor is cancelled, and we can now compute the expression p ( f ) 共 2m⫺1; 兲 ⬅2 ˜
冕
⬁
⫺⬁
d 2
⫻exp共 ⫺i 兲
ˆ (1) 共 兲 W ˆ (2) 共 兲兴 m 关W
2 g (1) 共 兲 g (2) 共 兲
,
共5.4兲 by using a standard FFT routine, and obtain P ( f ) (2m⫺1; )⬅p ( f ) (2m⫺1; )⫺p ( f ) (2m⫺1;0). P ( f ) (2m; ) 关Eq. 共4.5兲兴 can be calculated in a similar way. The probability for zero jumps 关Eq. 共4.6兲兴 can be computed as follows. Substituting Eq. 共5.2兲 into Eq. 共4.6兲 gives P ( f ) 共 0; 兲 ⫽1⫺ ˜ ⫻
冎 册
冋
关¯x ⫺x j 共 兲兴 2
V. NUMERICAL RESULTS AND DISCUSSION
To illustrate the two-level kinetics, we consider a symmetric reaction in which the reactant and product free energy is the same, ⌬G 12⫽0. The minima of the two potential energy surfaces are equal, i.e., E 1 ⫺E 2 ⫽0, and the position of the minima are d 1 ⫽0.05 and d 2 ⫽⫺0.05. We choose the coupling constant J⫽0.003 cm⫺1, the temperature T⫽300 K, and the vibronic coupling ⫽5000 cm⫺1 . With this choice of parameters, the Marcus rate becomes k⫽4.41⫻103 s⫺1 . The regularization parameter is set to 20 ⫽ ˜ 2 /10, and the relaxation rate ⌳ is varied between 0.01k and 10k. The numerical calculations require some care. The integrand in Eq. 共4.7兲 does not vanish at long times. We therefore rewrite Eq. 共4.7兲 by subtracting and adding the asymptotic value of the integrand C ( j) under the integral, i.e.,
d e i
冋
冕
⬁
⫺⬁
d 2sin共 /2兲 2 exp共 i /2兲
ˆ (1) 共 兲 W
g (1) 共 兲
⫹
ˆ (2) 共 兲 W
g (2) 共 兲
册
.
共5.5兲
The integrand is regular for →0, and this expression can be easily computed using numerical integration. By symmetry, the two waiting time distributions coincide: W (1) ( )⫽W (2) ( )⬅W( ). Figure 1 shows a logarithmic plot of W( ) for fast, intermediate and slow relaxation, ⌳⫽10k 共solid兲, ⌳⫽k 共dashed兲, ⌳⫽0.05k 共dotted兲, and ⌳⫽0.01k 共dashed-dotted兲, respectively. We clearly see the expected exponential decay of W( ) 关Eq. 共4.16兲兴 in the limit of fast relaxation, and the increasingly nonexponential behavior for slower relaxation rates.
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7422
J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
Chernyak, Schulz, and Mukamel
FIG. 2. Probability P ( f ) (n; ) of having n jumps at time . The four panels represent different relaxation rates ⌳ as indicated in each panel. The times k are 0.77 共squares兲, 1.53 共circles兲, 2.30 共triangles兲, 3.83 共diamonds兲, and 5.75 共crosses兲.
Figure 2 shows the probability P ( f ) (n; ) of having n jumps for different relaxation rates ⌳ 共as indicated in each panel兲. The various curves correspond to five values of k , represented by squares (k ⫽0.77), circles 共1.53兲, triangles 共2.30兲, diamonds 共3.83兲, and crosses 共5.75兲. The connecting lines are added to guide the eye. The fast relaxation (⌳⫽10k) limit gives a Poisson distribution, as expected. With decreasing relaxation rate, a strong deviation from Poisson statistics can be clearly discerned. The most pronounced feature is a peak at zero jumps. This can be understood as follows. If a jump occurs and the relaxation rate is fast, then the wavepacket produced at the crossing point is quickly equilibrated. However, in the case of slow relax-
ation, this wavepacket has a long lifetime at the crossing point, so that the system can jump back and forth several times. Thus, for slow relaxation, we have to think of the system as having a certain probability P ( f ) (0; ) that a jump does not occur, and a probability 1⫺ P ( f ) (0; ) that a jump does occur. Due to the long lifetime of the wavepacket at the crossing point, the probability that a jump does occur represents a broad range of the number of jumps n, so that the probability P ( f ) (n; ) for exactly n jumps to occur is fairly small in comparison to P ( f ) (0; ). The increasing deviation from Poisson statistics with decreasing ⌳ is further illustrated in Fig. 3, which shows P ( f ) (n; ) as a function of time k for different values of ⌳
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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
Single-molecule spectroscopy
7423
FIG. 3. Probability P ( f ) (n; ) of having n jumps at time , plotted vs . The four panels represent different relaxation rates ⌳ as indicated in each panel. The number of jumps n are represented by dash-dot-dotted (n⫽0), solid (n⫽1), dashed (n⫽2), dotted (n⫽3), and dash-dotted lines (n⫽4), respectively. The curve for n⫽0 has been reduced in the ordinate direction by a factor of 5.
as in the previous figure. Different numbers of jumps n are represented by dash-dot-dotted (n⫽0), solid (n⫽1), dashed (n⫽2), dotted (n⫽3), and dash-dotted lines (n⫽4), respectively. The curves for n⫽0 have been reduced by a factor of 5. With decreasing ⌳ the temporal maximum of P ( f ) (n; ) shifts to earlier times, and the function acquires a long tail. With increasing n, the maximum of the curves shifts to later times. ACKNOWLEDGMENTS
The support of the National Science Foundation through Grants No. CHE-9526125 and No. PHY94-15583, and the
Air Force office of scientific research is gratefully acknowledged. APPENDIX A: DISTRIBUTIONS OF NUMBERS OF FLIPS
The expressions derived in Sec. III 关Eqs. 共3.21兲, 共3.24兲, and 共3.19兲兴 give the probability P s of a stochastic trajectory ( j) of the Smoluchowski in terms of the Green functions G mn equations. In this appendix we apply these expressions to compute the probability P ( f ) (n; ) to have n jumps during the time interval . P ( f ) is defined by the time-ordered integrals:
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7424
J. Chem. Phys., Vol. 111, No. 16, 22 October 1999
P ( f ) 共 n; 兲 ⫽
冕 冕
d
0
⫻
j0
n
兺 ...j
n
0
d n⫺1 . . .
冕
2
0
Chernyak, Schulz, and Mukamel
d1
P s 共 j n ,..., j 0 ; n ,..., 1 兲 .
mn 共 兲 ⫽⫺ 共A1兲
n
Combining Eqs. 共A1兲 and 共3.21兲 yields P ( f ) 共 n; 兲 ⫽
冕
⬁
⫺⬁
d ˆ (0) 共 兲 W 2 j 0 . . . j n j n j n⫺1
兺
Tr兵 exp关 ⫺iL共 兲 兴˜ 其 ⫽1⫺ 共A2兲
The summation allows us to interpret Eq. 共A2兲 as a product of superoperators acting in the space of operators generated by 兩 i 典具 j 兩 . This can be obtained directly from Eq. 共A2兲. We will obtain it, however, in an alternative indirect way where the superoperator structure arises more naturally. To that end we introduce the operators L⬅ 兺 j L j and v ⬅ 兺 i j v i j which act in the total space of population wavepackets. Defining a family of operators L( )⬅L⫹ v parametrized by , expanding the operator exp关⫺iL( ) 兴 in powers of , and making use of Eqs. 共A1兲 and 共2.5兲 we obtain ⬁
Tr兵 exp关 ⫺iL共 兲 兴˜ 其 ⫽
兺
n⫽0
n P ( f ) 共 n; 兲 .
共A3兲
This serves as a generating function for the distribution of jumps, and we immediately have P ( f ) 共 n; 兲 ⫽
1 n Tr兵 exp关 ⫺iL共 兲 兴˜ 其 兩 ⫽0 . n! n
共A4兲
The r.h.s of Eq. 共A4兲 can be evaluated in the following way: The set of wavepackets (m) (q, )⬅ mm (q, ) which describe exp关⫺iL( ) 兴˜ can be obtained by solving the equation
(m) 共 q, 兲 (0) (m) ⫽L m 共 q, 兲 ⫹ d
兺n mn¯ 共 q⫺q mn 兲
⫻关 (n) 共 q, 兲 ⫺ (m) 共 q, 兲兴 ,
共A5兲
with the initial condition (n) (q,0)⫽˜ (n) (q). For ⫽1 Eq. 共A5兲 yields Eq. 共3.5兲. Since the Smoluchowski operators (0) conserve the trace, time integration of Eq. 共A5兲 gives Lm Tr兵 exp关 ⫺iL共 兲 兴˜ 其 ⫽1⫹
共A7兲
(m) where we have introduced the notation mn ( )⬅ mn ( ). Equation 共A7兲 can be solved by introducing the superoperators Eq. 共3.27兲. Substituting the solution of Eq. 共A7兲 into Eq. 共A6兲 we obtain
2 0
⫻exp共 ⫺i 兲 .
兺k mk G nk(m)共 兲
⫻ 关 km 共 兲 ⫺ mk 共 兲兴 ,
ˆ ( j n⫺1 ) 共 兲 . . . W ˆ ( j 1) 共 兲 D ¯ (0) ⫻W j 1 j 0共 兲 j j j j n n⫺2
1 ˜ ⫹ i mn
mn 冕 d ⬘ 关 (n) 共 q mn , ⬘ 兲 ⫺ (m) 共 q mn , ⬘ 兲兴 . 兺 mn 0
共A6兲 The r.h.s. of Eq. 共A6兲 only involves the values of the (n) ( ⬘ )⬅ (n) (q mn , ⬘ ) . wavepackets at the crossing points mn ( j) These can be expressed in terms of the Green functions G mn using a standard solution of Eq. 共A5兲. In the frequency domain this yields the following equation:
冕
d 1⫺e ⫺i mn 共 1⫺ 兲 ⫺⬁ 2 2 mnm ⬘ n ⬘ ⬁
兺
⫺1 ⫻ 关 A 共 兲 ⫺ B 共 兲兴 mn,m ⬘ n ⬘˜ m ⬘ n ⬘ .
共A8兲
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