Tunneling in the presence of fluctuations: the stochastic liouville equation approach

Chemical Physics 115 (1987) 391-397 North-Holland, Amsterdam

391

TUNNELING IN THE PRESENCE OF FLUCTUATIONS: THE STOCHASTIC LIOUVILLE EQUATION APPROACH Alessandro LAM1 and Giovanni VILLANI Istituto di Chimica Quantistica ed Energetica Molecolare del CNR, Via Risorgimento

35, 56100 Pisa, Itab

Received 4 February 1987

The stochastic Liouville equation is used to discuss the role of fluctuations in tunneling dynamics. Emphasis is given to the case where the matrix elements of the hamiltonian are continuous-time Markov chains, since in this case the computational scheme is shown to be very simple, being based on matrix manipulations. The utility of the method for arbitrary correlation times is illustrated in two examples concerning two-level proton tunneling. In the first the role of a fluctuating electric field created by the solvent is discussed and the results are interpreted in terms of “motional narrowing”. In the second example the influence of a strong electromagnetic field (which determines jumps between two potential energy curves) is analyzed for a concrete case, i.e. the tropolone molecule.

1. Introduction

Quantum tunneling has been recognized to be of fundamental importance in many physical [l-5], chemical [6-141 and biological processes [15]. The theoretical treatment of tunneling in isolated systems is well known. It consists in building up the suitable non-stationary state that at time t = 0 is localized in one part of the system and then following (analytically if possible or numerically) the time evolution dictated by the time-dependent S&r&linger equation. A very interesting and to some extent unsolved problem concerns the role of the fluctuations which the tunneling system undergoes and which may have different origins. For example they may be due to the medium in which the system under study is dissolved, or to the vibrations of the molecules bringing the tunneling system. A further

interesting case concerns jumps between two potential energy curves due to photon emission and absorption. We develop here a theoretical approach based on the Liouville stochastic equation introduced by Kubo [16,17], which in our opinion may reveal very useful in a number of such problems. In the following we will be concerned with a two-level model, for the sake of notational simplicity. It may be easily extended to many levels with the only complication of having bigger matrices to handle. The paper is organized as follows. Section 2 contains a detailed description of the model in its generality. In section 3 we present and discuss some numerical examples having in mind the specific features of the proton tunneling problem. Section 4 is devoted to a few concluding remarks.

2. Two-state model with noise The typical scenario for tunneling involves motion in a double-well potential containing many states. The two-state model can only mimic the behaviour of actual systems at low temperature but its importance resides mainly in the fact that it contains much of the basic ingredients necessary to build up models for more complicated systems. Working in the non-stationary basis of localized states, 1R) and 1L) (right 0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A. Lami, G. Villani / Fluctuations in tunneling dynamics

392

and left) one can write the following hamiltonian for the isolated system:

where the energy difference A is included to treat the general case of an asymmetric double well. The perturbation V is responsible for the tunneling. As is well known, if one is able to prepare the system in a localized state, say 1R), at time t = 0, then the probability of finding the system in 1R) at time t is: P,(t)

=q-*[q*

cos*(qt/2)

+A* sin*(qt/2)],

(2)

where q =

(A* + 4V*)-I’*.

(3)

Let us simulate the effect of the environment by assuming that the matrix elements of the hamiltonian undergo fluctuations:

v+

ER(f) V-to(t)

e(t)

(4)

A+c,(t)

with (G(f))

= (Q(f))

= (W>

= 0.

The problem becomes that of solving a Liouville equation with stochastic matrix elements li=All

(5)

where

I A=

0

0

0

!

-i

I/+

-&(t)]

v+ e(t)

e(t)

-P-+wl

v+ e(t)

4t) - dt) -A

-P+Wl

0

v+ e(t)

-[v+wl

\

-4,(t) -A, /

(6)

- ,%

If the stochastic vector r(t) is a Markov process then it can be shown [16,18] that the joint process (u, r) is also markovian and its probability density P(u, r, t) (i.e. the probability density that at time t the density matrix will be that specified by the vector u for the realization of H indicated by r) satisfies the equation:

Wu, r, t) at

r, t) +

= - CA,,(r)&u,P(u,

Y

CL.”

J++(u,r, t),

p, v=l,...,

4,

where I@ is the operator which appears in the master equation for the transition probability density of i.e. Il(r,

t 1r,,

to) = bQFn(r,

t ) r,,

to).

r,

(8)

A. Lami, G. Villani / Fluctuations in tunneling dynamics

393

The marginal averages, defined as

m,(r, t>= /d’(u, r, t> du,

(9)

can be shown to obey the simpler equation:

aq(r, t> = - CA,,(r)m,(r,

at

t) + J@m,(r, t>

(LO)

P

After solving eq. (lo), the required averaged density matrix elements can be obtained as

(IG~(~))= /ml(r, t>dry (~~~(1))= /m2(r,t) dr, (Gus) = /mg(rlt>dry (Gus) = /m,(r, t>dr.

01)

We suppose that N matrix elements of H, i.e. N components of the vector r, can jump independently among n values. Then we have N X n realizations of the stochastic process simulating fluctuations. The master equation for its probability density is:

where Wki is the probability of transition k +j per unit time (here k and j mean r, and 9). Coming back to eq. (10) we note that we have N X n equations for each marginal average (which are four): am,(.i, t) = C4p(j)m,(jy t> + C wjkm,(kT t). 03) at P

k

Notice the different role of matrices A and W in the 4 x N X n coupled differential equations. A couples different m characterized by the same realization of r, whereas W couples different realizations of the same m. The choice of the matrix elements of H to be treated as stochastic variables, as well as the exact form of W, depends on the physics of the problem, as we will see in section 3. The relevant parameters are the tunneling period T and the correlation time of detuning fluctuations, rc. The former is (see eq. (2)) T= 2~(4V~ + A2)-“2.

(14)

The latter can be obtained in a straightforward manner (see the appendix). The interesting thing in the present model is that it can be solved easily for any correlation time, i.e. for rc -=s
394

A. Lumi, Q Villani / Fluctuations in tunneling dynamics

3. Some examples

that since r assumes only four discrete values the integral becomes a sum. For example

In this section we apply the above theory to two different problems. Let us first consider proton tunneling in a solvent. As is well known the dipole moment d has opposite directions in the localized states, whereas the extradiagonal matrix elements are negligible if the potential energy barrier is sufficiently high and/or extended: (15) The solvent creates a fluctuating electric field which perturbs the tunneling system. We can model this situation by assuming that only the diagonal matrix elements of H are stochastic. The two-component vector r is assumed to jump between the four realizations:

(16) with a probability: -3

1 1 1

l-3 1

1

hu30))

- ~0)

-A)(4’)

= za2 ,-/O-1’)

c j=l

?(A

0.

(20)

Due to the linear nature of the problem the average density matrix elements can be written as linear combinations of exponent& of eigenvalues of a symmetric matrix multiplied by the time. The eigenvalue(s) having the real part equal to zero give(s) the asymptotic stationary distribution PRR= PLL = 1/2,

PRL = PLR = O.

(21)

The inverse of the next smaller (absolute value) real part gives the time necessary to relaxation. In fig. 1 we plot such time as a function of the logarithm of the correlation time, for two different values of the jump a. The four realizations of r are assumed to have the same initial probability and both times are in units of the tunneling period T. The interesting thing is the non-monotonic behaviour of the relaxation time as a function of correlation time. The present model predicts that fluctuations have weak effects on the relaxation time if their correlation times are either too short or too long with respect to the tunneling period.

(17)

2

I 30-

and where a is the energy jump and f is the probability per unit time of making a jump. As is shown in the appendix, the correlation. function for the energy difference is, in this case, (((4)

=

1\ 1

l-3

4

- &‘)

-A)))

20%I

(18)

‘1

-

IO-

and the correlation time rc = l/f.

(19)

As previously mentioned the’ marginal averages obey a system of sixteen differential equations which can be easily solved by a matrix diagonalization. The average density matrix elements can be obtained from eqs. (ll), taking into account

,

I

-2

I

I

I

0

2

4 l“9%/

T)

Fig. 1. Relaxation time as a function of the logarithm of the correlation time, for two different values of the energy jump a (for further details see text).

A. Lami, G. Wlani / Flucluations in tunneling

this idea to the proton tunneling of tropolone for which the coupling V for the ground state and the first T+‘TT* is given in the literature [28], but we want to stress that the same treatment can work successfully for isomerization tunneling (for example in stilbene). Let us show the result of the calculations remembering that here A = 0 (i.e. the tunneling is symmetric). In fig. 2 we give the time behaviour for the right average population for three different values of the probability of jump per unit time, f. The values of V for the ground and excited states are assumed to be 1 and 11 cm-‘, respectively [28]. The initial distribution of V is taken

In the first limit the system sees only the average hamiltonian, whereas in the second limit the actual behaviour is a superposition of those corresponding to the four realizations of r (i.e. of the hamiltonian). This is the analog of “motional narrowing” in the lineshape problem [15,17]. To our knowledge there is no discussion on this interesting point in the literature concerning tunneling, although precise spectroscopic measures in the weak temperature regime are available (as an example see ref. [23] and references therein). We consider here a further problem concerning tunneling in presence of a large number of photons which make the system to jump between two electronic states characterized by different barriers. The absorption and emission of photons is considered instantaneous with respect to tunneling. Furthermore they occur at random times, with a certain probability per unit time (depending linearly on the number of photons). If the photons have exactly the energy to match the lower localized states on the two surfaces, one can simulate the whole process by a two-level problem in which the coupling V jumps randomly between the two values corresponding to the two different electronic states. The underlying tacit assumption is that the coherence between the ground and the excited state is lost in a very short time. We apply

I

395

dynamics

P,(V)

= 6(V-

1).

The time is in units of the tunneling period for the averaged value of the coupling, i.e. 6 cm-’ (this time unit corresponds to = 3.2 X lo-*’ s). Looking at fig. 2 one observes that even the fluctuations of the coupling determine the damping of oscillations around the stationary state in which right and left have equal populations. The effect is maximum when the frequency of jumps is of the same order of the tunneling frequency. The values of the transition probability used in the picture correspond to intensities which may be estimated to be in the range 0.1-10 GW/cm’.

I

I

I

I

I

2

4

6

8

IO

1

12

t, ,T

c 14

Fig. 2. Average population localized on the right as a function of time for three different values of the probability of jump f per unit time (for further details see text).

396

A. hmi,

G. Villani / Fluchtations in tunneling dynamics

4. Conclusion

Appendix

We have presented a general and computationally convenient method, founded on the Liouville stochastic equation [16,X3], for studying the evolution towards the stationary state of systems undergoing fluctuating perturbations. The basic assumption is that certain numbers of matrix elements of the hamiltonian are markovian stochastic processes. In the case that they are continous-time Markov chains, i.e. they are allowed to jump among a set of discrete values, the only calculation to do in order to have the average density matrix is the diagonalization of a square matrix of order where n is the number of states innxNXn,, cluded, N is the number of matrix elements of the hamiltonian assumed to be stochastic and n, is the number of values each element of H can assume. The above approach gives a powerful and simple tool for studying in a non-perturbative manner the influence of the surroundings on an actual system. We have illustrated the approach treating the simpler system exhibiting tunneling, i.e. a two-level system. In a first example we have considered energies fluctuating between two values in order to simulate the role of the solvent. The relevant result is that the analog of motional narrowing can occur. The second example was concerned with a system which can jump between two electronic states characterized by different energy barriers between localized states. This seems appropriate to treat not only proton tunneling (as in the example discussed) but also heavy groups tunneling (i.e. isomerization) in presence of strong electromagnetic fields. As a conclusion we note that the most appropriate use of the present approach is to mimic actual systems in which jumps occur among a discrete set of values, as in the example of photon absorption/emission between two electronic states. However, it may reveal very useful in simulating noise in general, since it permits easy calculations for arbitrary correlation times. The usual assumption of gaussian noise becomes rapidly too involved to be used as the number of levels increases (the two-level case can be solved analytically [29]).

The autocorrelation function of a fluctuating quantity x(t) is defined as ((x(0x(0>>

= (xG)xG’)>

- (x(t)>(xV)>* (A-1)

The autocorrelation function of the energy difference between the two localized states (here E(t) = cL(t) + A - cR(t)) can be calculated from the initial distribution of E, P,,(E) and the conditional transition probabilities at time r = t ’ - t, since (E(t)E(t’)) = (E(O)E(T)). One has (E(O)E(r))

= Cn,(j

I ‘)EjE,‘~(E~),

(A.21

j.k

where the double sum is over all the realizations of E. For the first example discussed in the paper (fig. 1) one has (A-3)

since the four realizations have the same probability at time t = 0. The conditional transition probabilities satisfy the equations

ay’k)=~w(jlz)n,(zlk),

(A.4

I

which can be easily solved by matrix diagonalization. In fact one has

(A-5) where eWc = w( diagonal).

64.6)

In our case

w,f 4 i -:1 -:l-3

1

1

1

11’i

A. Lami, G. Villani / Fluctuations in tunneling dynamics

1 2 1 I11

CEl

1 -1 -1 1

-1 -1

I121P.H. Cribb, Chem. Phys. 88 (1984) 47.

-1 1 1 1

(A-7)

-1 1 1

and from eqs. (A.2) and (AS) we obtain ((E(O)E(7)))

397

= 2a2 exp( -jk).

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