Abstract: Delayed Luminescence (DL) is the long-term „ultraweak" afterglow of biological systems after exposure to light illumination. It displays hyperbolic-like relaxation with sometimes remarkable hyperbolic oscillations around the decay function. In this paper we demonstrate that the hyperbolic relaxation of biological systems is a characteristic active response of an ergodic coherent state, while the hyperbolic-like oscillations are consequences of the coupling of at least two modes of coherent states with a necessary difference in the characteristic damping frequencies. The oscillations describe the energy transfer dynamics between the modes under study. DL provides a powerful non-invasive tool for investigating energy- and information-transfer within biological tissues.

Delayed Luminescence (DL) is the long-term afterglow of biological systems after illumination with white or monochromatic light in the spectral range of at least 400 to 800 nm. The intensity is (with a few up to some hundred thousand photons/()) about 20 orders of magnitude lower than the common fluorescence of phosphorescence. The decay time is dependent on the tissue under investigation, but also on the conditions of illumination such as intensity, duration, and spectral distribution. DL lasts from seconds to hours. DL was observed for the first time by Strehler and Arnold [1] on green plants. With the increasing sensitivity of photocounters, it has turned out that all living systems display distinct DL, independent of whether they are photosynthetic or not. There are three characteristic properties of DL which all biological systems have in common [2]:

1. The spectral relaxation function I(t), where I is the spectral intensity at time t after illumination, is rather independent of the frequency of the light under investigation.
2. I(t) does not follow an exponential decay function, but a hyperbolic-like relaxation function of the type I(t) µ 1/ (1+ l t), where the constant l is rather independent of the light frequency under study.

3. The photocount statistics (PCS), that is the probability p(n,D t) of registering n photons in a preset time interval D t << 1/l , follows at least down to D t < 10^-5 s, a Poissonian distribution

(1)

where is the mean value of n over D t.

In living systems, DL approaches for t >> 1/l so-called „biophoton emission," which is a permanent „ultraweak" photon emission from all living systems in the same spectral range as DL. Biophoton emission displays an intensity of from a few to up to some hundred photons/(). Biophoton emission is also subject to a Poissonian PCS [2, 3].

In recent papers we have shown that a Poissonian PCS together with a hyperbolic relaxation of an excited state is under ergodic conditions sufficient for a fully coherent state [4, 5, 6].

However, one remarkable feature of DL remained unexplained, i.e. the fact that living systems, and only living systems, sometimes display hyperbolic oscillations of the type sin ( ln(1+ l t)+j ) around the hyperbolic-like relaxation function, where j is a fixed phase [2, 7, 8, 9]. Fig. 1 shows a typical example.

Fig.1: DL of Acetabularia Acetabulum after 3s white-light illumination of a tungsten lamp (150W). Courtesy of Rafael Gaya-Moreno (IIB, Neuss)

In this paper we would like to propose an explanation for this phenomenon, again in terms of coherent states.

Mehta and Sudarshan [10] presented an explicit solution of the time evolution of a coherent state for the most general form of the Hamiltonian H(t) that keeps coherent states coherent. It can be expressed as

(2)

where w (t) and ß(t) are real functions, and f(t) is a complex function of time t. a and a+ are the annihilation operator and the creation operator, respectively. The normalized operators a and a+ are subject to

; (3)

The coherent state /a > is defined as an eigenstate of the annihilation operator according to

a /a > = a /a >, (4)

where the eigenvalue a represents the complex field amplitude of the coherent electromagnetic field.

The solution to (2) can be written as [10]

, (5a)

where

(5b)

This solution (5a) is the result of the equation of motion [10]:

(6)

From the physical point of view the problem is thus solved in principle, and all the possible solutions of (5a) under definite boundary conditions are known.

Providing that biological systems make use of these solutions (5a), we have to introduce some evident biological constraints in order to find more definitely reasonable solutions under the conditions of biological regulation and optimization.

One of the most general functions of biological organization is homeostasis, which is that the energy of the system is kept constant even over a considerable time period under external perturbations such as stress, heat, light absorption, or cold. This means that

(7)

Insertion of (6) into the second term II of (7) results in

(8)

There is ample indication that I and II have to vanish independently, since a should not be influenced by external classical energy sources in the case of biological regulation, as for instance in the example of mitotic figures [11] (Fig. 2).

Fig.2: Mitotic figures (left) follow the field patterns of cavity resonator waves (right) under the boundary conditions of the cell under observation. The spatial pattern follows, as usual, classical electrodynamics. The time-behaviour has to be described in terms of coherent states.

Actually, if the molecules have to follow the field pattern of cavity resonator waves which are adjusted to the boundary conditions of the cell and display the dynamics of coherent states, it is necessary that they be independent of classical energy sources over considerably long-term intervals. The temporal stability of the field amplitudes a (t) can be provided only by

(9)

The solution to (9) is trivial for constant w . It simply describes a frequency-stable oscillation that cannot give rise to hyperbolic relaxation. However, for time-dependent w (t), we obtain the solution

(10)

where (11)

This type of damping does not only agree with the observed hyperbolic-like relaxation function of DL, but provides at the same time the frequency stability of every oscillation also for time-independent w within the coherent system [6, 12].

The validity of (9) then requires

(12)

which has for constant and the solution

(13)

However, in the case that ß(t) is involved in homeostatic regulation, a more general solution to (12) can be written in the form

(14)

where m is a parameter.

Thus we obtain from (12)

(15)

Under stationary conditions we arrive at and

(16)

However, also for we approach the optimum zone of , where highly sensitive fluctuations around this turning point are evident. At the same time we obtain

(17)

arriving again at the hyperbolic-like relaxation of DL.

It is worthwhile to note that these solutions can explain the phenomenon of DL quantitatively, but they do not describe the physical mechanism of homeostatic regulation. They cannot explain the appearance of oscillations around the hyperbolic relaxation according to (17).

For this information we have to go back to equ. (6). It can be shown that when taking into account chaotic states or only one coherent state it is impossible to get the imaginary terms of

therefore excluding oscillations.

However, by coupling coherent states and with a difference between and , we arrive at oscillations that are in agreement with the observed hyperbolic-like oscillations around the decay function. Let us, for simplicity, start with the most straightforward equation (18)

(18a)

(18b)

These equations provide that the coherent states remain coherent, since f₁(t) and f₂(t) according to (6) simply take the values

(19a)

and (19b)

These coupled oscillations satisfy the boundary conditions

(20a)

(20b)

We confine to real values of k and may expect

After straightforward calculation, we arrive at

(21)

A solution for (21) takes the form

(22a)

, (22b)

where

(23)

For , we then obtain finally

(24)

This solution describes surprisingly accurately the observed oscillations around the hyperbolic-like relaxation of DL. Fig. 3 displays a typical example.

Fig.3: The solution of coupled coherent states fits the time-dependence of DL. It is described as
,
where I₀(t) is shown in Fig.3a (A=151432, l =20.6), I_R(t) in Fig.3b( A=9000, k = 1.8, l ₁=0.015, l ₂=4.0, j =p /2) and I_RR(t) in Fig.3c. This example displays the evaluation of the DL of Fig.1.

We would like to note that this analysis may provide a powerful tool for analyzing biological tissues in terms of DL. We have already shown that there are significant differences between the oscillations of normal tissue and tumor tissue which grow on the same leaf of a plant [2]. There is also evidence that seeds of different germination capacity display different DL [13].

The possible mechanism behind that difference may be able to be explained in terms of phase-conjugation effects [14].

1. B.L. Strehler and W. Arnold, Light production by green plants. J.Gen.Physiol.34 (1951), 809-820.
2. F.A.Popp, B.Ruth, W.Bahr, J.Böhm, P.Graß, G.Grolig, M.Rattemeyer, H.G.Schmidt and P.Wulle: Emission of Visible and Ultraviolet Radiation by Active Biological Systems. Collective Phenomena(Gordon and Breach), 3 (1981), 187-214.
3. B.Ruth and F.A.Popp: Experimentelle Untersuchungen zur ultraschwachen Photonenemission biologischer Systeme. Z.Naturforsch. 31C (1976), 741-745.
4. F.A.Popp, K.H.Li, and Q.Gu (eds.): Recent Advances in Biophoton Research and its Applications. World Scientific, Singapore-New Jersey 1992.
5. K.H.Li and F.A.Popp: Non-exponential decay law of radiation systems with coherent rescattering. Phys.Lett. 93A (1983), 262-266.
6. F.A.Popp and K.H.Li: Hyperbolic Relaxation as a Sufficient Condition of a Fully Coherent Ergodic Field. Intern.J.Theor.Phys. 32 (1993), 1573-1583.
7. W.B.Chwirot, R.S.Dygdala, and S.Chwirot: Optical coherence of white-light-induced photon emission from microsporocytes of Larix europea. Cytobios 44 (1985), 239-246.
8. W.B.Chwirot, R.S.Dygdala, and S.Chwirot: Quasi-monochromatic-light-induced photon emission from microsporocytes of larch showing oscillation decay behaviour predicted by the electromagnetic model of differentiation. Cytobios 47 (1987), 137-146.
9. D.V.Parkhomtchouk and M.Yamamoto: Super-Delayed Luminescence from Biological Tissues. J.Intl.Soc.Life Info.Sci. 18 (2000), 413-417.
10. C.L.Mehta and E.C.G. Sudarshan, Time Evolution of Coherent States. Phys.Lett. 22 (1966), 574-576.
11. F.A.Popp: Photon-storage in biological systems, in: Electromagnetic Bio-Information, pp.123-149. Eds. F.A.Popp, G.Becker, W.L.König, and W.Peschka. Urban & Schwarzenberg, München-Baltimore 1979.
12. R.P.Bajpai, S.Kumar, and V.A. Sivadasan: Biophoton Emission in the evolution of a squeezed state of a frequency stable damped oscillator. Applied Mathematics and Computation 93 (1998), 277-288.
13. J.E.M. Souren, E.Boon-Niermeyer, and R. van Wijk: Germination capacity of tomatoes seeds and ultra-weak photon-induced delayed luminescence, in, Biophotonics and Coherent Systems. Eds. L.Beloussov, F.A.Popp, V.Voeikov, and R.van Wijk. Moscow University Press, 2000, pp.419-430.
14. F.A.Popp and J.J.Chang: Mechanism of interaction between electromagnetic fields and living organisms. Science in China C43 (2000), 507-518.