## 1. Introduction

So-called macroscopic quantum effects(MQE) refer to a quantum phenomenon that occurs on a macroscopic scale. Such effects are obviously different from the microscopic quantum effects at the microscopic scale as described by quantum mechanics. It has been experimentally demonstrated [1-17] that macroscopic quantum effects are the phenomena that have occurred in superconductors. Superconductivity is a physical phenomenon in which the resistance of a material suddenly vanishes when its temperature is lower than a certain value, Tc, which is referred to as the critical temperature of superconducting materials. Modern theories [18-21] tell us that superconductivity arises from the irresistible motion of superconductive electrons. In such a case we want to know “How the macroscopic quantum effect is formed? What are its essences? What are the properties and rules of motion of superconductive electrons in superconductor?” and, as well, the answers to other key questions. Up to now these problems have not been studied systematically. We will study these problems in this chapter.

## 2. Experimental observation of property of macroscopic quantum effects in superconductor

(1) Superconductivity of material. As is known, superconductors can be pure elements, compounds or alloys. To date, more than 30 single elements, and up to a hundred alloys and compounds, have been found to possess the characteristics [1-17] of superconductors. When

How can this phenomenon be explained? After more than 40 years’ effort, Bardeen, Cooper and Schreiffier proposed the new idea of Cooper pairs of electrons and established the microscopic theory of superconductivity at low temperatures, the BCS theory [18-21],in 1957, on the basis of the mechanism of the electron-phonon interaction proposed by Frohlich [22-23]. According to this theory, electrons with opposite momenta and antiparallel spins form pairs when their attraction, due to the electron and phonon interaction in these materials, overcomes the Coulomb repulsion between them. The so-called Cooper pairs are condensed to a minimum energy state, resulting in quantum states, which are highly ordered and coherent over the long range, and in which there is essentially no energy exchange between the electron pairs and lattice. Thus, the electron pairs are no longer scattered by the lattice but flow freely resulting in superconductivity. The electron pairs in a superconductive state are somewhat similar to a diatomic molecule but are not as tightly bound as a molecule. The size of an electron pair, which gives the coherent length, is approximately 10^{−4} cm. A simple calculation shows that there can be up to 10^{6} electron pairs in a sphere of 10^{−4} cm in diameter. There must be mutual overlap and correlation when so many electron pairs are brought together. Therefore, perturbation to any of the electron pairs would certainly affect all others. Thus, various macroscopic quantum effects can be expected in a material with such coherent and long range ordered states. Magnetic flux quantization, vortex structure in the type-II superconductors, and Josephson effect [24-26] in superconductive junctions are only some examples of the phenomena of macroscopic quantum mechanics.

(2) The magnetic flux structures in superconductor. Consider a superconductive ring. Assume that a magnetic field is applied at T >Tc, then the magnetic flux lines
^{-15} Wb is the flux quantum, representing the flux of one magnetic flux line. This means that the magnetic fluxes passing through the hole of the ring can only be multiples of

An experiment conducted in 1961 surely proves this to be so, indicating that the magnetic flux does exhibit discrete or quantized characteristics on a macroscopic scale. The above experiment was the first demonstration of the macroscopic quantum effect. Based on quantization of the magnetic flux, we can build a “quantum magnetometer” which can be used to measure weak magnetic fields with a sensitivity of 3×10^{-7} Oersted. A slight modification of this device would allow us to measure electric current as low as 2.5×10^{-9} A.

(3) Quantization of magnetic-flux lines in type-II superconductors. The superconductors discussed above are referred to as type-I superconductors. This type of superconductor exhibits a perfect Maissner effect when the external applied field is higher than a critical magnetic value
_{3}Sn compounds in which the magnetic field partially penetrates inside the material when the external field

(4) The Josephson phenomena in superconductivity junctions [24-26]. As it is known in quantum mechanics, microscopic particles, such as electrons, have a wave property and that can penetrate through a potential barrier. For example, if two pieces of metal are separated by an insulator of width of tens of angstroms, an electron can tunnel through the insulator and travel from one metal to the other. If voltage is applied across the insulator, a tunnel current can be produced. This phenomenon is referred to as a tunneling effect. If two superconductors replace the two pieces of metal in the above experiment, a tunneling current can also occur when the thickness of the dielectric is reduced to about 30

Evidently, this is due to the long-range coherent effect of the superconductive electron pairs. Experimentally, it was demonstrated that such an effect could be produced via many types of junctions involving a superconductor, such as superconductor-metal-superconductor junctions, superconductor-insulator- superconductor junctions, and superconductor bridges. These junctions can be considered as superconductors with a weak link. On the one hand, they have properties of bulk superconductors, for example, they are capable of carrying certain superconducting currents. On the other hand, these junctions possess unique properties, which a bulk superconductor does not. Some of these properties are summarized in the following.

(A) When a direct current (dc) passing through a superconductive junction is smaller than a critical value Ic, the voltage across the junction does not change with the current. The critical current Ic can range from a few tens of μA to a few tens of mA.

(B) If a constant voltage is applied across the junction and the current passing through the junction is greater than Ic, a high frequency sinusoidal superconducting current occurs in the junction. The frequency is given by υ=2eV/h, in the microwave and far-infrared regions of (5-1000)×10^{9}Hz. The junction radiates a coherent electromagnetic wave with the same frequency. This phenomenon can be explained as follows: The constant voltage applied across the junction produces an alternating Josephson current that, in turn, generates an electromagnetic wave of frequency, υ. The wave propagates along the planes of the junction. When the wave reaches the surface of the junction (the interface between the junction and its surrounding), part of the electromagnetic wave is reflected from the interface and the rest is radiated, resulting in the radiation of the coherent electromagnetic wave. The power of radiation depends on the compatibility between the junction and its surrounding.

(C) When an external magnetic field is applied over the junction, the maximum dc current, Ice, is reduced due to the effect of the magnetic field. Furthermore, I_{c} changes periodically as the magnetic field increases. The

(D) When a junction is exposed to a microwave of frequency, υ, and if the voltage applied across the junction is varied, it can be seen that the dc current passing through the junction increases suddenly at certain discrete values of electric potential. Thus, a series of steps appear on the dc I − V curve, and the voltage at a given step is related to the frequency of the microwave radiation by nυ=2eVn/h(n=0,1,2,3…). More than 500 steps have been observed in experiments.

Josephson first derived these phenomena theoretically and each was experimentally verified subsequently. All these phenomena are, therefore, called Josephson effects [24-26]. In particular, (1) and (3) are referred to as dc Josephson effects while (2) and (4) are referred to as ac Josephson effects. Evidently, Josephson effects are macroscopic quantum effects, which can be explained well by the macroscopic quantum wave function. If we consider a superconducting junction as a weakly linked superconductor, the wave functions of the superconducting electron pairs in the superconductors on both sides of the junction are correlated due to a definite difference in their phase angles. This results in a preferred direction for the drifting of the superconducting electron pairs, and a dc Josephson current is developed in this direction. If a magnetic field is applied in the plane of the junction, the magnetic field produces a gradient of phase difference, which makes the maximum current oscillate along with the magnetic field, and the radiation of the electromagnetic wave occur. If a voltage is applied across the junction, the phase difference will vary with time and results in the Josephson effect. In view of this, the change in the phase difference of the wave functions of superconducting electrons plays an important role in Josephson effect, which will be discussed in more detail in the next section.

The discovery of the Josephson effect opened the door for a wide range of applications of superconductor theory. Properties of superconductors have been explored to produce superconducting quantum interferometer–magnetometer, sensitive ammeter, voltmeter, electromagnetic wave generator, detector, frequency-mixer, and so on.

## 3. The properties of boson condensation and spontaneous coherence of macroscopic quantum effects

### 3.1. A nonlinear theoretical model of theoretical description of macroscopic quantum effects

From the above studies we know that the macroscopic quantum effect is obviously different from the microscopic quantum effect, the former having been observed for physical quantities, such as, resistance, magnetic flux, vortex line, and voltage, etc.

In the latter, the physical quantities, depicting microscopic particles, such as energy, momentum, and angular momentum, are quantized. Thus it is reasonable to believe that the fundamental nature and the rules governing these effects are different.

We know that the microscopic quantum effect is described by quantum mechanics. However, the question remains relative to the definition of what are the mechanisms of macroscopic quantum effects? How can these effects be properly described?

What are the states of microscopic particles in the systems occurring related to macroscopic quantum effects? In other words, what are the earth essences and the nature of macroscopic quantum states? These questions apparently need to be addressed.

We know that materials are composed of a great number of microscopic particles, such as atoms, electrons, nuclei, and so on, which exhibit quantum features. We can then infer, or assume, that the macroscopic quantum effect results from the collective motion and excitation of these particles under certain conditions such as, extremely low temperatures, high pressure or high density among others. Under such conditions, a huge number of microscopic particles pair with each other condense in low-energy state, resulting in a highly ordered and long-range coherent. In such a highly ordered state, the collective motion of a large number of particles is the same as the motion of “single particles”, and since the latter is quantized, the collective motion of the many particle system gives rise to a macroscopic quantum effect. Thus, the condensation of the particles and their coherent state play an essential role in the macroscopic quantum effect.

What is the concept of condensation? On a macroscopic scale, the process of transforming gas into liquid, as well as that of changing vapor into water, is called condensation. This, however, represents a change in the state of molecular positions, and is referred to as a condensation of positions. The phase transition from a gaseous state to a liquid state is a first order transition in which the volume of the system changes and the latent heat is produced, but the thermodynamic quantities of the systems are continuous and have no singularities. The word condensation, in the context of macroscopic quantum effects has its’ special meaning. The condensation concept being discussed here is similar to the phase transition from gas to liquid, in the sense that the pressure depends only on temperature, but not on the volume noted during the process, thus, it is essentially different from the above, first-order phase transition. Therefore, it is fundamentally different from the first-order phase transition such as that from vapor to water. It is not the condensation of particles into a high-density material in normal space. On the contrary, it is the condensation of particles to a single energy state or to a low energy state with a constant or zero momentum. It is thus also called a condensation of momentum. This differs from a first-order phase transition and theoretically it should be classified as a third order phase transition, even though it is really a second order phase transition, because it is related to the discontinuity of the third derivative of a thermodynamic function. Discontinuities can be clearly observed in measured specific heat, magnetic susceptibility of certain systems when condensation occurs. The phenomenon results from a spontaneous breaking of symmetries of the system due to nonlinear interaction within the system under some special conditions such as, extremely low temperatures and high pressures. Different systems have different critical temperatures of condensation. For example, the condensation temperature of a superconductor is its critical temperature

From the above discussions on the properties of superconductors, and others we know that, even though the microscopic particles involved can be either Bosons or Fermions, those being actually condensed, are either Bosons or quasi-Bosons, since Fermions are bound as pairs. For this reason, the condensation is referred to as Bose-Einstein condensation[33-36] since Bosons obey the Bose-Einstein statistics. Properties of Bosons are different from those of Fermions as they do not follow the Pauli exclusion principle, and there is no limit to the number of particles occupying the same energy levels. At finite temperatures, Bosons can distribute in many energy states and each state can be occupied by one or more particles, and some states may not be occupied at all. Due to the statistical attractions between Bosons in the phase space (consisting of generalized coordinates and momentum), groups of Bosons tend to occupy one quantum energy state under certain conditions. Then when the temperature of the system falls below a critical value, the majority or all Bosons condense to the same energy level (e.g. the ground state), resulting in a Bose condensation and a series of interesting macroscopic quantum effects. Different macroscopic quantum phenomena are observed because of differences in the fundamental properties of the constituting particles and their interactions in different systems.

In the highly ordered state of the phenomena, the behavior of each condensed particle is closely related to the properties of the systems. In this case, the wave function

In the absence of any externally applied field, the Hamiltonian of a given macroscopic quantum system can be represented by the macroscopic wave function

Here H’=H presents the Hamiltonian density function of the system, the unit system in which m=h=c=1 is used here for convenience. If an externally applied electromagnetic field does exist, the Hamiltonian given above should be replaced by

or, equivalently

where

where

Since charge is invariant under the transformation and neutrality is required for the Hamiltonian, there must be (Q_{1} + Q_{2} + + Q_{n}) = 0 in such a case. Furthermore, since

If we rewrite Eq. (1) as the following

We can see that the effective potential energy,

rather than at

In this case the macroscopic quantum state is the stable state of the system. This shows that the Hamiltonian of a normal state differs from that of the macroscopic quantum state, in which the two ground states satisfy

In order to make the expectation value in a new ground state zero in the macroscopic quantum state, the following transformation [16-17] is done:

so that

After this transformation, the Hamiltonian density of the system becomes

Inserting Eq. (4) into Eq. (7), we have

Consider now the expectation value of the variation

After the transformation Eq. (6), it becomes

where the terms

Then Eq. (9) can be written as

Obviously, two sets of solutions,

If the displacement is very small, i.e.

Its’ solution attenuates exponentially indicating that the ground state,

Therefore, the transition from the state

In the presence of an electromagnetic field with a vector potential

Since

We can see that the effective interaction energy of

and is in agreement with that given in Eq. (4). Therefore, using the same argument, we can conclude that the spontaneous symmetry breakdown and the second-order phase transition also occur in the system. The system is changed from the ground state of the normal phase,

### 3.2.The features of the coherent state of macroscopic quantum effects

Proof that the macroscopic quantum state described by Eqs. (1) - (2) is a coherent state, using either the second quantization theory or the solid state quantum field theory is presented in the following paragraphs and pages.

As discussed above, when

It is a time-independent nonlinear Schrödinger equation (NLSE), which is similar to the GL equation. Expanding
_{p}

where

A new field

The transformation between the fields

where

From Eq. (6) we now have

From Eq. (6), we can obtain the following relationship between the annihilation operator a_{p} of the new field
_{p} of the

where

Therefore, the new ground state

Thus we have

According to the definition of the coherent state, equation (25) we see that the new ground state

where

By reconstructing a quasiparticle-operator-free new formulation of the Bogoliubov-Valatin transformation parameter dependence [55], W. S. Lin et al [56] demonstrated that the BCS state is not only a coherent state of single-Cooper-pairs, but also the squeezed state of the double-Cooper- pairs, and reconfirmed thus the coherent feature of BCS superconductive state.

### 3.3. The Boson condensed features of macroscopic quantum effects

We will now employ the method used by Bogoliubov in the study of superfluid liquid helium 4He to prove that the above state is indeed a Bose condensed state. To do that, we rewrite Eq. (16) in the following form [12-17]

Since the field

where

Because the condensed density

where

the following relations can be obtained

where

while

where

We will now study two cases to illustrate the concepts.

(A) Let

Using this concept, we can obtain the following form from Eqs. (32) and (34)

From Eq. (32), we know that

We can then prove that

where

Now, inserting Eqs. (30), (37)-(38) and

where

Both

This is the condensed density of the ground state

These correspond to the energy spectra of

(B) In the case of Mp=0, a similar approach can be used to arrive at the energy spectrum corresponding to

Based on experiments in quantum statistical physics, we know that the occupation number of the level with an energy of

where

As can be seen from Eqs. (27) and (28), the number of particles is extremely large when they lie in condensed state, that is to say:

Because

Substituting Eq. (41) into Eq. (47), we can see that:

which is the ground state of the condensed phase, or the superconducting phase, that we have known. Thus, the density of states,

In the last few decades, the Bose-Einstein condensation has been observed in a series of remarkable experiments using weakly interacting atomic gases, such as vapors of rubidium, sodium lithium, or hydrogen. Its’ formation and properties have been extensively studied. These studies show that the Bose-Einstein condensation is a nonlinear phenomenon, analogous to nonlinear optics, and that the state is coherent, and can be described by the following NLSE or the Gross-Pitaerskii equation [57-59]:

where

where H’=H, the nonlinear parameters of
_{0} and a_{1} are the transverse (y, z) and the longitudinal (x) condensate sizes (without self-interaction) respectively, (Integrations over y and z have been carried out in obtaining the above equation).

It is not surprising to see that Eq. (48) is exactly the same as Eq. (15), corresponding to the Hamiltonian density in Eq. (49) and, where used in this study is naturally the same as Eq. (1). This prediction confirms the correctness of the above theory for Bose-Einstein condensation. As a matter of fact, immediately after the first experimental observation of this condensation phenomenon, it was realized that the coherent dynamics of the condensed macroscopic wave function could lead to the formation of nonlinear solitary waves. For example, self-localized bright, dark and vortex solitons, formed by increased (bright) or decreased (dark or vortex) probability density respectively, were experimentally observed, particularly for the vortex solution which has the same form as the vortex lines found in type II-superconductors and superfluids. These experimental results were in concordance with the results of the above theory. In the following sections of this text we will study the soliton motions of quasiparticles in macroscopic quantum systems, superconductors. We will see that the dynamic equations in macroscopic quantum systems do have such soliton solutions.

## 3.4 Differences of macroscopic quantum effects from the microscopic quantum effects

From the above discussion we may clearly understand the nature and characteristics of macroscopic quantum systems. It would be interesting to compare the macroscopic quantum effects and microscopic quantum effects. Here we give a summary of the main differences between them.

Concerning the origins of these quantum effects; the microscopic quantum effect is produced when microscopic particles, which have only a wave feature are confined in a finite space, or are constituted as matter, while the macroscopic quantum effect is due to the collective motion of the microscopic particles in systems with nonlinear interaction. It occurs through second-order phase transition following the spontaneous breakdown of symmetry of the systems.

From the point-of-view of their characteristics, the microscopic quantum effect is characterized by quantization of physical quantities, such as energy, momentum, angular momentum, etc. wherein the microscopic particles remain constant. On the other hand, the macroscopic quantum effect is represented by discontinuities in macroscopic quantities, such as, the resistance, magnetic flux, vortex lines, voltage, etc. The macroscopic quantum effects can be directly observed in experiments on the macroscopic scale, while the microscopic quantum effects can only be inferred from other effects related to them.

The macroscopic quantum state is a condensed and coherent state, but the microscopic quantum effect occurs in determinant quantization conditions, which are different for the Bosons and Fermions. But, so far, only the Bosons or combinations of Fermions are found in macroscopic quantum effects.

The microscopic quantum effect is a linear effect, in which the microscopic particles and are in an expanded state, their motions being described by linear differential equations such as the Schrödinger equation, the Dirac equation, and the Klein- Gordon equations.

On the other hand, the macroscopic quantum effect is caused by the nonlinear interactions, and the motions of the particles are described by nonlinear partial differential equations such as the nonlinear Schrödinger equation (17).

Thus, we can conclude that the macroscopic quantum effects are, in essence, a nonlinear quantum phenomenon. Because its’ fundamental nature and characteristics are different from those of the microscopic quantum effects, it may be said that the effects should be depicted by a new nonlinear quantum theory, instead of quantum mechanics.

## 4. The nonlinear dynamic natures of electrons in superconductors

### 4.1. The dynamic equations of electrons in superconductors

It is quite clear from the above section that the superconductivity of material is a kind of nonlinear quantum effect formed after the breakdown of the symmetry of the system due to the electron-phonon interaction, which is a nonlinear interaction.

In this section we discuss the properties of motion of superconductive electrons in superconductors and the relation of the solutions of dynamic equations in relation to the above macroscopic quantum effects on it. The study presented shows that the superconductive electrons move in the form of a soliton, which can result in a series of macroscopic quantum effects in the superconductors. Therefore, the properties and motions of the quasiparticles are important for understanding the essences and rule of superconductivity and macroscopic quantum effects.

As it is known, in the superconductor the states of the electrons are often represented by a macroscopic wave function,

as mentioned above, where

in the absence of any external field. If the system is subjected to an electromagnetic field specified by a vector potential

where e*=2e,

in the absence and presence of an external fields respectively, and

Equations (52) - (54) are just well-known the Ginzburg-Landau (GL) equation [48-54] in a steady state, and only a time-independent Schrödinger equation. Here, Eq. (52) is the GL equation in the absence of external fields. It is the same as Eq. (15), which was obtained from Eq. (1). Equation (54) can also be obtained from Eq. (2). Therefore, Eqs. (1)-(2) are the Hamiltonians corresponding to the free energy in Eqs. (50)- (51).

From equations (52) - (53) we clearly see that superconductors are nonlinear systems. Ginzburg-Landau equations are the fundamental equations of the superconductors describing the motion of the superconductive electrons, in which there is the nonlinear term of

### 4.2. The dynamic properties of electrons in steady superconductors

We first study the properties of motion of superconductive electrons in the case of no external field. Then, we consider only a one-dimensional pure superconductor [62-63], where

and where

This is a well-known wave packet-type soliton solution. It can be used to represent the bright soliton occurred in the Bose-Einstein condensate found by Perez-Garcia et. al. [64]. If the signs of

The energy of the soliton, (56), is given by

We assume here that the lattice constant, r_{0}=1. The above soliton energy can be compared with the ground state energy of the superconducting state, Eground=

From the above discussion, we can see that, in the absence of external fields, the superconductive electrons move in the form of solitons in a uniform system. These solitons are formed by a nonlinear interaction among the superconductive electrons which suppresses the dispersive behavior of electrons. A soliton can carry a certain amount of energy while moving in superconductors. It can be demonstrated that these soliton states are very stable.

### 4.3. The features of motion of superconductive electrons in an electromagnetic field and its relation to macroscopic quantum effects

We now consider the motion of superconductive electrons in the presence of an electromagnetic field

For bulk superconductors, J is a constant (permanent current) for a certain value of

where Ueff is the effective potential of the superconductive electron in this case and it is schematically shown in Fig. 2. Comparing this case with that in the absence of external fields, we found that the equations have the same form and the electromagnetic field changes only the effective potential of the superconductive electron. When

Where E is a constant of integration which is equivalent to the energy, the lower limit of the integral,

It can be seen from Fig. 3 that the denominator in the integrand in Eq. (64) approaches zero linearly when u=u_{1}=
_{2}=

where g= u_{0}−u_{1} and satisfies

It can be seen from Eq. (65) that for a large part of sample, u_{1} is very small and may be neglected; the solution u is very close to u_{0}. We then get from Eq. (65) that

Substituting the above into Eq. (61), the electromagnetic field

For a large portion of the superconductor, the phase change is very small. Using

Equations (67) and (68) are analytical solutions of the GL equation.(63) and (64) in the one-dimensional case, which are shown in Fig. 3. Equation (67) or (65) shows that the superconductive electron in the presence of an electromagnetic field is still a soliton. However, its amplitude, phase and shape are changed, when compared with those in a uniform superconductor and in the absence of external fields, Eq. (66). The soliton here is obviously influenced by the electromagnetic field, as reflected by the change in the form of solitary wave. This is why a permanent superconducting current can be established by the motion of superconductive electrons along certain direction in such a superconductor, because solitons have the ability to maintain their shape and velocity while in motion.

It is clear from Fig.4 that

Recently, Garadoc-Daries et al. [68], Matthews et al. [69] and Madison et al.[70] observed vertex solitons in the Boson-Einstein condensates. Tonomure [71] observed experimentally magnetic vortexes in superconductors. These vortex lines in the type-II-superconductors are quantized. The macroscopic quantum effects are well described by the nonlinear theory discussed above, demonstrating the correctness of the theory.

We now proceed to determine the energy of the soliton given by (67). From the earlier discussion, the energy of the soliton is given by:

which depends on the interaction between superconductive electrons and electromagnetic field.

From the above discussion, we understand that for a bulk superconductor, the superconductive electrons behave as solitons, regardless of the presence of external fields. Thus, the superconductive electrons are a special type of soliton. Obviously, the solitons are formed due to the fact that the nonlinear interaction

## 5. The dynamic properties of electrons in superconductive junctions and its relation to macroscopic quantum effects

### 5.1. The features of motion of electron in S-N junction and proximity effect

The superconductive junction consists of a superconductor (S) which contacts with a normal conductor (N), in which the latter can be superconductive. This phenomenon refers to a proximity effect. This is obviously the result of long- range coherent property of superconductive electrons. It can be regarded as the penetration of electron pairs from the superconductor into the normal conductor or a result of diffraction and transmission of superconductive electron wave. In this phenomenon superconductive electrons can occur in the normal conductor, but their amplitudes are much small compare to that in the superconductive region, thus the nonlinear term

while that on the N side of the junction is

Thus, the expression for

In the S region, we have obtained solution of (69) in the previous section, and it is given by (65) or (67) and (68). In the N region, from Eqs. (70)- (71) we can easily obtain

where

here

### 5.2. The Josephson effect in S-I-S and S-N-S as well as S-I-N-S junctions

A superconductor-normal conductor -superconductor junction (S-N-S) or a superconductor-insulator-superconductor junction (S-I-S) consists of a normal conductor or an insulator sandwiched between two superconductors as is schematically shown in Fig.6a．The thickness of the normal conductor or the insulator layer is assumed to be L and we choose the z coordinate such that the normal conductor or the insulator layer is located at

The electrons in the superconducting regions (
_{1} from Eq.(66), we have [73-74]

The electrons in the superconducting regions (
_{1}=u_{0}.

From Eq.(71), we have

where

If we substitute Eqs.(64) - (67) into Eq.(73), the phase shift of wave function from an arbitrary point x to infinite can be obtained directly from the above integral, and takes the form of:

For the S-N-S or S-I-S junction, the superconducting regions are located at

According to the results in (70) - (71) and the above similar method, the change of the phase in the I or N region of the S-N-S or S-I-S junction may be expressed as [75-76]

where

Near the critical temperature (T<Tc), the current passing through a weakly linked superconductive junction is very small (

where

where

Thus

where

Equation (78) is the well-known example of the Josephson current. From Section I we know that the Josephson effect is a macroscopic quantum effect. We have seen now that this effect can be explained based on the nonlinear quantum theory. This again shows that the macroscopic quantum effect is just a nonlinear quantum phenomenon.

From Eq. (79) we can see that the Josephson critical current is inversely proportional to sin (
_{c})^{2}．

Finally, it is worthwhile to mention that no explicit assumption was made in the above on whether the junction is a potential well (

We now study Josephson effect in the superconductor -normal conductor-insulator-superconductor junction (SNIS) is shown schematically in Fig. 6b. It can be regarded as a multilayer junction consists of the S-N-S and S-I-S junctions. If appropriate thicknesses for the N and I layers are used (approximately 20 °A– 30 °A), the Josephson effect similar to that discussed above can occur in the SNIS junction. Since the derivations are similar to that in the previous sections, we will skip much of the details and give the results in the following. The Josephson current in the SNIS junction is still given by

but, where

It can be shown that the temperature dependence of

## 6. The nonlinear dynamic-features of time- dependence of electrons in superconductor

### 6.1. The soliton solution of motion of the superconductive electron

We studied only the properties of motion of superconductive electrons in steady states in superconductors in section 2.3.2, and which are described by the time-independent GL equation. In such a case, the superconductive electrons move as solitons. We ask, “What are the features of a time-dependent motion in non-equilibrium states of a superconductor?” Naturally, this motion should be described by the time-dependent Ginzburg-Landau (TDGL) equation [48-54,77] in this case. Unfortunately, there are many different forms of the TDGL equation under different conditions. The one given in the following is commonly used when an electromagnetic field

In certain situations, the following forms of the TDGL equation are also used.

TDGL equation (83) can be written in the following form when

Where

For convenience, let

If we let

here

Substituting Eq. (88) into Eq. (86), we get:

Now let

where

and where

Substituting Eqs. (92) and (93) into Eq. (89), we have:

Since

Clearly in the case discussed,

This shows that
_{0}=0 in Eq. (96) is kept, and it can be shown that this soliton solution is stable in such a case. Therefore, we choose g_{0}=0 and obtain the following from Eq. (91):

Thus, we obtain from Eq. (95) that

Substituting Eq. (98) into Eqs. (92) - (93), we obtain:

Finally, substituting the Eq. (99) into Eq. (96), we can get

This is also a soliton solution, but its shape，amplitude and velocity have been changed relatively as compared to that of Eq. (87). It can be shown that Eq. (102) does indeed satisfy Eq. (85). Thus, equation (85) has a soliton solution. It can also be shown that this solition solution is stable.

### 6.2. The properties of soliton motion of the superconductive electrons

For the solution of Eq. (102), we may define a generalized time-dependent wave number,

The usual Hamilton equations for the superconductive electron (soliton) in the macroscopic quantum systems are still valid here and can be written as [80-81]

This means that the frequency ω still represents the meaning of Hamiltonian in the case of nonlinear quantum systems. Hence,

These relations in Eqs. (103)-(104) show that the superconductive electrons move as if they were classical particles moving with a constant acceleration in the invariant electric-field, and that the acceleration is given by

From the above studies we see that the time-dependent motion of superconductive electrons still behaves like a soliton in non-equilibrium state of superconductor. Therefore, we can conclude that the electrons in the superconductors are essentially a soliton in both time-independent steady state and time-dependent dynamic state systems. This means that the soliton motion of the superconductive electrons causes the superconductivity of material. Then the superconductors have a complete conductivity and nonresistance property because the solitons can move over a macroscopic distances retaining its amplitude, velocity, energy and other quasi- particle features. In such a case the motions of the electrons in the superconductors are described by a nonlinear Schrödinger equations (52), or (53) or (80) or (82) or (84). According to the soliton theory, the electrons in the superconductors are localized and have a wave-corpuscle duality due to the nonlinear interaction, which is completely different from those in the quantum mechanics. Therefore, the electrons in superconductors should be described in nonlinear quantum mechanics[16-17].

## 7. The transmission features of magnetic-flux lines in the Josephson junctions

### 7.1. The transmission equation of magnetic-flux lines

We have learned that in a homogeneous bulk superconductor, the phase

where d’ is the thickness of the junction. Because the voltage V and magnetic field

In this case, the total current in the junction is given by

In the above equation,
_{n} =V/R(V ) if the resistance in the junction is R(V ) and a voltage V is applied at two ends of the junction),

where

Equation (107) is the equation satisfied by the phase difference. It is a Sine-Gordon equation (SGE) with a dissipative term. From Eq.(105), we see that the phase difference

### 7.2. The transmission features of magnetic-flux lines

Assuming that the resistance R in the junction is very high, so that
_{0} = 0, equation (107) reduces to

Define

which is the 1D Sine-Gordon equation. If we further assume that

it becomes

where

A kink soliton solution can be obtained as follows

From the Josephson relations, the electric potential difference across the junction can be written as

where

We can then determine the magnetic flux through a junction with a length of L and a cross section of 1 cm2. The result is

Therefore, the kink (

This result shows clearly that magnetic flux in superconductors is quantized and this is a macroscopic quantum effect as mentioned in Section 1. The transmission of the quantum magnetic flux through the superconductive junctions is described by the above nonlinear dynamic equation (107) or (108).The energy of the soliton can be determined and it is given by

However, the boundary conditions must be considered for real superconductors. Various boundary conditions have been considered and studied. For example, we can assume the following boundary conditions for a 1D superconductor,

where h and g are the general Jacobian elliptical functions and satisfy the following equations

with a’, b’, and c’ being arbitrary constants. Coustabile et al. also gave the plasma oscillation, breathing oscillation and vortex line oscillation solutions for the SG equation under certain boundary conditions. All of these can be regarded as the soliton solution under the given conditions.

Solutions of Eq.(108) in two and three-dimensional cases can also be found[80-81]. In two- dimensional case, the solution is given by

where

In addition, P_{i}, q_{i} and

where X, Y, and T are similarly defined as in the 2D case given above, and

here

We now discuss the SG equation with a dissipative term

In terms of these new parameters, the 1D SG equation (107) can be rewritten as

The analytical solution of Eq.(113) is not easily found. Now let

Equation (113) then becomes

This equation is the same as that of a pendulum being driven by a constant external moment and a frictional force which is proportional to the angular displacement. The solution of the latter is well known, generally there exists an stable soliton solution[80-81]. Let

Expand Y as a power series of

and so on. Substituting these

where

where snF is the Jacobian sine function. Introducing the symbol cscF = 1/snF, the solution can be written as

This is a elliptic function. It can be shown that the corresponding solution at

It can be seen from the above discussion that the quantum magnetic flux lines (vortex lines) move along a superconductive junction in the form of solitons. The transmission velocity

That is, the transmission velocity of the vortex lines depends on the current
_{0} is, the faster the transmission velocity will be; and when I_{0} is finite, the greater the

## 8. Conclusions

We here first reviewed the properties of superconductivity and macroscopic quantum effects, which are different from the microscopic quantum effects, obtained from some experiments. The macroscopic quantum effects occurred on the macroscopic scale are caused by the collective motions of microscopic particles, such as electrons in superconductors, after the symmetry of the system is broken due to nonlinear interactions. Such interactions result in Bose condensation and self-coherence of particles in these systems. Meanwhile, we also studied the properties of motion of superconductive electrons, and arrived at the soliton solutions of time-independent and time-dependent Ginzburg-Landau equation in superconductor, which are, in essence, a kind of nonlinear Schrödinger equation. These solitons, with wave-corpuscle duality, are due to the nonlinear interactions arising from the electron-phonon interaction in superconductors, in which the nonlinear interaction suppresses the dispersive effect of the kinetic energy in these dynamic equations, thus a soliton states of the superconductive electrons, which can move over a macroscopic distances retaining the energy, momuntum and other quasiparticle properties in the systems, are formed. Meanwhile, we used these dynamic equations and their soliton solutions to obtain, and explain, these macroscopic quantum effects and superconductivity of the systems. Effects such as quantization of magnetic flux in superconductors and the Josephson effect of superconductivity junctions,thus we concluded that the superconductivity and macroscopic quantum effects are a kind of nonlinear quantum effects and arise from the soliton motions of superconductive electrons. This shows clearly that studying the essences of macroscopic quantum effects and properties of motion of microscopic particles in the superconductors has important significance of physics.

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