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The Master Equation

The purpose of FieldApp is to acquaint the user with the notion of a quantum field. One of the basic problems of quantum field theory is analyzing how a single mode of a given field state will evolve in a thermal reservoir.

Consider a field which is trapped inside some cavity. An infinite number of field modes exist inside this cavity. Let us focus on one particular mode. Let us suppose that this mode is in contact with a thermal reservoir. The mode and reservoir taken together form an isolated system held at some constant temperature, , where T is the Kelvin temperature, is Boltzmann's constant, and is a fundamental temperature. Assume that whatever energy the mode dumps into this reservoir does not affect because the reservoir is large.

The derivation of the equation which governs this system's evolution is quite complex and can be found in a number of advanced quantum mechanics texts.[12] Before stating this equation, it is helpful to picture what this equation should imply. The single mode of the cavity field is a harmonic oscillator. The reservoir which we shall place this mode in contact with will be modeled by an infinite number of harmonic oscillators. Imagine a large number of springs bunched together in a small region of space. Focus on a single spring and try to picture how it will be affected by all the other springs. This single spring will be slowly damped by the other springs. A single mode in contact with a thermal reservoir is exactly analogous to this situation. Indeed, Enrico Fermi used this same argument to vindicate his interpretation of how this system would evolve.

This intuitive idea is mathematically described by the master equation:

 

where is the density matrix for the quantum field and

is the central decay constant, and Q is the cavity's quality factor. Therefore, a large value of Q implies a small amount of damping, while a small Q implies a large amount of damping. If the Q is infinite, then is zero, and field's density matrix does not change in time. is a result from statistical mechanics. It is the average photon number, n, of the mode for the temperature, .

Again is a density matrix which has the same properties as the one for the two--state atom, i.e., it is hermitian, the diagonal elements are real numbers, and the trace of the diagonal elements is equal to one. The definition of an expectation value is also the same. There is only one significant difference between this density matrix and that of the two--state atom. The two--state atom's density matrix had four elements. This density matrix has an infinite number of elements. FieldApp handles this problem by truncating the field's density matrix. This truncation will be discussed later in the section on FieldApp's algorithm.

It is necessary, as with the two--state atom, to find an equation for the arbitrary element of the Master equation's density matrix. This new form can be derived using Dirac notation and the matrix element. In general, if is any operator, then the matrix element of is . We want to find the arbitrary matrix element of which would equal

 

For any given equation, this task would be either very difficult or impossible; however, every operator in the master equation is either an or . Both <n| and |m> are arbitrary elements of the field. eq:q5 and eq:q6 define how the creation and destruction operators affect |n>. Using these two basic definitions, the equation of motion for an arbitrary element of the master equation's density matrix can be found.

As an example, let us find in terms of the matrix elements of . All of these derivations make use of the identity operator

where the sum is over every element of the |k> basis. In this case, the basis would be all the states of the quantum field. Since the identity operator has no effect on a state or operator, it may be substituted into a given equation at any place. The utility of this operator shall become apparent. Two copies of the identity operator will be used in analyzing the current operator. One will be designated the k basis and the other the l basis. Each set of functions is equivalent---only the designations differ:

Invoking eq:q5 and eq:q6 yields

where is the Kronecker delta; it is equal to zero for all values except except o=p. Therefore, the delta above selects: n=k+1 and k=l-1. These equations imply that n=l. Using this result along with the other two equations and simplifying, one obtains

This result should not be too surprising because is the operator.

In this same manner each piece of the master equation can be analyzed. The definitions of the creation and destruction operator are the key to fulfilling this analysis. The ultimate result is:

 

eq:q9 illustrates a useful result about this density matrix. The rate of change of the element, , is related to ,, and . All of these elements share the diagonal on which exists. The individual diagonals of the density matrix are coupled; however, they are independent of each other. Each diagonal forms a manifold of coupled ordinary differential equations. Therefore, one can neglect all the diagonals unnecessary for the calculation of expectation values without any loss in generality. This result is quite handy because it greatly simplifies FieldApp's algorithm, and no error in the density matrix will result from this simplification.

eq:q9 is the heart of FieldApp's algorithm. The off--diagonal elements could be complex. Therefore, it is instructive and necessary to break this arbitrary element down into real and imaginary parts. Let where . Thus,

since the derivative is a linear operator. A similar substitution into eq:q9 forms

 

From eq:q10, one can easily find the equations of motion for and

  

eq:q11 and eq:q12 have intriguing forms. eq:q11 is essentially the same as eq:q9. This similarity implies that all the physics of the system lies in the real part of the elements of the density matrix.

The solution to eq:q11 is not at all initially obvious; however, the solution to eq:q12 is an exponential. This solution may at first seem to be a problem since above the diagonal n>m, and the exponential decays; however, below the diagonal n<m, and the exponential seems to increase; however, the exponential decays nonetheless because the density matrix is hermitian, i.e., . The extra minus sign will force eq:q12 to decay. On the diagonal, there are no imaginary parts. Therefore, the imaginary parts of the field's density matrix simply decay.

If initially the field's density matrix has no imaginary parts, then there will never be any imaginary parts of the density matrix. Various initial conditions for the field shall be discussed, and each is, in general, complex; however, without any loss of generality one can neglect the imaginary parts of these conditions. This omission implies an assumption about the initial position of the field in phase space. By removing the imaginary terms, one initially sets the magnetic field equal to zero and maximizes the electric field. This assumption does not alter any of the fundamental physics of the quantum field and will be utilized because it simplifies and speeds up FieldApp's algorithm without sacrificing any generality.



next up previous contents index
Next: Photon Statistics Up: Theory Previous: The Harmonic Oscillator



Andy Antonelli
Wed May 17 14:34:24 EDT 1995