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Algorithm and Parameters

eq:q11 is central to FieldApp's algorithm which is solved with a Runge--Kutta Four method. The Numerical option under the Parameters item allows the user to change the number of time steps between data registration as well as the size of the time step. FieldApp allows the user to set a time manually, or the program can adjust the time step according to the values of temperature and .

The density matrix is assumed to be initially real---the imaginary parts of this matrix are ignored. This assumption cuts the number of equations that must be solved in half. Even with this assumption, the complete density matrix cannot be solved because it has an infinite number of elements. Therefore, the density matrix is truncated to a finite matrix with n diagonal elements. Under the Field in Parameters, the Field Number dialog allows the user to specify n. In some situations, an error will occur because of this truncation. If a result observed in FieldApp seems contradictory to the theory, then try adding more field states and rerun the simulation. Truncation error is usually only a problem at high temperatures and large average photon numbers.

In order to obtain the expectation values calculated by FieldApp, the two diagonals above and below the central diagonal must be calculated; indeed, the two above and below are the same because the matrix is hermitian and real. This trick can be used because the master equation couples a diagonal only to itself. The number of equations is ultimately reduced from an infinite number to 3n-1.

The Field Type option in Field changes the initial state of the density matrix. It can be initialized to: Fock, thermal, coherent, or squeezed coherent state. Dialog boxes prompt the user for information about the characteristics of these states. The Fock state requires a state number. A thermal state is defined in terms of a temperature. This temperature may differ from the temperature of the reservoir. Both the coherent and squeezed coherent states have an average photon number field, and the squeezed coherent state also requires a squeezing parameter.

The quantum field is analyzed with the two windows below the field window. Both have the same options. All of the expectation values and variances discussed in the theory section can be analyzed with FieldApp ,i.e.,<N>, <q>, <p>, , , and . Under Variances the option to plot the ratio of the variance of the photon number divided by the average photon number exists.

Figure: Examples of E vs z, the Wigner Function, and q, vs t

The options under Spatial and Uncertainties provide visualizations that utilize both the averages and variances of p and q. An example of each of these options can be seen in Figure gif. The variance in p and q is used to give an approximate range about the average values of p and q in which measurements of p and q could exist. The p, vs t and q, vs t show this range by generating a series of vertical lengths whose length is twice the uncertainty of the given variable. The midpoint of these segments is the average value of the variable. The spatial plots show the electric and magnetic fields inside the cavity. This plot randomly generates points in a cavity scaled from zero to in the range provided by the uncertainties of p and q about their averages. FieldApp is also able to plot the Wigner function. This plot is the quantum mechanical phase space. When FieldApp is running an ellipse is created, the center of this ellipse is the average value of position and momentum. The ellipse itself represents the distribution of position and momentum about this average value. The ellipse is only visible when the simulation is running. Neither the ellipse data or the spatial data may be archived.

next up previous contents index
Next: Simulations Up: About FieldApp Previous: Introduction

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