Unitary expansion of the time evolution operator

We propose an expansion of the unitary evolution operator, associated with a given Schrödinger equation, in terms of a finite product of explicit unitary operators. In this manner, this unitary expansion can be truncated at the desired level of approximation, as shown in the given examples.

Figure 1

Raman transition inside a cavity that is quasiresonant with the transition between the lower and higher levels of a $\Lambda$ atom.

Figure 2

Survival probability of $|g;0\rangle$ as a function of $\tau =\omega t.$ Solid line, exact solution; dashed line, $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1;$ dot-dashed line, first Born approximation; dotted line, $\mathrm{Û}\approx {\mathrm{Û}}_0.$ ${\omega }_0/\omega =0.6;g/\omega =0.5.$

Figure 3

Probability of $|g;0\rangle$ making a transition to the state $|e;1\rangle$ as a function of $\tau =\omega t.$ Solid line, exact solution; dashed line, $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1;$ dot-dashed line, first Born approximation; dotted line, $\mathrm{Û}\approx {\mathrm{Û}}_0.$ ${\omega }_0/\omega =0.6;g/\omega =0.5.$

Figure 4

Survival probability of $|g;0\rangle$ as a function of $\tau =\omega t.$ Solid line, exact solution; dashed line, $\mathrm{Û}\approx e^{-i{\mathrm{Ĥ}}_{0}t}{e}^{-i\lambda {\mathrm{Ŵ}}_1(t)};$ dot-dashed line, $\mathrm{Û}\approx e^{-i{\mathrm{Ĥ}}_{0}t}{e}^{-i\lambda {\mathrm{Ŵ}}_1(t)}{e}^{-i\lambda {\mathrm{Ŵ}}_2(t)};$ dotted line, $\mathrm{Û}\approx e^{-i{\mathrm{Ĥ}}_{0}t}.$ ${\omega }_0/\omega =1;g/\omega =0.8.$

Figure 5

Probability for $|g;0\rangle$ to make a transition to the state $|e;1\rangle$, as a function of $\tau =\omega t.$ Solid line, exact solution; dashed line, $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1;$ dot-dashed line, $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1{\mathrm{Û}}_2$; dotted line, $\mathrm{Û}\approx {\mathrm{Û}}_0.$ ${\omega }_0/\omega =1;g/\omega =0.8.$

Figure 6

Survival probability of $|g;0\rangle$ as a function of $\tau =\omega t.$ Solid line, exact solution is superposed with $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1;$ dotted line, $\mathrm{Û}\approx {\mathrm{Û}}_0.$ ${\omega }_0/\omega =0.5;g/\omega =2.$

Figure 7

Probability of $|g;0\rangle$ making a transition to the state $|e;1\rangle$ as a function of $\tau =\omega t.$ Solid line, exact solution is superposed with $\mathrm{Û}\approx {\mathrm{Û}}_0{\mathrm{Û}}_1;$ dotted line, $\mathrm{Û}\approx {\mathrm{Û}}_0.$ ${\omega }_0/\omega =0.5;g/\omega =2.$