Introduction to Optical Lattices and Excited Bands (and All That)

Abstract

This chapter provides an introduction to the physics in excited bands of optical lattices. We will start by briefly discussing general features of the physics in optical lattices in Sect. 2.1. In Sect. 2.2 we review properties of single particles in periodic potentials and introduce the p and d orbitals in excited bands.

“And God said, “Let there be light,” and there was light. And God saw that light was good. Some time later, there were optical lattices; and then it was even better.”

—Adapted from a famous book.

Appendix: p-Band Hamiltonian Parameters in the Harmonic Approximation

For further reference, we compute here the various coupling constants in the harmonic approximation. As discussed before, under this assumption the Wannier functions are taken as Hermite polynomials, and therefore (2.48) and (2.47) can be obtained from computation of simple Gaussian integrals. Here

$$\begin{aligned} \begin{array}{ccl} U_{xx} &{}=&{}\displaystyle {U_0\int dx\;\left( \frac{\sqrt{2}}{\pi ^{1/4}x_0^{3/2}}\right) ^4x^4\;e^{-2x^2/x_0^{2}}\int dy\;\left( \frac{1}{\pi ^{1/4}y_0^{1/2}}\right) ^4e^{-2y^2/y_0^{2}}} \\ &{}=&{} \displaystyle {U_0 \left( \frac{\sqrt{2}}{\pi ^{1/4}x_0^{3/2}}\right) ^4 \frac{3}{4}\frac{\sqrt{\pi }}{2^{5/2}}x_0^5 \left( \frac{1}{\pi ^{1/4}y_0^{1/2}}\right) ^4\sqrt{\frac{\pi }{2}}y_0 = U_0\left( \frac{3}{8\pi }\frac{1}{x_0y_0}\right) .} \end{array} \end{aligned}$$

(2.72)

Analogous calculation yields $U_{yy}=U_0\left( \frac{3}{8\pi }\frac{1}{x_0y_0}\right)$.

We now compute $U_{xy}$:

$$\begin{aligned} \begin{array}{ccl} U_{xy} &{}=&{} \displaystyle {U_0\int \;dx\left( \frac{\sqrt{2}}{\pi ^{1/4}x_0^{3/2}}\right) ^2x^2\;e^{-x^2/x_0^{2}} \left( \frac{1}{\pi ^{1/4}x_0^{1/2}}\right) ^2e^{-x^2/x_0^{2}} \times }\\ &{} &{} \displaystyle { \;\;\;\;\;\int \;dy\left( \frac{\sqrt{2}}{\pi ^{1/4}y_0^{3/2}}\right) ^2y^2\; e^{-y^2/y_0^{2}} \left( \frac{1}{\pi ^{1/4}y_0^{1/2}}\right) ^2e^{-y^2/y_0^{2}}}\\ &{}=&{} \displaystyle {U_0\int \;dx \left( \frac{\sqrt{2}}{\pi ^{1/2}x_0^{2}}\right) ^2x^2e^{-2x^2/x_0^2} \int \;dy \left( \frac{\sqrt{2}}{\pi ^{1/2}y_0^{2}}\right) ^2y^2e^{-2y^2/y_0^2}}\\ \\ &{}=&{}\displaystyle {U_0\left( \frac{1}{8\pi }\frac{1}{x_0y_0}\right) }, \end{array} \end{aligned}$$

(2.73)

from where it follows that $U_{xx} = U_{yy} = 3 U_{xy}$. Notice, however, that the relation $U_{\alpha \alpha }/U_{\alpha \beta } = 3$ is only true in the harmonic approximation, and that this is the case regardless of the wave vectors of the lattice $k_x$ and $k_y$. In fact, it is very surprising that the coupling constants in the harmonic approximation do not even depend on the values of the lattice vector, but only on the lattice amplitudes $V_x$ and $V_y$ ³⁵ [16]. This is not the case, however, when the Hamiltonian parameters are computed with use of the lattice Wannier functions.

Now according to Eq. (2.47), we use Eqs. (2.27) and (2.28) to compute the tunneling coefficients as

$$\begin{aligned} \begin{array}{ccc} -t_{xx} &{}=&{} \displaystyle {\left( \frac{\sqrt{2}}{\pi ^{1/4}x_0^{3/2}}\right) ^2V_x\int \,dx \,x(x + d)\sin ^2x\,e^{-x^2/2x_0^2}e^{-(x +d)^ 2/2x_0^2}} \\ &{} &{}\displaystyle {+ \left( \frac{\sqrt{2}}{\pi ^{1/4}x_0^{3/2}}\right) ^2\int \,dx \frac{d}{dx}(xe^{-x^2/2x_0^ 2})\frac{d}{dx}\left( (x+d)e^{(x+d)^2/2x_0^ 2}\right) }. \end{array} \end{aligned}$$

(2.74)

d is used here as the lattice constant, and we have already used that the integral in the y-direction yields 1. In the same way,

$$\begin{aligned} \begin{array}{ccc} -t_{xy} &{}=&{}\displaystyle { \left( \frac{1}{\pi ^{1/4}y_0^{1/2}}\right) ^2V_y\int \,dy\,\sin ^2y\,e^{-y^2/2x_0^2}e^{-(y +d)^2/2y_0^2}}\\ \\ &{} &{}\displaystyle {+\left( \frac{1}{\pi ^{1/4}y_0^{1/2}}\right) ^2\int \,dy\frac{d}{dy}e^{-y^2/2y_0^2}\frac{d}{dy}e^{-(y +d)^2/2y_0^ 2}}. \end{array} \end{aligned}$$

(2.75)

Fig. 2.10

Comparison between the values of the couplings obtained from analytical and numerical computations as a function of V. It is shown in a that the harmonic approximation fails to reproduce the results obtained numerically for the tunneling coefficients when tunneling occurs in the direction of the node. In b we show the results for the interaction coefficients. In particular the estimates obtained from the harmonic approximation are always larger than the values of the couplings computed numerically

Fig. 2.11

Ratio $U_{xx}/U_{xy}$ for different values of the amplitude of the optical potential. Notice here that $U_{xx}/U_{xy}$ is always larger than 3 for numerical computations with the lattice Wannier functions

The expressions for $t_{yx}$ and $t_{yy}$ are obtained by making $x\rightarrow y$ and $y\rightarrow x$ with $x_0 \rightarrow y_0$ and $y_0\rightarrow x_0$ (Figs. 2.10 and 2.11).