Layer-by-layer assembly of multilayer optical lattices: A theoretical proposal

Abstract

We propose methods for synthesizing multilayer optical lattices of cold atoms in a layer-by-layer manner, to unlock the potential of optical lattices in simulating novel multilayer systems. The scheme combines compressed Gaussian beams of far red-detuned laser and an optical device with modified Michelson interferometers as key components. The compressed Gaussian beam makes possible standalone and purely two-dimensional optical lattices, which are the building blocks of target multilayer optical lattices. The modified Michelson interferometers split one compressed Gaussian beam into several beams, in a manner allowing delicate control over the vertical distances between the outgoing beams and their propagation directions. We illustrate the proposal with the displaced dice lattice, which is a trilayer lattice that maps to the dice lattice when projected to the same layer. Both the dice model and its interesting variants may be realized. We also discuss extensions to other interesting multilayer lattices such as the twisted multilayer lattices.

Appendices

Appendix A: Transformation of the Gaussian beam by the cylindrical Keplerian telescope

The beam shaper in our scheme is a cylindrical Keplerian telescope. In this section, we list the mathematical expressions for the Gaussian beams input to and output from the cylindrical Keplerian telescope, by summarizing well-known formulae for the transformations of Gaussian beams by paraxial optical elements [51, 52].

We refer to Fig. 1b for the setup. The apertures of $\hbox {L}_{1}$ and $\hbox {L}_{2}$ are commonly much larger than the diameter of the input Gaussian laser beam, which is determined by the output aperture of the laser device and typically in the order of 1 mm [51]. The diffraction of the laser beam by the lenses $\hbox {L}_{1}$ and $\hbox {L}_{2}$ should therefore be minor effects and is ignored in this work.

We consider the propagation of the laser beam parallel to the x axis, according to Fig. 1b of the main text. Before entering the beam shaper, the complex amplitude of the electric field of standard $\hbox {TEM}_{00}$ mode Gaussian laser beam can be written as [51]

$$\begin{aligned} E(x,y,z)=\frac{A_{0}}{\rho (x)}\text {exp}\left\{ -ik\left[ x+\frac{y^2+z^2}{2q_{1}(x)}\right] +i\phi (x)\right\} ,\nonumber \\ \end{aligned}$$

(A1)

where $k=2\pi /\lambda$ is the wave number, $A_{0}/\rho (x)$ is the amplitude of the electric field along the x axis ($y=z=0$), $\rho (x)=\rho _{0}\sqrt{1+(\frac{\lambda x}{\pi \rho _{0}^{2}})^2}$ is the x-coordinate dependent radius of the disk-like laser beam profile. $\rho _{0}$ is the waist radius at the waist plane ($x_{0}=0$ here) of the input Gaussian beam. Usually, $\rho _{0}$ is much larger than the wavelength of the laser (i.e., $\rho _{0}\gg \lambda$), so that $\rho (x)$ increases slowly as x departs from the waist of the beam. A parameter characterizing the increase of $\rho (x)$ is the Rayleigh range $z_{R}$, which is determined by $\rho (x_{0}+z_{R})=\sqrt{2}\rho (x_{0})$. From the above definition of $\rho (x)$, we see that

$$\begin{aligned} z_{R}=\pi \rho _{0}^{2}/\lambda . \end{aligned}$$

(A2)

A section of the beam centering at the waist and with a length 2$z_{R}$ is usually taken as the range over which the expansion of the Gaussian beam is small and negligible. In terms of the Rayleigh range, $\rho (x)=\rho _{0}\sqrt{1+(\frac{x}{z_{R}})^2}$. The phase factor $\phi (x)=\text {arctan}(\frac{\lambda x}{\pi \rho _{0}^{2}})=\text {arctan}(\frac{x}{z_{R}})$. Twice of the Rayleigh range (i.e., $2z_{R}$), or sometimes the Rayleigh range $z_{R}$ itself, is also called the confocal parameter of the Gaussian beam [52].

Being neither plane wave nor spherical wave, the Gaussian beam is characterized by the $q_{1}(x)$ parameter defined as [51]

$$\begin{aligned} \frac{1}{q_{1}(x)}=\frac{1}{R(x)}-i\frac{\lambda }{\pi \rho (x)^{2}}. \end{aligned}$$

(A3)

$R(x)=x[1+(\frac{z_{R}}{x})^{2}]$ is the x-dependent curvature radius of the equiphase surface. The waist of the Gaussian beam at the $x=x_{0}=0$ plane is special and has an infinite curvature, namely $R(0)=\infty$ and $q_{1}(0)=iz_{R}$. The beam profile determined by Eq. (A1) on the x plane is related to the beam profile on the $x=0$ waist plane, through a free propagation of x distance in the free space. Namely, the field amplitude on the $x=0$ plane,

$$\begin{aligned} E(0,y,z)=\frac{A_{0}}{\rho _{0}}\exp \left\{ -ik\frac{y^2+z^2}{2q_{1}(0)}\right\} =\frac{A_{0}}{\rho _{0}} \exp \left\{ -\frac{y^2+z^2}{\rho _{0}^{2}}\right\} , \nonumber \\ \end{aligned}$$

(A4)

is transformed to

$$\begin{aligned}&E(x,y,z)e^{ikx} \nonumber \\&\quad =\frac{A_{0}}{\rho _{0}}\frac{1}{A+\left[ B/q_{1}(0)\right] }\exp \left\{ -ik\frac{y^2+z^2}{2q_{1}(x)}\right\} ,\quad \quad \end{aligned}$$

(A5)

where

$$\begin{aligned} q_{1}(x)=\frac{Aq_{1}(0)+B}{Cq_{1}(0)+D}. \end{aligned}$$

(A6)

The four coefficients are the components of the ABCD matrix for a free-space propagation of distance x, which is known to be [51, 52]

$$\begin{aligned} \begin{pmatrix} A &{}\quad B \\ C &{}\quad D \end{pmatrix}=\begin{pmatrix} 1 &{}\quad x \\ 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

(A7)

Substituting the above coefficients into Eqs. (A5) and (A6), and defining the phase factor $\phi (x)=\arctan (\frac{x}{z_{R}})$, we reproduce Eq. (A1).

The free space as an optical medium is isotropic and transforms the Gaussian beam identically in both the y direction and the z direction. In an anisotropic medium (i.e., an astigmatic medium), however, the transformation matrix may be different along different directions. In this case, we treat the two directions independently and get two independent factors of the electric field amplitude, which are then multiplied together to give the overall electric field amplitude of the transformed laser beam [52].

The cylindrical Keplerian telescope defined by Fig. 1b of the main text, the beam shaper in our proposal, happens to be an astigmatic optical device. It reshapes only the z axis amplitude distribution of the input Gaussian beam. The transformation of the y axis amplitude distribution of the input Gaussian beam is just like the transformation in the free space. This anisotropic reshaping is conveniently represented by introducing two new q parameters, $q_{2}$ and $q_{3}$, so that the electric field amplitude of the output laser beam becomes

$$\begin{aligned} \frac{A^{\prime }_{0}}{\sqrt{\rho _{2}(x)\rho _{3}(x)}}\text {exp}\left\{ -ik\left[ x+\frac{y^2}{2q_{2}(x)}+\frac{z^2}{2q_{3}(x)}\right] +i\phi _{23}(x)\right\} ,\nonumber \\ \end{aligned}$$

(A8)

If the loss of the laser in passing through the telescope is negligible, we may set $A^{\prime }_{0}=A_{0}$. $\rho _{2}(x)$ and $\rho _{3}(x)$ are associated to $q_{2}$ and $q_{3}$, just as $\rho (x)$ is related to $q_{1}$ according to Eq. (A3). Before passing the beam shaper, we have $q_{2}(x)=q_{3}(x)=q_{1}(x)$. After passing the beam shaper, $q_{2}(x)=q_{1}(x)$ still holds. However, $q_{3}(x)$ is different from $q_{1}(x)$ and transforms from $q_{1}(0)$ according to the overall ABCD matrix of the telescope system in the z direction. The phase factor $\phi _{23}(x)$ is the average of the phase factors related to the transformations in the y and the z directions,

$$\begin{aligned} \phi _{23}(x)=\frac{\phi _{2}(x)+\phi _{3}(x)}{2}. \end{aligned}$$

(A9)

$\phi _{2}(x)=\arctan (\frac{\lambda x}{\pi \rho _{0}^{2}})$. $\phi _{3}(x)$ is related to the components of the overall ABCD matrix of the cylindrical Keplerian telescope in the z direction through [52]

$$\begin{aligned} e^{-i\phi _{3}(x)}=\frac{A+[B/q_{3}(0)]}{|A+[B/q_{3}(0)]|}, \end{aligned}$$

(A10)

where $q_{3}(0)=q_{2}(0)=q_{1}(0)=iz_{R}$.

The overall ABCD matrix of a composite optical system is the matrix multiplication of the ABCD matrices of the optical elements therein in sequence. We consider an input Gaussian laser beam with the waist at the $x=0$ plane at a distance $l_{1}$ to the left of $\hbox {L}_{1}$, and consider the output laser beam at a distance $l_{2}$ to the right of $\hbox {L}_{2}$. The telescope system is equivalent to a composition of five optical elements: a free space of length $l_{1}$, $\hbox {L}_{1}$ of focal length $f_{1}$, a free space of length $f_{1}+f_{2}\equiv L$, $\hbox {L}_{2}$ of focal length $f_{2}$, and a free space of length $l_{2}$. The ABCD matrix for a thin lens whose focal length is f is known as [51, 52]

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 \\ -\frac{1}{f} &{}\quad 1 \end{pmatrix}. \end{aligned}$$

(A11)

Together with the ABCD matrix for a free-space propagation for a length of l, defined by setting $x=l$ in Eq. (A7), we get the overall ABCD matrix for the z-axis transformation of the cylindrical Keplerian telescope by multiplying the five ABCD matrices together in sequence as

$$\begin{aligned} \begin{pmatrix} A &{} \quad B \\ C &{} \quad D \end{pmatrix}=\begin{pmatrix} -\frac{1}{M} &{}\quad L-\frac{l_{1}}{M}-Ml_{2} \\ 0 &{}\quad -M \end{pmatrix}, \end{aligned}$$

(A12)

where $M=f_{1}/f_{2}$ is the magnification of the telescope in the z direction. The q parameter at the waist of the incoming laser beam is $q_{1}(0)=i\pi \rho ^{2}_{0}/\lambda =iz_{R}$. After passing through the cylindrical Keplerian telescope, the q parameter in the z direction becomes

$$\begin{aligned} q_{3}(l_{2})=\frac{Aq_{1}(0)+B}{Cq_{1}(0)+D} =\frac{q_{1}(0)+l_{1}+M^{2}l_{2}-ML}{M^{2}}.\quad \quad \end{aligned}$$

(A13)

By writing

$$\begin{aligned} \frac{1}{q_{3}(l_{2})}=\frac{1}{R_{3}(l_{2})}-i\frac{\lambda }{\pi \rho _{3}^{2}(l_{2})}, \end{aligned}$$

(A14)

and comparing the two sides of Eq. (A14), we get

$$\begin{aligned}&R_{3}(l_{2})=\frac{A^{2}z_{R}^{2}+B^{2}}{BD}, \end{aligned}$$

(A15)

$$\begin{aligned}&\frac{\pi \rho _{3}^{2}(l_{2})}{\lambda }=\frac{A^{2}z_{R}^{2}+B^{2}}{ADz_{R}}. \end{aligned}$$

(A16)

For fixed $l_{1}$, the position of the waist for the z-axis amplitude distribution of the output laser beam is determined by $R_{3}({\tilde{l}}_{2})=\infty$, which gives

$$\begin{aligned} {\tilde{l}}_{2}=\frac{L-\frac{l_{1}}{M}}{M}. \end{aligned}$$

(A17)

Substituting into the expression for $\rho _{3}(l_{2})$, we get the waist size $\rho _{0z}$ for the amplitude distribution along the z axis for the output laser beam as

$$\begin{aligned} \rho _{0z}=\rho _{3}({\tilde{l}}_{2})=\frac{\rho (0)}{M}=\frac{\rho _{0}}{M}. \end{aligned}$$

(A18)

Taking the two waist size parameters $\rho _{0}$ and $\rho _{0z}$ as references, the laser beam is compressed by M times in the z direction. Therefore, the cylindrical Keplerian telescope transforms the disk-like beam profile of the input Gaussian beam to the straight filamentary beam profile of the output beam. Equivalently, the rod-like input Gaussian laser beam becomes the slab-like (or, sheet-like) output laser beam.

Note that, the waist of the amplitude distribution along the y direction is not changed by the telescope system and is still located at the plane $l_{1}$ to the left of $\hbox {L}_{1}$. We may want to change the waist position for the amplitude distribution of the output laser beam along y to the same position as that along z. Or alternatively, we may want to make the output laser beam have a uniform distribution along the y direction. This objective can be realized by replacing the simple cylindrical telescope used in Fig. 1b of the main text by more sophisticated optical systems [73, 74]. However, because of the excellent unidirectionality of laser, one may take as a very good approximation to ignore the departure from the unidirectional propagation along y. This should be allowed in particular in the laboratory where the optical distances are of limited range. Therefore, we adhere in this work to the simple cylindrical telescope shown in Fig. 1b of the main text. The waist size parameters $\rho _{0}$ and $\rho _{0z}$ are taken to measure the width and thickness of the beam profile of the output laser beam.

Besides technical limitations, and to facilitate the accurate control over the interlayer distances of the multilayer optical lattices, the reduced Rayleigh range for the amplitude distribution along the z axis also requires us to relax the restriction in the thickness of the slab-like output laser beam. Because as $\rho _{0z}$ decreases, the far-field divergence angle of the beam along z increases as $1/\rho _{0z}$ [51], the unidirectionality of the laser beam is weakened. As a result, the spatial extent, over which the thickness in the z-axis amplitude distribution of the laser beam could be regarded approximately constant, decreases monotonically with $\rho _{0z}$. Quantitatively, the range (along x, centering at the waist) over which the thickness of the laser along the z direction may be regarded as constant is approximately twice the Rayleigh range, $2z_{R}=2\pi \rho _{0z}^{2}/\lambda$. If $\rho _{0z}\simeq \lambda$ is achieved, the spatial region over which a uniform 2D optical lattice may be defined hosts only about 100 unit cells of the optical lattice. For $\rho _{0z}\simeq 5\lambda$, on the other hand, the approximately uniform region of the resulting multilayer optical lattice may have more than $6\times 10^{4}$ unit cells. This should be enough for most quantum simulations.

Finally, we comment on the position of the z-axis waist of the compressed Gaussian beam. According to Eq. (A17), we have ${\tilde{l}}_{2}\simeq f_{2}$ for $l_{1}\simeq 0$ and $M\gg 1$. In the layer-by-layer scheme for the multilayer optical lattices, we wish to have the z-axis waist positions of the various beams approximately coincide with each other at the center of the designated location for the optical lattice. To be able to insert the optical elements in Fig. 3 in between $\hbox {L}_{2}$ of the cylindrical telescope and the location for the optical lattice, ${\tilde{l}}_{2}$ and thus $f_{2}$ should be large enough. For practical reasons, $f_{1}$ cannot be too large. Therefore, the magnification M cannot be large. This means that, to achieve the desired compression ratio along the z axis (i.e., about 200), we have to cascade two or more telescopes. In the last telescope, we can choose its $f_{2}$ (magnification factor) larger (smaller) than those of the preceding telescopes. For example, we can cascade two cylindrical telescopes of magnification 10 and a third cylindrical telescope with a magnification of 2 to achieve a total magnification of 200. In the last telescope, we may choose $f_{2}\simeq 10$ cm, which gives ${\tilde{l}}_{2}\simeq 15$ cm for $l_{1}\simeq 0$. By minimizing the sizes of the optical elements in the device, we should be able to bring the z-axis waist positions of the various sheet-like compressed beams to the center of the designated spatial region [e.g., within the circle of Fig. 3a]. Besides increasing $f_{2}$, another method is to add a proper cylindrical lens to each MMI and refocus the outgoing beams, so that their z-axis waists are at or close to the center of the optical lattice. Still another, more radical, strategy is to use diffractionless laser beams (in particular, the Bessel beams [75,76,77,78]) with a thin-line beam profile. This strategy not only resolves the issue of small ${\tilde{l}}_{2}$ but may also enlarge the area of the ensuing multilayer optical lattice. It however requires more analysis as regards how to achieve this novel non-diffracting thin-line beam with sufficient intensity, which we leave to future studies.

Appendix B: The 2D limit of an individual layer

In terms of the compressed Gaussian beam output from the cylindrical telescope, we may form a quasi-2D optical lattice with reduced thickness characterized by $\rho _{0z}$. In applying to our layer-by-layer assembling of multilayer optical lattices, it is crucial to know the upper limit of $\rho _{0z}$ required to bring the thickness of the resulting optical lattice down to the 2D limit.

Physically, if the input Gaussian laser beam is not compressed in the z axis, each 2D site of the ensuing 2D optical lattice is in fact a quasi one-dimensional (q-1D) tube [19], for which the spectrum is a quasi-continuous 1D spectrum. As we compress the z-axis breadth (i.e., $\rho _{0z}$) of the laser beam, the q-1D spectrum becomes more and more discrete. Let us label the bound states in each site (i.e., potential well) of the 2D optical lattice by $\epsilon _{0}$, $\epsilon _{1}$, $\epsilon _{2}$, $\ldots$, in an order of increasing with the subscript. Then, the site should be considered as a real zero-dimensional point rather than a q-1D tube when $\epsilon _{1}-\epsilon _{0}>E_{R}=h^{2}/(2m\lambda ^{2})$. $E_{R}$ is the recoil energy of the optical lattice. If this condition is satisfied, only the lowest bound state of each quantum well is occupied. A tight-binding model for the cold atoms loaded to the optical lattice may be constructed by retaining only the lowest bound state for each site of the 2D lattice.

To be concrete, let us consider the optical dipole potential defined in Eqs. (1)–(4) of the main text for a single layer of the displaced dice lattice, which is a triangular lattice. By redefining the origin of coordinates, we may set the phase factors to zero. The optical dipole potential is written as

$$\begin{aligned} U(x,y,z)=-V_{0}\left\{ 1+4\cos \frac{bx}{2}\left[ \cos \frac{bx}{2}+\cos \frac{\sqrt{3}by}{2}\right] \right\} g^{2}(z),\nonumber \\ \end{aligned}$$

(B1)

where $b=4\pi /(\sqrt{3}a)$ is the magnitude of the reciprocal lattice vector, $g(z)=\exp (-z^{2}/\rho _{0z}^{2})$. We consider a red-detuned laser, so that [7]

$$\begin{aligned} V_{0}=\frac{\epsilon _{0}}{4}\alpha _{0}'E_{0}^{2}>0. \end{aligned}$$

(B2)

$(x,y,z)=(0,0,0)$ is clearly the bottom of one potential well. For simplicity, we focus on this potential well. Since we are interested in the 2D limit of the optical potential, we assume sufficiently deep potential wells and that the low-lying bound states are confined close to the bottom of the potential well. According to this assumption, we expand U(x, y, z) to the leading order polynomials of x, y, and z close to $(x,y,z)=(0,0,0)$. We get

$$\begin{aligned} U(x,y,z)\simeq & {} -9V_{0}+\frac{8\pi ^{2}V_{0}}{a^{2}}(x^{2}+y^{2})+\frac{18V_{0}}{\rho _{0z}^{2}}z^{2} \nonumber \\= & {} -9V_{0}+U(x)+U(y)+U(z). \end{aligned}$$

(B3)

The above potential energy term plus the kinetic energy term of the cold atom define an anisotropic 3D harmonic oscillator. We choose $\{{\hat{H}}_{x},{\hat{H}}_{y},{\hat{H}}_{z}\}$ as the complete set of commuting observables, where

$$\begin{aligned} {\hat{H}}_{\alpha }=\frac{p_{\alpha }^{2}}{2m}+U(\alpha ), \end{aligned}$$

(B4)

$\alpha =x,y,z$, and m is the mass of the cold atoms in the optical lattice. The eigen-spectrum of the above anisotropic harmonic oscillator is

$$\begin{aligned}&E_{n_{x}n_{y}n_{z}}=-9V_{0}+(n_{x}+n_{y}+1)\hbar \omega _{xy}\nonumber \\&\quad +\left( n_{z}+\frac{1}{2}\right) \hbar \omega _{z}, \end{aligned}$$

(B5)

where $n_{\alpha }$ ($\alpha =x,y,z$) are nonnegative integers quantifying the quantization of the energy spectrum. $\omega _{xy}=\frac{4\pi }{a}\sqrt{\frac{V_{0}}{m}}$ and $\omega _{z}=\frac{6}{\rho _{0z}}\sqrt{\frac{V_{0}}{m}}$ are separately the angular frequencies for the center-of-mass motion of the cold atom in the xy plane and along the z direction.

In order for the single layer to be in the 2D limit, the separation between the lowest energy bound state and the second lowest energy bound state should be larger than the recoil energy of the optical lattice. We therefore require $\hbar \omega _{xy}>E_{R}$ and $\hbar \omega _{z}>E_{R}$. For our purpose, considering the continuous compression of $\rho _{0z}$ from $\rho _{0}$, we originally have $\rho _{0z}\gg a$ and correspondingly $\omega _{z}/\omega _{xy}=\frac{3}{2\pi }\frac{a}{\rho _{0z}}\ll 1$. We assume the $\hbar \omega _{xy}>E_{R}$ condition is always satisfied for all $\rho _{0z}$. As we compress $\rho _{0z}$ continuously, the 2D limit is attained when $\hbar \omega _{z}>E_{R}$. Suppose $\hbar \omega _{xy}=\gamma E_{R}$ ($\gamma \gg 1$), the condition $\hbar \omega _{z}>E_{R}$ amounts to

$$\begin{aligned} \rho _{0z}<\frac{3\gamma }{2\pi }a. \end{aligned}$$

(B6)

For our triangular lattice, $a=\frac{2}{3}\lambda$. For $\gamma >5\pi \simeq 15.71$, $\rho _{0z}=5\lambda$ satisfies the above constraint and the 2D limit is attained. For $^{40}$K cold atoms, red-detuned laser with a wavelength 1064 nm or 1030 nm were used in experiments [12, 67,68,69,70]. Correspondingly, $\rho _{0z}=5\lambda$ amounts to $\rho _{0z}\approx 5$ $\mu$m.

From the definition of the angular frequency, $\hbar \omega _{xy}=\gamma E_{R}>15.71E_{R}$ is equivalent to $V_{0}=\frac{\gamma ^{2}}{18}E_{R}>13.71E_{R}$. This is an intermediate value for the lattice depth. However, it is $9V_{0}$ from the minima to the maxima of the optical dipole potential defined by Eq. (B1). Therefore, $9V_{0}$ is the overall lattice depth, which should satisfy $9V_{0}>123.39E_{R}$. As a result, we need to work with a very deep optical lattice to attain the 2D limit for an individual triangular optical lattice. Luckily, this depth is within the reach of experiments [7, 79, 80]. Clearly, for even deeper optical dipole potential, $\rho _{0z}$ larger than 5$\lambda$ may also lead to the 2D limit. Conversely, if we compress the Gaussian beam thinner, with $\rho _{0z}<5\lambda$, a shallower optical potential may also attain the 2D limit. For example, if we consider $\rho _{0z}=3\lambda$, $9V_{0}>44.5E_{R}$ is enough to get the 2D limit for the triangular optical lattice.

Finally, we note that the analysis in this section is based on the optical dipole potential of a single layer. In a multilayer optical lattice containing several contiguous layers, such as the displaced dice lattice consisting of three consecutive layers, the superposition of the optical potentials for the various layers may possibly change the actual depth of the resulting multilayer optical dipole potential. From Fig. 2 of the main text for the lattice geometry and the expressions of the optical dipole potentials of the various layers, this superposition tends to reduce the actual depth of the optical potential for the displaced dice lattice. From the analysis carried out later in “Appendix D”, the superposition does not change $\omega _{z}$ in the leading order approximation. Therefore, the above criterion for the 2D limit is still applicable. However, the overall trilayer optical dipole potential confining the cold atoms in the displaced dice lattice is actually shallower than the potential of a single layer. The shorter the distances between the three layers, the shallower the trilayer optical dipole potential becomes. For other multilayer optical lattices synthesized through the layer-by-layer approach, the depth of the overall multilayer optical dipole potential may also be larger than the depth of a single layer. At the same time, the critical depth of the lattice for attaining the 2D limit may also change quantitatively.

Appendix C: Reflection of a ray by the Fresnel bimirror with about 90$^{\circ }$ intersection angle

In what follows, we call the special Fresnel bimirror with a 90$^{\circ }$ intersection angle as the right-angle bimirror. In Fig. 3 of the main text, the reflection plane of the ray of light is assumed to be perpendicular to the intersection edge of the right-angle bimirror. In this case, the retroreflected light propagates along the direction exactly opposite to that of the incident ray of light. This is always the case once the intersection edge of the right-angle bimirror is perpendicular to the incident ray of light. The lateral shift (i.e., the distance) between the incident and the retroreflected rays of light depends on the incident angle and the distance from the point of incidence to the intersection edge. As the orientation of the intersection edge has a $2\pi$ rotational freedom in the plane perpendicular to the incident ray of light, the lateral shift of the retroreflected ray can also occur at any direction away from the incident ray of light.

By twisting the intersection edge of the right-angle bimirror away from the plane perpendicular to the incident ray of light by a small angle $\theta$, the reflection plane of the incident ray of light is also slightly deviated from perpendicular to the intersection edge. In this case, we may twist the propagation direction of the retroreflected ray away from exactly the reverse direction of the incident ray. Here, the intersection edge at an angle $\theta$ to the plane perpendicular to the incident ray of light also has a $2\pi$ rotational freedom, which may be characterized by an angle $\varphi$. As regards the retroreflected ray of light, we expect there to be a corresponding freedom in the deviation of its propagation direction with respect to the direction exactly opposite to the incident ray.

Instead of plotting the full optical path, which is clumsy and not easy to show clearly the various interesting situations, we will express the reflection off the bimirror in terms of vector algebra. Because of the simplicity of the reflection by plane mirrors, exact analytical expressions for the outgoing reflected ray may be obtained. Here, we assume the sizes of the bimirrors to be much larger than the $\rho _{0}$ and $\rho _{0z}$ of the laser beams, so diffractions by the bimirrors are ignored.

For generality, we consider a Fresnel bimirror whose intersection angle $\alpha$ is close to but may be slightly different from 90$^{\circ }$. We notice that the reflected ray is uniquely determined by the position and orientation of the $\alpha$-angle bimirror, and the propagation direction and point of incidence of the incident ray. In terms of vector algebra, these conditions are contained in the two equations for the two mirror planes and the equation for the incident ray.

Fig. 4

Fresnel bimirror with an intersection angle $\alpha$ between the two mirrors $\hbox {M}_{1}$ and $\hbox {M}_{2}$. The Cartesian coordinate is defined by taking the center of the intersection edge as the origin O, and the intersection edge as the y axis. The xy plane bisects the bimirror

To proceed, we firstly define the coordinate system in reference to the bimirror shown in Fig. 4 of this section. We consider a 3D Cartesian coordinate. We assume the intersection edge of the bimirror is along the y axis, so that the planes perpendicular to the intersection edge are parallel to the xz plane. An arbitrary perpendicular plane of the intersection edge crosses with the two mirrors at two lines, which join each other at a point of the intersection edge and subtend an angle of $\alpha$. We assume that the bimirror is divided equally by the xy plane containing the intersection edge. We define the x axis to run opposite to the open-mouth direction of the $\alpha$-angle bimirror, so that the incident ray propagates along a direction very close to the positive x direction (i.e., close to $\hat{{\mathbf {x}}}$, the unit vector along the $+x$ direction). We set the origin O of the coordinate, (0, 0, 0), at the middle of the intersection edge. The two mirrors are assumed to be large enough so that the interesting incident rays are all reflected twice by the $\alpha$-angle bimirror before leaving it. Finally, we assume that the crossing point between the incident ray and the first mirror (e.g., $\hbox {M}_{1}$) is close to the intersection edge.

Under the above conditions, the equations for the two mirror planes are

$$\begin{aligned} {\mathbf {r}}\cdot \hat{{\mathbf {n}}}_{i}=0, \end{aligned}$$

(C1)

where $i=1,2$ labels the two mirrors, $\hat{{\mathbf {n}}}_{1}=(\sin \frac{\alpha }{2},0,\cos \frac{\alpha }{2})$ and $\hat{{\mathbf {n}}}_{2}=(\sin \frac{\alpha }{2},0,-\cos \frac{\alpha }{2})$ are unit normal vectors for $\hbox {M}_{1}$ and $\hbox {M}_{2}$.

In the application of the main text, we fixed $\alpha =\pi /2$ and the incident ray propagates along the positive x direction (i.e., $\hat{{\mathbf {x}}}$). In this case, the incident ray is exactly retroreflected, up to a lateral shift in the z direction. Here, we assume the propagation direction of the incident ray may be slightly away from the $\hat{{\mathbf {x}}}$ direction by a small angle $\theta$. To characterize this direction, we introduce a spherical polar coordinate taking $\hat{{\mathbf {x}}}$ as the polar axis, $\theta$ as the polar angle, and define the azimuthal angle $\varphi$ as the angle with respect to the $\hat{{\mathbf {y}}}$ direction in the yz plane. In this polar coordinate, the propagation of the incident ray is along the unit vector

$$\begin{aligned} \hat{{\mathbf {u}}}=(\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi ). \end{aligned}$$

(C2)

Without losing generality, we assume that the incident ray shines at a point P on $\hbox {M}_{1}$ of a distance $d>0$ away from the intersection edge. The coordinate of P is written as

$$\begin{aligned} {\mathbf {r}}_{1}=\left( -d\cos \frac{\alpha }{2},y_{1},d\sin \frac{\alpha }{2}\right) . \end{aligned}$$

(C3)

For simplicity, we set $y_{1}=0$. The equation of the incident ray is thus

$$\begin{aligned} ({\mathbf {r}}-{\mathbf {r}}_{1})\times \hat{{\mathbf {u}}}={\mathbf {0}}. \end{aligned}$$

(C4)

The reflected ray leaves $\hbox {M}_{1}$ also at P. Assuming the unit vector along its propagation direction to be $\hat{{\mathbf {u}}}_{1}$, the law of reflection dictates that

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{{\mathbf {u}}}\cdot \hat{{\mathbf {n}}}_{1}=-\hat{{\mathbf {u}}}_{1}\cdot \hat{{\mathbf {n}}}_{1}, \\ \hat{{\mathbf {u}}}_{1}\cdot (\hat{{\mathbf {u}}}\times \hat{{\mathbf {n}}}_{1})=0, \\ \end{array}\right. } \end{aligned}$$

(C5)

where the first equation ensures the reflection angle to be equal to the incident angle, and the second equation says that the reflected ray lies in the same plane containing the incident ray and the unit normal vector of $\hbox {M}_{1}$ passing P. Solution to Eq. (C5) gives

$$\begin{aligned} \hat{{\mathbf {u}}}_{1}=\hat{{\mathbf {u}}}-2(\hat{{\mathbf {u}}}\cdot \hat{{\mathbf {n}}}_{1})\hat{{\mathbf {n}}}_{1}. \end{aligned}$$

(C6)

From $\hat{{\mathbf {u}}}\cdot \hat{{\mathbf {u}}}=\hat{{\mathbf {n}}}_{1}\cdot \hat{{\mathbf {n}}}_{1}=1$, we have $\hat{{\mathbf {u}}}_{1}\cdot \hat{{\mathbf {u}}}_{1}=1$. So $\hat{{\mathbf {u}}}_{1}$ is the unit vector along the propagation direction of the first reflected ray. The equation for the first reflected ray leaving $\hbox {M}_{1}$ at P is thus

$$\begin{aligned} ({\mathbf {r}}-{\mathbf {r}}_{1})\times \hat{{\mathbf {u}}}_{1}={\mathbf {0}}. \end{aligned}$$

(C7)

The coordinate ${\mathbf {r}}_{2}$ of the crossing point between the first reflected ray and $\hbox {M}_{2}$ is determined by solving the following simultaneous equations

$$\begin{aligned} {\left\{ \begin{array}{ll} ({\mathbf {r}}-{\mathbf {r}}_{1})\times \hat{{\mathbf {u}}}_{1}={\mathbf {0}}, \\ {\mathbf {r}}\cdot \hat{{\mathbf {n}}}_{2}=0. \\ \end{array}\right. } \end{aligned}$$

(C8)

The solution ${\mathbf {r}}={\mathbf {r}}_{2}=(x_{2},y_{2},z_{2})$ to the above equations is also the point at which the second reflected ray leaves $\hbox {M}_{2}$ and the whole bimirror. Similar to the first reflection, the unit vector $\hat{{\mathbf {u}}}_{2}$ along the propagation direction of the second reflected ray is determined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{{\mathbf {u}}}_{1}\cdot \hat{{\mathbf {n}}}_{2}=-\hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {n}}}_{2}, \\ \hat{{\mathbf {u}}}_{2}\cdot (\hat{{\mathbf {u}}}_{1}\times \hat{{\mathbf {n}}}_{2})=0, \\ \end{array}\right. } \end{aligned}$$

(C9)

which gives

$$\begin{aligned} \hat{{\mathbf {u}}}_{2}=\hat{{\mathbf {u}}}_{1}-2(\hat{{\mathbf {u}}}_{1}\cdot \hat{{\mathbf {n}}}_{2})\hat{{\mathbf {n}}}_{2}. \end{aligned}$$

(C10)

The equation for the second (i.e., final) reflected ray outgoing from the bimirror is

$$\begin{aligned} ({\mathbf {r}}-{\mathbf {r}}_{2})\times \hat{{\mathbf {u}}}_{2}={\mathbf {0}}. \end{aligned}$$

(C11)

The explicit expressions for ${\mathbf {r}}_{2}=(x_{2},y_{2},z_{2})$ and $\hat{{\mathbf {u}}}_{2}=(u_{2x},u_{2y},u_{2z})$ are

(C12)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{2x}=\cos \theta \cos (2\alpha )-\sin \theta \sin \varphi \sin (2\alpha ), \\ u_{2y}=\sin \theta \cos \varphi , \\ u_{2z}=\cos \theta \sin (2\alpha )+\sin \theta \sin \varphi \cos (2\alpha ). \\ \end{array}\right. } \end{aligned}$$

(C13)

The lateral shift related to ${\mathbf {r}}_{2}-{\mathbf {r}}_{1}$ is linearly proportional to d and could be tuned continuously. To characterize the change in the propagation direction, we study the direction cosines of $\hat{{\mathbf {u}}}_{2}$ with regards to the direction of $\hat{{\mathbf {u}}}$ and $\hat{{\mathbf {u}}}\times \hat{{\mathbf {y}}}$, which turn out to be

$$\begin{aligned} \hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {u}}}=-1+2[1-\sin ^{2}\alpha (1-\sin ^{2}\theta \cos ^{2}\varphi )], \end{aligned}$$

(C14)

and

$$\begin{aligned} \hat{{\mathbf {u}}}_{2}\cdot (\hat{{\mathbf {u}}}\times \hat{{\mathbf {y}}})=\sin (2\alpha )(\sin ^{2}\theta \sin ^{2}\varphi +\cos ^{2}\theta ). \end{aligned}$$

(C15)

From these formulae, we uncover the conditions for the following two especially interesting cases.

In the first case, which is the case we focused on in the main text, the reflected ray propagates in exactly the opposite direction to the incident ray. This is satisfied if

$$\begin{aligned} \hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {u}}}=-1, \end{aligned}$$

(C16)

which leads to $\alpha =\pi /2$ and $\varphi =\pm \pi /2$. That is, the ray incident on an $\alpha =\pi /2$ right-angle bimirror propagates along a direction parallel to the xz-plane and close to $\hat{{\mathbf {x}}}$ (i.e., $\theta$ is small so that the incident ray undergoes two reflections off the bimirror).

In the second case, the reflected ray propagates almost opposite to the direction of the incident ray but deflected a little bit away from that opposite direction in the plane perpendicular to $\hat{{\mathbf {u}}}\times \hat{{\mathbf {y}}}$. This case is determined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {u}}}+1=\epsilon , \\ \hat{{\mathbf {u}}}_{2}\cdot (\hat{{\mathbf {u}}}\times \hat{{\mathbf {y}}})=0, \\ \end{array}\right. } \end{aligned}$$

(C17)

where $0<\epsilon \ll 1$. For $\theta \simeq 0$ that we focus on, the only solution to the second equation of Eq. (C17) for $0<\alpha <\pi$ is $\alpha =\pi /2$. Substituting $\alpha =\pi /2$ into Eq. (C14), we have

$$\begin{aligned} \hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {u}}}+1=2\sin ^{2}\theta \cos ^{2}\varphi . \end{aligned}$$

(C18)

Then, the first equation of Eq. (C17) is satisfied once $\theta$ is small but nonzero and $\varphi \ne \pm \pi /2$. For our purpose of using the bimirror, it is enough to fix $\varphi =0$ or $\varphi =\pi$, so that $\hat{{\mathbf {u}}}$ is parallel to the xy plane. In this case, suppose the rotation angle from $\hat{{\mathbf {u}}}$ to $\hat{{\mathbf {u}}}_{2}$ is $\pi +{\tilde{\theta }}$, we have

$$\begin{aligned}&\hat{{\mathbf {u}}}_{2}\cdot \hat{{\mathbf {u}}}\nonumber \\&\quad =\cos (\pi +{\tilde{\theta }})=-\cos {\tilde{\theta }}=2\sin ^{2}\theta -1=-\cos (2\theta ).\nonumber \\ \end{aligned}$$

(C19)

Therefore, rotating the incident ray in the xy plane by an angle of $\theta$ away from $\hat{{\mathbf {x}}}$ leads to a rotation of the reflected ray by an angle of 2$\theta$ in the xy plane, compared to the direction of exact retroreflection. This rotation is also easy to see by substituting $\alpha =\pi /2$ and $\sin \varphi =0$ into the explicit expression for $\hat{{\mathbf {u}}}_{2}$ in Eq. (C13) and comparing it with the definition of $\hat{{\mathbf {u}}}$. The angle ${\tilde{\theta }}=2\theta$ may be called as the twist angle. The twist angle ${\tilde{\theta }}$ enables a continuous control over the twisting angle between consecutive layers of the multilayer optical lattice constructed in the layer-by-layer manner.

For both of the two cases considered above, the right-angle bimirror may be replaced by an isosceles right-angle triangular prism (in brief, a right-angle prism) made of low-loss glass with a high refractive index. To see this, notice that the above analyses may be considered as corresponding to the two total internal reflections inside the interior of the right-angle prism. $\hbox {M}_{1}$ and $\hbox {M}_{2}$ of the right-angle bimirror are interpreted here as the two mutually perpendicular rectangular faces of the prism, and the incident ray enters the right-angle prism through the rectangular bevel face in between the two perpendicular faces. The actual incident ray is connected to the ray incident on the $\hbox {M}_{1}$ face at P by a refraction at the bevel face of the right-angle prism. The actual outgoing ray is connected to the ray leaving the $\hbox {M}_{2}$ face at ${\mathbf {r}}_{2}$ by another refraction at the bevel face of the right-angle prism. For the first case studied above, it is easy to see that the retroreflected ray still reverses the direction of the incident ray. In simple words, two parallel rays are still parallel rays upon refraction by a planar dielectric interface. For the second case, the actual twisting angle is changed from 2$\theta$ to 2$\theta '$. The actual twisting angle 2$\theta '$ is related to the twisting angle 2$\theta$ within the right-angle prism by Snell’s law

$$\begin{aligned} \sin \theta '=n\sin \theta , \end{aligned}$$

(C20)

where n is the refractive index of the glass making up the right-angle prism. The refractive index for the air has been taken as 1.

The above analyses deal with a single ray of light. In considering the reflection of a laser beam, we regard it as an assembly of nearly parallel rays. The reflection of a compressed Gaussian beam, whose long axis of the beam profile is within the xy plane, by a right-angle bimirror or a right-angle prism follows the same rules explained above.

Finally, we notice that, rotating the incident ray (or, the incident line beam) in the xy plane with respect to the $\hat{{\mathbf {x}}}$ direction is equivalent to fixing the incident ray to propagate along $\hat{{\mathbf {x}}}$ and rotating the intersection edge of the right-angle bimirror in the xy plane by the same angle in the opposite sense (i.e., clockwise versus counterclockwise, and vice versa). In experimental implementations, it seems more convenient to control the bimirrors. Therefore, it is preferable to perform the rotations by acting upon the bimirrors, rather than upon the incident rays.

Appendix D: Estimation of tight-binding parameters for ultracold atoms in the displaced dice lattice

We show in this section that by tuning the amplitudes of the electric field vectors of the laser beams and the z-axis coordinates of the three layers, we may realize the following family of tight-binding models for cold atoms in the displaced dice lattice

$$\begin{aligned} {\hat{H}}= & {} \sum \limits _{\langle i,j\rangle ,\sigma }(t_{ba}b^{\dagger }_{i\sigma }a_{j\sigma }+\text {H.c.}) +\sum \limits _{\langle i,j\rangle ,\sigma }(t_{bc}b^{\dagger }_{i\sigma }c_{j\sigma }+\text {H.c.}) \nonumber \\&+\sum \limits _{i,\sigma }(\varepsilon _{a}a^{\dagger }_{i\sigma }a_{i\sigma } +\varepsilon _{b}b^{\dagger }_{i\sigma }b_{i\sigma } +\varepsilon _{c}c^{\dagger }_{i\sigma }c_{i\sigma }). \end{aligned}$$

(D1)

Equation (D1) is an equivalent formulation of Eq. (6) of the main text, by merging the two on-site energy terms of the latter. The parameters in Eq. (D1) follow the same definitions for Eq. (6) of the main text. To validate this model, the intralayer (i.e., intrasublattice) nearest-neighboring (NN) hopping amplitudes and the NN hopping amplitudes between the A layer and the C layer should be much smaller than the BA interlayer and BC interlayer NN hopping amplitudes.

To justify the above approximation, we have to make a reasonable estimation over the magnitudes of the relevant NN hopping amplitudes (6 in total). We suppose all three layers of the displaced dice lattice have reached the 2D limit and the on-site interactions have been tuned to zero by the Feshbach resonance. We also assume that fermionic cold atoms occupy only the lowest-energy bound states of the potential wells. In this case, a reasonable order of magnitude estimation over the relevant hopping amplitudes follows by taking the ground state eigenvectors of the approximate harmonic oscillator models for each quantum well as the local Wannier orbitals and substitute them into the definition of the hopping amplitudes. A merit of this approach, despite less accurate, is that all hopping amplitudes may be evaluated in a fully analytical manner. Having an explicit analytical expression for the hopping amplitudes in terms of the parameters of the optical lattice greatly facilitates a qualitative understanding over the model parameters of the tight-binding model.

For the displaced dice lattice defined in the main text, the single-body Hamiltonian is

$$\begin{aligned} {\hat{H}}_{0}=-\frac{\hbar ^{2}}{2m}\varvec{\nabla }^{2}+V(x,y,z), \end{aligned}$$

(D2)

where m is the mass of the cold fermionic alkali-metal element such as $^{40}$K or $^{6}$Li. V is the sum of the optical dipole potentials of all three layers

$$\begin{aligned} V(x,y,z)=U_{A}(x,y,z)+U_{B}(x,y,z)+U_{C}(x,y,z),\quad \end{aligned}$$

(D3)

where the subindices A, B, and C indicate the three triangular layers. Here, following the discussions of the main text, we do not include the interlayer interference terms in the optical potential. Taking the minimum of one potential well of the B layer as the origin of coordinate, $(x,y,z)=(0,0,0)$, we have

$$\begin{aligned} U_{B}(x,y,z)= & {} -V_{B}\left[ 1+4\cos \frac{bx}{2}\left( \cos \frac{bx}{2}+\cos \frac{\sqrt{3}by}{2}\right) \right] g^{2}(z), \end{aligned}$$

(D4a)

$$\begin{aligned} U_{A}(x,y,z)= & {} U_{B}(x,y,z)|_{x\rightarrow x-\frac{a}{\sqrt{3}},y\rightarrow y,z\rightarrow z-z_{A};V_{B}\rightarrow V_{A}}, \end{aligned}$$

(D4b)

$$\begin{aligned} U_{C}(x,y,z)= & {} U_{B}(x,y,z)|_{x\rightarrow x+\frac{a}{\sqrt{3}},y\rightarrow y,z\rightarrow z-z_{C};V_{B}\rightarrow V_{C}},\nonumber \\ \end{aligned}$$

(D4c)

where the Gaussian factor $g(z)=\exp {(-z^{2}/\rho _{0z}^{2})}$, the length of the reciprocal lattice vector $b=4\pi /(\sqrt{3}a)$.

Generally, the hopping amplitude between two NN sites (i.e., two NN minima of the optical potential) located at ${\mathbf {r}}_{0}=(x_{0},y_{0},z_{0})$ and ${\mathbf {r}}_{1}=(x_{1},y_{1},z_{1})$ is [1, 81, 82]

$$\begin{aligned} t_{NN}({\mathbf {r}}_{0}{\mathbf {r}}_{1})=\int d^{3}{\mathbf {r}}\psi ^{*}_{0}({\mathbf {r}}-{\mathbf {r}}_{0}){\hat{H}}_{0}\psi _{0}({\mathbf {r}}-{\mathbf {r}}_{1}). \end{aligned}$$

(D5)

In this definition, the layer (sublattice) index is implicitly included in the reference coordinates ${\mathbf {r}}_{0}$ and ${\mathbf {r}}_{1}$. $\psi _{0}({\mathbf {r}}-{\mathbf {r}}_{0})$ is the Wannier function for the local orbital at ${\mathbf {r}}_{0}$.

For a qualitative estimation of the hopping amplitudes, we take $\psi _{0}$ as the ground-state wave function for a quantum well in the harmonic approximation. In “Appendix B” analyzing the condition for attaining the 2D limit, we focused on the optical dipole potential for a single layer. In the present displaced dice lattice containing three layers, the bottom of the potential well in one layer is also influenced by the optical potentials of the other two layers. Therefore, the approximate harmonic oscillator model for the quantum well in each layer should also be supplemented with the contribution from the other two layers. For the B-layer, we consider the potential well whose minimum is at (0, 0, 0). For the A-layer, we consider the potential well whose minimum is at $(a/\sqrt{3},0,z_{A})$. For the C-layer, we consider the potential well whose minimum is at $(-a/\sqrt{3},0,z_{C})$.

First, we consider the potential well of the A-layer at $(a/\sqrt{3},0,z_{A})$. It is easy to see that $U_{A}(a/\sqrt{3},0,z_{A})=-9V_{A}$ and $U_{B}(a/\sqrt{3},0,z_{A})=U_{C}(a/\sqrt{3},0,z_{A})=0$. Therefore, in contrast to $U_{A}$ which attains its minimum, $U_{B}$ and $U_{C}$ both attain their maxima at $(a/\sqrt{3},0,z_{A})$. To get an approximate model close to $(a/\sqrt{3},0,z_{A})$, we expand the optical dipole potentials of all three layers into the polynomials of the relative coordinates ${\tilde{x}}=x-a/\sqrt{3}$, ${\tilde{y}}=y$, and ${\tilde{z}}=z-z_{A}$. To the leading order of these relative coordinates, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} U_A({\tilde{x}},{\tilde{y}},{\tilde{z}})\simeq -9V_{A}+\frac{8\pi ^{2}V_{A}}{a^{2}}({\tilde{x}}^{2}+{\tilde{y}}^{2}) +\frac{18V_{A}}{\rho _{0z}^{2}}{\tilde{z}}^{2}, \\ U_B({\tilde{x}},{\tilde{y}},{\tilde{z}})\simeq -\frac{4\pi ^{2}V_{B}}{a^{2}}e^{-2z_{A}^{2}/\rho _{0z}^{2}}({\tilde{x}}^{2}+{\tilde{y}}^{2}), \\ U_C({\tilde{x}},{\tilde{y}},{\tilde{z}})\simeq -\frac{4\pi ^{2}V_{C}}{a^{2}}e^{-2(z_{A}-z_{C})^{2}/\rho _{0z}^{2}}({\tilde{x}}^{2}+{\tilde{y}}^{2}). \\ \end{array}\right. } \end{aligned}$$

(D6)

Therefore, for a potential well of $U_{A}$, the presence of $U_{B}$ and $U_{C}$ reduces the strength of the harmonic oscillator in the xy plane from $\frac{8\pi ^{2}V_{A}}{a^{2}}$ to

$$\begin{aligned} \frac{8\pi ^{2}V_{A}}{a^{2}}\left[ 1-\frac{V_{B}}{2V_{A}}e^{-2z_{A}^{2}/\rho _{0z}^{2}} -\frac{V_{C}}{2V_{A}}e^{-2(z_{A}-z_{C})^{2}/\rho _{0z}^{2}}\right] \equiv \frac{8\pi ^{2}{\bar{V}}_{A}}{a^{2}},\nonumber \\ \end{aligned}$$

(D7)

where we have defined the reduced parameter ${\bar{V}}_{A}$. To the leading order of ${\tilde{z}}$, the harmonic oscillator along the z direction is unchanged. Effectively, the presence of $U_B$ and $U_C$ enhances the relative strength of the approximate harmonic confinement along the z direction. The ground state energy of this anisotropic harmonic oscillator model is

$$\begin{aligned} \varepsilon _{0A}=-9V_{A}+\hbar \omega _{xyA}+\frac{1}{2}\hbar \omega _{zA}, \end{aligned}$$

(D8)

where the angular frequencies are

$$\begin{aligned} \omega _{xyA}=\frac{4\pi }{a}\sqrt{\frac{{\bar{V}}_{A}}{m}}, \; \omega _{zA}=\frac{6}{\rho _{0z}}\sqrt{\frac{V_{A}}{m}}. \end{aligned}$$

(D9)

The analysis for the B layer and the C layer are the same. For the potential well of the B layer centering at (0, 0, 0), the effective anisotropic harmonic oscillator potential is

$$\begin{aligned} U_B({\tilde{x}},{\tilde{y}},{\tilde{z}})\simeq -9V_{B}+\frac{8\pi ^{2}{\bar{V}}_{B}}{a^{2}}({\tilde{x}}^{2}+{\tilde{y}}^{2}) +\frac{18V_{B}}{\rho _{0z}^{2}}{\tilde{z}}^{2},\nonumber \\ \end{aligned}$$

(D10)

where the relative coordinates are simply ${\tilde{x}}=x$, ${\tilde{y}}=y$, and ${\tilde{z}}=z$. The reduced oscillator strength in the xy plane is characterized by

$$\begin{aligned} {\bar{V}}_{B}=V_{B}\left[ 1-\frac{V_{A}}{2V_{B}}e^{-2z_{A}^{2}/ \rho _{0z}^{2}}-\frac{V_{C}}{2V_{B}}e^{-2z_{C}^{2}/\rho _{0z}^{2}}\right] .\quad \end{aligned}$$

(D11)

The ground state energy of this anisotropic harmonic oscillator model is

$$\begin{aligned} \varepsilon _{0B}=-9V_{B}+\hbar \omega _{xyB}+\frac{1}{2}\hbar \omega _{zB}, \end{aligned}$$

(D12)

where the angular frequencies are

$$\begin{aligned} \omega _{xyB}=\frac{4\pi }{a}\sqrt{\frac{{\bar{V}}_{B}}{m}}, \omega _{zB}=\frac{6}{\rho _{0z}}\sqrt{\frac{V_{B}}{m}}. \end{aligned}$$

(D13)

For the potential well of the C layer centering at $(-a/\sqrt{3},0,z_{C})$, the effective anisotropic harmonic oscillator potential is

$$\begin{aligned} U_C({\tilde{x}},{\tilde{y}},{\tilde{z}})\simeq -9V_{C}+\frac{8\pi ^{2}{\bar{V}}_{C}}{a^{2}}({\tilde{x}}^{2}+{\tilde{y}}^{2}) +\frac{18V_{C}}{\rho _{0z}^{2}}{\tilde{z}}^{2}, \quad \quad \end{aligned}$$

(D14)

where the relative coordinates are ${\tilde{x}}=x+a/\sqrt{3}$, ${\tilde{y}}=y$, and ${\tilde{z}}=z-z_{C}$. The reduced oscillator strength in the xy-plane is characterized by

$$\begin{aligned} {\bar{V}}_{C}=V_{C}[1-\frac{V_{A}}{2V_{C}}e^{-2(z_{C}-z_{A})^{2}/\rho _{0z}^{2}}-\frac{V_{B}}{2V_{C}}e^{-2z_{C}^{2}/\rho _{0z}^{2}}].\nonumber \\ \end{aligned}$$

(D15)

The ground state energy of this anisotropic harmonic oscillator model is

$$\begin{aligned} \varepsilon _{0C}=-9V_{C}+\hbar \omega _{xyC}+\frac{1}{2}\hbar \omega _{zC}, \end{aligned}$$

(D16)

where the angular frequencies are

$$\begin{aligned} \omega _{xyC}=\frac{4\pi }{a}\sqrt{\frac{{\bar{V}}_{C}}{m}}, \omega _{zC}=\frac{6}{\rho _{0z}}\sqrt{\frac{V_{C}}{m}}. \end{aligned}$$

(D17)

The wave function of the lowest bound state is written generally as

$$\begin{aligned} \psi _{0}({\mathbf {r}}-{\mathbf {r}}_{0})=N_{0}e^{-\frac{\alpha _{xy}^{2}}{2}[(x-x_{0})^{2}+(y-y_{0})^{2}] -\frac{\alpha _{z}^{2}}{2}(z-z_{0})^{2}},\nonumber \\ \end{aligned}$$

(D18)

where $\alpha _{xy}=\sqrt{m\omega _{xy}/\hbar }$, $\alpha _{z}=\sqrt{m\omega _{z}/\hbar }$, and the normalization factor $N_{0}=\alpha _{xy}\sqrt{\alpha _{z}}/\pi ^{\frac{3}{4}}$. m, $a=\frac{2\lambda }{3}$, and $\rho _{0z}$ are the same for all three triangular layers. The angular frequencies are however dependent on the layer index through the above definitions. Correspondingly, according to the position of ${\mathbf {r}}_{0}$ and ${\mathbf {r}}_{1}$, we should put the layer index (A, B, or C) to $\omega _{xy}$, $\omega _{z}$, and other related quantities.

We are now ready to evaluate the hopping amplitudes. For the single-body Hamiltonian defined by Eq. (D2) and the local orbital defined by Eq. (D18), the NN hopping amplitudes defined by Eq. (D5) may be calculated analytically. For the displaced dice lattice, there are six independent NN hopping amplitudes. For the intralayer NN hopping amplitude in the B layer, we take ${\mathbf {r}}_{0}=(0,0,0)$ and ${\mathbf {r}}_{1}=(\frac{\sqrt{3}}{2},\frac{1}{2},0)a$. The result for the integral is

$$\begin{aligned}& t_{NN}\left( BB\right) = -\frac{\hbar ^{2}}{2m}\left( \frac{1}{4}\alpha _{xyB}^{4}a^{2}-\alpha _{xyB}^{2}-\frac{1}{2} \alpha _{zB}^{2}\right) \exp \left( -\frac{1}{4}\alpha _{xyB}^{2}a^{2}\right) \nonumber \\&\quad -3\left\{ \left[ 1-\frac{2}{3}\exp \left( -\frac{b^{2}}{4\alpha _{xyB}^{2}}\right) \right] V_{B}+\left[ 1+\frac{1}{3}\exp \left( -\frac{b^{2}}{4\alpha _{xyB}^{2}}\right) \right] \cdot \right. \nonumber \\&\quad \cdot \left[ V_{A}\exp \left( -\frac{2\alpha _{zB}^{2}\rho _{0z}^{2}}{2+\alpha _{zB}^{2}\rho _{0z}^{2}}\frac{z_{A}^{2}}{\rho _{0z}^{2}}\right) \right. \nonumber \\&\quad \left. \left. +V_{C}\exp \left( -\frac{2\alpha _{zB}^{2}\rho _{0z}^{2}}{2+\alpha _{zB}^{2}\rho _{0z}^{2}}\frac{z_{C}^{2}}{\rho _{0z}^{2}}\right) \right] \right\} \cdot \nonumber \\&\quad \cdot \sqrt{\frac{\alpha _{zB}^{2}\rho _{0z}^{2}}{2+\alpha _{zB}^{2}\rho _{0z}^{2}}}\exp \left( -\frac{1}{4}\alpha _{xyB}^{2}a^{2}\right) . \end{aligned}$$

(D19)

A crucial qualitative feature of the result is the existence of a common exponential factor with the exponent proportional to the square of a, which is the distance between ${\mathbf {r}}_{0}$ and ${\mathbf {r}}_{1}$. Because the three layers of the displaced dice lattice are all triangular lattices and their relative positions are similar, we could get the NN hopping amplitudes within the A layer and those within the C layer by suitable substitutions of indices. Without writing out the explicit expressions, the relevant substitutions are simply

(D20)

(D21)

Therefore, the NN intralayer hopping amplitudes in all three layers have an exponential factor, whose negative exponent is proportional to $a^{2}$, the square of the distance between the NN potential wells.

Now, we calculate the three NN interlayer hopping amplitudes. For the NN hopping amplitude between the B layer and the A layer, we choose ${\mathbf {r}}_{0}=(0,0,0)$ on the B layer and ${\mathbf {r}}_{1}=(\frac{a}{\sqrt{3}},0,z_{A})$ on the A layer. The complete expression for the hopping amplitude is too lengthy and so we split it into four parts

$$\begin{aligned} t_{NN}(BA)=t^{K}_{BA}+t^{A}_{BA}+t^{B}_{BA}+t^{C}_{BA}, \end{aligned}$$

(D22)

where the first term is the matrix element of the kinetic energy term of ${\hat{H}}_{0}$, the remaining three terms are related separately to the three components of the potential energy term of ${\hat{H}}_{0}$. In addition, we define the following abbreviations for the composite quantities

(D23)

When $V_{A}$ and $V_{B}$ are at most slightly different (i.e., $V_{A}\simeq V_{B}$), which is the case that we focus on, $\eta _{21}\simeq 1$, $\zeta _{21}\simeq 1$, $\alpha _{xy21}\simeq \alpha _{xyB}^{2}$, and $\alpha _{z21}\simeq \alpha _{zB}^{2}$. The results for the four terms of $t_{NN}(BA)$ are

$$\begin{aligned} t^{K}_{BA}= & {} -\frac{\hbar ^{2}}{2m}\eta _{21}\sqrt{\zeta _{21}} \left\{ \frac{\left( \eta _{21}\alpha _{xy21}\right) ^{2}}{4}\left( \frac{a}{\sqrt{3}}\right) ^{2} \right. \nonumber \\&\left. +\frac{\left( \zeta _{21}\alpha _{z21}\right) ^{2}}{4}z_{A}^{2}-\eta _{21}\alpha _{xy21}-\frac{\zeta _{21}\alpha _{z21}}{2}\right\} \cdot \nonumber \\&\cdot \exp \left[ -\frac{\eta _{21}\alpha _{xy21}}{4}\left( \frac{a}{\sqrt{3}}\right) ^{2}-\frac{\zeta _{21}\alpha _{z21}}{4}z_{A}^{2}\right] . \end{aligned}$$

(D24)

$$\begin{aligned} t^{A}_{BA}= & {} -3V_{A}\eta _{21}\sqrt{\frac{2\alpha _{z21}\rho _{0z}^{2}}{4+\left( \alpha _{zB}^{2}+\alpha _{zA}^{2}\right) \rho _{0z}^{2}}}\cdot \nonumber \\&\cdot \left\{ 1+\frac{2}{3}e^{\frac{-b^{2}}{2\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) }} \left[ \cos \frac{4\pi \alpha _{xyB}^{2}}{3\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) } \right. \right. \nonumber \\&\left. \left. +2\cos \frac{2\pi \alpha _{xyB}^{2}}{3\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) }\right] \right\} \cdot \\&\cdot \exp \left[ -\frac{\eta _{21}\alpha _{xy21}}{4}\left( \frac{a}{\sqrt{3}}\right) ^{2} -\frac{\left( 4+\alpha _{zA}^{2}\rho _{0z}^{2}\right) \alpha _{zB}^{2}}{8+2\left( \alpha _{zB}^{2}+\alpha _{zA}^{2}\right) \rho _{0z}^{2}}z_{A}^{2}\right] .\end{aligned}$$

(D25)

$$\begin{aligned} t^{B}_{BA}= & {} t^{A}_{BA}|_{V_{A}\rightarrow V_{B},\alpha _{xyA}\leftrightarrow \alpha _{xyB},\alpha _{zA}\leftrightarrow \alpha _{zB}}. \end{aligned}$$

(D26)

$$\begin{aligned} t^{C}_{BA}= & {} -3V_{C}\eta _{21}\sqrt{\frac{2\alpha _{z21}\rho _{0z}^{2}}{4+\left( \alpha _{zB}^{2}+\alpha _{zA}^{2}\right) \rho _{0z}^{2}}}\cdot \nonumber \\&\cdot \left\{ 1+\frac{2}{3}e^{\frac{-b^{2}}{2\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) }} \left[ \cos \frac{2\pi \left( \alpha _{xyB}^{2}-\alpha _{xyA}^{2}\right) }{3\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) } \right. \right. \nonumber \\&\left. +2\cos \frac{2\pi \left( \alpha _{xyB}^{2}+2\alpha _{xyA}^{2}\right) }{3\left( \alpha _{xyB}^{2}+\alpha _{xyA}^{2}\right) }\right] \}\cdot \nonumber \\&\cdot \exp \left[ -\frac{\eta _{21}\alpha _{xy21}}{4}\left( \frac{a}{\sqrt{3}}\right) ^{2} \right. \nonumber \\&\left. -\frac{\left( \alpha _{z21}\rho _{0z}z_{A}\right) ^{2}+4\left( \alpha _{zB}z_{C}\right) ^{2}+4\alpha _{zA}^{2}\left( z_{C}-z_{A}\right) ^{2}}{8+2\left( \alpha _{zB}^{2}+\alpha _{zA}^{2}\right) \rho _{0z}^{2}}\right] .\nonumber \end{aligned}$$

(D27)

In comparison with the three NN intralayer hopping amplitudes, the exponential factors have two changes. Firstly, the in-plane exponent is now proportional to $(\frac{a}{\sqrt{3}})^{2}$ instead of $a^{2}$. Secondly, there is additional exponential decay associated with the difference in the z-coordinates (i.e., depending on $z_{A}$ and $z_{C}$).

For the NN hopping amplitude between the B layer and the C layer, we choose ${\mathbf {r}}_{0}=(0,0,0)$ on the B layer and ${\mathbf {r}}_{1}=(-\frac{a}{\sqrt{3}},0,z_{C})$ on the C layer. The complete expression for the hopping amplitude is again split into four parts

$$\begin{aligned} t_{NN}(BC)=t^{K}_{BC}+t^{A}_{BC}+t^{B}_{BC}+t^{C}_{BC}, \end{aligned}$$

(D28)

where the first term is the matrix element of the kinetic energy term of ${\hat{H}}_{0}$, the remaining three terms are related separately to the three components of the potential energy term of ${\hat{H}}_{0}$. The four terms of $t_{NN}(BC)$ turn out to be very similar in expression to the four terms of $t_{NN}(BA)$, and could be obtained from the expressions shown above by the following substitutions

$$\begin{aligned} {\left\{ \begin{array}{ll} t^{K}_{BC}=t^{K}_{BA}|_{A\rightarrow C}, \\ t^{A}_{BC}=t^{C}_{BA}|_{A\rightarrow C,C\rightarrow A}, \\ t^{B}_{BC}=t^{B}_{BA}|_{A\rightarrow C}, \\ t^{C}_{BC}=t^{A}_{BA}|_{A\rightarrow C}. \\ \end{array}\right. } \end{aligned}$$

(D29)

From this similarity, $t_{NN}(BC)$ show similar dependencies as $t_{NN}(BA)$ on the parameters of the optical dipole potential.

Finally, for the NN hopping amplitude between the A layer and the C layer, we choose ${\mathbf {r}}_{0}=(\frac{a}{\sqrt{3}},0,z_{A})$ on the A layer and ${\mathbf {r}}_{1}=(\frac{a}{2\sqrt{3}},\frac{a}{2},z_{C})$ on the C layer. The complete expression for the hopping amplitude is also split into four parts

$$\begin{aligned} t_{NN}(AC)=t^{K}_{AC}+t^{A}_{AC}+t^{B}_{AC}+t^{C}_{AC}, \end{aligned}$$

(D30)

where the first term is the matrix element of the kinetic energy term of ${\hat{H}}_{0}$, the remaining three terms are related separately to the three components of the potential energy term of ${\hat{H}}_{0}$. These four terms may also be obtained from the four terms for $t_{NN}(BA)$ by substitution of parameters as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} t^{K}_{AC}=t^{K}_{BA}|_{B\rightarrow C,z_{A}\rightarrow z_{A}-z_{C}}, \\ t^{A}_{AC}=t^{A}_{BA}|_{B\rightarrow C,z_{A}\rightarrow z_{A}-z_{C}}, \\ t^{C}_{AC}=t^{A}_{AC}|_{A\leftrightarrow C}=t^{A}_{BA}|_{B\rightarrow A, A\rightarrow C,z_{A}\rightarrow z_{A}-z_{C}}, \\ t^{B}_{AC}=t^{C}_{BA}|_{V_{C}\rightarrow V_{B},\alpha _{xyB}\rightarrow \alpha _{xyC},\alpha _{zB}\rightarrow \alpha _{zC},z_{A}\leftrightarrow z_{A}-z_{C}}. \\ \end{array}\right. } \end{aligned}$$

(D31)

In comparison with $t_{NN}(BA)$ and $t_{NN}(BC)$ whose dominant terms have an exponent proportional to $-z_{A}^{2}$ and $-z_{C}^{2}$, $t_{NN}(AC)$ is exponentially smaller because the dominant terms have an exponent proportional to $-(z_{A}-z_{C})^{2}$. For example, $(z_{A}-z_{C})^{2}=4z_{A}^{2}$ in the special case $z_{A}=-z_{C}$, which amount to a fourth-power decay in the hopping amplitude. On the other hand, we have shown that the interlayer NN hopping amplitudes have an exponent proportional to $(\frac{a}{\sqrt{3}})^{2}$ which is $\frac{1}{3}$ of the corresponding exponent for the three intralayer NN hopping amplitudes. This shows that the intralayer NN hopping amplitudes have a component that is third power smaller than the corresponding factor for the interlayer NN hopping amplitudes. Combining these two trends, it is possible to tune $z_{A}$ and $-z_{C}$ to certain medium values, so that all the four kinds of hopping amplitudes that are neglected in the tight-binding model are much smaller (e.g., smaller by about one to two orders of magnitude) than $t_{NN}(BA)$ and $t_{NN}(BC)$ that are retained in the model.

As an order-of-magnitude comparison among the six NN hopping amplitudes, let us set $V_{A}=V_{B}=V_{C}$ and $z_{A}=-z_{C}=z_{0}$. For simplicity, we also suppress the layer indices on $\alpha _{xy}$ and $\alpha _{z}$. The magnitudes of the various hopping amplitudes are determined by the dominant exponential factors therein. From the above explicit expressions, the dominant exponential factors are $\exp [-(\alpha _{xy}a)^{2}/4]$ for $t_{NN}(\alpha \alpha )$ ($\alpha =A,B,C$), are $\exp [-(\alpha _{xy}a)^{2}/12-(\alpha _{z}z_{0})^{2}/4]$ for $t_{NN}(BA)$ and $t_{NN}(BC)$, and is $\exp [-(\alpha _{xy}a)^{2}/12-(\alpha _{z}z_{0})^{2}]$ for $t_{NN}(AC)$. To justify the tight-binding model of Eq. (D1), we require the exponents for $t_{NN}(BA)$ and $t_{NN}(BC)$ to be at least two orders of magnitude larger than those for the other four NN hopping amplitudes. This amounts to

$$\begin{aligned} \exp \left[ -\frac{(\alpha _{xy}a)^{2}}{6}+\frac{(\alpha _{z}z_{0})^{2}}{4}\right] <10^{-2}, \end{aligned}$$

(D32)

and

$$\begin{aligned} \exp \left[ -\frac{3(\alpha _{z}z_{0})^{2}}{4}\right] <10^{-2}. \end{aligned}$$

(D33)

Eq. (D33) leads to $\exp [-\frac{(\alpha _{z}z_{0})^{2}}{4}]<(0.01)^{1/3}\simeq 0.2154$, and correspondingly $\exp [\frac{(\alpha _{z}z_{0})^{2}}{4}]>(100)^{1/3}\simeq 4.642$. Substituting $\exp [\frac{(\alpha _{z}z_{0})^{2}}{4}]=5$ into Eq. (D32), we get $\exp [-\frac{(\alpha _{xy}a)^{2}}{12}]<\sqrt{0.002}\simeq 0.0447$. This inequality sets a lower bound to the strength of the optical dipole potential, which turns out to be ${\bar{V}}_{\alpha }>1.004E_{R}$ or equivalently $\hbar \omega _{xy\alpha }>4.251E_{R}$ ($\alpha =A,B,C$). This constraint is weaker than that inferred in “Appendix B” for the 2D limit. It should always be satisfied for the considered deep lattices. For known strengths of the optical dipole potential and the $\rho _{0z}$ parameter, $\exp [\frac{(\alpha _{z}z_{0})^{2}}{4}]\simeq 5$ or larger may easily be fulfilled by tuning $z_{0}$.

Suppose we have tuned the parameters so that the BA and BC NN interlayer hopping amplitudes dominate among the six NN hopping amplitudes. The tight-binding model has four parameters [$\varepsilon _{ab}$, $\varepsilon _{cb}$, $t_{ba}=t_{NN}(BA)$, and $t_{bc}=t_{NN}(BC)$]. Correspondingly, we also have control over four parameters of the optical dipole potential ($V_{A}/V_{B}$, $V_{C}/V_{B}$, $z_{A}-z_{B}=z_{A}$, $z_{C}-z_{B}=z_{C}$). Therefore, we may tune the four parameters in the tight-binding model freely within the scope of applicability of the model.

We mention in passing that, all the tight-binding parameters obtained above may be reexpressed in terms of strengths of the optical potential (i.e., $V_{i}$, $i=A,B,C$), the geometric parameters (i.e., $\rho _{0z}$ and $z_{i}$, $i=A,B,C$) measured in units of the wavelength $\lambda$, and the recoil energy $E_{R}$. In particular, it is conventional to express all the energies in units of the recoil energy. The conversion is easy to make through the following relations:

(D34)

where $\gamma _{0}=\rho _{0z}/\lambda$, $i=A,B,C$. We will not write out the explicit expressions of the tight-binding parameters in terms of this alternative parameter set.