Topological Spin Models in Rydberg Lattices

Abstract

We show that resonant dipole–dipole interactions between Rydberg atoms in a triangular lattice can give rise to artificial magnetic fields for spin excitations. We consider the coherent dipole–dipole coupling between np and ns Rydberg states and derive an effective spin-1/2 Hamiltonian for the np excitations. By breaking time-reversal symmetry via external fields, we engineer complex hopping amplitudes for transitions between two rectangular sub-lattices. The phase of these hopping amplitudes depends on the direction of the hop. This gives rise to a staggered, artificial magnetic field which induces non-trivial topological effects. We calculate the single-particle band structure and investigate its Chern numbers as a function of the lattice parameters and the detuning between the two sub-lattices. We identify extended parameter regimes where the Chern number of the lowest band is $C=1$ or $C=2$.

This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.

Appendices

Appendix 1: Dipole Matrix Elements

We evaluate the matrix elements of the electric-dipole-moment operator $\widehat{\boldsymbol{d}}$ of an individual atom via the Wigner–Eckert theorem [64, 65] and find

$$ \left\langle n{l}_{j^{\prime}}^{\prime }{m}^{\prime }|\widehat{\boldsymbol{d}}|n{l}_jm\right\rangle \kern0.5em =\mathcal{D}\sum \limits_{q=-1}^1{C}_{jm1q}^{j^{\prime }{m}^{\prime }}{\boldsymbol{\epsilon}}_q, $$

(23)

where ${C}_{jm1q}^{j^{\prime }{m}^{\prime }}$ are Clebsch–Gordan coefficients and the spherical unit vectors ${\boldsymbol{\epsilon}}_q$ in Eq. (23) are defined as

$$ {\epsilon}_1=-\frac{{\mathbf{e}}_x-i{\mathbf{e}}_y}{\sqrt{2}},\kern1em {\epsilon}_0={\mathbf{e}}_z,\kern1em {\epsilon}_{-1}=\frac{{\mathbf{e}}_x+i{\mathbf{e}}_y}{\sqrt{2}}. $$

(24)

The reduced dipole matrix element is [64, 65]

$$ {\displaystyle \begin{array}{cc}\hfill \mathcal{D}=& {\left(-1\right)}^{\mathrm{j}+{\mathrm{l}}^{\prime }-1/2}\sqrt{2j+1}\sqrt{2l+1}\hfill \\ {}\hfill & \left\{\begin{array}{ccc}{l}^{\prime }& l& 1\\ {}j& {j}^{\prime }& 1/2\end{array}\right\}{C}_{10l0}^{l^{\prime }0}e\left\langle {n}^{\prime }{l}^{\prime }|r| nl\right\rangle, \hfill \end{array}} $$

(25)

where the $3\times 2$ matrix in curly braces is the Wigner $6-j$ symbol, e is the elementary charge, and $\left\langle {n}^{\prime }{l}^{\prime }|r| nl\right\rangle$ is a radial matrix element.

Appendix B: Rubidium Parameters

Here we calculate the strength of the dipole–dipole interaction for rubidium atoms and estimate the magnitude of the level shifts required for realizing our model. For $n{s}_{1/2}\leftrightarrow n{p}_{3/2}$ transitions in rubidium with principal quantum number $n=70$, the reduced dipole moment $\mathcal{D}$ in Eq. (25) is given by

$$ \mathcal{D}\approx 2909e{a}_0, $$

(26)

where e is the elementary charge and ${a}_0$ is the Bohr radius. It follows that the strength of the dipole–dipole coupling ${V}_0$ in Eq. (7) for $a=20\kern0.166667em \upmu \mathrm{m}$ is

$$ {V}_0/\hbar \approx 2\pi \times 1.03\kern0.166667em \mathrm{MHz}. $$

(27)

The lifetime of the $n{s}_{1/2}$ and $n{p}_{3/2}$ states at temperature $T=300\mathrm{K}$ and for $n=70$ is ${T}_{\mathrm{s}}\approx 151.6\kern0.166667em \upmu \mathrm{s}$ and ${T}_{\mathrm{p}}\approx 191.3\kern0.166667em \upmu \mathrm{s}$, respectively [66]. Note that these values take into account the lifetime reduction due to blackbody radiation. The hopping rates vary with the lattice parameters but are typically of the order of ${V}_0$. It follows that in principle many coherent hopping events can be observed before losses due to spontaneous emission set in. This finding is consistent with the experimental observations in Ref. [42]. Note that the magnitude of ${V}_0$ can be increased by reducing the size of the lattice constant a or by increasing n.

Next we discuss the requirements for reducing the general Hamiltonian in Eq. (2) to our model in Eq. (14). First, we note that the level shifts induced between Zeeman sub-states must be large compared to ${V}_0$ and hence of the order of $10\kern0.166667em \mathrm{MHz}$. Shifts of this magnitude can be realized with weak magnetic fields [67] or AC stark shifts [42]. Furthermore, the fine structure splitting between the $n{s}_{1/2}$ and $n{p}_{3/2}$ manifolds is $\Delta {E}_{\mathrm{FS}}\approx 2\pi \times 10.8\kern0.166667em \mathrm{GHz}$ [68], which is much larger than ${V}_0$, and hence, it is justified to neglect off-resonant terms in Eq. (3). Finally, we note that the energy difference between the $n{p}_{3/2}$ manifold and the nearby $n{p}_{1/2}$ manifold is approximately $285\kern0.166667em \mathrm{MHz}$ [68], which is also much larger than ${V}_0$. It follows that the $n{p}_{1/2}$ states can be safely neglected.

Appendix C: k-space Hamiltonian

The k-space Hamiltonian can be obtained by considering the single-excitation sub-space ${\mathcal{E}}_1$ spanned by the basis states

$$ \mid \alpha \left\rangle ={S}_{\alpha}^{+}\mid 0\right\rangle, $$

(28)

where $\mid \alpha \Big\rangle$ denotes one p excitation at site $\alpha$ and $\mid 0\Big\rangle$ is the “vacuum” state with zero excitations, i.e. the atoms at all lattice sites are in state $\mid {s}_{1/2}1/2\Big\rangle$. In order to solve the eigenvalue equation

$$ {H}_{\mathrm{eff}}\mid \psi \left\rangle =E\mid \psi \right\rangle $$

(29)

with $\mid \psi \Big\rangle \in {\mathcal{E}}_1$, we describe the lattice in Fig. 1 by a rectangular Bravais lattice with a two-atomic basis. More specifically, the direct lattice points are given by the $\mathcal{R}$ atoms such that the basis is comprised of one $\mathcal{R}$ atom at $\mathbf{0}$ and one $\mathcal{B}$ atom at $\left(\mathbf{a}+\mathbf{b}\right)/2$. According to Bloch’s theorem [69], we can solve Eq. (29) with the Ansatz

$$ \mid \psi \left\rangle =\sum \limits_{\alpha }{u}_{\alpha}\mid \alpha \right\rangle, $$

(30)

where the coefficients ${u}_{\alpha }$ can be written as

$$ {u}_{\alpha }=\left\{\begin{array}{cc}{\psi}_{\mathcal{R}}{e}^{\mathrm{i}\mathbf{k}\cdotp \mathbf{U}\left(\alpha \right)},\hfill & \alpha \in \mathcal{R},\hfill \\ {}{\psi}_{\mathcal{B}}{e}^{\mathrm{i}\mathbf{k}\cdotp \mathbf{U}\left(\alpha \right)},\hfill & \alpha \in \mathcal{B},\hfill \end{array}\right. $$

(31)

and $\mathbf{k}$ is a point in the first Brillouin zone of the direct lattice. The vector $\mathbf{U}\left(\alpha \right)$ in Eq. (31) is the Bravais lattice point associated with site $\alpha$,

$$ \mathbf{U}\left(\alpha \right)=\left\{\begin{array}{cc}{\mathbf{R}}_{\alpha },\hfill & \alpha \in \mathcal{R},\hfill \\ {}{\mathbf{R}}_{\alpha }-\left(\mathbf{a}+\mathbf{b}\right)/2,\hfill & \alpha \in \mathcal{B}.\hfill \end{array}\right. $$

(32)

With Eqs. (30) and (31), Eq. (29) can be reduced to the following matrix equation for the amplitudes ${\psi}_{\mathcal{R}}$ and ${\psi}_{\mathcal{B}}$,

$$ \mathcal{H}\left(\mathbf{k}\right)\left(\begin{array}{c}{\psi}_{\mathcal{R}}\hfill \\ {}{\psi}_{\mathcal{B}}\hfill \end{array}\right)=E\left(\begin{array}{c}{\psi}_{\mathcal{R}}\hfill \\ {}{\psi}_{\mathcal{B}}\hfill \end{array}\right), $$

(33)

where the $2\times 2$ matrix $\mathcal{H}\left(\mathbf{k}\right)$ is the k-space Hamiltonian. We find $\mathcal{H}\left(\mathbf{k}\right)$ using the software package MATHEMATICA [70] for each set of lattice parameters a and b. In general, the resulting expressions are too complicated to display here. In the special case of nearest-neighbour interactions only, we find

$$ {\left[\mathcal{H}\left(\mathbf{k}\right)\right]}_{11}=-{V}_0\cos \left(\mathbf{k}\cdotp \mathbf{a}\right)+\hbar \Delta, $$

(34a)

$$ {\displaystyle \begin{array}{c}{\left[\mathcal{H}\left(\mathbf{k}\right)\right]}_{12}=\frac{V_0\sqrt{48}}{{\left[1+{\left(b/a\right)}^2\right]}^{3/2}}\left({e}^{-\mathrm{i}\left(\mathbf{k}\cdotp \mathbf{a}+\mathbf{k}\cdotp \mathbf{b}\right)}{e}^{-2\mathrm{i}\alpha }+{e}^{-2\mathrm{i}\alpha}\right.\hfill \\ {}\hfill \left.+{e}^{-\mathrm{i}\mathbf{k}\cdotp \mathbf{a}}{e}^{2\mathrm{i}\alpha }+{e}^{-\mathrm{i}\mathbf{k}\cdotp \mathbf{b}}{e}^{2\mathrm{i}\alpha}\right),\hfill \end{array}} $$

(34b)

$$ {\left[\mathcal{H}\left(\mathbf{k}\right)\right]}_{22}=-\frac{1}{3}{V}_0\cos \left(\mathbf{k}\cdotp \mathbf{a}\right), $$

(34c)

where $\cos \left(\alpha \right)=1/{\left[1+{\left(b/a\right)}^2\right]}^{1/2}$ and ${\left[\mathcal{H}\left(\mathbf{k}\right)\right]}_{21}={\left[\mathcal{H}\left(\mathbf{k}\right)\right]}_{12}^{\ast }$.