Quantum simulation of the Haldane phase using open shell molecules

Abstract

Dipolar molecules in optical traps are a versatile platform for studying many-body phases of quantum matter in the presence of strong and long-range interactions. The dipolar interactions in such setups can be enabled by microwave driving opposite parity rotational levels of the molecules. We find that the regime where the $N=0,J=1/2,F=1$ state is coupled to the $N=1,J=3/2,F=2$ manifold with circularly polarized microwaves, in the presence of a small magnetic field, can lead to spin-1 quantum magnetic Hamiltonians, due to the decoupling between electron spin and orbit, that is unique to the $^2Σ$ ground state molecules. We demonstrate that in one dimension, the phase diagram associated with this Hamiltonian, computed via tensor network methods, hosts the celebrated Haldane phase. We find that the Haldane phase persists even in the presence of SU(3) correction terms that break the SU(2) algebra of the Hamiltonian. We discuss the feasibility of the proposed scheme for $^2Σ$ molecules with large rotational constants such as the directly laser cooled molecule MgF for future experiments.

Figures (9)

• Figure 1: (a) Schematic of the experimental setup showing diatomic dipolar molecules loaded into a one-dimensional lattice of spacing $a$. The depth of the lattice is high such that the molecules are pinned to their sites and hopping is suppressed. The molecules are subjected to circularly-polarized microwaves $\mathbf{E}$ of frequency $\omega$ produced by a microwave horn propagating in the $z$-direction, and a static magnetic field $\mathbf{B}$, also in the $z$-direction. (b) Phase diagram showing $\log_{10}P_\varrho$ as a function of Zeeman shift $\epsilon_z/V_{dd}$ and Rabi frequency $\Omega_R/V_{dd}$ (bottom left). The bottom right panel shows an expanded view of the rectangular region in the bottom left panel with rotated axes $(\epsilon_z \cos\eta + \Omega_R \sin\eta)/{V_{dd}}$ (horizontal) and $(-\epsilon_z \sin\eta + \Omega_R \cos\eta)/{V_{dd}}$ (vertical), where $\eta = \arccos(1/\sqrt{1+5.45^2})$. The blue data points correspond to the topologically non-trivial Haldane phase and the gray background, and the red data points in the bottom right panel, correspond to topologically trivial phases. In the bottom left panel, red data points have been omitted for the sake of clarity. Using the bottom left panel, the Haldane phase exists for a maximum $\epsilon_z \approx 1.1 \, V_{dd} \approx 55\,\mathrm{kHz}$ ($B \approx 40\,\mathrm{mG}$) and $\Omega_R \approx 7\,V_{dd} \approx 350 \,\mathrm{kHz}$
• Figure 2: Relevant rotational and hyperfine structure of a $^2\Sigma$ molecule. The lowest rotational state is $N=0$ and has a parity of +, while the first excited rotational state is $N=1$ of the opposite parity. The energy difference between them is $\omega_0$. The sublevel structure is denoted by the quantum numbers $\mathinner{|{J,F,m_F}\rangle}$ leading to the states depicted. The microwave drive is right circularly polarized ($\sigma^+$), is detuned from the transition by $\delta$ and leads to a Rabi frequency of the drive $\Omega_R$. Although the analysis is applicable to any $^2\Sigma$ molecule, we perform numerical simulations for the $^{24}$MgF molecule and the frequencies are indicated for this molecular species Doppelbauer_MgF_2022.
• Figure 3: Microwave dressed energy levels $\varepsilon^{(-)}_n - \omega/2$ (left) and $\varepsilon^{(+)}_n + \omega/2$ (right) in Eqs. \ref{['eq:quasienergy1']}-\ref{['eq:quasienergy3']}, as functions of the Zeeman shift $\epsilon_z$ for fixed Rabi frequency $\Omega_R = 2.45\,|\delta|$. The top (bottom) panels correspond to a blue (red) detuned microwave $\omega > \omega_0$ ($\omega < \omega_0$).
• Figure 4: Phase diagram of $H_{\text{eff}}$ in Eq. \ref{['eq:H_eff']} as a function of $J_{zz}/|J_{xx}|$ and $D/|J_{xx}|$ for $h^z_i = 0$ (left panels) and $h^z_i = 0.1 J_{xx}$ (right panels), with $J_{xx} < 0$ and $\mathbb V_{\text{SU(3)}} = 0$ in both cases. The colormaps show the string order parameters $\mathcal{S}^x_{|i-j|}$ (bottom panels) and $\mathcal{S}^z_{|i-j|}$ (top panels) obtained using finite DMRG with a chain size $L = 100$ over a distance of $j - i = 80$, maximum bond dimension $\chi_{\text{max}} = 100$ and the maximum interaction range of 4. Regions I, II, III and IV in (a) denote the experimentally accessible regions in the phase diagram. Table \ref{['tab:ExpParams']} summarizes the experimental parameter values for detuning and Zeeman shift, and the initial state of the dipolar molecules required to access these regions.
• Figure 5: Phase diagram of $H_{\text{eff}}$ in Eq. \ref{['eq:H_eff']} as a function of $J_{zz}/|J_{xx}|$ and $D/|J_{xx}|$ for $h^z_i = 0$ (left panels) and $h^z_i = 0.1 J_{xx}$ (right panels), with $J_{xx} < 0$ and $\mathbb V_{\text{SU(3)}} = 0$ in both cases. The colormaps show the entanglement entropy $S$ (top panels) and $P_\varrho$ (bottom panels). The plots are obtained with finite DMRG with the same set of parameters as Fig. \ref{['fig:SOP']}.