Bose-Hubbard physics in synthetic dimensions from interaction Trotterization

Activating transitions between a set of atomic internal states has emerged as an elegant scheme by which lattice models can be designed in ultracold atomic gases. In this approach, the internal states can be viewed as fictitious lattice sites defined along a synthetic dimension, hence offering a powerful method by which the spatial dimensionality of the system can be extended. Interparticle collisions generically lead to infinite-range interactions along the synthetic dimensions, which a priori precludes the design of Bose-Hubbard-type models featuring on-site interactions. In this paper, we solve this obstacle by introducing a protocol that realizes strong and tunable “on-site” interactions along an atomic synthetic dimension. Our scheme is based on pulsing strong intraspin interactions in a fast and periodic manner, hence realizing the desired on-site interactions in a digital (Trotterized) manner. We explore the viability of this protocol by means of numerical calculations, which we perform on various examples that are relevant to ultracold-atom experiments. This general method, which could be applied to various atomic species by means of fast-response protocols based on Fano-Feshbach resonances, opens the route for exploration of strongly correlated matter in synthetic dimensions.

Figure 1

(a) A synthetic dimension is spanned by $M$ internal states of an atom. “Hopping” matrix elements along the synthetic dimension, ${\mathrm{\Omega }}_m$, are controlled by laser coupling, while intraspin interactions $U_{mm}$ are tuned by Fano-Feshbach resonances. The scheme assumes that interspin interactions $U_{m\ne n}$ (infinite-range interactions along the synthetic dimension) can be made small compared with all other energy scales. (b) The Trotterization scheme consists in pulsing intraspin interactions in a successive and time-periodic manner: In the first step, one activates $U_{11}$ only, then $U_{22}$ only, etc. The drive frequency $\omega =2\pi /T$ is assumed to be the largest frequency in the system: $\hslash \omega \gg {\mathrm{\Omega }}_m,U_{mm}$.

Figure 2

(a) Structure of hyperfine (Zeeman) levels of ${}^7\mathrm{Li}$ atoms. We propose to use the four lowest Zeeman states as a synthetic dimension of size $M=4$. (b) Variation of the scattering length $a_{1,1}$ with the magnetic field, exhibiting a Feshbach resonance at $B=738$ G. The singlet and triplet scattering lengths $a_{\text{s}}$ and $a_{\text{t}}$ are shown as dashed and dotted lines, respectively. (c) Scheme of the permutation between internal states used to engineer an effective interaction $U_{\text{eff}}=U_{11}/M$ between atoms of identical internal state only. The red curves indicate the exchange of a pair of spin states using a resonant radio-frequency $\pi$ pulse.

Figure 3

Time evolution of the renormalized (a) $\mathrm{\Delta }{n}_D$ and (b) $\mathrm{\Delta }{O}_p$ calculated in $l=M/2$ for fixed $M=N=10$ and $U_{\text{eff}}/J_{\text{eff}}=1.0$ and different values of $\omega$ (in units of $J_{\text{eff}}$). The inset in (a) shows the quadratic growth of $\mathrm{\Delta }{n}_D$ for different periods. The times $t$ and $\tau$ are expressed in units of $1/J_{\text{eff}}$, and the frequency $\omega$ is expressed in units of $J_{\text{eff}}$.

Figure 4

Time evolution of (a) $\mathrm{\Delta }n\left(t\right)$, (b) $n_D\left(t\right)$, and (c) $O_p(M/2,t)$ for fixed $M=N=10$ and $\omega =20\pi J_{\text{eff}}$ and different values of $U_{\text{eff}}/J_{\text{eff}}$. The time $t$ is expressed in units of $1/J_{\text{eff}}$.

Figure 5

Time evolution of the rung current ${\mathcal{J}}_r$ and chiral current ${\mathcal{J}}_c$ for a synthetic ladder with a density $N/2M=0.5$ with $M=N=8$, $\omega =20\pi J_{\text{eff}}$, $\theta =0.9\pi$, and $U_{\text{eff}}/J_{\text{eff}}=4$ and different values of $J_{\text{real}}/J_{\text{eff}}$. The local rung current is evaluated using the four central sites. The time $t$ is expressed in units of $1/J_{\text{eff}}$, the currents are expressed in units of $J_{\text{eff}}$, and the chosen initial state corresponds to the ground state of Eq. (20).

Figure 6

Time evolution of the staggered magnetization $m_s$ for different values of $U_{\text{eff}}/J_{\text{eff}}$. In all the results we fixed $M=N=8$ and $\omega =20\pi J_{\text{eff}}$, and the time $t$ is expressed in units of $1/J_{\text{eff}}$. The chosen initial state corresponds, $|n_1,n_2,...,n_M\rangle =|0,2,0,2,....,0,2\rangle$; see Eq. (23). Solid curves show the dynamics associated with the Trotter sequence, while dotted curves show that generated by the effective Bose-Hubbard Hamiltonian in Eq. (8), using the same initial state.