Simulating the Haldane phase in trapped-ion spins using optical fields

We propose to experimentally explore the Haldane phase in spin-one XXZ antiferromagnetic chains using trapped ions. We show how to adiabatically prepare the ground states of the Haldane phase, demonstrate their robustness against sources of experimental noise, and propose ways to detect the Haldane ground states based on their excitation gap and exponentially decaying correlations, nonvanishing nonlocal string order, and doubly degenerate entanglement spectrum.

Figure 1

Phase diagram of the spin-one XXZ antiferromagnetic (AFM) Hamiltonian [Eq. (1)]. The first derivation in this text (as well as the previous derivation [40]) cover $0\le \lambda \le 2$ (in blue), whereas the second derivation in this text covers the whole positive $\lambda \ge 0$ plane (in blue and red).

Figure 2

${}^{171}{\mathrm{Yb}}^{+}$ ground states used for modeling the spin-one particle (in green). The beat frequencies between non-co-propagating Raman beams (in red) drive the $\delta$ detuned transitions between $\left|-1\right.〉\to \left|0\right.〉$ and $\left|1\right.〉\to \left|0\right.〉$ (in blue), generating the spin-laser interaction in Eq. (2).

Figure 3

Generating the $D$ term. Generating the $D$ term in Eq. (1) is done in two ways: (1) (a) Applying two additional co-propagating Raman beams, corresponding to a ${\mathrm{\Delta }}_D$ detuned transition between $\left|0^{\prime }\right.〉⟷\left|0\right.〉$ results in an ac Stark shift $\frac{{\mathrm{\Omega }}_D^2}{4{\mathrm{\Delta }}_D}\left|0\right.〉\left.〈0\right|$. This could alternatively be generated using a ${\mathrm{\Delta }}_D$ detuned microwave driving field. (2) (b) Within the non-co-propagating Raman beams (red) (Fig. 2), we regard the $\left|0\right.〉$ state as $D^{\prime }$ shifted, and move to the interaction picture with respect to the $D^{\prime }$-shifted bare energy structure. Thus, we are left with $-D^{\prime }\left|0\right.〉\left.〈0\right|$, which is $D^{\prime }{F}_z^2$. In our later derivation (which covers the entire $\lambda \ge 0$ plane), we will impose a $2D^{\prime }$ detuning rather than a $D^{\prime }$ detuning.

Figure 4

Generating ${\mathrm{\Omega }}^{\prime }{F}_{z}cos\theta$ for the $\lambda$-generating trick. There are two ways of generating this term: (1) (a) Additional copropagating Raman beams (red) perform the $\pm {\mathrm{\Delta }}_z$ detuned transitions between $\left|\pm 1\right.〉\leftrightarrow \left|0\right.〉$, respectively, with a Rabi frequency of ${\mathrm{\Omega }}_z$. These effective transitions can alternatively be realized using detuned microwave driving fields (blue). Therefore, we obtain the desired term from the ac Stark shifts where ${\mathrm{\Omega }}^{\prime }cos\theta ={\mathrm{\Omega }}_z^2/{\mathrm{\Delta }}_z$. (2) (b) Adding detunings to the non-co-propagating Raman beams (red) (Fig. 2), such that ${\delta }^{\prime }=\delta +{\mathrm{\Omega }}^{\prime }cos\theta$. In this way we are left with the desired term, after moving to the interaction picture with respect to the detuned bare energy structure.

Figure 5

Generating ${\mathrm{\Omega }}^{\prime }{F}_{\alpha }sin\theta$ for the $\lambda$-generating trick. (a) We apply carrier transitions using additional co-propagating Raman beams (red), or alternatively microwave driving fields (blue), while keeping the imposed detunings for generating the D term [Fig. 3] and the ${\mathrm{\Omega }}^{\prime }{F}_{z}cos\theta$ term [Fig. 4]. Here, ${\mathrm{\Omega }}_{\mathrm{car}}={\mathrm{\Omega }}^{\prime }sin\theta \sqrt{2}$, and $\alpha ={\alpha }_1-{\alpha }_{-1}$ is determined by the initial phase difference of the two effective transitions. The same carrier transitions, only without detunings, are used in our later approach as dressing fields that transform the system into the dressed state basis (b), thus protecting it from the magnetic noise.

Figure 6

Adiabatic path using the symmetry breaking perturbation. When crossing a second-order phase transition the energy gap is closed, and the adiabatic approximation is invalid. We break the symmetries of the Hamiltonian [Eq. (1)] and thus, go around the phase transition, keeping a finite energy gap during the whole path.

Figure 7

Generating ${\mathrm{\Omega }}^{\prime }{S}_{\alpha }sin\theta$ for the $\lambda$-generating trick. There are two ways to realize this term of the $\lambda$-generating trick: (1) (a) to engineer ${\mathrm{\Omega }}^{\prime }{S}_{y}sin\theta$, we apply additional co-propagating laser Raman beams, which perform the two Raman transitions between $\left|\mp 1\right.〉\leftrightarrow \left|0\right.〉$ with $\pm {\delta }_{\lambda }=\pm \left(\frac{{\mathrm{\Omega }}_{\mathrm{car}}}{\sqrt{2}}-{\mathrm{\Omega }}^{\prime }cos\theta \right)$ detunings, $\pm \frac{\pi }{2}$ initial phases, and the same effective Rabi frequency ${\mathrm{\Omega }}_y=\sqrt{2}{\mathrm{\Omega }}^{\prime }sin\theta$, respectively; (2) (b) to engineer $-{\mathrm{\Omega }}^{\prime }sin\theta S_x$ we apply a $z$-polarized radio-wave driving field, on resonance with the dressed-state energy structure [Fig. 5 (b)], and with a Rabi frequency of ${\mathrm{\Omega }}_z=2{\mathrm{\Omega }}^{\prime }sin\theta$. Regardless of method, we obtain the desired term by moving to the rotating frame of the dressed energy structure, and using RWA while assuming ${\mathrm{\Omega }}^{\prime }\ll \sqrt{2}\mathrm{\Omega }$.