Analog quantum simulation of generalized Dicke models in trapped ions

We propose the analog quantum simulation of generalized Dicke models in trapped ions. By combining bicromatic laser interactions on multiple ions we can generate all regimes of light-matter coupling in these models, where here the light mode is mimicked by a motional mode. We present numerical simulations of the three-qubit Dicke model both in the weak field (WF) regime, where the Jaynes-Cummings behavior arises, and the ultrastrong coupling (USC) regime, where a rotating-wave approximation cannot be considered. We also simulate the two-qubit biased Dicke model in the WF and USC regimes and the two-qubit anisotropic Dicke model in the USC regime and the deep-strong coupling regime. The agreement between the mathematical models and the ion system convinces us that these quantum simulations can be implemented in the laboratory with current or near-future technology. This formalism establishes an avenue for the quantum simulation of many-spin Dicke models in trapped ions.

Figure 1

Simulation of the Dicke model with three trapped ions in the WF regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\delta }^r=0$, and ${\delta }^b=-2\pi \times 125\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow ,\downarrow \rangle$. Dotted magenta lines: Dicke model; dashed blue lines: ion system.

Figure 2

Simulation of the Dicke model with three trapped ions in the USC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\delta }^r=-2\pi \times 100\text{Hz}$, and ${\delta }^b=-2\pi \times 2.7\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow ,\downarrow \rangle$. Dotted magenta lines: Dicke model; dashed blue lines: ion system.

Figure 3

Simulation of the biased Dicke model in a two trapped-ion system in the WF regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,h=\mathrm{\Omega }\eta /2,{\delta }^r=0$, and ${\delta }^b=-2\pi \times 125\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: biased Dicke model; dashed blue lines: ion system.

Figure 4

Simulation of the biased Dicke model in a two trapped-ion system in the WF regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,h=5\mathrm{\Omega }\eta /2,{\delta }^r=0$, and ${\delta }^b=-2\pi \times 125\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: biased Dicke model; dashed blue lines: ion system.

Figure 5

Simulation of the biased Dicke model in a two trapped-ion system in the USC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,h=\mathrm{\Omega }\eta /2,{\delta }^r=-2\pi \times 100\text{Hz}$, and ${\delta }^b=-2\pi \times 2.7\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: biased Dicke model; dashed blue lines: ion system.

Figure 6

Simulation of the biased Dicke model in a two trapped-ion system in the USC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\mathrm{\Omega }=2\pi \times 50\text{kHz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,h=5\mathrm{\Omega }\eta /2,{\delta }^r=-2\pi \times 100\text{Hz}$, and ${\delta }^b=-2\pi \times 2.7\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: biased Dicke model; dashed blue lines: ion system.

Figure 7

Simulation of the anisotropic Dicke model in a two trapped-ion system in the USC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\mathrm{\Omega }}^r=2\pi \times 50\text{kHz},s=3,{\delta }^r=-2\pi \times 100\text{Hz}$, and ${\delta }^b=-2\pi \times 7.7\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: anisotropic Dicke model; dashed blue lines: ion system.

Figure 8

Simulation of the anisotropic Dicke model in a two trapped-ion system in the USC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\mathrm{\Omega }}^r=2\pi \times 50\text{kHz},s=5,{\delta }^r=-2\pi \times 100\text{Hz}$, and ${\delta }^b=-2\pi \times 12.7\text{kHz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: anisotropic Dicke model; dashed blue lines: ion system.

Figure 9

Simulation of the anisotropic Dicke model in a two trapped-ion system in the DSC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\mathrm{\Omega }}^r=2\pi \times 50\text{kHz},s=3,{\delta }^r=-2\pi \times 112\text{Hz}$, and ${\delta }^b=-2\pi \times 7238\text{Hz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: anisotropic Dicke model; dashed blue lines: ion system.

Figure 10

Simulation of the anisotropic Dicke model in a two trapped-ion system in the DSC regime with parameters $\nu =2\pi \times 3\text{MHz},{\omega }^0=2\pi \times {10}^{14}\text{Hz},\eta =0.05,\mathrm{\Gamma }=\mathrm{\Omega }\eta /100,{\mathrm{\Omega }}^r=2\pi \times 50\text{kHz},s=5,{\delta }^r=-2\pi \times 187\text{Hz}$ and ${\delta }^b=-2\pi \times 12063\text{Hz}$. Initial state: $|1,\downarrow ,\downarrow \rangle$. Dotted magenta lines: anisotropic Dicke model; dashed blue lines: ion system.