Experimental Realization of a Quantum Phase Transition of Polaritonic Excitations

We report an experimental realization of the Jaynes-Cummings-Hubbard model using the internal and radial phonon states of two trapped ions. An adiabatic transfer corresponding to a quantum phase transition from a localized insulator ground state to a delocalized superfluid (SF) ground state is demonstrated. The SF phase of polaritonic excitations characteristic of the interconnected Jaynes-Cummings (JC) system is experimentally explored, where a polaritonic excitation refers to a combination of an atomic excitation and a phonon interchanged via a JC coupling.

Figure 1

(a) Conceptual schematic of the JCH system with two ions. Two ions are illuminated with an excitation laser nearly resonant to the red-sideband transition with which the JC coupling arises and dressed atoms or polaritons are formed. Intersite hopping ($\kappa$) is naturally incorporated from Coulomb couplings, and along with effective on-site repulsion between polaritons due to the JC coupling a Hubbard-type model is formed. (b) Energy levels of ${}^{40}\mathrm{Ca}^{+}$ relevant to motional cooling and induction of the JC coupling.

Figure 2

Measured and simulated quantum dynamics of the JCH system with two ions. Populations of the excited state of one ion and the other are plotted in (a) and (b), respectively, with circles. Each point obtained is the average of 50 experiments. Curves are numerically simulated results, which are multiplied by 0.8 to consider population quenching from $D_{5/2}$ to $S_{1/2}$ in the relatively long detection time of 80 ms due to a stray intensity from the 854-nm beam.

Figure 3

(a) Variation of the average internal-state populations during the adiabatic transfer. Each point obtained is the average of 50 experiments. The inset shows the time dependence of the JC coupling coefficient $2g/2\pi$ (the solid curve with the vertical axis on the left) and the detuning $\Delta /2\pi$ (the dashed curve with the vertical axis on the right). (b) Time dependence of the eigenenergies obtained by diagonalyzing the instantaneous Hamiltonians based on the pulse parameters used in the experiment. The lowest three curves corresponds, from the lowest to the third lowest, to $|{\psi }_{\mathrm{atI}}⟩\to |{\psi }_{\mathrm{phSF}}⟩$, $(1/\sqrt{2})(|g_1⟩|e_2⟩+|e_1⟩|g_2⟩)\otimes {\widehat{a}}_r^{†}|0{⟩}_1|0{⟩}_2\to |g_1⟩|g_2⟩\otimes {\widehat{a}}_c^{†}{\widehat{a}}_r^{†}|0{⟩}_1|0{⟩}_2$, and $(1/\sqrt{2})(|g_1⟩|e_2⟩-|e_1⟩|g_2⟩)\otimes {\widehat{a}}_r^{†}|0{⟩}_1|0{⟩}_2\to |g_1⟩|g_2⟩\otimes (1/\sqrt{2}){\widehat{a}}_c^{†2}|0{⟩}_1|0{⟩}_2$, respectively. (c),(d) Measurement of the average phonon numbers of collective modes before the adiabatic transfer. Results of spectroscopy over radial red- and blue-sideband transitions before the adiabatic transfer are shown in (c) and (d), respectively. Each point obtained is the average of 50 experiments, and the red curves are the results of fitting with multiple Gaussians. The label $-{\omega }_{\mathrm{c}.\mathrm{m}.}$ ($+{\omega }_{\mathrm{c}.\mathrm{m}.}$) indicates the red- (blue-) sideband resonance of the center-of-mass (c.m.) mode, and $-{\omega }_{\mathrm{rock}}$ $+{\omega }_{\mathrm{rock}}$) the red- (blue-) sideband resonance of the rocking mode. (e),(f) Measurement of the average phonon numbers of collective modes after the adiabatic transfer, in a similar way to (c) and (d). Results of spectroscopy over radial red- and blue-sideband transitions after the adiabatic transfer are shown in (e) and (f), respectively.

Figure 4

Estimated experimental and calculated values for the excitation number variances (atomic, phonon, and total) during the adiabatic transfer. The points are experimental values and the curves are calculated from exact ground states against the actual time dependence of the experimental parameters. (a) Values for the atomic excitation number variance $\Delta {\widehat{N}}_{a,1}^2$ (the circles and the solid curve) and the phonon-number variance $\Delta {\widehat{N}}_{p,1}^2$ (the triangles and the dashed curve). Each point obtained is the average of 50 experiments. (b) Values for the total excitation number variance $\Delta {\widehat{N}}_1^2$. Instead of directly estimating the experimental values for this quantity, the upper and lower bounds are estimated along with their errors, and shown as the circles and triangles (see text for details).