Quantum phases of interacting phonons in ion traps

The vibrations of a chain of trapped ions can be considered, under suitable experimental conditions, as an ensemble of interacting phonons, whose quantum dynamics is governed by a Bose-Hubbard Hamiltonian. In this work we study the quantum phases which appear in this system, and show that thermodynamical properties, such as critical parameters and critical exponents, can be measured in experiments with a limited number of ions. In addition to that, interacting phonons in trapped ions offer us the possibility to access regimes which are difficult to study with ultracold bosons in optical lattices, such as models with attractive or site-dependent phonon-phonon interactions.

Figure 1

(a) Mean phonon number $⟨n_j⟩$ and (b) fluctuations $\delta n_j$ at each ion in the phonon ground state in an array of microtraps. Number of ions $N=50$ and total phonon number $N_{\text{ph}}=50$.

Figure 2

(a) Mean phonon number and (b) fluctuations, at each ion in the phonon ground state in a linear Paul trap. $N=50$, $N_{\text{ph}}=50$.

Figure 3

Correlation functions (a) $C_{i_0,j}^{aa}$ and (b) $C_{i_0,j}^{nn}$ as a function of coordinate $j$, at the superfluid phase $\left(U/t=0.1\right)$ in an array of ion microtraps. We choose $i_0=26$ (center of the chain), $N=N_{\text{ph}}=50$. The dotted and solid lines are numerical results and fittings in the region where the functions show algebraic decay.

Figure 4

The evolution of the exponent $\alpha$ of $C_{i_0,j}^{aa}$ as a function of $U/t$ in the regime of superfluid phase, in an array of ion microtraps $\left(i_0=26,N=N_{\text{ph}}=50\right)$. The dotted line is from numerical data and the solid line the fitting result from the Luttinger liquid, that is, $\alpha \approx A\sqrt{\frac{U/t}{n_0}}$, where the coefficient $A\approx 1.68$.

Figure 5

Correlation functions (a) $C_{i_0,j}^{aa}$ and (b) $C_{i_0,j}^{nn}$ at the superfluid phase $U/t=0.2$ in a linear Paul trap $\left(N=N_{\text{ph}}=50\right)$. The dotted lines are numerical data, and the solid lines are fittings in the region where the correlations decay algebraically.

Figure 6

The evolution of the parameters $\alpha$ in $C_{i_0,j}^{aa}$ with $U/t$. The dotted line is the values from the correlation, and the solid line is the fitting data from $\alpha \approx 0.215\sqrt{\frac{U/t}{n_0}}$, corresponding to the Luttinger-liquid theory.

Figure 7

Correlation functions (a) $C_{i_0,j}^{aa}$ and (b) $C_{i_0,j}^{nn}$ in a linear Paul trap when $U/t=2$ $\left(N=N_{\text{ph}}=50\right)$. Both of them show the coexistence of superfluid and Mott-insulator phases. In (a) and (b), the insets (i) show the exponential decay in the region of the Mott phase and the insets (ii) show the occupation number and fluctuations at the same parameters. The exponents $\alpha$ of the algebraic decay are also given in the figures. The dotted and solid lines are numerical and fitting data, respectively.

Figure 8

(a) Correlation functions $C_{i_0,j}^{aa}$ at $U/t=3$ and $C_{i_0,j}^{nn}$ at $U/t=2.4$ decay exponentially with the coordinate $j$ at the Mott-insulator phase for an array of microtraps $\left(N=N_{\text{ph}}=50,i_0=26\right)$. (b) The inverse of the correlation lengths in $C_{i_0,j}^{aa}$ and $C_{i_0,j}^{nn}$. The circle markers represent $C_{i_0,j}^{aa}$ and the square markers $C_{i_0,j}^{nn}$. The solid and dashed lines are the fitting data, respectively. (c) The inverse of the correlation lengths in $C_{i_0,j}^{aa}$ for different size of chains: $N=60\left(\text{dagger}\right)$, $N=80\left(\text{circle}\right)$, and $N=100\left(\text{triangle}\right)$, which is a finite scaling to support the robustness of the critical point. The extrapolations in (b) and (c) show that the critical point in the microtraps is $U_c/t\approx 1.55$.

Figure 9

The exponential decay correlations $C_{i_0,j}^{aa}$ and $C_{i_0,j}^{nn}$ at the Mott phase $U/t=2.8$ for a linear Paul trap $\left(N=N_{\text{ph}}=50,i_0=26\right)$.

Figure 10

The correlation $C_{i_0,j}^{aa}$ at the Mott phase $U/t=6$ in an array of microtraps (circle markers) and a linear Paul trap (square markers), respectively. $N=N_{\text{ph}}=50$, $i_0=26$. The inset shows the power-law decay with exponent $\alpha \approx 3$ for the microtraps only.

Figure 11

(a) Densities of phonons in an array of microtraps with $N=50$, $N_{\text{ph}}=25$. (b) Fluctuations at the same conditions. (c) The correlation function $C_{i_0,j}^{aa}$ $\left(i_0=26\right)$ at Tonks-gas phase $U/t=10$ with exponent $\alpha \approx 0.54$, where the dotted and solid lines are numerical and fitting data, respectively. (d) Evolution of exponent $\alpha$ of $C_{i_0,j}^{aa}$ with the interaction $U/t$, which would approach $\alpha \approx 0.58$.

Figure 12

(a) and (b) show densities and fluctuations of phonons, respectively, in a linear Paul trap with $N=50$ and $N_{\text{ph}}=50$. (c) The correlation $C_{i_0,j}^{aa}$ at the Tonks-gas phase $U/t=6$, whose exponent is $\alpha \approx 0.48$. (d) Evolution of the exponents $\alpha$ of $C_{i_0,j}^{aa}$, approaching $\alpha \approx 0.53$. In (c) the dotted and solid lines represent numerical and fitting data, respectively.

Figure 13

The evolution of the parameter $⟨O⟩$ with the interaction $U/t$ in an array of microtraps (solid line) and a linear Paul trap (dashed line). $N=50$, $N_{\text{ph}}=25$. The inset is the same transition in the logarithmic scale, which shows in detail the transition approaching zero.

Figure 14

(a) Density of phonons and (b) phonon number fluctuations in an array of microtraps with $N=10$, $N_{\text{ph}}=10$, and negative on-site interactions. (c) Density of phonons and (d) phonon number fluctuations in a linear Paul trap under the same conditions.

Figure 15

Evolution of the order parameter $⟨O⟩$ with the ratio $U/t$ for an array of ion microtraps (solid line) and a linear Paul trap (dashed line). $N=N_{\text{ph}}=10$.

Figure 16

The phonon density $⟨n_j⟩$ and phonon number fluctuations $\delta n_j$ in an array of ion microtraps, with the on-site interactions defined by Eq. (23), $U/t=40$, $N=50$, and $N_{\text{ph}}=2N=100$.

Figure 17

The correlation function $⟨{\sigma }_{i_0}^{+}{\sigma }_j⟩$ defined by Eq. (24). $i_0=26$, $N=50$, $N_{\text{ph}}=100$, $U/t=40$.

Figure 18

The energy gap $\Delta E$ between the ground state and the first excited state as a function of $U_{\text{even}}/U_{\text{odd}}$, $U_{\text{odd}}/t=40$. For simplicity, we consider here a small system of ions $N=6$.