Mesoscopic spin-boson models of trapped ions

Trapped ions arranged in Coulomb crystals provide us with the elements to study the physics of a single spin coupled to a boson bath. In this work, we show that optical forces allow us to realize a variety of spin-boson models, depending on the crystal geometry and the laser configuration. We study in detail the ohmic case, which can be implemented by illuminating a single ion with a traveling wave. The mesoscopic character of the phonon bath in trapped ions induces effects such as the appearance of quantum revivals in the spin evolution.

Figure 1

(Color online) Scheme for the implementation of spin-boson couplings in Coulomb chains.

Figure 2

Phonon bath properties of a Coulomb chain with $N=50$. (a) Dots: Exact axial vibrational spectrum. Continuous line: Approximation ${\omega }_n^{1\mathrm{D}}\simeq \overline{\delta \omega }n$. (b) Spectral density of spin-boson couplings induced by a standing wave (with $k{\overline{z}}_1=1$) or a traveling wave (with $2V_{0}k\overline{z}∕{\omega }_z=1$). The continuous lines are obtained with the exact vibrational spectrum, by substituting the $\delta$ function in $J\left(\omega \right)$ by a constant function in each interval $({\omega }_n,{\omega }_{n+1})$, of height $({\omega }_{n+1}-{\omega }_n{)}^{-1}$. The dashed lines are the approximations of Eqs. (4, 5).

Figure 3

(a) Transition from the underdamped (thick line, $\Delta =10{\omega }_z$, $\alpha =2\times {10}^{-3}$) to the overdamped regime (thin line, $\Delta =3{\omega }_z$, $\alpha =4\times {10}^{-3}$), in the high-temperature limit of a spin-boson model with $N=50$ ions, $T=250{\omega }_z$, $ϵ=0$. (b) NIBA time kernels corresponding to these two regimes.

Figure 4

Spin evolution in the mesoscopic ohmic spin-boson model, in a Coulomb chain with $N=50$ ions, $T=0$, $\Delta =10{\omega }_z$, $ϵ=0$, and different dissipation strengths.

Figure 5

(a) Evolution of the decay rate in the strong dissipation regime. $N=50$, $\Delta =10{\omega }_z$. (b) Axial trapping frequency of the trap that is required as a function of dissipation strength in units of the recoil energy, with different values of $N$. $E_R=k^2∕2m$, with $k$ the wave vector in Eq. (3). For example, ${\mathrm{Be}}^{+}$, $2\pi ∕k=313\mathrm{nm}$, yields $E_R=225\mathrm{kHz}$.