Tailoring Bose-Einstein-condensate environments for a Rydberg impurity

Experiments have demonstrated that the excitation of atoms embedded in a Bose-Einstein condensate to Rydberg states is accompanied by phonon creation. Here we provide the theoretical basis for the description of phonon-induced decoherence of the superposition of two different Rydberg states. To this end, we determine Rydberg-phonon coupling coefficients using a combination of analytical and numerical techniques. From these coefficients, we calculate bath correlation functions, spectral densities, and reorganization energies. These quantities characterize the influence of the environment and form essential inputs for followup open quantum system approaches. We find that the amplitude of bath correlations scales like the power law ${\nu }^{-6}$ with the principal quantum number $\nu$, while reorganization energies scale exponentially, reflecting the extreme tunability of Rydberg atomic properties.

Figure 1

Sketch of a Rydberg impurity in a BEC environment. (Left) A single Rydberg impurity in either of two selected internal states ($s,p$) couples to a large volume (blue or orange) of an embedding BEC (green), exciting phonons with wave vector $\mathbf{q}$ (red arrows). The shape of the coupling volume strongly differs in the two Rydberg electronic states. (Right) The system-environment coupling potential $U$ has a peculiar oscillatory long-range character. For illustration, we sketch $U$ for an impurity in state $|s\rangle$, cutoff at large $\left|U\right|$ for better visibility, along an arbitrary axis, here $z$.

Figure 2

Overview of Rydberg-phonon coupling constants for a single impurity with $\nu =40$. (a) The coupling ${\kappa }_q^{\left(ss\right)}$ of state $|s\rangle$ to a phonon with wave number $q$ is isotropic due to the symmetry of the Rydberg wave function. The right axis shows the phonon dispersion relation (9) for orientation. (b) In contrast, the transition coupling ${\kappa }_{q,10}^{\left(sp\right)}$ between $|s\rangle$ and $|p\rangle$ is anisotropic, since it involves $|p\rangle$ states. (c) Isotropic component ${\kappa }_{q,00}^{\left(pp\right)}$ of coupling in the state $|p\rangle$. (d) Anisotropic component ${\kappa }_{q,20}^{\left(pp\right)}$ of coupling in the state $|p\rangle$. (e) Transition coupling ${\kappa }_q^{\left(sp\right)}$ in the $q_x, q_z$ plane. (f) The difference of couplings $\mathrm{\Delta }{\kappa }_{\mathbf{q}}={\kappa }_q^{\left(pp\right)}-{\kappa }_q^{\left(ss\right)}$ will be most relevant for Rydberg decoherence.

Figure 3

Phonon correlation functions, defined in Eq. (28), for coupling to a Rydberg impurity with principal quantum number $\nu =40$. (a) $\mathrm{Re}\left[C^{\left(zz\right)}\left(\tau \right)\right]$ (black) and $\mathrm{Im}\left[C^{\left(zz\right)}\left(\tau \right)\right]$ (red dashed); (b) $C^{\left(yy\right)}\left(\tau \right)$ with the same line styles. Other correlations vanish: $C^{\left(zy\right)}\left(\tau \right)=C^{\left(yz\right)}\left(\tau \right)=0$.

Figure 4

Spectral properties of BEC phonon environment for a Rydberg impurity in $\nu =120$. (a) Spectral density $J^{\left(z\right)}\left(\omega \right)$ and (b) $J^{\left(y\right)}\left(\omega \right)$. These are defined by Eqs. (29) and (30) or equivalently Eqs. (31) and (32).

Figure 5

Tuning of the condensate environment. (a) Variation of the bath auto-correlation function $C^{\left(zz\right)}\left(0\right)$ (black diamonds) as a function of principal quantum number $\nu$ along with the power-law fit $C^{\left(zz\right)}\left(0\right)=1.25\times {10}^{11}{\nu }^{-6}{\mathrm{MHz}}^2$ as a red dashed line. We show both linear (black) and logarithmic (red) axes. (b) The reorganization energy ${\lambda }^{\left(z\right)}$, see Eq. (33), with the exponential fit ${\lambda }^{\left(z\right)}=5.76\times {10}^3{e}^{-0.06\times \nu }$ GHz.

Figure 6

(a) Coupling constant ${\kappa }_q^{\left(ss\right)}$ (red solid) of state $|s\rangle$ to a phonon with wave number $q$ for principal quantum number $\nu =40$ as in Fig. 2 and the fit (black dashed) $g_0{f}^{\left(ss\right)}/\sqrt{\mathcal{V}}$ from (B16) with the parameters in Table 1. (b) Isotropic part of the coupling constant ${\kappa }_{q,00}^{\left(pp\right)}$ of state $|p\rangle$ with its the fit $g_0{f}_1^{\left(pp\right)}/\sqrt{\mathcal{V}}$ and (c) the anisotropic part ${\kappa }_{q,20}^{\left(pp\right)}$ with the fit $g_0{f}_2^{\left(pp\right)}/\sqrt{\mathcal{V}}$ from (B18). (d) The transition coupling ${\kappa }_q^{\left(sp\right)}$ between $|s\rangle$ and $|p\rangle$ to a phonon with wave number $q$ and its fit $g_0{f}^{\left(sp\right)}/\sqrt{\mathcal{V}}$ from (B17).