Characterizing high-$n$ quasi-one-dimensional strontium Rydberg atoms

The production of high-$n$, $n\sim 300$, quasi-one-dimensional (quasi-1D) strontium Rydberg atoms through two-photon excitation of selected extreme Stark states in the presence of a weak dc field is examined using a crossed laser-atom beam geometry. The dipolar polarization of the electron wave function in the product states is probed using two independent techniques. The experimental data are analyzed with a classical trajectory Monte Carlo simulation employing initial ensembles that are obtained with the aid of quantum calculations based on a two-active-electron model. Comparisons between theory and experiment highlight different characteristics of the product quasi-1D states, in particular, their large permanent dipole moments, $\sim$1.0 to 1.2$n^{2}e{a}_0$, where $e$ is the electronic charge and $a_0$ is the Bohr radius. Such states can be engineered using pulsed electric fields to create a wide variety of target states.

Figure 1

Schematic diagram of the apparatus. The inset shows the two-photon excitation scheme employed.

Figure 2

(a) Evolution of the measured excitation spectrum of strontium with increasing applied dc field $F_{\mathrm{dc}}$ for $M=0$ states in the vicinity of $n\sim 274$ [thick red (dark grey) lines]. The thin solid black lines show the calculated Stark energy level structure for singly excited strontium near $n\sim 50$, and the thick blue (light grey) lines denote calculated excitation spectra. The applied field is normalized to the crossing field $F_{\mathrm{cross}}$ (see text), i.e., $F_0=F_{\mathrm{dc}}/F_{\mathrm{cross}}$. The energy axis is scaled such that $E_0=1$ corresponds to the energy difference between neighboring $n$ and ($n-1$) manifolds. (b) Field dependence of the calculated average scaled dipole moment for low-$L$ states and selected extreme blue-shifted (uphill) and red-shifted (downhill) Stark states.

Figure 3

Classical trajectories of an excited electron for (a) strontium in the absence of external fields, (b),(c) strontium with $F_{\mathrm{dc}}=0.4$ V/cm $\simeq 0.08F_{\mathrm{cross}}$, and (d),(e) hydrogen with $F_{\mathrm{dc}}=5$ V/cm $\simeq F_{\mathrm{cross}}$. (b),(d) and (c),(e) relate to blue-shifted and red-shifted orbits, respectively. The initial conditions correspond to the $n=50$ $D$-state ($L=2$) and the dynamics is generated by the SAE model potential Eq. (9). $\vec{\ell }$ is parallel to the $y$ axis and $F_{\mathrm{dc}}$ is directed along the $z$ axis.

Figure 4

(a) Quantum probability distribution ${\rho }_{\mathrm{qm}}\left(k\right)$ [Eq. (11)] of the parabolic quantum number $k(=-n{A}_z)$ for the $M=0$ “52$D$” state as a function of applied dc field normalized as in Fig. 2. (b)–(d) Cuts through the distribution in (a) for the selected values of $F_{\mathrm{dc}}$ indicated. These are compared to the classical distribution ${\rho }_{\mathrm{cl}}\left(k\right)$ (see text).

Figure 5

(a) Coulomb potential as modified by a dc field $F_{\mathrm{dc}}$. (b) Contour plot of the energy surface $E=-1/r+z{F}_{\mathrm{dc}}$. A low-$\ell$ trajectory on the downhill side (red line) is rapidly ionized over the Stark saddle, whereas one on the uphill side (green line) remains remote from the saddle and is not ionized.

Figure 6

Ionization behavior [Eq. (13)] of low-$\ell$ (${\ell }_i\sim 0$) Rydberg states subject to a field step (see text). The electron is initially located at $z_i\simeq r_{i}cos{\theta }_i$. The electron energy and the field amplitude are given by $E_i=-1/\left(2n^2\right)$ and $F_{\mathrm{step}}=0.17/n^4$.

Figure 7

Illustration of circular wave-packet generation. (a),(c) Kepler ellipses (in green) elongated along the $z$ axis are transformed by the pump pulse $F_{\mathrm{pump}}$ to circular orbits. Depending on their orientation, the final circular orbit revolves clockwise or anticlockwise in the $xz$ plane. (b) Ellipses oriented at angle $\theta$ to the $z$ axis produce circular orbits but not in the $xz$ plane.

Figure 8

Calculated expectation values $\langle x\left(t\right)\rangle$ and $\langle z\left(t\right)\rangle$ for the Bohr wave packet ($n=50$) generated by a pump pulse with $F_{\mathrm{pump}}\simeq 50$ V/cm and $T_{\mathrm{pump}}\simeq 52$ ps [Eq. (15)]. The initial state is the “52$D$” state photoexcited in the presence of a weak dc field $F_{\mathrm{dc}}$. The solid lines show the results of quantum simulations using the TAE model [Eq. (6)] and the dashed lines the results of CTMC simulations [Eqs. (1) and (9)] employing the hybrid-$\ell$ distribution as the initial condition.

Figure 9

Measured ionization probabilities for “312$D$” states excited in the dc fields, indicated as a function of the scaled amplitude of a probe field $F_{\mathrm{step}}$ of 10-ns duration, $T_p\sim 2.3T_n$. Symbols denote data points recorded with the probe field applied parallel () and antiparallel () to the dc electric field, $F_{\mathrm{dc}}$. Calculated ionization probabilities for the “312$D$” states obtained using the hybrid-$\ell$ and the parabolic-$k$ distributions are shown by the black and light-blue (gray) lines, respectively. Solid (dashed) lines are for the probe field parallel (antiparallel) to $F_{\mathrm{dc}}$. The error bars indicate the 10% uncertainty in $F_{\mathrm{step}}$.

Figure 10

(a)–(c) Survival probabilities measured for “312$D$” states produced in the dc field indicated and subject to a 5 mV$$cm${}^{-1}$, 80-ns-long pump pulse as a function of the time delay between the end of the pump pulse and application of a 6-ns-long probe pulse $F_{\mathrm{step}}(=120$ mV$$cm${}^{-1}$ when probing along the $x$ axis and 105 mV$$cm${}^{-1}$ when probing along the $z$ axis), together with the corresponding simulations employing the hybrid-$\ell$ (d)–(f) and the parabolic-$k$ (g)–(i) distributions. The black (light blue/gray) lines correspond to application of the probe pulse in the $z$ ($x$) directions. The applied dc fields are (a),(d),(g) $F_{\mathrm{dc}}=0$, (b),(e),(h) $300\mu$V$$cm${}^{-1}$, and (c),(f),(i) $500\mu$V$$cm${}^{-1}$.

Figure 11

(a) The asymmetry, i.e., the differences in survival probability, between positive and negative probe pulses derived from Fig. 9. (b) Peak amplitude of the oscillations in survival probability observed when probing along the $z$ axis following creation of circular wave packets, taken from Fig. 10. $▴$ : measurements, —–: calculations using the hybrid-$\ell$ ensemble, and - - - : using the parabolic-$k$ distribution. The error bars correspond to stray fields of $\sim 50\mu$V$$cm${}^{-1}$.