Observation of Stückelberg oscillations in dipole-dipole interactions

We have observed Stückelberg oscillations in the dipole-dipole interaction between Rydberg atoms with an externally applied radio-frequency field. The oscillating rf field brings the interaction between cold Rydberg atoms in two separated volumes into resonance. We observe multiphoton transitions when varying the amplitude of the rf field and the static electric field offset. The angular momentum states we use show a quadratic Stark shift, which leads to a fundamentally different behavior than linearly shifting states. Both cases are studied theoretically using the Floquet approach and are compared. The amplitude of the sidebands, related to the interaction strength, is given by the Bessel function in the linearly shifting case and by the generalized Bessel function in the quadratically shifting case. The oscillatory behavior of both functions corresponds to Stückelberg oscillations, an interference effect described by the semiclassical Landau-Zener-Stückelberg model. The measurements prove coherent dipole-dipole interaction during at least $0.6\mu \text{s}$.

Figure 1

(Color online) Energy level diagrams of state $|1⟩$ and $|2⟩$ as a function of electric field. State $|1⟩$ has no Stark shift, state $|2⟩$ has a linear Stark shift in Fig. (a) and a quadratic Stark shift in Fig. (b). Due to the coupling between the states, the crossing becomes an avoided crossing, depicted in the inset. When an oscillating field is applied, depicted as a dashed purple line, this translates as an oscillating energy for state $|2⟩$, depicted on the right-hand side as a dash-dotted red line. The orange dotted lines depict the average energy during the oscillation and the corresponding field; in (a) this field is simply $F_S$, the average field, but in (b) it is different and we define it as the effective field $F_{\text{eff}}$, given in Eq. (25).

Figure 2

(Color online) The sidebands of state $|2⟩$ for the linear case in an oscillating field of 8 MHz for various rf amplitudes. The intensity of a line represents the population of the respective sideband, where the original state, or the $n=0$ sideband, is colored blue and indicated with an “o.” The red dashed lines depict the borders of the classically allowed region. We used $k=2\pi \times 60\text{MHz}/\left(\text{V}/\text{cm}\right)$ and $W_0=2\pi \times 25\text{MHz}$.

Figure 3

(Color online) The sidebands of state $|2⟩$ for the quadratic case in an oscillating field of 8 MHz for different rf amplitudes. The intensity of a line symbolizes the population of that sideband. The $n=0$ sideband is depicted in blue and indicated with an “o.” The red dashed lines depict the position of the asymptotes of the distribution of occurring energies. We used the values corresponding to the experimental values of the left resonance, as mentioned at Eq. (52).

Figure 4

(Color online) Depicted in black is the sideband population of state $|2⟩$ as a function of the sideband energy [Eqs. (19, 24)] for various frequencies. $\omega$ is $2\pi$ times the denoted frequency. We used again $k=2\pi \times 60\text{MHz}/\left(\text{V}/\text{cm}\right)$ and $W_0=2\pi \times 25\text{MHz}$ and field values of $F_S=0.2\text{V}/\text{cm}$ and $F_{\text{rf}}=0.3\text{V}/\text{cm}$. The red dashed lines depict the distribution of classically occurring energies, calculated with the approach of Eq. (26).

Figure 5

(Color online) Similar as Fig. 4. We used the experimental values for the left resonance [Eq. (52)] and field values of $F_S=0.2\text{V}/\text{cm}$ and $F_{\text{rf}}=0.45\text{V}/\text{cm}$. The distribution of occurring energies (red dashed lines) is calculated with Eq. (26). Note the occurrence of three turning points.

Figure 6

(Color online) As Fig. 5, but at a lower frequency and zoomed in around the middle asymptote. The blue dotted curve depicts the moving average of the sideband population depicted in black.

Figure 7

(Color online) Calculated interaction strength as a function of static and oscillating field for the linear case. The vertical lines depict the $n\omega$ resonances, where $n$ varies from $-6$ (the one most right) to 6 (the one most left) and $\omega$ is fixed to $2\pi \times 4\text{MHz}$. The darker the color the stronger the interaction. Destructive interference between the two states is depicted by the curved lines in red (solid) and green (shaded), according to Stückelberg theory described in Sec. 2E. The dashed blue lines give the boundaries of the classically allowed region; in other words these lines depict field combinations where the crossing [Eq. (6)] occurs at an extremum of the rf oscillation.

Figure 8

(Color online) Calculated interaction strength as a function of static and oscillating field for the quadratic case. The quarter circles in varying gray depict the $n\omega$ resonances. Here $n$ runs from $-25$ (in the upper right corner) to 3 (in the lower left corner); the frequency of the rf field is here $\omega =2\pi \times 8\text{MHz}$. The darker the color the stronger the interaction. Destructive Stückelberg interference between the two states is depicted by the curved lines in red (solid), green (shaded), and orange (checkerboard pattern) (Sec. 2E). The dashed blue lines give all field combinations where the crossing is reached during an extremum of the rf oscillation.

Figure 9

(Color online) The energies of states $|a⟩$ and $|b⟩$ as a function of time during the rf oscillation for the case of linear shifting Stark states. Indicated are the times where the two states cross, $t_1$, $t_2$, and $t_3$, and the matrices that describe the evolution of the wave function at the indicated time intervals: ${\mathbf{M}}^T$, ${\mathbf{G}}_{12}$, $\mathbf{M}$, and ${\mathbf{G}}_{23}$. The green (shaded) and red (solid) colored areas indicate the phase evolution of the state $|b⟩$, ${\Theta }_{ij}$, which is contained in the $G_{ij}$ matrix.

Figure 10

(Color online) The population of state $|b⟩$ as a function of the static field $F_S$ and the rf amplitude $F_{\text{rf}}$ [Eq. (45)] for $N=3$, where red (dark areas) is a large population and white is a population of 0.

Figure 11

(Color online) Similar to Fig. 9; now for the case of quadratically shifting Stark states and large rf amplitudes, such that both the $+F_{\text{quad}}$ and the $-F_{\text{quad}}$ crossing occur. At $t_1$ and $t_4$ the field is at the $+F_{\text{quad}}$ crossing and at $t_2$ and $t_3$ the field is at the $-F_{\text{quad}}$ crossing. Furthermore, it is clearly visible that ${\Theta }_{12}$ and ${\Theta }_{34}$ are equal, they cover the section between $+F_{\text{quad}}$ and $-F_{\text{quad}}$.

Figure 12

(Color online) The population of state $|b⟩$ as a function of the static field $F_S$ and the rf amplitude $F_{\text{rf}}$ [Eq. (49)] for $N=3$, where red (dark areas) is a large population and white is a population of 0.

Figure 13

(Color online) The two-atom energy levels of the $41d+49s$ and $42p+49p$ system. In the inset we have zoomed in around the two relevant crossings, which become avoided crossings due to the dipole-dipole interaction.

Figure 14

(Color online) The $49p$ fraction in arbitrary units, depicted in black, is measured as a function of static field for different rf amplitudes. The red dashed and blue dash-dotted resonance lines correspond to respectively Eqs. (52, 53) and run from the $+3$ photon transition (lower left corner) to the $-14$ or $-11$ photon transition (upper right corner). The 0-photon transition is depicted with thicker lines. The resonance peaks are expected where the green dotted lines cross with the red dashed and blue dash-dotted lines, indicated with green dots.

Figure 15

(Color online) The $49p$ fraction as a function of rf amplitude at zero static field. The red dashed and blue dotted vertical lines depict the expected positions of the even-photon transitions for both transitions given in Eq. (52, 53).

Figure 16

(Color online) The $49p$ fraction measured for different mixing angles [Eq. (56)]. From left to right we have increasing $F_{\text{rf}}$ and decreasing $F_S$. The effective field is kept constant at the n-photon resonance for each plot. The red dashed lines show the result of the simulation discussed in Sec. 3C. The blue dash-dotted lines depict the squared generalized Bessel function [Eq. (23)] directly, which can be seen as a small transfer approximation. The green dotted vertical lines depict the boundaries of the classically allowed region and correspond to the lower blue dashed lines in Fig. 8.