Simulations of the dipole-dipole interaction between two spatially separated groups of Rydberg atoms

The dipole-dipole interaction among ultracold Rydberg atoms is simulated. We examine a general interaction scheme in which two atoms excited to the $x$ and $x^{\prime }$ states are converted to $y$ and $y^{\prime }$ states via a Förster resonance. The atoms are arranged in two spatially separated groups, each consisting of only one species of atom. We monitor the state mixing by recording the fraction of atoms excited to the $y^{\prime }$ state as the distance between the two groups is varied. With zero detuning a many-body effect that relies on always resonant interactions causes the state mixing to have a finite range. When the detuning is greater than zero, another many-body effect causes a peak in the state mixing when the two groups of atoms are some distance away from each other. To obtain these results it is necessary to include multiple atoms and solve the full many-body wave function. These simulation results are supported by recent experimental evidence. These many-body effects, combined with appropriate spatial arrangement of the atoms, could be useful in controlling the energy exchange among the atoms.

Figure 1

(a) Energy level diagram for the resonant energy exchange of the $x{x}^{\prime }y{y}^{\prime }$ system. (b) The geometry studied consists of two groups of $x$ and $x^{\prime }$ atoms separated by a distance $d$.

Figure 2

(Color online) Fraction of initial $x^{\prime }$ atoms in the $y^{\prime }$ state as a function of the separation between two spherical groups of randomly placed atoms for four different times. The detuning is zero and the total number of atoms is 12. The times are: $1\mu \text{s}$ (solid blue), $4\mu \text{s}$ (dot-dashed red), $7\mu \text{s}$ (dotted yellow), and $10\mu \text{s}$ (dashed green). (a) When the number of $x$ and $x^{\prime }$ atoms are equal (6 each), the state mixing persists to large distances. The inset shows data out to $250\mu \text{m}$ showing that the $y^{\prime }$ fraction eventually reaches zero as the Rabi period increases. For the case of (b) $3x^{\prime }$ atoms and $9x$ atoms and the case of (c) $1x^{\prime }$ atom and $11x$ atoms, the fraction of atoms in the $y^{\prime }$ state drops to zero with a shorter separation (around $50\mu \text{m}$).

Figure 3

(Color online) Fraction of atoms in the $y^{\prime }$ state as a function of the separation between two spherical groups of randomly placed atoms after an interaction time of $5\mu \text{s}$. The detuning $\Delta \approx 4\text{MHz}$ and the total number of atoms is 12. The most prominent feature is the peak in the $y^{\prime }$ fraction that occurs away from overlap for configurations with 3 to $6x^{\prime }$ atoms.

Figure 4

Fraction of atoms in the $y^{\prime }$ state after an interaction time of $5\mu \text{s}$ as a function of the separation between the two groups of Rydberg atoms with volumes defined by excitation lasers with Gaussian beam profiles. The detuning is $\Delta \approx 2\text{MHz}$ and the total number of atoms is 12. The location of the peaks in the $y^{\prime }$ fraction away from overlap at about $20\mu \text{m}$ and the sharp decrease in the state mixing from about $40–50\mu \text{m}$ compare well to the data in Ref. [2].

Figure 5

(Color online) Fraction of atoms in the $y^{\prime }$ state as a function of the separation between two spherical groups of randomly placed atoms after an interaction time of $5\mu \text{s}$. In this case, there are $3x^{\prime }$ and $9x$ atoms. The solid blue line is for $\Delta =0$ and is from the same data as Fig. 2b. The same simulation was run with the always resonant interactions removed and the result is shown with the dashed red line. While the $y^{\prime }$ fraction is smaller, it also persists to large separations, implicating the always resonant interactions as the cause for the finite range of the state mixing at zero detuning.

Figure 6

(a) One $x$ atom at a distance $d$ from two $x^{\prime }$ atoms, which are separated by a distance $R$. We hold the distance $R$ fixed and consider what happens as $d$ is varied. (b) The maximum probability of the three atoms being in the one of the states $|{\varphi }_{e1}⟩$ or $|{\varphi }_{e2}⟩$ vs the separation $d$ in units of $R$.

Figure 7

Two $x$ atoms and two $x^{\prime }$ atoms arranged so that the separation between any $x{x}^{\prime }$ pair is always $d$. The pair of $x$ atoms and the pair of $x^{\prime }$ atoms each comprise a “group” of atoms. The separation within the pairs is $R$ and is held fixed. (a) The two groups of atoms at their smallest separation; all of the atoms are in the same plane. This view is along the axis of separation. (b) A view of the atoms from the “side.” From this view, atoms 3 and 4 are on top of each other and the distance $R$ between them is foreshortened.

Figure 8

(Color online) Fraction of atoms in the $y^{\prime }$ state as a function of the separation between two amorphous spherical groups of atoms. The detuning $\Delta \approx 4\text{MHz}$. Shown here are the cases for which the number of $x$ and $x^{\prime }$ atoms are equal for different total numbers of atoms. The peak in the $y^{\prime }$ fraction away from overlap moves to larger separations and becomes smaller as the total number of atoms is increased. Unless a sufficient number of atoms is included in the calculation, the peak is not observed.

Figure 9

(Color online) (a) Geometry for a linear array of atoms where $d$ is the distance between the lines and $s$ is the spacing between the atoms. (b) Fraction of atoms in the $y^{\prime }$ state averaged over $10\mu \text{s}$ as a function of the separation $d$ between two lines of 5 atoms with $s=10\mu \text{m}$. The separation $d$ starts at $1\mu \text{m}$ and is increased in steps of $0.01\mu \text{m}$. As before, there is a peak in the $y^{\prime }$ fraction away from the overlap at about $25\mu \text{m}$. (c) The coefficients $\left|c_{0,j}\right|$ linking each dressed state to the initial state as a function of position. Only two of the 252 dressed states are significantly coupled to the initial state; they are shown in solid blue and dashed red (the remaining 250 coefficients are shown in dotted lines). One dressed state (solid blue) is weakly coupled to the initial state at small separations but evolves to be nearly identical to the initial state at large separations. Another dressed state (dashed red) is strongly coupled to the initial state at small separations and evolves into a state that is nearly totally decoupled from the initial state at large separations. Where these two dressed states cross, they are both equally coupled to the initial state and coupled to many other excited states. At this crossing, there is a strong peak in the $y^{\prime }$ fraction shown in (b).

Figure 10

(Color online) Fraction of atoms in the $y^{\prime }$ state averaged over $10\mu \text{s}$ as a function of the separation and the detuning. The atoms are arranged as in Fig. 9a. Lighter shades of gray correspond to larger fractions of atoms in the $y^{\prime }$ state. The location of the peak in the $y^{\prime }$ fraction away from overlap can be seen as a light band that curves from the upper-left to the lower-right. The resolution of this data is $1\mu \text{m}$ and about 0.07 MHz.

Figure 11

(Color online) Dressed state coefficients $\left|c_{0,j}\right|$ and $y^{\prime }$ fractions plotted together for the cases: (a) two lines of two atoms, (b) one line of two atoms and one line of three atoms, (c) two lines of three atoms, and (d) two lines of four atoms. With five or fewer total atoms, as in (a) and (b), the $y^{\prime }$ fraction peaks at overlap. With six total atoms, as in (c), a peak in the $y^{\prime }$ fraction away from overlap just begins to form near a $6\mu \text{m}$ separation. As more atoms are included, as in (d) and Fig. 9, the peak becomes more pronounced and moves to larger separations.