Dissipation-induced dipole blockade and antiblockade in driven Rydberg systems

We study theoretically and experimentally the competing blockade and antiblockade effects induced by spontaneously generated contaminant Rydberg atoms in driven Rydberg systems. These contaminant atoms provide a source of strong dipole-dipole interactions and play a crucial role in the system's behavior. We study this problem theoretically using two different approaches. The first is a cumulant expansion approximation, in which we ignore third-order and higher connected correlations. Using this approach for the case of resonant drive, a many-body blockade radius picture arises, and we find qualitative agreement with previous experimental results. We further predict that as the atomic density is increased, the Rydberg population's dependence on Rabi frequency will transition from quadratic to linear dependence at lower Rabi frequencies. We study this behavior experimentally by observing this crossover at two different atomic densities. We confirm that the larger density system has a smaller crossover Rabi frequency than the smaller density system. The second theoretical approach is a set of phenomenological inhomogeneous rate equations. We compare the results of our rate-equation model to the experimental observations [E. A. Goldschmidt et al., Phys. Rev. Lett. 116, 113001 (2016)] and find that these rate equations provide quantitatively good scaling behavior of the steady-state Rydberg population for both resonant and off-resonant drives.

Figure 1

Theoretical three-level system. The $g$ and $s$ states are coupled via a classical laser with Rabi frequency $\mathrm{\Omega }$ and detuning $\delta$, while the $s$ and $p$ states interact via a dipole-dipole interaction ${\sum }_{i\ne j}{\sigma }_i^{sp}{\sigma }_j^{ps}$. There are three decay processes: $s\to g,p\to g$, and $s\to p$, with decay rates of ${\gamma }_s,{\gamma }_p,{\gamma }_R$, respectively.

Figure 2

Approximate divergence diagrams for cumulant expansion in 3D. (a) Divergence diagram on resonance $\left(\delta =0\right)$. Most of the low Rabi frequencies on resonance are convergent, although they become less stable as the interactions are increased. Sufficiently large Rabi frequencies are also convergent, where steady-state populations begin to saturate. (b) Divergence diagram for $\mathrm{\Omega }/{\gamma }_s=0.4$. A very narrow region near resonance is convergent for sufficiently small interaction strengths. The outer edges of the divergent region grow approximately quadratically in detuning in this parameter regime.

Figure 3

Illustration of how fitting the Rydberg population as a function of decoherence is used to approximate the Rydberg population in divergent regions. (a) $n_{3\mathrm{D}}{C}_3/{\gamma }_s=1250,\mathrm{\Omega }/{\gamma }_s=0.4$. The orange line corresponds to a quartic fit of the blue circles where ${\gamma }_d/{\gamma }_s>0.1$. The purple square denotes the population for ${\gamma }_d=0$, which is convergent for these parameters. (b) Relative error of Rydberg population extracted from a ${\gamma }_d/{\gamma }_s>0.1$ fit compared to actual Rydbeg population at ${\gamma }_d=0$, denoted ${\langle {\sigma }^{ss}\rangle }_{\mathrm{F}}$ and ${\langle {\sigma }^{ss}\rangle }_0$, respectively, for several choices of parameters just outside of the divergent region. The inset shows the parameters used in the divergence diagram on resonance, where the orange $\times$'s denote the parameters used and the blue line separates the convergent and divergent regions, with the top right corresponding to the divergent region.

Figure 4

Correlations between $s$ and $p$ atoms for $\mathrm{\Omega }/{\gamma }_s=0.4$. The blue dots are from the cumulant expansion while the orange line corresponds to exact calculations for just two atoms separated a distance $r$. These are plotted in (a) 1D for $n_{1\mathrm{D}}^3{C}_3/{\gamma }_s=800$ and (b) 3D for $n_{3\mathrm{D}}{C}_3/{\gamma }_s=800$.

Figure 5

Ratio of the many-body blockade radius $r_b$ to the two-body blockade radius $r_b^{\left(2\right)}$ as a function of interaction strength. (a) 1D system with ${\gamma }_d=0$ for all points. (b) 3D system with examples of both ${\gamma }_d=0$ and ${\gamma }_d\ne 0$.

Figure 6

Steady-state $s$ state population dependence on interaction strength for $\mathrm{\Omega }/{\gamma }_s=0.4$. (a) 1D system with best fit power-law with exponent of $-0.055$. (b) 3D system with best fit power-law exponent of $-1/5$.

Figure 7

Scaling exponent $b$ from fit of $\langle {\sigma }^{ss}\rangle =a{\mathrm{\Omega }}^b$ for $\mathrm{\Omega }/{\gamma }_s=0.05,0.1$ as a function of interaction strength. (a) 1D system. (b) 3D system with data only up to $n_{3\mathrm{D}}{C}_3/{\gamma }_s=200$ due to finite-size effects.

Figure 8

Resonant pumping rate as a function of Rabi frequency for two atomic densities, where $f$ corresponds to the fractional density of atoms initially in the driven ground state and $f=100\%$ corresponds to a density of $57\mu {\mathrm{m}}^{-3}$. Blue circles (red squares) are from experimental data for $f=25\%\left(f=100\%\right)$ and lines are from fits of Eq. (12). The empty blue circle and red square correspond to the fitted crossover Rabi frequency ${\mathrm{\Omega }}_c$ for each density, denoting the crossover from quadratic to linear scaling in Rabi frequency. Error bars represent the one standard deviation from exponential fits to extract the pumping rates.

Figure 9

Example of the time dependence of the $s$ population ensemble average ${\langle {\sigma }^{ss}\rangle }_{\text{ave}}$ on resonance in units of ${\gamma }_s^{-1}$. While neither a steady state nor a limit cycle is reached quickly, ${\langle {\sigma }^{ss}\rangle }_{\text{ave}}$ very quickly approaches an approximate steady-state value, whose time average we will denote by $\langle {\sigma }^{ss}\rangle$ for simplicity.

Figure 10

Examples of near-Lorentzian line shapes from inhomogeneous rate equations at $n_{3\mathrm{D}}{C}_3/{\gamma }_s=5000$ for several Rabi frequencies. Error bars indicate standard error from five random distributions of atoms and the lines are best-fit Lorentzians.

Figure 11

Resonant steady-state $s$ population scaling as a function of Rabi frequency and interaction strength. Different points of the same color and symbol correspond to different Rabi frequencies and error bars represent one standard deviation from the Lorentzian fits. (a) Theoretical rate-equation results, where $f=100\%$ corresponds to $n_{3\mathrm{D}}{C}_3/{\gamma }_s=5000$. The solid line is a linear fit with a slope of 3. (b) Experimental resonant pumping rate results from Ref. [22] where $f=100\%$ corresponds to $n_{3\mathrm{D}}{\beta }_3=6612$. The solid line is a linear fit with a slope of 3. Note that our definition of $\mathrm{\Omega }$ differs from the reference by a factor of 2.

Figure 12

Steady-state linewidth scaling as a function of Rabi frequency and interaction strength. Different points of the same color and symbol correspond to different Rabi frequencies and error bars represent one standard deviation from the Lorentzian fits. (a) Theoretical rate-equation results, where $f=100\%$ corresponds to $n_{3\mathrm{D}}{C}_3/{\gamma }_s=5000$. The solid line is a linear fit with a slope of 0.5 and $y$ intercept of 1.3 (the bare linewidth). (b) Experimental linewidth results from Ref. [22] where $f=100\%$ corresponds to $n_{3\mathrm{D}}{\beta }_3=6612$. The solid line is a linear fit with a slope of 1.8. Note that our definition of $\mathrm{\Omega }$ differs from the reference by a factor of 2.

Figure 13

(a) Scaling coefficient for resonant population as a function of ${({\gamma }_p/{\gamma }_R)}^{1/2}$. The coefficient ${\alpha }_r$ is extracted from fitting the resonant population according to $\langle {\sigma }^{ss}\rangle ={\alpha }_r(\mathrm{\Omega }/{\gamma }_s)/{(n_{3D}{C}_3/{\gamma }_s)}^{1/2}$ for fixed ${\gamma }_R,{\gamma }_p$, where error bars correspond to one standard deviation from the fits. The solid line is a linear fit with a slope of 1.5. (b) Scaling coefficient for the widths as a function of ${({\gamma }_R/{\gamma }_p)}^{1/2}$. The coefficient ${\alpha }_w$ is extracted from fitting the widths according to $\mathrm{\Gamma }/{\gamma }_s={\alpha }_w(\mathrm{\Omega }/{\gamma }_s){(n_{3D}{C}_3/{\gamma }_s)}^{1/2}$ for fixed ${\gamma }_R,{\gamma }_p$, where error bars (not visible) correspond to one standard deviation from the fits. The solid line is a linear fit with a slope of 0.7.

Figure 14

Relative error of cumulant expansion (CE) with respect to quantum trajectories (QT). Error bars denote standard error from quantum trajectories sampling. The total sampling time is $4500{\gamma }_s^{-1}$.

Figure 15

Example line shapes for several implementations of decoherence, where lines correspond to Lorentzian fits. (a) Lattice implementations of the rate equations. (b) Gaussian distribution implementation of the rate equations.

Figure 16

Experimental excitation and measurement scheme. The ground-state manifold is initially populated with a fraction $f$ in the $|g\rangle =|F=2,m_F=-2\rangle$ state and $1-f$ in the $|F=2,m_F=2\rangle$ state using three applications of microwave rapid adiabatic passage. The $|F=2,m_F=-2\rangle$ state is then driven via an off-resonant two-photon transition through the $5p_{1/2}$ state to the $18s_{1/2}$ state with intermediate detuning $\mathrm{\Delta }$, two-photon detuning $\delta$, and lower and upper Rabi frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$.