Photon propagation through dissipative Rydberg media at large input rates

We study the dissipative propagation of quantized light in interacting Rydberg media under the conditions of electromagnetically induced transparency. Rydberg blockade physics in optically dense atomic media leads to strong dissipative interactions between single photons. The regime of high incoming photon flux constitutes a challenging many-body dissipative problem. We experimentally study in detail the pulse shapes and the second-order correlation function of the outgoing field and compare our data with simulations based on two novel theoretical approaches well-suited to treat this many-photon limit. At low incoming flux, we report good agreement between both theories and the experiment. For higher input flux, the intensity of the outgoing light is lower than that obtained from theoretical predictions. We explain this discrepancy using a simple phenomenological model taking into account pollutants, which are nearly stationary Rydberg excitations coming from the reabsorption of scattered probe photons. At high incoming photon rates, the blockade physics results in unconventional shapes of measured correlation functions.

Figure 1

(a) Three-level rEIT scheme, where $|g\rangle$, $|e\rangle$, and $|s\rangle$ are ground, excited, and long-lived Rydberg states, respectively. State $|e\rangle$ decays spontaneously at rate $\gamma$, while ${\gamma }_{\mathrm{ss}}$ describes the decoherence of $|s\rangle$. (b) Schematic representation of the theoretical model in position space at two time instances. A photon pulse, incident on the medium with velocity $c$, propagates as Rydberg polaritons with a group velocity $v_g$ inside the medium. However, due to Rydberg blockade, only one polariton per $r_b$ can propagate without loss. All other photons are scattered (represented by the solid wavy red arrow) at the beginning of the medium whenever a polariton is already inside the medium within $r_b$. There are additional losses in the medium (represented by the dashed wavy red arrow) due to the finite width of the EIT transparency window. (c) and (d) Output pulse shapes as a function of time predicted by the theory (see Sec. 2), for two different choices of incoming rates $R_{\mathrm{in}}$, blockade time ${\tau }_b$, and EIT filtering time ${\tau }_{\mathrm{EIT}}$: (c) A time trace for ${\tau }_{\mathrm{EIT}}={\tau }_b/5$ and $R_{\mathrm{in}}=10/{\tau }_b$, which gives rise to a train of photons [33]. (d) A time trace for ${\tau }_{\mathrm{EIT}}={\tau }_b/2$ and $R_{\mathrm{in}}=3/{\tau }_b$, which are closer to the parameters accessible in current experiments and in this work. Instead of well-separated humps, the intensity exhibits oscillations with the peaks corresponding to the photon humps in (c).

Figure 2

Time traces of the output pulse for a uniform cloud with an input Tukey pulse of the same shape as the one used in the experiment. Comparison between MPS (dashed lines) and effective hard-sphere model (solid lines) without free parameters. The atomic cloud is taken as uniform in both models with $L=47.2\mu \text{m}$, chosen to be consistent with the length of the experimental cloud (described in Sec. 3). Other parameters are $\text{OD}=33$, $\gamma /2\pi =6.065$ MHz, $\mathrm{\Omega }/2\pi =10$ MHz, and $C_6/2\pi =1.87820\times {10}^{14}\mathrm{Hz}$ $\mu {\mathrm{m}}^6$ for the $n=111$ Rydberg state. Decoherence in the Rydberg level with full width ${\gamma }_{\mathrm{ss}}/2\pi =40$ kHz, which we extract from the transmission at low incoming rates. MPS simulations used $N=60$ and bond dimensions $D=80,120$, and 160 for rates 0.6 ph/$\mu \mathrm{s}=0.078/{\tau }_b$, 4.2 ph/$\mu \mathrm{s}=0.55/{\tau }_b$, and 10.5 ph/$\mu \mathrm{s}=1.4/{\tau }_b$, respectively (here ph stands for photons). Note good agreement at initial times. Also note that the agreement is better for lower incoming rates than for higher ones. We do not see multiple humps because the blockade time ${\tau }_b=0.13\mu \mathrm{s}$ and filtering time ${\tau }_{\text{EIT}}=\sqrt{\text{OD}}/{\gamma }_{\text{EIT}}=0.11\mu \mathrm{s}$ are comparable, and because the rise time of the pulse $t_{\mathrm{rise}}=0.8\mu \mathrm{s}$ is much greater than ${\tau }_b$.

Figure 3

As a function of the input photon rate, the figure shows (a) the experimentally measured output photon rate in the steady-state region of the output pulse [see Figs. 4 and 4], (b) transmission of the weak Gaussian test pulse through the medium, and (c) the number of detected ions. Lines connecting points are only a guide to the eye.

Figure 4

(a) Schematic of the experimental setup showing the probe and control beams focused into the atomic cloud, as well as the detectors for probe photons (SPCMs) and ions (MCP). (b) Illustration of the pulse sequence for a single experimental run. (c) and (d) Output pulse shapes (solid lines) observed in the experiment for input photon rates $R_{\mathrm{in}}=4.2$ ph/$\mu \mathrm{s}=0.55/{\tau }_b$ and $R_{\mathrm{in}}=71.8$ ph/$\mu \mathrm{s}=9.3/{\tau }_b$, respectively. Dashed lines depict the input pulses, whose values are divided by factor of 10 in (c) and 100 in (d) for easier viewing. Also shown are the input and output pulses of the weak Gaussian test pulse following the main probe pulse. The main distortion observed in the outgoing probe pulses is the appearance of the initial hump, which becomes more pronounced for higher input photon rates. The orange-shaded regions indicate the timing window we analyze to obtain the steady-state outgoing photon rate.

Figure 5

Second-order correlation function measured for different photon rates between 0.2 ph/$\mu \mathrm{s}=0.026/{\tau }_b$ and 10.1 ph/$\mu \mathrm{s}=1.3/{\tau }_b$ within the time window marked by orange shading in Fig. 4. The figure suggests that the height of the bunching feature saturates at high input rates. We confirmed that the MPS toy model reproduces qualitatively this saturation for the chosen parameters, but since the model does not provide a more quantitative description of the system and we do not have measurements for the higher input rates, we refrain from showing and discussing in detail these results.

Figure 6

Comparison of the experimental photon output with the output simulated using MPS. (a)–(c) Time traces for various input photon rates: (a) 0.6 ph/$\mu \mathrm{s}=0.078/{\tau }_b$, (b) 4.2 ph/$\mu \mathrm{s}=0.55/{\tau }_b$, and (c) 10.5 ph/$\mu \mathrm{s}=1.4/{\tau }_b$. From (a), we see that the reference pulse is well described using a Tukey function. (d) Steady-state output as a function of the input rate. Experimental data and theoretical curves are shown with solid and dashed lines, respectively. In (a)–(c) blue (dark) curves indicate input pulses while the orange (light) ones depict the output. The input pulses indicated with $÷10$ have been divided by a factor of 10 for easier viewing. The Rydberg interaction is modeled by a sum of five exponentials as described in Appendix pp3-s1. In (a)–(c), MPS density matrix simulations use time step $0.01/\gamma$, $N=60$ effective atoms, and bond dimensions $D$ equal to (a)–(b) 100, (c) 180. The steady-state results in (d) are from quantum jump MPS simulations with time step $0.01/\gamma$, number of effective atoms $N=70$, and bond dimensions dependent on the input rate as shown in Fig. 11 in Appendix pp3-s2. In the experimental results shown in (d) the error bars are plotted but are smaller than the data points.

Figure 7

Experimental second-order correlation function (solid) compared with its MPS-simulated counterpart (dashed) for different input photon rates. Experimental parameters used in MPS simulations are as in all other figures except ${\gamma }_{ss}/2\pi =120$ kHz, because the $g^{\left(2\right)}$ data were taken in slightly different experimental conditions. The Rydberg interaction is modeled by three exponentials, as described in Appendix pp3-s1. MPS density matrix simulations used time step $0.01/\gamma$, $N=60$, and bond dimension $D=140$ for input rates 0.4 ph/$\mu \mathrm{s}=0.052/{\tau }_b$ and 2.9 ph/$\mu \mathrm{s}=0.38/{\tau }_b$, and $D=180$ for the input rate 7.7 ph/$\mu \mathrm{s}=1/{\tau }_b$.

Figure 8

(a) Illustration of radiation trapping of scattered probe photons in the atomic cloud. Reabsorption of probe photons is possible within the larger control beam. The two three-level atoms that we zoom into schematically represent a process where the left atom emits a photon (red arrow), which is then absorbed by the right atom. (b) Level scheme for the effective model we introduce to incorporate the pollutant atoms in our numerics.

Figure 9

(a) Population decay in function of time for a single excitation placed at the center of the elongated 3D atomic cloud. For details see main text. (b) Comparison of experimental data for 7.1 ph/$\mu \mathrm{s}$ with MPS pollution model for different decay rates. The labels in the legend denote $\{{\gamma }_{sp},{\gamma }_{pg}\}/(2\pi 100\mathrm{kHz}$). (c) and (d) Intensity output time traces from the experiment, MPS pollution model, and standard MPS (depicted by blue-solid, orange-dashed, and green-dot-dashed lines, respectively), for input rates of (c) 4.2 ph/$\mu \mathrm{s}=0.55/{\tau }_b$ and (d) 7.1 ph/$\mu \mathrm{s}=0.92/{\tau }_b$. The pollution toy model with ${\gamma }_{sp}/2\pi ={\gamma }_{pg}/2\pi =100$ kHz shows better agreement with the experimental time traces than the original MPS model with ${\gamma }_{ss}/2\pi =40$ kHz. Correlation function $g^{\left(2\right)}$ for (e) 3.0 ph/$\mu \mathrm{s}=0.39/{\tau }_b$ and (f) 7.7 ph/$\mu \mathrm{s}=1/{\tau }_b$ for the toy model with pollution decay rates as above, compared with experiment and the original MPS model with ${\gamma }_{ss}/2\pi =120$ kHz. All other parameters as in the rest of this paper. The Rydberg interaction is modeled by three exponentials as described in Appendix pp3-s1. MPS density matrix simulations used time step $0.01/\gamma$, $N=60$, and bond dimensions (c) (original) $D=100$, (pollution) $D=120$; (d) (original) $D=140$, (pollution) $D=180$; (e) (original) $D=140$, (pollution) $D=200$; (f) (original) $D=180$, (pollution) $D=260$.

Figure 10

(a) The Rydberg interaction is truncated at $V=20\gamma$ and is approximated using a sum of interactions with exponential form. The specific interaction being modeled has $C_6/2\pi =1.8782\times {10}^{14}\mathrm{Hz}\mu {\mathrm{m}}^{-6}$. Note that we only require a good approximation at the atomic positions, and in between the potential may take arbitrary values. Here we plot the approximate potential only at the positions of the atoms assuming a spin chain with interatomic distance $a=2.5\mu \mathrm{m}$. (b) The number of exponentials used leads to only minor differences in the time trace shown here for an input photon rate of 10.4 ph/$\mu \mathrm{s}$. Parameters used in MPS simulations: $\gamma /2\pi =6.065$ MHz, $\text{OD}=33$, ${\gamma }_{\mathrm{ss}}/2\pi =40$ kHz, $N=60$, $D=100$. The atomic cloud has a Gaussian distribution $n\left(z\right)=exp[-z^2/\left(2{\sigma }^2\right)]$ with $\sigma =23.6\mu \mathrm{m}$.

Figure 11

Convergence of the predicted steady-state output photon rate with bond dimension $D$ of the MPS used in the quantum jump trajectories. Parameters used in MPS simulations: $\gamma /2\pi =6.065$ MHz, $\text{OD}=33$, ${\gamma }_{\mathrm{ss}}/2\pi =40$ kHz, $N=70$. The Rydberg interaction is modeled by five exponentials as described in Appendix pp3. The atomic cloud has a Gaussian distribution $n\left(z\right)=exp[-z^2/\left(2{\sigma }^2\right)]$ with $\sigma =23.6\mu \mathrm{m}$. The vertical bars denote the statistical errors $s$ in the averaging of intensities over all trajectories.

Figure 12

Convergence of the time traces with bond dimension of the MPS for the input rate of 4.2 ph/$\mu \mathrm{s}$ (left) and 10.4 ph/$\mu \mathrm{s}$ (right). Higher input photon rates require larger bond dimension, where for an input rate of 4.2 ph/$\mu \mathrm{s}$ the time trace has already converged for $D=100$, while at input rate of 10.4 ph/$\mu \mathrm{s}$ convergence is not seen until $D=180$. Parameters used in MPS simulations: $\gamma /2\pi =6.065$ MHz, $\text{OD}=33$, ${\gamma }_{\mathrm{ss}}/2\pi =40$ kHz, $N=60$. The Rydberg interaction is modeled by five exponentials as described in Appendix pp3-s1. The atomic cloud has a Gaussian distribution $n\left(z\right)=exp[-z^2/\left(2{\sigma }^2\right)]$ with $\sigma =23.6\mu \mathrm{m}$.

Figure 13

Convergence of the time trace of the MPS simulation with the number of effective atoms $N$ used in the spin model. Here we show the time trace for an input rate of 10.4 ph/$\mu \mathrm{s}$. The simulations are all done for maximum bond dimension 100. Parameters used in MPS simulations: $\gamma /2\pi =6.065$ MHz, $\text{OD}=33$, ${\gamma }_{\mathrm{ss}}/2\pi =40$ kHz. The Rydberg interaction is modeled by five exponentials as described in Appendix pp3-s1. The atomic cloud has a Gaussian distribution $n\left(z\right)=exp[-z^2/\left(2{\sigma }^2\right)]$ with $\sigma =23.6\mu \mathrm{m}$.