Driven-dissipative dynamics of a strongly interacting Rydberg gas

We study the nonequilibrium many-body dynamics of a cold gas of ground-state alkali-metal atoms weakly admixed by Rydberg states with laser light. On a time scale shorter than the lifetime of the dressed states, effective dipole-dipole or van der Waals interactions between atoms can lead to the formation of strongly correlated phases, such as atomic crystals. Using a semiclassical approach, we study the long-time dynamics where decoherence and dissipative processes due to spontaneous emission and blackbody radiation dominate, leading to heating and melting of atomic crystals as well as particle losses. These effects can be substantially mitigated by performing active laser cooling in the presence of atomic dressing.

Figure 1

Sketch of the proposed setup. Rydberg-dressed ground-state atoms confined in the $x$-$y$ plane, for example, by an optical lattice (not shown), are polarized perpendicular to the plane by a dc electric field, ${\mathbf{E}}_{\mathrm{dc}}$, or an ac microwave field, ${\mathbf{E}}_{\mathrm{mw}}\left(t\right)$. Atoms can be in either the dressed ground state $|\tilde{g}\rangle$ (small particles) with a dipole moment ${\mathbf{d}}_{\tilde{g}}$ or one of the intermediate Rydberg states $|m\rangle$ (large particle) with a dipole moment ${\mathbf{d}}_m$ (see text), resulting in strong dipole-dipole interactions.

Figure 2

Qualitative sketch of the energy levels (black lines), lasers (thin solid arrows), and decay paths (wiggly arrows) for (a) the dc electric field dressing scheme and (b) the ac microwave dressing scheme. In both schemes the ground state $|g\rangle$ is weakly dressed with a Rydberg state $|r_F\rangle$ ($|r\rangle$) using a far detuned laser with Rabi frequency ${\Omega }_r$ and detuning ${\Delta }_r\gg {\Omega }_r$. In the long-time limit cascaded decay from the Rydberg states $|r\rangle$, $|s\rangle$, or $|r_F\rangle$ (wiggly arrows) will populate intermediate Rydberg states $|m_{\left(F\right)}\rangle$ due to spontaneous emission and blackbody radiation. In panel (a) a static electric field, ${\mathbf{E}}_{\mathrm{dc}}=F{\mathbf{e}}_z$, polarizes the atoms, leading to new Stark-split eigenstates, indicated with an index $F$. Both, the Rydberg state $|r_F\rangle$ and intermediate states $|m_F\rangle$ will obtain an intrinsic dipole moment ${\mathbf{d}}_r$ and ${\mathbf{d}}_m$, respectively (thick arrows). In panel (b) a near resonant ac microwave field, ${\mathbf{E}}_{\mathrm{mw}}\left(t\right)$, with Rabi frequency ${\Omega }_s$ couples two Rydberg states $|r\rangle$ and $|s\rangle$ and induces an oscillating dipole moment proportional to ${\mathbf{d}}_{rs}$. The intermediate Rydberg states $|m\rangle$ have no dipole moment.

Figure 3

Single trajectory of a semiclassical molecular dynamics simulation studying the nonequilibrium and melting dynamics of a 2D Rydberg-dressed crystal as a function of time $t$. Heating due to decay from the Rydberg states and population of intermediate states $|m_{\left(F\right)}\rangle$ leads to a rapid increase of the mean kinetic energy, $E_{\mathrm{kin}}\left(t\right)$ (solid line, left axis), and a reduction of the particle number due to losses (dotted line, right axis). In panel (a) the dipole moment in the Rydberg state is created using the dc electric field dressing scheme of Sec. 2b and the same atomic parameters as given there. Initially we choose a density $n_{2\mathrm{D}}=1\mu {\text{m}}^{-2}$. Two insets show histograms of the momentum distribution at $t=27\mu$s and $t=40\mu$s. In panel (b) the dipole moment in the Rydberg state is created using the ac microwave dressing scheme of Sec. 2c with the same atomic parameters as given in the text. The initial density is $n_{2\mathrm{D}}=0.2\mu {\text{m}}^{-2}$. In both panels the simulation time corresponds to the effective lifetime of the dressed ground-state atoms, ${\tau }_{\tilde{g}}=125\mu$s or ${\tau }_{\tilde{g}}=12.8$ ms, respectively. The thin dashed line is the initial melting temperature, $T_M$, of the crystal.

Figure 4

Energies ${\omega }_{\alpha }$, states $|\alpha \rangle$ (solid black lines), lasers (solid arrows), and decay paths (wiggly arrows) for (a) the dc electric field and (b) the ac microwave dressing scheme. Additionally to Fig. 2 a lower-lying excited state $|e\rangle$ is coupled in both schemes to the ground state $|g\rangle$ using a near-resonant laser with Rabi frequency ${\Omega }_e$. State $|e\rangle$ decays directly to the ground state with a rate ${\Gamma }_e$ yielding a closed cycle for laser cooling. Cascaded decay from the Rydberg states $|r\rangle$, $|r_F\rangle$, and $|s\rangle$ (not shown) will populate intermediate Rydberg states $|m_{\left(F\right)}\rangle$ and $|m_{\left(F\right)}^{\prime }\rangle$ according to the decay rates ${\Gamma }_{\alpha \beta }$ (see text).

Figure 5

Energy eigenvalues $E\left(\mathbf{r}\right)$ (dressed BO potential surfaces) of Rydberg-dressed atoms confined in a 2D geometry obtained by diagonalizing the Hamiltonian of Eq. (16). Here, $\mathbf{r}=r(\cos \varphi ,\sin \varphi ,0)$ is the 2D coordinate in the plane with $z=0$. Atoms are polarized by the dc electric field dressing scheme of Sec. 4b1 and ${\Delta }_r=2{\Omega }_r$. Energy surfaces are labeled using arrows: $E_{\tilde{g}\tilde{g}}$ (solid line), $E_{\tilde{g}\tilde{r}+\tilde{r}\tilde{g}}$ (dash-dotted line), and $E_{\tilde{r}\tilde{r}}$ (dashed line). At the Condon point $r=r_c$ an avoided crossing leads to a rapid change of the ground-state interaction potential $E_{\tilde{g}\tilde{g}}$: For $r>r_c$ atoms prepared in the dressed ground state $|\tilde{g}\rangle$ are weakly interacting, $E_{\tilde{g}\tilde{g}}\sim {({\Omega }_r/2{\Delta }_r)}^4{d}_0^2/r^3$, while for $r<r_c$ the potential inherits the character of the Rydberg-Rydberg interaction, $E_{\tilde{g}\tilde{g}}\sim d_0^2/r^3$.

Figure 6

Illustration of the ac microwave dressing scheme. (Left) Energies ${\omega }_{\alpha }$, states $|\alpha \rangle$ (solid lines), and lasers (solid arrows) of the system described by the Hamiltonian of Eq. (20). The unitary transformation $U_{rs}$ of Eq. (22) diagonalizes the Rydberg subspace $\left\{\right|r\rangle ,|s\rangle \}$. (Right) In a rotating frame the new states $|+\rangle$ and $|-\rangle$ with energies $E_{\pm }$ are separated by an energy difference $\sqrt{{\Omega }_s^2+{\Delta }_s^2}$ and individually coupled to the ground state with Rabi frequencies ${\Omega }_{\pm }$ according to the Hamiltonian of Eq. (25).

Figure 7

(a) Energy eigenvalues $E\left(\mathbf{r}\right)$ (dressed BO potential surfaces) of Rydberg-dressed atoms confined in a 2D geometry obtained by diagonalizing the Hamiltonian of Eq. (26). Here, $\mathbf{r}=r(\cos \varphi ,\sin \varphi ,0)$ is the 2D coordinate in the plane with $z=0$. Atoms are polarized by the ac microwave dressing scheme of Sec. 4b1 and ${\Delta }_r=4{\Omega }_r$, ${\Omega }_s=1.5{\Delta }_r$ and ${\Delta }_s=0$. Energy surfaces are labeled using arrows. The energies $E_{++}^{\mathrm{MW}}\left(r\right)$ and $E_{--}^{\mathrm{MW}}\left(r\right)$ are strongly affected by the dipole interaction and at the Condon point $r_c^{\mathrm{MW}}=g\left(\alpha \right)r_c$ an avoided crossing leads to a rapid change of the ground-state interaction potential $E_{\tilde{g}\tilde{g}}^{\mathrm{MW}}\left(r\right)$ (thick line): For $r>r_c$ atoms prepared in the dressed ground state $|\tilde{g}\rangle$ are weakly interacting, $E_{\tilde{g}\tilde{g}}^{\mathrm{MW}}\sim f\left(\alpha \right){({\Omega }_r/2{\Delta }_r)}^4{d}_0^2/r^3$, while for $r<r_c$ the potential inherits the character of the Rydberg-Rydberg interaction, $E_{\tilde{g}\tilde{g}}^{\mathrm{MW}}\sim d_0^2/r^3$. The figure shows that the Condon radius is slightly shifted to larger values compared to the dc dressing scheme of Fig. 5. Panel (b) shows the dependence of the interaction strength, $f\left(\alpha \right)$ (solid line), and the Condon radius, $g\left(\alpha \right)$ (dashed line), defined in Eq. (28) on $\alpha$. There exists a region $\Delta \alpha$ where $f\left(\alpha \right)>1$ and $g\left(\alpha \right)<1.5$.

Figure 8

Contour plots of the 3D potentials $V_{3\mathrm{D}}\left(\mathbf{r}\right)$ and $V_{3\mathrm{D}}^{\mathrm{mol}}\left(\mathbf{r}\right)$ of Eqs. (29) and (30) are shown in panels (a) and (b), respectively, in units of $V_0=2{\Delta }_r{\left({\Omega }_r/2{\Delta }_r\right)}^4$. Here, $\mathbf{r}=(\rho \cos \varphi ,\rho \sin \varphi ,z)$ is the relative distance between two atoms, with $\rho$ the in-plane radial coordinate and $z$ the transversal coordinate. Brighter regions represent stronger repulsive interactions. Two saddle points (circles) located at $({\rho }_{☆},\pm z_{☆})$ separate the repulsive from the attractive short-range region.

Figure 9

Relative decay rates ${\Gamma }_{r,n\ell }/{\Gamma }_r$ of the $|r\rangle =|16d_{5/2},m_j=-5/2\rangle$ state of ${}^{85}$Rb. Green (dark gray) and gray (light gray) bars indicate decay rates to $|np\rangle$ and $|nf\rangle$ states, respectively. In both cases we summed over $j$ and $m_j$. The top and bottom panels show the relative decay rates for $T=0$ K and $T=300$ K, respectively. The total decay rate of ${\Gamma }_{16d}=2\pi \times$ 51 kHz (43 kHz) for $T=300$ K ($T=0$) agrees well with calculations carried out in [73]. Note that in the top and bottom panels the $y$ axes are cut at 5$\%$ and 10$\%$, respectively. For states with a higher decay rate percentage numbers on the left side of the bars indicate their value.

Figure 10

Stark map: Atomic energy levels $E_{\alpha }\left(F\right)$ of ${}^{85}$Rb as a function of the field strength $F$ around the state $|16d,m=0\rangle$. Note that we plot only states with magnetic quantum number $m=0$. States with an angular quantum number $\ell >3$ have a negligible quantum defect and show a linear Stark effect, while $s$, $p$ and $d$ states show a quadratic. The energy of the state which for $F\to 0$ connects to $|16d,m=0\rangle$ (thick line) is well separated in energy from neighboring states up to a field strength of $F\approx 300$ kV/m.

Figure 11

Electric dipole moment $d_{\alpha }=\langle {\alpha }_F|{\widehat{d}}_z|{\alpha }_F\rangle$ of the Stark split states $|{\alpha }_F\rangle$ of a ${}^{85}$Rb atom in an electric dc field with strength $F=3$ kV/cm. States are numbered with increasing energy, for example, the state $|16d,m=0\rangle$ corresponds to $\alpha =1017$. The figure shows that depending on the state the dipole moment $d_{\alpha }$ can be positive (parallel) or negative (antiparallel), leading to a repulsive or attractive interaction.

Figure 12

Time evolution of the atomic dipole moment $d\left(t\right)$ according to population of various internal states governed by Eq. (40) for a ${}^{85}$Rb atom. In panel (a) the atom is polarized using the dc electric field dressing scheme while in panel (b) it is polarized with the ac microwave dressing scheme using the same parameters as given in Sec. 6a. In both cases the dipole moment of the dressed ground state is $d_{\tilde{g}}=30$ D. In the case of the dc electric field dressing scheme [panel (a)] the dipole moment fluctuates between large positive and negative values due to population of intermediate states, while in the case of the ac microwave dressing scheme [panel (b)] the dipole moment jumps between $d\left(t\right)=0$ and $d\left(t\right)=30$ D.

Figure 13

Single kinetic energy trajectories, $E_{\mathrm{kin}}$, as a function of time $t$ for a single Rydberg-dressed ${}^{85}$Rb atom (thin black lines). We observe an increase in kinetic energy due to photon recoil due to decay events from the Rydberg state. The thick line is a linear fit of 50 single trajectories, which gives a mean heating rate of $2\pi \times 106.3\text{kHz}/\text{ms}$.

Figure 14

dc electric field case. We numerically investigate the heating rate (top panel) and the remaining particle number after a time ${\tau }_{\tilde{g}}=125\mu$s (bottom panel) as a function of the density $n_{2\mathrm{D}}$ for different laser cooling parameters $\beta$ and lattice depths $V_0$. Parameters see text.

Figure 15

ac microwave field case. We numerically investigate the heating rate (top panel) and the remaining particle number after a time ${\tau }_{\tilde{g}}=12.8$ ms (bottom panel) as a function of the density $n_{2\mathrm{D}}$ for different laser cooling parameters $\beta$ and lattice depths $V_0$. Parameters see text.