Dissipative dynamics of atomic and molecular Rydberg gases: Avalanche to ultracold plasma states of strong coupling

When formed with sufficient density, a Rydberg gas of principal quantum number n₀ > 30 undergoes a spontaneous avalanche to form an ultracold plasma [23, 32, 46]. Collisional rate processes combine with ambipolar hydrodynamics to govern the properties of the evolving plasma. For a molecular Rydberg gas, neutral fragmentation, occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering. Neutral dissociation combined with radial expansion in a shaped distribution of charged particles, can give rise to striking effects of self-assembly and spatial correlation [54, 57].

The formation of a molecular ultracold plasma requires the conditions of local temperature and density afforded by a high mach-number skimmed supersonic molecular beam. Such a beam propagates at high velocity in the laboratory, with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame [58]. The low-temperature gas in a supersonic molecular beam differs in three important ways from the atomic gas laser-cooled in a magneto-optical trap (MOT).

The milli-Kelvin temperature of the gas of ground-state NO molecules entrained in a beam substantially exceeds the sub-100 micro-Kelvin temperature of laser-cooled atoms in a MOT. However, the evolution to plasma tends to erase this distinction, and the two further characteristics that distinguish a beam offer important advantages for ultracold plasma physics: Charged-particle densities in a molecular beam can exceed those attainable in a MOT by orders of magnitude. A great many different chemical substances can be seeded in a free-jet expansion, and the option this affords to form other molecular ultracold plasmas, introduces interesting and potentially important new degrees of freedom governing the dynamics of their evolution.

Seeded in a skimmed supersonic molecular beam, nitric oxide forms different phase-space distributions in the longitudinal (propagation) and transverse coordinate dimensions. As it propagates in z, the NO molecules reach a terminal laboratory velocity, u_∥, of about 1400 m s⁻¹, which varies with the precise seeding ratio.

The distribution of v_∥, narrows to define a local temperature, T_∥, of approximately 0.5 K. The beam forms a Gaussian spatial distribution in the transverse coordinates, x and y. In this plane, the local velocity, v_⊥(r) is defined for any radial distance almost entirely by the divergence velocity of the beam, u_⊥(r). Phase-space sorting cools the temperature in the transverse coordinates, T_⊥ to a value as low as ∼5 mK [58].

The stagnation pressure and seeding ratio determine the local density distribution as a function of z. For example, expanding from a stagnation pressure of 500 kPa with a 1:10 seeding ratio, a molecular beam propagates 2.5 cm to a skimmer and then 7.5 cm to a point of laser interaction, where it contains NO at a peak density of $1.6\times {10}^{14}$ cm⁻³.

Here, crossing the molecular beam with a laser beam tuned to the transition sequence, ${\rm{X}}{}^{2}{{\rm{\Pi }}}_{1/2}\,N^{\prime\prime} =1\mathop{\longrightarrow }\limits^{{\omega }_{1}}{\rm{A}}{}^{2}{{\rm{\Sigma }}}^{+}\,N^{\prime} \,=0\mathop{\longrightarrow }\limits^{{\omega }_{2}}{n}_{0}f(2)$ forms a Gaussian ellipsoidal volume of Rydberg gas in a single selected principal quantum number, n₀, orbital angular momentum, ℓ = 3, NO⁺ core rotational quantum number, ${N}^{+}=2$ and total angular momentum neglecting spin, N = 1.

A typical ω₁ pulse energy of 2 μJ and a Gaussian width of 0.2 mm serves to drive the first step of this sequence in a regime of linear absorption. Overlapping this volume by an ω₂ pulse with sufficient fluence to saturate the second step forms a Rydberg gas ellipsoid with a nominal peak density of 5 × 10¹² cm⁻³ [46, 58]. Fluctuations in the pulse energy and longitudinal mode of ω₁ cause the real density to vary. For certain experiments, we find it convenient to saturate the ω₁ transition, and vary the density of Rydberg gas by delaying ω₂. An ω₁–ω₂ delay, Δt, reduces the Rydberg gas density by a precise factor, ${e}^{-{\rm{\Delta }}t/\tau }$, where τ is the 200 ns radiative lifetime of NO ${\rm{A}}{}^{2}{{\rm{\Sigma }}}^{+}\,N^{\prime} =0$ [59, 60].

The density distribution of a Rydberg gas defines a local mean nearest neighbor distance, or Wigner–Seitz radius of ${a}_{{ws}}\,={\left(3/4\pi \rho \right)}^{1/3}$, where ρ refers to the local Rydberg gas density. For example, a Rydberg gas with a density of ρ₀ = 0.5 × 10¹² cm⁻³ forms an Erlang distribution [61] of nearest neighbor separations with a mean value of $2{a}_{{ws}}=1.6\,\mu {\rm{m}}$.

A semi-classical model [38] suggests that 90 percent of Rydberg molecule pairs separated by a critical distance, ${r}_{c}=1.8\cdot 2{n}_{0}^{2}{a}_{0}$ or less undergo Penning ionization within 800 Rydberg periods. We can integrate the Erlang distribution from r = 0 to the critical distance r = r_c for a Rydberg gas of given n₀, to define the local density of Penning electrons (ρ_e at t = 0) produced by this prompt interaction, for any given initial local density, ρ₀ by the expression:

$$\begin{eqnarray}&&{\rho }_{e}({\rho }_{0},{n}_{0})=\displaystyle \frac{0.9}{2}\cdot 4\pi {\rho }_{0}^{2}{\int }_{0}^{{r}_{c}}{r}^{2}{{\rm{e}}}^{-\tfrac{4\pi }{3}{\rho }_{0}{r}^{3}}{\rm{d}}r.\end{eqnarray} \tag{ 2 }$$

Evaluating this definite integral yields an equation in closed form that predicts the Penning electron density for any particular initial Rydberg density and principal quantum number.

$$\begin{eqnarray}&&{\rho }_{e}({\rho }_{0},{n}_{0})=\displaystyle \frac{0.9{\rho }_{0}}{2}(1-{{\rm{e}}}^{-\tfrac{4\pi }{3}{\rho }_{0}{r}_{c}^{3}}).\end{eqnarray} \tag{ 3 }$$

Prompt Penning ionization acts on the portion of the initial nearest-neighbor distribution in the Rydberg gas that lies within r_c. When a molecule ionizes, its collision partner relaxes to a lower principal quantum number, $n^{\prime} \lt {n}_{0}/\sqrt{2}$. This close-coupled interaction disrupts the separability of Rydberg orbital configurations in the Penning partner. This causes mixing with core penetrating states that are strongly dissociative. Penning partners are thus very likely to dissociate, leaving a spatially isolated distribution of ions. We refer to the spatial correlation that results as a Penning lattice [62]. The extent of this effect varies depending on the local density and the selected initial principal quantum number. Figure 1 shows the degree to which Rydberg gases with initial principal quantum numbers from 30 to 80 form a Penning lattice for an initial density of 1 × 10¹² cm⁻³.

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The electrons produced by prompt Penning ionization start an electron impact avalanche. The kinetics of this process are well described by a set of coupled rate equations that account for state-to-state electron-Rydberg inelastic scattering, electron-impact ionization and three-body ion-electron recombination [40, 50–52] using detailed rate coefficients, k_ij, k_i,ion and ${k}_{i,{tbr}}$ validated by MD simulations [41].

$$\begin{eqnarray}\begin{array}{rcl}-\displaystyle \frac{d{\rho }_{i}}{{dt}} & = & \sum _{j}{k}_{{ij}}{\rho }_{e}{\rho }_{i}-\sum _{j}{k}_{{ji}}{\rho }_{e}{\rho }_{j}\\ & & +{k}_{i,{ion}}{\rho }_{e}{\rho }_{i}-{k}_{i,{tbr}}{\rho }_{e}^{3}\end{array}\end{eqnarray} \tag{ 4 }$$

and,

$$\begin{eqnarray}&&\displaystyle \frac{d{\rho }_{e}}{{dt}}=\sum _{i}{k}_{i,{ion}}{\rho }_{e}^{2}-\sum _{i}{k}_{i,{tbr}}{\rho }_{e}^{3}.\end{eqnarray} \tag{ 5 }$$

The relaxation of Rydberg molecules balances with collisional ionization to determine an evolving temperature of avalanche electrons to conserve total energy per unit volume.

$$\begin{eqnarray}&&{E}_{{tot}}=\displaystyle \frac{3}{2}{k}_{B}{T}_{e}(t){\rho }_{e}(t)-R\sum _{i}\displaystyle \frac{{\rho }_{i}(t)}{{n}_{i}^{2}}.\end{eqnarray} \tag{ 6 }$$

Here, for simplicity, we neglect the longer-time effects of Rydberg predissociation and electron-ion dissociative recombination [51].

Such calculations show that the conversion from Rydberg gas to plasma occurs on a timescale determined largely by the local Penning electron density, or Penning fraction, ${P}_{f}={\rho }_{e}/{\rho }_{0}$, which depends on the local density of Rydberg molecules and their initial principal quantum number.

Avalanche times predicted by coupled rate equation calculations range widely. For example, in a model developed for experiments on xenon, simulations predict that a Rydberg gas with n₀ = 42 at a density of 8.8 × 10⁸ cm⁻³ (P_f = 6 × 10⁻⁵) avalanches with a half time of 40 μs [37]. At an opposite extreme, rate equations estimate that a Rydberg gas of NO with n₀ = 60 at a density of $1\times {10}^{12}\,{\mathrm{cm}}^{-3}$ (P_f = 0.3) rises to plasma in about 2 ns [51].

Selective field ionization (SFI) probes the spectrum of binding energies in a Rydberg gas. Applied as a function of time after photoexcitation, SFI maps the evolution from a state of selected initial principal quantum number, n₀, to plasma [54]. Figure 2 shows SFI spectra taken at a sequence of delays after the formation of 49f(2) Rydberg gases of varying density.

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Here, we can see that a 49f(2) Rydberg gas with an estimated initial density ρ₀ = 3 × 10¹¹ cm⁻³ relaxes to plasma on a timescale of about 500 ns. Observations such as these agree well with the predictions of coupled rate-equation calculations. This variation in relaxation dynamics with ρ₀ and n₀ simply reflect corresponding variations in the density of prompt Penning electrons available to initiate the avalanche to plasma.

Figure 3 illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum number match when plotted as a function of time scaled by the ultimate plasma frequency and fraction of prompt Penning electrons. The dashed line gives an approximate account of the scaled rate of avalanche under all conditions of Rydberg gas density and initial principal quantum number in terms of the simple sigmoidal function:

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{e}}{{\rho }_{0}}=\displaystyle \frac{a}{b+{e}^{-c\tau }},\end{eqnarray} \tag{ 7 }$$

where,

$$\begin{eqnarray}&&\tau =t{\omega }_{e}{P}_{f}^{3/4},\end{eqnarray} \tag{ 8 }$$

in which ω_e is the plasma frequency after avalanche, P_f is the fraction of prompt Penning electrons, and a = 0.000 62, b = 0.000 82 and c = 0.075 are empirical coefficients.

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As outlined above, the local density and principal quantum number together determine the rate at which a Rydberg gas avalanches to plasma. Our experiment crosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameter TEM₀₀ ω₁ laser beam to produce a Gaussian ellipsoidal distribution of molecules excited to the A ${}^{2}{{\rm{\Sigma }}}^{+}$ $v=0,\,N^{\prime} =0$ intermediate state. A larger diameter ω₂ pulse then drives a second step that forms a Rydberg gas in a single ${n}_{0}f(2)$ state with the spatial distribution of the intermediate state.

We model this shaped Rydberg gas as a system of 100 concentric ellipsoidal shells of varying density [53]. Coupled rate equations within each shell describe the avalanche to plasma. This rate process proceeds from shell to shell with successively longer induction periods, determined by the local density as detailed above. The rising conversion of Rydberg molecules to ions plus neutral dissociation products conserves the particle number in each shell. We assume that local space charge confines electrons to shells, conserving quasi-neutrality. Electrons exchange kinetic energy at the boundaries of each shell, which determines a single plasma electron temperature.

The upper frame of figure 4 shows contours of NO⁺ ion density after 100 ns obtained from a shell-model coupled rate-equation simulation of the avalanche of a Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state, $50f(2)$ and a density of 2 × 10¹¹ cm⁻³. Here, we simulate a relaxation that includes channels of predissociation at every Rydberg level and redistributes the energy released to electrons, which determines a uniform rising electron temperature for all shells.

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For comparison, the lower frame plots curves describing the ion density of each shell as a function of time from 20 to 100 ns, as determined by applying equation (7) for the local conditions of initial Rydberg gas density. This numerical approximation contains no provision for predissociation. Coupled rate-equation simulations for uniform volumes show that predissociation depresses yield to some degree, but has less effect on the avalanche kinetics [51]. Therefore, we can expect sets of numerically estimated shell densities, scaled to agree with the simulated ion density at the elapsed time of 100 ns to provide a reasonable account of the earlier NO⁺ density profiles as a function of time.

For each time step, the difference, ρ₀–ρ_e defines the neutral population of each shell. We assign a fraction of this population to surviving Rydberg molecules, such that the total population of NO* as a function of time agrees with the prediction of the shell-model simulation, as shown in figure 5. We consider the balance of this neutral population to reflect NO* molecules that have dissociated to form N(⁴S) + O(³P). Figure 4 plots these surviving Rydberg densities as a function of radial distance for each evolution time. At the initial density of this simulation, note at each time step that a higher density of Rydberg molecules encloses the tail of the ion density distribution in x.

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We regard the ions as initially stationary. The release of electrons creates a radial electric potential gradient, which gives rise to a force, $-e{\rm{\nabla }}{\phi }_{k,j}(t)$, that accelerates the ions in shell j in direction k according to [63]:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{-e}{m^{\prime} }{\rm{\nabla }}{\phi }_{k,j}(t) & = & \displaystyle \frac{\partial {u}_{k,j}(t)}{\partial t}\\ & = & \displaystyle \frac{{k}_{B}{T}_{e}(t)}{m^{\prime} {\rho }_{j}(t)}\displaystyle \frac{{\rho }_{j+1}(t)-{\rho }_{j}(t)}{{r}_{k,j+1}(t)-{r}_{k,j}(t)},\end{array}\end{eqnarray} \tag{ 9 }$$

where ${\rho }_{j}(t)$ represents the density of ions in shell j.

The instantaneous velocity, ${u}_{k,j}(t)$ determines the change in the radial coordinates of each shell, ${r}_{k,j}(t)$,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {r}_{k,j}(t)}{\partial t}={u}_{k,j}(t)={\gamma }_{k,j}(t){r}_{k,j}(t),\end{eqnarray} \tag{ 10 }$$

which in turn determines shell volume and thus its density, ρ_j(t). The electron temperature supplies the thermal energy that drives this ambipolar expansion. Ions accelerate and T_e falls according to:

$$\begin{eqnarray}&&\displaystyle \frac{3{k}_{B}}{2}\displaystyle \frac{\partial {T}_{e}(t)}{\partial t}=-\displaystyle \frac{m^{\prime} }{\sum _{j}{N}_{j}}\sum _{k,j}{N}_{j}{u}_{k,j}(t)\displaystyle \frac{\partial {u}_{k,j}(t)}{\partial t},\end{eqnarray} \tag{ 11 }$$

where we define an effective ion mass, $m^{\prime}$, that recognizes the redistribution of the electron expansion force over all the NO⁺ charge centers by resonant ion-Rydberg charge exchange, which occurs with a very large cross section [40].

$$\begin{eqnarray}&&m^{\prime} =\left(1+\displaystyle \frac{{\rho }_{j}^{* }(t)}{{\rho }_{j}(t)}\right)m,\end{eqnarray} \tag{ 12 }$$

in which ${\rho }_{j}^{* }(t)$ represents the instantaneous Rydberg density in shell j.

The initial avalanche in the high-density core of the ellipsoid leaves few Rydberg molecules, so this term has little initial effect. Rydberg molecules predominate in the lower-density wings. There, momentum sharing by charge exchange assumes a greater importance.

We see this most directly in the ω₂ absorption spectrum of transitions to states in the ${n}_{0}f(2)$ Rydberg series, detected as the long-lived signal that survives a flight time of 400 μs to reach the imaging detector. The balance between the rising density of ions and the falling density of Rydberg molecules depends on the initial density of electrons produced by prompt Penning ionization. As clear from equation (3), this Penning fraction depends sensitively on the principal quantum number, and for all principal quantum numbers, on the initial Rydberg gas density.

Figure 6 shows a series of ω₂ late-signal excitation spectra for a set of initial densities. Here, we see a clear consequence of the higher-order dependence of Penning fraction—and thus the NO⁺ ion—NO* Rydberg molecule balance—on n₀, the ω₂-selected Rydberg gas initial principal quantum number. This Penning-regulated NO⁺ ion—NO* Rydberg molecule balance appears necessary as a critical factor in achieving the long ultracold plasma lifetime required to produce this signal. We are progressing in theoretical work that explains the stability apparently conferred by this balance.

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Avalanche in the core of the Rydberg gas ellipsoid forms a gas of energetic electrons. This electron gas expands, beginning a process that radially accelerates core cations and quenches the electron kinetic energy. NO⁺ cations follow electrons into the wings of the NO* Rydberg gas. There, resonant ion-Rydberg charge exchange, which occurs on a distance scale at least 1000 times larger than the Langevine cross section [64], far outpaces Rydberg–Rydberg Penning interactions. The recurring charge exchange between NO⁺ ions and NO* Rydberg molecules redistributes the ambipolar force of the expanding electron gas [40, 65], further quenching the electron temperature and equalizing the ion and Rydberg velocities.

This momentum matching effectively channels electron energy through ion motion into the overall ±x motion of gas volumes in the laboratory. The internal kinetic energy of the plasma, which at this point is defined almost entirely by the ion-Rydberg relative motion, falls. Spatial correlation develops, and over a period of 500 ns, the system forms the plasma/high-Rydberg quasi-equilibrium dramatically evidenced by the SFI results in figure 2.

Electron energy released in the avalanche of the Rydberg gas to plasma thus redistributes, causing the ellipsoidal gas to bifurcate [54, 57], forming very long-lived, separating charged-particle distributions. We capture the electron signal from these recoiling volumes on an imaging detector as pictured in figure 7. Here, momentum matching preserves density and enables ions and Rydberg molecules to relax to positions that minimize potential energy, building spatial correlation.

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This classical description of avalanche and relaxation forms an important point of reference from which to interpret our experimental observations. The laser-crossed molecular beam illumination geometry creates a Rydberg gas with a distinctively shaped high-density spatial distribution. This initial condition has an evident effect on the evolution dynamics.

We have developed semi-classical models that explicitly consider the coupled rate and hydrodynamic processes governing the evolution from Rydberg gas to plasma using a realistic, ellipsoidal representation of the ion/electron and Rydberg densities [53]. We emphasize that no combination of initial conditions can produce a simulation that conforms classically with the state of arrested relaxation we observe experimentally.

Remarkable experimental images show that spontaneous avalanche to plasma splits the core of an ellipsoidal Rydberg gas of nitric oxide. As ambipolar expansion quenches the electron temperature of this core plasma, long-range, resonant charge transfer from ballistic ions to frozen Rydberg molecules in the wings of the ellipsoid quenches the ion-Rydberg molecule relative velocity distribution. This sequence of steps gives rise to an extraordinary mechanics of self-assembly, in which the kinetic energy of initially formed hot electrons and ions drives an observed separation of plasma volumes. These dynamics redistribute ion momentum, channeling electron energy into a reservoir of mass-transport. This starts a process that evidently anneals separating volumes to a state of ultracold, correlated ions, electrons and Rydberg molecules.

We have devised a three-dimensional spin model to describe this arrested state of the ultracold plasma in terms of two, three and four-level dipole–dipole energy transfer interactions (spin flip-flops), together with Ising interactions that arise from the concerted pairwise coupling of resonant pairs of dipoles [55].

The Hamiltonian includes the effects of onsite disorder owing to (1) the distribution of high-Rydberg states populated in the ensemble and (2) the unique electrostatic environment of every dipole. Extending ideas developed for simpler systems [66, 67], one can make a case that this disorder gives rise to slow dynamics [56], including an arrest in the relaxation of NO Rydberg molecules to predissociating states of lower principal quantum number.

Systems of realistic dimension ought to thermalize by energy transfer that spreads from rare but inevitable ergodic volumes (Griffiths regions) [68–71]. However, a feature in the self-assembly of the molecular ultracold plasma may preclude destabilization by rare thermal domains: Whenever the quenched plasma develops a delocalizing Griffiths region, the local predissociation of relaxing NO molecules promptly proceeds to deplete that region to a void of no consequence.

In summary, the classical dynamics of avalanche and bifurcation appear to create a quenched condition of low temperature and high disorder in which dipole–dipole interactions drive self-assembly to a localized state purified by the predissociation of thermal regions. We suggest that this state of the quenched ultracold plasma offers an experimental platform for studying quantum many-body physics of disordered systems.