Ionization of Rydberg atoms embedded in ultracold plasma due to electron–atom interaction Open Access

Ultracold plasma (UCP) is experimentally obtained by resonant photoionization of laser-cooled atoms. Since the resonant processes do not provide additional kinetic energy, the plasma thus created is still at a very low temperature. The ions are typically around the temperature $∼$ 10  $μ$K, whereas the electrons have a substantially higher temperature of the order of $∼$ 100 mK.¹ Similar to moderately hot plasma that has an intermediate temperature range of $102$– $105$ K,² UCPs are also partially ionized plasmas. The other end of the temperature spectrum is the high-temperature plasmas, which are also known as High Energy Density Plasmas (HEDPs). They are fully ionized plasmas, and their behavior has been extensively studied as they play a key role in military applications and nuclear fusion research and technology.² Though UCPs and HEDPs are radically different in density and temperature, their characteristics overlap for low screening strength (⁠ $κ$⁠) and low coupling constant (⁠ $Γ$⁠),^2–4 which paves the way for multi-scale theories at the interface of atomic physics, plasma physics, and condensed matter physics. As a result, well-diagnosed experiments on ultracold plasma and its theoretical understanding can provide insights into the dynamics of HED plasma, ionosphere dynamics, and internal mechanisms within the core of white dwarfs and gas-giant planets.

Several research groups have modeled the evolution of ultracold plasma using hydrodynamical equations, which could explain phenomena such as ultrafast electron cooling, plasma expansion, and the appearance of strongly correlated regime.^4,6 Most of these consider the UCP as a dilute plasma since the mean inter-atomic distance is large enough, and the interactions between the constituents are very few. On the other hand, hot plasmas (HEDPs) have been analyzed using the classical, non-equilibrium thermodynamical approach, and the basic plasma properties such as quasi-neutrality, Debye shielding, and plasma oscillations are all explained thereon. Similarly, extending this classical approach to UCP has provided insights into some of their collective behavior, such as electron–electron correlation, the temperature dynamics of the electrons, the properties of expanding plasma cloud, etc.^2–6 However, to account for the transport properties, electrical conductivity, and ionization characteristics, a quantum treatment to the UCP system is required. This is because the treatment of electron–atom collision is not trivial within a classical approach.⁷ 

One feature commonly encountered in UCP is the presence of neutral atoms in high-lying Rydberg states. There are several phenomena that create atoms in Rydberg states inside an UCP. In the first case, some ions recombine with electrons through three-body recombination and get captured in Rydberg states instead of the ground state.⁴ In the second case, the photoionizing laser is a pulse and, hence, will have a spectral width. Atoms that interact with the tail end of this spectrum will only reach Rydberg states instead of continuum.^5,6 In a third case, as in experiments described in Ref. 5, the atoms are specifically excited to Rydberg states in order to study their effects. In all cases, these Rydberg atoms interact with other Rydberg atoms and also electrons from ionized atoms. These interactions cause the Rydberg atoms to ionize further, resulting in an increase in plasma density over and above the initial density. The phenomena are explained either using a fully classical approach using Coulomb interaction⁶ or a semiclassical approach, which accounts for quantum effects such as screening and diffraction. It is important to note that both moderately hot plasmas and UCPs are partially ionized, and quantum effects become important in accounting for the electron–atom interaction in dense matter. The bound state interactions, which may be particularly important for such systems, are not trivial within an analytical approach. Hence, we investigate the quantum nature of the interaction of free electrons with Rydberg atoms and compute the probability of ionization of the Rydberg atoms.

We begin by looking at the interaction potential in the moderately hot plasma. Moderately hot plasmas, like the ultracold plasma, are partially ionized, and the interaction with bound states plays a significant role in both cases. Rosmej et al. treated these bound states using a quantum statistical approach and derived the scattering potential for the electron–atom interaction in a plasma medium. Here, the bound states were considered to be atoms, and the plasma medium consisted of electrons, atoms, and ions. This approach was used to calculate virial coefficients in alkali plasmas.^7,8 It was further helpful in determining the effect of screening on the plasma conductivity for noble gas plasmas.⁹ Dzhumagulova et al.⁷ used these interaction potentials to calculate momentum transfer cross section for the Helium semiclassical plasma. Compared to moderately hot plasma, UCP has a longer Debye radius (⁠ $rD$⁠), hence a smaller screening parameter (⁠ $κ$⁠), along with a smaller degeneracy parameter. These three parameters define the nature of interaction potential for all constituents of any plasma, which, in turn, can be used to understand the dynamics of the overall plasma.

In the case of moderately hot plasma, the interaction potential is obtained using three components, viz., Hartree–Fock, polarization, and exchange potential. All of these involve contribution from partial ionization, screening effect due to remaining plasma, and are independent of kinetic energies. Hence, the same approach can be used for UCP as well, as all these situations are present in UCP also. In this context, we calculate a screened Hartree–Fock term using the approach used by Karakhtanov⁹ to compute interaction potential for Hydrogen plasma. Following the same approach, we replace the Coulomb interaction for the isolated system with a Debye potential for Cesium atoms. As the ions and neutral atoms in UCP are even slower than HEDP and moderately hot plasma, the adiabatic approximation of stationary ions used by Karakhtanov⁹ is even better suited for UCPs. For calculating polarization potential, we use the method as given by Paikeday and Alexander¹⁰ and Paikeday and Longstreet¹¹ In general, the polarization potential depends on the energy of free electrons, but Paikeday showed that dependence is very weak. Therefore, we neglect electron energy dependence. Exchange potential is calculated by using the method of Mittleman and Watson,¹² and we obtain a generic “optical potential” by adding the above three components and study quantum-mechanical scattering from this potential. We obtain scattering cross sections for electrons scattered by this potential and compare with the experimental results of Vanhaecke et al.,⁵ Redmer et al.,⁸ Ramazanov et al.,¹³ and Rosmej et al.¹⁴ They have created a dense sample of Rydberg atoms within an ultracold plasma and monitored ionization of these Rydberg atoms.

A theoretical explanation for this was provided by Thomas Pohl using the classical, Monte Carlo simulation of evolution of electron temperature.⁶ We use quantum-mechanical scattering and show that it plays a significant role, especially at lower temperatures (⁠ $∼1 mK$⁠) and larger interaction times (⁠ $∼100 μs$⁠). We analytically calculate the appropriate optical potential for the Rydberg atoms, mainly using the screened polarization potential, and compute the cross sections for their ionization due to electron scattering. The polarizability of the Rydberg atoms has been taken care of using the static non-relativistic dipole polarizability for hydrogen-like species in the (n,l) state.¹⁵ Our results agree qualitatively and quantitatively with the experimentally measured values of the percentage ionization of Rydberg atoms as measured by Vanhaecke et al. These experiments also indicate that the probability of ionization is much lower when the atoms occupy energy levels lower than n = 30 compared to when the value of n is much higher. Our calculations also show the reason for this behavior.

This communication is organized as follows: (i) We first provide an overview of existing theoretical models on electron–atom scattering within a plasma medium; (ii) identify the conditions of UCP under which the potentials used by us are valid; (iii) calculate the full form of potentials for Rydberg atoms embedded within a UCP; (iv) obtaining the cross section for electron scattering from a single Cesium atom, which causes it to ionize; and (v), finally, multiply with electron and atom densities to compare with the experimentally measured values of “percentage ionization.”

Here, Z represents the atomic number and n and l are the principal and azimuthal quantum numbers, respectively.

The theoretical methods mentioned earlier are used to compute the interaction potential for lower-lying states. However, in the case of atoms in the Rydberg levels, both the electron densities and polarizabilities differ drastically, resulting in different behavior for polarization, thus the interaction potentials.

Using the above-mentioned method, we compute the total optical potential for our system of Rydberg atoms embedded in Cesium ultracold plasma. Since the electron density for the Cesium atom varies for different Rydberg levels, the expressions for screened Hartree–Fock, Polarization, and Exchange potential change with the Rydberg levels. The results for n = 20, 32, and 40 are shown in Fig. 1. Here, Figs. 1(a)–1(c) show the terms screened Hartree–Fock, screened polarization potential, and the screened exchange potential, respectively. Figure 1(d) is the total of these terms, which is called optical potential.

From Fig. 1, we observe that the Hartree–Fock potential is dominant, which is expected. Based on the Green function technique, it has been shown earlier^7,14 that although for an unscreened case (without plasma medium), the polarization potential has the main contribution in the optical potential at large distances, but when we include the plasma effects and calculate screened optical potential, the Hartree–Fock term is dominant. The Hartree–Fock and the total optical potential are attractive at smaller distances but are repulsive at larger distances, as discussed in Ref. 7. The reason for this is that at smaller distances, the nuclear interaction dominates, and the electronic contribution to the interaction potential will not be that significant. However, at somewhat larger distances, the electronic contribution is of a higher order of magnitude than the nuclear contribution to the potential, giving rise to a repulsive potential.¹⁹ This repulsive potential does not take into account the electron–electron correlation effects and seems to increase with the higher Rydberg states of the Cesium atom. The higher the Rydberg state, the more the electron density functional stretches away from the nucleus, leading to more electronic contribution to the potential. Moreover, the polarization potential becomes more attractive with a higher Rydberg state due to the polarizability, which increases as $n6$ for alkali atoms. It is important to mention that while our potential plots seem to increase for $r∼10ao$⁠, our Hartree–Fock as well as optical potential asymptotically converges to 0 for $r∼30ao$ as obtained in Ref. 7. It can also be noticed that both the Hartree–Fock potential and exchange potential undergo small fractional change with a change in n. On the other hand, the polarization potential increases to a large extent as the value of n increases. The reason for this is that, in the Rydberg state, the ionic core density scarcely changes, while the excited electron has a lower density, so that the contribution of $I1$⁠, $I2$⁠, and $I3$ in the Hartree–Fock potential is practically unaffected. However, the polarization of the atom increases as $n6$ when excited to shell level n,¹⁵ hence causing a large fractional change in polarization potential.

In particular, the potential for Rydberg atom–electron interaction is our main concern for this work, which we obtain by computing the relevant Hartree–Fock, polarization, and exchange potentials and adding them to make the Optical Potential, as shown in the figure earlier.

We have employed screened polarization potential to study the Rydberg levels of the Cesium plasma as it simplifies the calculations. The attractive polarization potential for Rydberg atoms increases with the shell number.

The effect of Rydberg levels on the phase shift and the cross section has been plotted as a function of electron momentum (k $ao$⁠) in Fig. 2. It can be noted that slow-moving (low temperature) electrons in the plasma medium have a larger cross section. Additionally, it is clear from the plots that s-wave cross sections are dominant over the p-wave and the d-wave cross sections. It was expected because, for low-energy scattering, only l = 0 scattering is important. An incoming particle having a momentum $p=ℏk$ and angular momentum $L=lℏ$ is at a distance l/k from the center of potential when it is at the distance of closest approach. Now, for low-energy scattering, the value of l/k is high, but the potential is small at larger distances; hence, the phase shift and the cross section values will be small. Moreover, the s-wave with no angular momentum does not face a centrifugal potential barrier $V′(r)=ℏ22μl(l+1)r2$⁠, which becomes significant for higher-order partial waves, thus suppressing its contribution to the scattering cross section. S-wave is spherically symmetric, and it overlaps with the optical potential more effectively and gives rise to a higher phase shift. Due to these underlying mechanisms, we have used the s-wave contribution only, to obtain the total cross sections. We can observe in Fig. 2 that with increasing Rydberg levels, the phase shift due to polarization potential increases for all the partial waves, i.e., the s-wave, the p-wave, and the d-wave. In other words, the cross section increases for all the partial waves. However, the low phase shift for even higher Rydberg excitation shows the absence of any bound states in Cesium, even as high as n = 100. This is unlike Helium, where the formation of bound states within UCP was predicted by Dzhumagulova et al.,⁷ based on the Levinson theorem.²¹ 

Spontaneous ionization of Rydberg atoms within a plasma^22–24 is attributed to the interaction of electrons of plasma with Rydberg atoms. Vanhaecke et al.⁵ studied the number of Rydberg atoms remaining un-ionized within the plasma and showed its dependency on the power of the photoionizing laser. In other words, increasing laser power ionizes more atoms, hence an increase in electrons, which, in turn, increases the number of ionized Rydbergs, resulting in a decrease in remaining Rydberg atoms. A theoretical modeling for this process using Monte Carlo technique was given by Pohl et al.,⁶ and they explained the phenomena as a result of the avalanche process.

We have derived an analytical expression for the Rydberg atom–electron interaction potential, using a modified Hartree–Fock potential, and computed the relevant cross section for the case of Cesium atoms present within a UCP. To verify our methodology, at first, we compared the electron impact scattering cross section for the ground state Cesium atom using the unscreened optical potential, as shown in Fig. 3. Although the experimental and theoretical values seem to be similar for higher energy (100–200 eV) scattering, the lower energy scattering cross sections have discrepancies in theoretical and experimental data. Our cross section calculations using optical potential align with the electron impact scattering cross section calculated using the Convergent closed-coupling (CCC) method by Bray and Stelbovics,²⁵ which has a good agreement with the experimental data. However, our method gives a slightly larger cross section for low-energy electron–atom scattering for the unscreened potential than the experimentally determined values by Jaduszliwer and Chan and Kauppila and Stein.²⁶ 

Here, $σ$ refers to the cross section for a single electron–atom interaction, $σtotal$ refers to the total cross section for all electron–atom interactions, integrated over Maxwell–Boltzmann distribution for the given electron temperature, k is the momentum parameter, $No$ is the initial number of electrons in the plasma taken to be of the order of $106$ for our calculations, and P is the laser power of the laser used for plasma formation, which determines the number of electrons. The quantity c in Eq. (2) depends on the properties of the laser, such as laser detuning and saturation parameter, and also on the specific atomic system.²⁷ We use it as a constant in the fitting parameter for our data. The effect of the temperature of the electron gas is incorporated as energies of electrons, following the Maxwell–Boltzmann distribution, characteristic of the plasma.²⁸ We normalize the electron numbers data from experiments into cross sections and compare them with the theoretically obtained values by us, as shown in the top portion of Fig. 4. Plots with unconnected symbols are experimental data from Ref. 5, whereas plots with lines and symbols are our data. We can see that our calculated values are very close to the experimental value, although there is a small variation. Perhaps an experiment tuned to directly measure the interaction cross section would be required to compare and find out if any additional physics needs that affect the cross section need to be incorporated in the computation.

The bottom plot of Fig. 4 shows a comparison between the number of electrons when Rydberg atoms are ionized (closed circles) and when they are not ionized (open circles). They indicate a significant increase in the number of electrons when Rydberg atoms are present.

Vanhaecke et al.⁵ attributed their experimental results to avalanche ionization due to collisions between plasma electrons and Rydberg atoms. These observations agree well with our cross section calculations, which indicate that much of the information is contained in the associated atomic processes as had been reported in multiple theoretical descriptions of electron–atom interaction in the plasma environment.^6,30–33 A characteristic of the avalanche process is that it depends upon the interaction time—a longer interaction time results in a higher percentage of electrons and ions. This has been obtained in Fig. 2, where we have shown that for low electron energies having a longer interaction time with the Rydberg atoms, the cross section values are higher. Second, it is also observed that the percentage of ionization is proportional to the plasma density but is dependent on the Rydberg density, ruling out the effect of autoionization of the Rydberg atoms. This agrees well with our cross section calculations because the expression for total cross section contains N, which is a number of plasma electrons, which is directly proportional to the plasma density but not the density of Rydberg atoms. A contradicting observation was also made in Vanhaecke's experiment that up to 100 cm⁻¹ above the ionization limit, the percentage ionization did not depend strongly on the plasma electron energy, which seems to contradict our model because in Fig. 2, our simplistic assumption of the single electron–atom cross section calculations depending on the energy of plasma electrons might not be the exact value for the total integrated cross sections, which determine the ionization efficiency for the complete system.

We have shown that the quantum-mechanical scattering cross section can be a new approach to studying the phenomena involving electron-Rydberg atom interactions. We show the consistency of our model with the experimental results and observe that the underlying atomic processes can be crucial to understanding the ionization of Rydberg atoms. More quantitatively, our calculations explain why ionization drastically reduces when the scattering length approaches the Bohr radius, which is in accordance with experimental results.

The author acknowledges the generous support of CSIR-HRDG, the ministry of Science and Technology, and the government of India for the award of JRF/SRF [Award No. 09/0414(13706)/2022-EMR-I] throughout the course of the research.

The authors have no conflicts to disclose.

Satyam Prakash: Formal analysis (equal); Investigation (lead); Methodology (lead); Writing – original draft (equal). Ashok S. Vudayagiri: Conceptualization (equal); Methodology (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

HEDPs are typically at temperatures higher than $103$ K, with hottest ones having densities of the order $1026$ cm⁻³. Plasmas at temperature $∼102$ K are called moderately hot plasmas; they are partially ionized and, therefore, are not at equilibrium. The high electron density in these plasmas, coupled with a high temperature, allows for a very high energy density and, subsequently, very high pressure. Due to the high kinetic energy environment, the Coulomb coupling $Γ=Z2e2/4πϵoKBT$ is usually low for HEDPs compared to the UCPs. Thus, HEDPs do not enter a strongly correlated regime as easily, whereas UCPs, on the other hand, easily attain a high coupling regime (⁠ $Γ>1$⁠). From Table I, we can see that the high energy density plasma shows high screening strength and a very low value of degeneracy parameter [ $Θ=(λdB/a)−2$], which indicates that the corresponding de-Broglie wavelength of electron, $λdB=h/2πmKBT$⁠, is much larger compared to Wigner–Seitz radius $a=(3/4πn)1/3$⁠. For the degeneracy values (⁠ $Θ<1$⁠), the HEDPs, or more precisely the moderately hot plasma, show quantum-mechanical scattering. As can be noted from the table, HEDPs show this behavior for density (⁠ $∼1021 cm−3$⁠) and temperature (⁠ $∼104$ K). Increasing the electron density while keeping the temperature constant at $8×103$ K gives the degeneracy parameter less than 1; hence, quantum-mechanical scattering is suitable for these density and temperature values.

Screening strength $κao$ is usually low for UCPs, and the degeneracy parameter is assumed to be much higher compared to HEDPs. UCPs are strongly coupled compared to the HEDPs. Although UCPs and HEDPs are radically different in density (⁠ $ne$⁠) and temperature (T), their characteristics overlap for low screening strength (⁠ $κao$⁠) and low coupling constant (⁠ $Γ$⁠).³ Our calculations show that for a density of the order of $1013cm−3$⁠, the UCPs have comparatively lesser degeneracy parameters for the electronic temperatures around 80 and 8 mK. It is worth mentioning that the theoretical parameters are closer to the actual UCPs in terms of temperature (⁠ $∼$ 100 mK), but they differ from the actual UCPs in terms of density (⁠ $1010$ cm⁻³). For instance, our system of Cesium UCP has a temperature of around 1 K and a density of $1010cm−3$⁠, so the screening parameter that we have used in our calculations is $κao=0.7 mm$⁠. On the other hand, as given in the table, a comparatively smaller degeneracy parameter and bigger screening parameter can be obtained for the respective values of temperature and density. A degeneracy parameter value of 0.05 in the table means a de-Broglie wavelength that is approximately four times larger than the Wigner-Seitz radius and, therefore, will exhibit quantum effects in electron–atom scattering. If we can achieve similar densities (⁠ $∼4×1013cm−3$⁠) at temperatures lower than 80 mK, we get the UCPs with even lower degeneracy parameters, for which the quantum-mechanical electron–atom interaction will be valid.