# Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling

Chloé Malbrunot, Tim Wolz, L. Nowak, Daniel Comparat

► To cite this version:

Chloé Malbrunot, Tim Wolz, L. Nowak, Daniel Comparat. Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling. Journal of Physics B: Atomic, Molecular and Optical Physics, 2022, 55 (4), pp.044003. 10.1088/1361-6455/ac5188 . hal-03625826

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Submitted on 5 Jun 2024

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# Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling

Chloé Malbrunot, Tim Wolz, L. Nowak, Daniel Comparat

► To cite this version:

Chloé Malbrunot, Tim Wolz, L. Nowak, Daniel Comparat. Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling. Journal of Physics B: Atomic, Molecular and Optical Physics, 2022, 55 (4), pp.044003. 10.1088/1361-6455/ac5188 . hal-03625826

HAL Id: hal-03625826

https://hal.science/hal-03625826v1

Submitted on 5 Jun 2024

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling

C Malbrunot<fnref n="1" />,*, T Wolz<fnref n="1" />, L Nowak<fnref n="1" /> and D Comparat<fnref n="2" />

E-mail: chloe.m@cern.ch

Received 30 November 2021, revised 25 January 2022

Accepted for publication 2 February 2022

Published 7 March 2022

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

We present results of simulations highlighting the performance of a cooling mechanism in a magnetic trap tailored for excited Rydberg atoms (or molecules) with high magnetic moment. Unlike previous work, the cooling is achieved through fast stimulated decay of inter-manifold transitions which optimizes the scheme. This mechanism is relevant to experiments with trapped antihydrogen atoms for which the trapping fraction is currently low. We find several-fold enhancements of the number of atoms trapped in the configurations probed.

Keywords: antihydrogen, Rydberg atoms, stimulated deexcitation, magnetic trap, cooling

(Some figures may appear in colour only in the online journal)

## 1. Introduction

Magnetically trapped cold hydrogen atoms have been subject of considerable interest in view of, for example, studies of Bose Einstein condensation [1] or high precision atomic spectroscopy [2]. Measurements in neutral atom traps on hydrogen at temperatures around 400 µK have allowed the determination of the 1s–2s two-photon transition with a fractional resolution of 2 parts in 10¹<fnref n="2" /> in the 1990s [3]. Further improvement, beyond the extraordinary precision of few parts in 10¹⁵ obtained in a beam of hydrogen [4], using atoms cooled in a magnetic trap are currently being discussed, together with other measurements including hydrogen quantum properties in a gravitational field [5].

More recently, these studies have attracted remarkable attention in the context of antihydrogen. Experiments at

CERN's antiproton decelerator with magnetically trapped antihydrogen atoms have enabled the first measurement of optical [6, 7] and microwave transitions [8, 9] in an atom made up entirely of antimatter paving the way to one of the most stringent tests of CPT symmetry. Further measurements, using a vertical neutral atom trap, are planned [10] in view of testing the effect of the Earth's gravitational field on antimatter, alongside other proposed experiments where the atoms are extracted away from the trap environment [11–13]. Because of the specific environment required for the synthesis of antihydrogen, the magnetic trap depth for the formed neutral atoms is limited to roughly 0.8 T which corresponds, for antihydrogen atoms in ground state, to a temperature of ~0.5 K. State-of-the-art techniques to form the anti-atoms, however, yield best formation temperatures of ~40 K which would correspond to a trapping efficiency of only ~10⁻⁴ [14]. Techniques to enhance this trapping fraction are thus highly relevant to the field. To this goal, activities are currently pursued to lower the temperatures of the constituents of antihydrogen, antiprotons and positrons. A combination of cooling techniques for charged particles are typically used (cyclotron, evaporative, and adiabatic cooling) to reach temperatures as low as ~4 K [15] and ~20 K [7] for antiprotons and positrons respectively. Recently, the suc-

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland</footnote>

<footnote n="2">² Université Paris-Saclay, CNRS, Laboratoire Aimé Cotton, 91405, Orsay, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
# Simulation of antihydrogen deexcitation in neutral atom traps for improved trapping and cooling

C Malbrunot<fnref n="1" />,*, T Wolz<fnref n="1" />, L Nowak<fnref n="1" /> and D Comparat<fnref n="2" />

E-mail: chloe.m@cern.ch

Received 30 November 2021, revised 25 January 2022

Accepted for publication 2 February 2022

Published 7 March 2022

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

We present results of simulations highlighting the performance of a cooling mechanism in a magnetic trap tailored for excited Rydberg atoms (or molecules) with high magnetic moment. Unlike previous work, the cooling is achieved through fast stimulated decay of inter-manifold transitions which optimizes the scheme. This mechanism is relevant to experiments with trapped antihydrogen atoms for which the trapping fraction is currently low. We find several-fold enhancements of the number of atoms trapped in the configurations probed.

Keywords: antihydrogen, Rydberg atoms, stimulated deexcitation, magnetic trap, cooling

(Some figures may appear in colour only in the online journal)

## 1. Introduction

Magnetically trapped cold hydrogen atoms have been subject of considerable interest in view of, for example, studies of Bose Einstein condensation [1] or high precision atomic spectroscopy [2]. Measurements in neutral atom traps on hydrogen at temperatures around 400 µK have allowed the determination of the 1s–2s two-photon transition with a fractional resolution of 2 parts in 10¹<fnref n="2" /> in the 1990s [3]. Further improvement, beyond the extraordinary precision of few parts in 10¹⁵ obtained in a beam of hydrogen [4], using atoms cooled in a magnetic trap are currently being discussed, together with other measurements including hydrogen quantum properties in a gravitational field [5].

More recently, these studies have attracted remarkable attention in the context of antihydrogen. Experiments at

CERN's antiproton decelerator with magnetically trapped antihydrogen atoms have enabled the first measurement of optical [6, 7] and microwave transitions [8, 9] in an atom made up entirely of antimatter paving the way to one of the most stringent tests of CPT symmetry. Further measurements, using a vertical neutral atom trap, are planned [10] in view of testing the effect of the Earth's gravitational field on antimatter, alongside other proposed experiments where the atoms are extracted away from the trap environment [11–13]. Because of the specific environment required for the synthesis of antihydrogen, the magnetic trap depth for the formed neutral atoms is limited to roughly 0.8 T which corresponds, for antihydrogen atoms in ground state, to a temperature of ~0.5 K. State-of-the-art techniques to form the anti-atoms, however, yield best formation temperatures of ~40 K which would correspond to a trapping efficiency of only ~10⁻⁴ [14]. Techniques to enhance this trapping fraction are thus highly relevant to the field. To this goal, activities are currently pursued to lower the temperatures of the constituents of antihydrogen, antiprotons and positrons. A combination of cooling techniques for charged particles are typically used (cyclotron, evaporative, and adiabatic cooling) to reach temperatures as low as ~4 K [15] and ~20 K [7] for antiprotons and positrons respectively. Recently, the suc-

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹ Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland</footnote>

<footnote n="2">² Université Paris-Saclay, CNRS, Laboratoire Aimé Cotton, 91405, Orsay, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>

<footnote>Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.</footnote>
cessful sympathetic cooling of positrons by laser-cooled beryllium ions has been demonstrated [16] opening a path toward antihydrogen formed at lower temperatures. Techniques to sympathetically cool antiprotons via laser-cooled anions or molecules are current topic of research [17, 18], but represent a great challenge due to the negative charge of the antiproton requiring a negative specie that can be laser-cooled. In both negative and positive charge cases, the bottleneck lies in the mass ratio between the laser-cooled specie and the charged particle that it should sympathetically cool. Another issue is the density ratio, which, if too large, can lead to capture by the other charged constituent of antihydrogen.

Once trapped in larger quantities, antihydrogen atoms can be further cooled in view of increasing the measurements precision. Demonstration of three-dimensional laser-cooling of the trapped antihydrogen 'cloud' through a one-dimensional optical access to the neutral atom trap has recently been reported [19]. Laser-cooling of hydrogen is notoriously difficult, the required wavelength to drive the 1s-2p transition lying in the vacuum ultraviolet region (121.6 nm). Latest generation pulsed Lyman-α lasers are able to generate a few hundreds nW [20] to µW [21] of average power leading to cooling times for antihydrogen of the order of hours. Several groups worldwide are working on developing more powerful pulsed and cw Lyman-α radiation also in view of improved precision in hydrogen spectroscopy [22, 23].

The Doppler limit for laser cooling of (anti)hydrogen is 2.4 mK, but so far roughly 50 mK in one dimension was demonstrated in antihydrogen (as well as a reduction in temperature in the other dimensions) mainly because of limited optical access and very low antihydrogen densities limiting the laser-driven transition rate and possible equilibration mechanism provided by collisions in three-dimensional cooling. Other cooling techniques applicable to the antihydrogen experimental conditions and which could be used, for example, in conjunction with Lyman-α cooling are thus also highly relevant. Targeting temperatures below what direct laser-cooling can achieve, the GBAR collaboration proposed the formation of antihydrogen ions ($\bar{H}^+$) [24] to benefit from a positively charged atom that can be sympathetically cooled via a laser-cooled ion before undergoing photo-detachment. Currently, this path, however drastically reduces the number of antimatter atoms which can be produced and thus measured. Other techniques acting directly on the antihydrogen atom are therefore highly sought after.

We propose and theoretically investigate here a method to increase the number of antihydrogen atoms trapped making use of the fact that they are initially produced in highly excited Rydberg states. The proposed cooling mechanism relies on rapidly modifying the magnetic moment of the antihydrogen at an appropriate position in the magnetic trap to enhance the number of trapped atoms in ground state.

This light-stimulated cooling mechanism in a magnetic trap was readily proposed by Cesar *et al* in reference [25] after earlier theoretical investigation, by other groups, of the same

cooling mechanism induced by spontaneously decaying atoms [26, 27]. Indeed, the presence of states with elevated magnetic dipoles $\mu \propto m$, where *m* is the magnetic quantum number, leads to an enhanced coupling to the magnetic trapping potential and consequently increased trapping capabilities compared to atoms in ground state. Those trapped atoms eventually spontaneously decay in the trap and through this process dissipate internal energy. If the energy dissipation happens where the kinetic energy of the atoms is minimal, this mechanism leads to an effective cooling and an overall increase in the final trapped fraction.

This so-called decay induced cooling has been theoretically investigated relying on phase space density [26] and kinetic Monte-Carlo approaches [27] highlighting also the relevant analytical relationships between trap parameters (depth, multipole order of the field), antihydrogen properties (temperature, quantum state) and the final temperature and trapping fractions. Cesar *et al* proposed the use of microwaves to enhance the described cooling effect via state mixing [25]. They envisioned driving, when the atoms are close to rest, the intra-manifold transitions ($n, m \rightarrow n, m-1$) from high toward low *m* states that exhibit high spontaneous emission rates. This led, for initial $m = n-1$ Rydberg states, to a few tens of percent (maximum 70% in the range of experimental parameters studied) improvement in the final trapping fraction when compared to spontaneous decay only.

Building on previous work on fast stimulated decay for antihydrogen [28], we propose here, to deexcite Rydberg circular states relying on THz radiation via the entire ($n, m \rightarrow n-1, m-1$) cascade to a circular state $n'$ that decays fast toward ground state. This mechanism allows to keep *m* maximal throughout the decay and minimize the number of states mixed thus accelerating the stimulated decay (the deexcitation time scales as the number of states mixed [28]) which is crucial to maximize the performance of the cooling.

We discuss the general concepts in section 2 before detailing the performed simulations in section 3. Results are presented and discussed in section 4.

## 2. General concepts

The (anti)hydrogen magnetic moment is given by $\mu = \mu_0 m$, where *m* is the magnetic quantum number and $\mu_0$ the Bohr magneton. Ignoring the spin orbit coupling for high *n*, the effective trap depth of a magnetic trap confining the atom can be written as $\mu_0 mB = k_B T$ with the effective temperature *T* and the Boltzmann constant $k_B$. Given that antihydrogen atoms are formed at much larger temperatures than the trap depth in ground state, high *m* Rydberg states lead to increased trapping capabilities.

The trap can hold onto excited antihydrogen atoms with temperatures verifying roughly $T < \mu_0 mB/k_B$ that would otherwise escape the trap in ground state.
cessful sympathetic cooling of positrons by laser-cooled beryllium ions has been demonstrated [16] opening a path toward antihydrogen formed at lower temperatures. Techniques to sympathetically cool antiprotons via laser-cooled anions or molecules are current topic of research [17, 18], but represent a great challenge due to the negative charge of the antiproton requiring a negative specie that can be laser-cooled. In both negative and positive charge cases, the bottleneck lies in the mass ratio between the laser-cooled specie and the charged particle that it should sympathetically cool. Another issue is the density ratio, which, if too large, can lead to capture by the other charged constituent of antihydrogen.

Once trapped in larger quantities, antihydrogen atoms can be further cooled in view of increasing the measurements precision. Demonstration of three-dimensional laser-cooling of the trapped antihydrogen 'cloud' through a one-dimensional optical access to the neutral atom trap has recently been reported [19]. Laser-cooling of hydrogen is notoriously difficult, the required wavelength to drive the 1s-2p transition lying in the vacuum ultraviolet region (121.6 nm). Latest generation pulsed Lyman-α lasers are able to generate a few hundreds nW [20] to µW [21] of average power leading to cooling times for antihydrogen of the order of hours. Several groups worldwide are working on developing more powerful pulsed and cw Lyman-α radiation also in view of improved precision in hydrogen spectroscopy [22, 23].

The Doppler limit for laser cooling of (anti)hydrogen is 2.4 mK, but so far roughly 50 mK in one dimension was demonstrated in antihydrogen (as well as a reduction in temperature in the other dimensions) mainly because of limited optical access and very low antihydrogen densities limiting the laser-driven transition rate and possible equilibration mechanism provided by collisions in three-dimensional cooling. Other cooling techniques applicable to the antihydrogen experimental conditions and which could be used, for example, in conjunction with Lyman-α cooling are thus also highly relevant. Targeting temperatures below what direct laser-cooling can achieve, the GBAR collaboration proposed the formation of antihydrogen ions ($\bar{H}^+$) [24] to benefit from a positively charged atom that can be sympathetically cooled via a laser-cooled ion before undergoing photo-detachment. Currently, this path, however drastically reduces the number of antimatter atoms which can be produced and thus measured. Other techniques acting directly on the antihydrogen atom are therefore highly sought after.

We propose and theoretically investigate here a method to increase the number of antihydrogen atoms trapped making use of the fact that they are initially produced in highly excited Rydberg states. The proposed cooling mechanism relies on rapidly modifying the magnetic moment of the antihydrogen at an appropriate position in the magnetic trap to enhance the number of trapped atoms in ground state.

This light-stimulated cooling mechanism in a magnetic trap was readily proposed by Cesar *et al* in reference [25] after earlier theoretical investigation, by other groups, of the same

cooling mechanism induced by spontaneously decaying atoms [26, 27]. Indeed, the presence of states with elevated magnetic dipoles $\mu \propto m$, where *m* is the magnetic quantum number, leads to an enhanced coupling to the magnetic trapping potential and consequently increased trapping capabilities compared to atoms in ground state. Those trapped atoms eventually spontaneously decay in the trap and through this process dissipate internal energy. If the energy dissipation happens where the kinetic energy of the atoms is minimal, this mechanism leads to an effective cooling and an overall increase in the final trapped fraction.

This so-called decay induced cooling has been theoretically investigated relying on phase space density [26] and kinetic Monte-Carlo approaches [27] highlighting also the relevant analytical relationships between trap parameters (depth, multipole order of the field), antihydrogen properties (temperature, quantum state) and the final temperature and trapping fractions. Cesar *et al* proposed the use of microwaves to enhance the described cooling effect via state mixing [25]. They envisioned driving, when the atoms are close to rest, the intra-manifold transitions ($n, m \rightarrow n, m-1$) from high toward low *m* states that exhibit high spontaneous emission rates. This led, for initial $m = n-1$ Rydberg states, to a few tens of percent (maximum 70% in the range of experimental parameters studied) improvement in the final trapping fraction when compared to spontaneous decay only.

Building on previous work on fast stimulated decay for antihydrogen [28], we propose here, to deexcite Rydberg circular states relying on THz radiation via the entire ($n, m \rightarrow n-1, m-1$) cascade to a circular state $n'$ that decays fast toward ground state. This mechanism allows to keep *m* maximal throughout the decay and minimize the number of states mixed thus accelerating the stimulated decay (the deexcitation time scales as the number of states mixed [28]) which is crucial to maximize the performance of the cooling.

We discuss the general concepts in section 2 before detailing the performed simulations in section 3. Results are presented and discussed in section 4.

## 2. General concepts

The (anti)hydrogen magnetic moment is given by $\mu = \mu_0 m$, where *m* is the magnetic quantum number and $\mu_0$ the Bohr magneton. Ignoring the spin orbit coupling for high *n*, the effective trap depth of a magnetic trap confining the atom can be written as $\mu_0 mB = k_B T$ with the effective temperature *T* and the Boltzmann constant $k_B$. Given that antihydrogen atoms are formed at much larger temperatures than the trap depth in ground state, high *m* Rydberg states lead to increased trapping capabilities.

The trap can hold onto excited antihydrogen atoms with temperatures verifying roughly $T < \mu_0 mB/k_B$ that would otherwise escape the trap in ground state.
Neglecting gravity, the equation of motion of an atom with mass $m_a$ and magnetic moment $\vec{\mu}$ in an inhomogeneous magnetic field $\vec{B}$ is determined by

$$m_a \frac{d^2 \vec{r}}{dt^2} = -\vec{\nabla} (\vec{B}(\vec{r}) \cdot \vec{\mu}). \quad (1)$$

The gas dynamics lead to higher probabilities to find the atoms close to rest when their potential (kinetic) energy is maximal (minimal), meaning that spontaneous emission is more likely to happen at the edge of the trap than in its center where the atoms' energy is mostly kinetic. A transition with $m \to m' < m$ lowers the effective trapping potential that is proportional to $m$. For atoms at (or close to) rest this dissipates the potential energy $(m - m') \mu_0 B$ and leads to a cooling of the atoms. This scheme can be viewed as similar to adiabatic cooling [e.g. [15]], a mechanism where the trap potential is modified. But here spontaneous decay acts on the internal degree of freedom of the atom rather than on the physical trap, which allows to increase the phase space density. Since, due to selection rules, spontaneous decay only decreases the magnetic moment by one unit for each decay process, the process is long (Rydberg lifetimes are of the order of ms) and can lead to trap spillage if the decay happens at an unfavourable place in the trap where

$$\mu_0 (m - m') B(\vec{r}) > \frac{1}{2} m_a \vec{v}^2. \quad (2)$$

The process is illustrated in figure 1(a).

This spontaneous emission cooling mechanism can be enhanced by stimulating a range of atomic transitions down to low $n$ levels. For the above-mentioned reasons, the decay must be stimulated when the atoms are close to rest. In a trap, slow atoms can be spectrally selected exploiting the linear Zeeman shift of the atomic levels along the varying magnetic field strength which is, for $\Delta m = 1$ transitions, 14 GHz T$^{-1}$. Thus, in a typical potential well of 0.75 T an initial detuning $\Delta\nu$ of the light sources of 10.5 GHz would be used to address only the initially hot atoms that are energetic enough to reach the trap edge, see figure 1(b). Once the most energetic atoms have been deexcited, the detuning can be decreased in order to address slower parts of the initial velocity distribution as illustrated in figure 1(c). Because the resonance condition covers the spectral width $\Gamma_L$, we can roughly estimate the number of energy resonance zones to be $Z = \Delta\nu/\Gamma_L$.

In a one dimensional trap, a particle of given total energy, would need a single 'oscillation' time (to be understood as trajectory length divided by the maximal velocity for a non harmonic potential) to reach a point of zero velocity. However, in a three-dimensional trapping field, particles typically take $Z^2$ times longer to reach the energy resonance zone where the velocity is small [29]. Therefore, the full detuning time, $t_{det}$, required to decrease the initial detuning to zero scales as $Z^3 = (\Delta\nu/\Gamma_L)^3$.

In general, narrow transition zones and thus small $\Gamma_L$ are desirable to best select velocities that are close to zero. Given that $t_{det} \propto \Gamma_L^{-3}$, the choice of $\Gamma_L$ will quickly be limited by the radiative lifetimes of the considered Rydberg excited states

that we aim at deexciting. If addressing long-lived enough states, the optimal choice of $\Gamma_L$ is in fact dictated by the trap depth in ground state which defines the maximum kinetic energy that the atom can gain during the deexcitation process in order to remain trapped in ground state. For our chosen example of a 0.75 T trap, equivalent to a depth of 500 mK in ground state, the atoms should thus roughly not gain more than the maximum trappable velocity of ~100 m s$^{-1}$ during the deexcitation phase. The trap size, field configuration and initial antihydrogen temperature then determine the optimal $\Gamma_L$ and Z.

### 3. Simulation

To assess the efficiency of THz induced cooling in a neutral atom trap, the kinematics of the atoms in the trap including atom-light interaction were simulated employing a kinetic Monte-Carlo method [29]. The interaction of the atoms with light is implemented relying on rate equations which is justified given the broadband and thus incoherent nature of the light in the low saturation limit. As mentioned above, we considered zero-field dipoles for spontaneous and stimulated rates, which is a valid approximation given the amplitude of the magnetic field considered here (0.75 T). We note that taking into account the increased rate in large magnetic fields would contribute to an improvement in the effect we aim to highlight. The trap configuration chosen for the simulation is a 9 mm diameter wide, three dimensional, linear cusp trap generated by two electromagnetic coils in anti-Helmholtz configuration. The choice of these dimensions was motivated by the experimental prospect of using non-superconducting or permanent magnets for a demonstration on hydrogen.

The temporal evolution of the quantum state distribution and the trap kinematics is simulated for a cloud of antihydrogen initialized at the trap center with a Maxwellian thermal center-of-mass velocity distribution. Given the CPU intensive nature of the simulation, we only simulated atoms with $v_{init} < v_{max,trap} = \sqrt{2\mu Bm/m_a}$. For similar reasons, all initialized atoms are considered to be low-field-seeking, the other ones being quickly ejected from the trap and thus irrelevant to the simulation. Potential losses due to spin flips in the (close to) zero field region at the trap center are neglected since they can be experimentally avoided with the addition of a constant field.

In line with previous studies [25–27], we constrain ourselves to the simulation of circular states for which $l = m = n - 1$ (as mentioned before, we consider positive $m$ only). Antihydrogen atoms are initially produced in a broad range of $(n, l, m)$ states [30–34]. However, by spontaneous emission, within the timescales needed for reaching the zone of stimulated transition (of the order of half a millisecond), the non-circular states rapidly decay (lifetimes scale as $l^2$) to lower lying maximum $m$ levels [35]. Once in circular states, the $(n, m) \to (n - 1, m - 1)$ dipoles being maximal, the circular nature of the states can be considered conserved.
Neglecting gravity, the equation of motion of an atom with mass $m_a$ and magnetic moment $\vec{\mu}$ in an inhomogeneous magnetic field $\vec{B}$ is determined by

$$m_a \frac{d^2 \vec{r}}{dt^2} = -\vec{\nabla} (\vec{B}(\vec{r}) \cdot \vec{\mu}). \quad (1)$$

The gas dynamics lead to higher probabilities to find the atoms close to rest when their potential (kinetic) energy is maximal (minimal), meaning that spontaneous emission is more likely to happen at the edge of the trap than in its center where the atoms' energy is mostly kinetic. A transition with $m \to m' < m$ lowers the effective trapping potential that is proportional to $m$. For atoms at (or close to) rest this dissipates the potential energy $(m - m') \mu_0 B$ and leads to a cooling of the atoms. This scheme can be viewed as similar to adiabatic cooling [e.g. [15]], a mechanism where the trap potential is modified. But here spontaneous decay acts on the internal degree of freedom of the atom rather than on the physical trap, which allows to increase the phase space density. Since, due to selection rules, spontaneous decay only decreases the magnetic moment by one unit for each decay process, the process is long (Rydberg lifetimes are of the order of ms) and can lead to trap spillage if the decay happens at an unfavourable place in the trap where

$$\mu_0 (m - m') B(\vec{r}) > \frac{1}{2} m_a \vec{v}^2. \quad (2)$$

The process is illustrated in figure 1(a).

This spontaneous emission cooling mechanism can be enhanced by stimulating a range of atomic transitions down to low $n$ levels. For the above-mentioned reasons, the decay must be stimulated when the atoms are close to rest. In a trap, slow atoms can be spectrally selected exploiting the linear Zeeman shift of the atomic levels along the varying magnetic field strength which is, for $\Delta m = 1$ transitions, 14 GHz T$^{-1}$. Thus, in a typical potential well of 0.75 T an initial detuning $\Delta\nu$ of the light sources of 10.5 GHz would be used to address only the initially hot atoms that are energetic enough to reach the trap edge, see figure 1(b). Once the most energetic atoms have been deexcited, the detuning can be decreased in order to address slower parts of the initial velocity distribution as illustrated in figure 1(c). Because the resonance condition covers the spectral width $\Gamma_L$, we can roughly estimate the number of energy resonance zones to be $Z = \Delta\nu/\Gamma_L$.

In a one dimensional trap, a particle of given total energy, would need a single 'oscillation' time (to be understood as trajectory length divided by the maximal velocity for a non harmonic potential) to reach a point of zero velocity. However, in a three-dimensional trapping field, particles typically take $Z^2$ times longer to reach the energy resonance zone where the velocity is small [29]. Therefore, the full detuning time, $t_{det}$, required to decrease the initial detuning to zero scales as $Z^3 = (\Delta\nu/\Gamma_L)^3$.

In general, narrow transition zones and thus small $\Gamma_L$ are desirable to best select velocities that are close to zero. Given that $t_{det} \propto \Gamma_L^{-3}$, the choice of $\Gamma_L$ will quickly be limited by the radiative lifetimes of the considered Rydberg excited states

that we aim at deexciting. If addressing long-lived enough states, the optimal choice of $\Gamma_L$ is in fact dictated by the trap depth in ground state which defines the maximum kinetic energy that the atom can gain during the deexcitation process in order to remain trapped in ground state. For our chosen example of a 0.75 T trap, equivalent to a depth of 500 mK in ground state, the atoms should thus roughly not gain more than the maximum trappable velocity of ~100 m s$^{-1}$ during the deexcitation phase. The trap size, field configuration and initial antihydrogen temperature then determine the optimal $\Gamma_L$ and $Z$.

### 3. Simulation

To assess the efficiency of THz induced cooling in a neutral atom trap, the kinematics of the atoms in the trap including atom-light interaction were simulated employing a kinetic Monte-Carlo method [29]. The interaction of the atoms with light is implemented relying on rate equations which is justified given the broadband and thus incoherent nature of the light in the low saturation limit. As mentioned above, we considered zero-field dipoles for spontaneous and stimulated rates, which is a valid approximation given the amplitude of the magnetic field considered here (0.75 T). We note that taking into account the increased rate in large magnetic fields would contribute to an improvement in the effect we aim to highlight. The trap configuration chosen for the simulation is a 9 mm diameter wide, three dimensional, linear cusp trap generated by two electromagnetic coils in anti-Helmholtz configuration. The choice of these dimensions was motivated by the experimental prospect of using non-superconducting or permanent magnets for a demonstration on hydrogen.

The temporal evolution of the quantum state distribution and the trap kinematics is simulated for a cloud of antihydrogen initialized at the trap center with a Maxwellian thermal center-of-mass velocity distribution. Given the CPU intensive nature of the simulation, we only simulated atoms with $v_{init} < v_{max,trap} = \sqrt{2\mu Bm/m_a}$. For similar reasons, all initialized atoms are considered to be low-field-seeking, the other ones being quickly ejected from the trap and thus irrelevant to the simulation. Potential losses due to spin flips in the (close to) zero field region at the trap center are neglected since they can be experimentally avoided with the addition of a constant field.

In line with previous studies [25–27], we constrain ourselves to the simulation of circular states for which $l = m = n - 1$ (as mentioned before, we consider positive $m$ only). Antihydrogen atoms are initially produced in a broad range of $(n, l, m)$ states [30–34]. However, by spontaneous emission, within the timescales needed for reaching the zone of stimulated transition (of the order of half a millisecond), the non-circular states rapidly decay (lifetimes scale as $l^2$) to lower lying maximum $m$ levels [35]. Once in circular states, the $(n, m) \to (n - 1, m - 1)$ dipoles being maximal, the circular nature of the states can be considered conserved.
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Illustration of cooling in a magnetic trap. The thin arrows are meant to highlight the sequence of events and do not represent the complex trajectories in the three-dimensional space. The numbers (0 or 1) refer to atoms in different quantum states either after a spontaneous emission process or after a stimulated deexcitation process to much deeper bound states. (a) Spontaneous decay case: a spontaneous emission ($m \rightarrow m-1$) occurring at the center of the trap does not lead to cooling, eventually leading to spillage from the trap (left side of the figure). On the contrary, when the spontaneous emission occurs at the edge of the trap and where the kinetic energy of the atom is minimal, the atom effectively cools (right side of the figure). (b) The decay can be forced selectively for atoms reaching the edge of the trap (at a field of 0.75 T in our case) using stimulated emission in a ($m_0 \rightarrow m_0-1 \rightarrow m_0-2 \rightarrow \dots \rightarrow m_1 \ll m_0$) cascade (the series of lasers required for this purpose is illustrated by a thick arrow, see figure 2 for details). The spectral selection of the atoms is highlighted by the gray band, requiring a laser detuning $\Delta\nu = 10.5$ GHz in our case. This stimulated emission drives atoms down to a state that quickly decays to ground state via spontaneous decay and leads to an important reduction of energy. (c) The detuning ($\Delta\nu$) is decreased with time ((b) $\rightarrow$ (c)) addressing atoms that were initially colder than in (b).</figcaption></figure>

Addressing circular states with a maximum principal quantum number $n = 30$ justifies a number of further simplifications. In particular, the diamagnetic term which leads to $l$-mixing and energy shifts is negligible. Further, we assumed that the internal states of the atom adiabatically follow the trapping magnetic field. We also neglected the effect of the electric fields required for charged-particle confinement in the currently used antihydrogen production scheme. These approximations hold best for circular states and limit the maximum Rydberg states considered to $n \sim 30–40$, above which those simplifications would be less valid.

Light sources with a top-hat spectral profile of width $\Gamma_L$ and central frequency $\nu$ stimulate individual $n \rightarrow n-1$ transitions from an initial level $n$ to a final level $n'$ that exhibits short spontaneous lifetimes. For circular $n = 30 \rightarrow n' = 29$ (respectively 13 $\rightarrow$ 12 and $6 \rightarrow$ 5) states, the transition frequency is $\nu = 256$ GHz (respectively 3.38 THz and 40.2 THz). The

light intensity is scaled such that transition rates $\Gamma_{n \rightarrow n-1}$ are constant. This is particularly important to avoid the creation of bottlenecks when deexciting a distribution of levels via a single fast spontaneous emission channel [28]. Since $\Gamma_{n \rightarrow n-1} \propto d_{n \rightarrow n-1}^2 \propto n^4$, the light intensity scaling implemented is $n^{-4}$. Hence, when driving 25 transitions, almost 50% of the power goes to the deepest transition while less than 1% of the power is assigned to drive the first $n \rightarrow n-1$ transition.

As explained in section 2, the number of zones can be chosen to be $Z = v_{max}/v_{GS}$, where $v_{max}$ and $v_{GS}$ are, respectively, the maximal velocity of trapped atoms in the ($n = 30, m = 29$) state and the maximal velocity of atoms trapped in ground state. Our trap dimensions and field strength lead to $Z = 5.4$. The detuning time in the three-dimensional trap is given by $t_{det} = t_{osc} \times Z^3$, where $t_{osc}$ is the one-dimensional ‘oscillation time’, $t_{osc} = 2 \times r/v_{max} \sim 20$ µs. This leads to $t_{det} = 2.9$ ms which is comparable to the spontaneous lifetime of the
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Illustration of cooling in a magnetic trap. The thin arrows are meant to highlight the sequence of events and do not represent the complex trajectories in the three-dimensional space. The numbers (0 or 1) refer to atoms in different quantum states either after a spontaneous emission process or after a stimulated deexcitation process to much deeper bound states. (a) Spontaneous decay case: a spontaneous emission ($m \rightarrow m-1$) occurring at the center of the trap does not lead to cooling, eventually leading to spillage from the trap (left side of the figure). On the contrary, when the spontaneous emission occurs at the edge of the trap and where the kinetic energy of the atom is minimal, the atom effectively cools (right side of the figure). (b) The decay can be forced selectively for atoms reaching the edge of the trap (at a field of 0.75 T in our case) using stimulated emission in a ($m_0 \rightarrow m_0 - 1 \rightarrow m_0 - 2 \rightarrow \dots \rightarrow m_1 \ll m_0$) cascade (the series of lasers required for this purpose is illustrated by a thick arrow, see figure 2 for details). The spectral selection of the atoms is highlighted by the gray band, requiring a laser detuning $\Delta\nu = 10.5$ GHz in our case. This stimulated emission drives atoms down to a state that quickly decays to ground state via spontaneous decay and leads to an important reduction of energy. (c) The detuning ($\Delta\nu$) is decreased with time ((b) $\rightarrow$ (c)) addressing atoms that were initially colder than in (b).</figcaption></figure>

Addressing circular states with a maximum principal quantum number $n = 30$ justifies a number of further simplifications. In particular, the diamagnetic term which leads to $l$-mixing and energy shifts is negligible. Further, we assumed that the internal states of the atom adiabatically follow the trapping magnetic field. We also neglected the effect of the electric fields required for charged-particle confinement in the currently used antihydrogen production scheme. These approximations hold best for circular states and limit the maximum Rydberg states considered to $n \sim 30–40$, above which those simplifications would be less valid.

Light sources with a top-hat spectral profile of width $\Gamma_L$ and central frequency $\nu$ stimulate individual $n \rightarrow n-1$ transitions from an initial level $n$ to a final level $n'$ that exhibits short spontaneous lifetimes. For circular $n = 30 \rightarrow n' = 29$ (respectively 13 $\rightarrow$ 12 and $6 \rightarrow$ 5) states, the transition frequency is $\nu = 256$ GHz (respectively 3.38 THz and 40.2 THz). The

light intensity is scaled such that transition rates $\Gamma_{n \rightarrow n-1}$ are constant. This is particularly important to avoid the creation of bottlenecks when deexciting a distribution of levels via a single fast spontaneous emission channel [28]. Since $\Gamma_{n \rightarrow n-1} \propto d_{n \rightarrow n-1}^2 \propto n^4$, the light intensity scaling implemented is $n^{-4}$. Hence, when driving 25 transitions, almost 50% of the power goes to the deepest transition while less than 1% of the power is assigned to drive the first $n \rightarrow n-1$ transition.

As explained in section 2, the number of zones can be chosen to be $Z = v_{max}/v_{GS}$, where $v_{max}$ and $v_{GS}$ are, respectively, the maximal velocity of trapped atoms in the ($n = 30, m = 29$) state and the maximal velocity of atoms trapped in ground state. Our trap dimensions and field strength lead to $Z = 5.4$. The detuning time in the three-dimensional trap is given by $t_{det} = t_{osc} \times Z^3$, where $t_{osc}$ is the one-dimensional ‘oscillation time’, $t_{osc} = 2 \times r/v_{max} \sim 20$ µs. This leads to $t_{det} = 2.9$ ms which is comparable to the spontaneous lifetime of the
$(n = 30, m = 29)$ state ($\tau \sim 2.3$ ms) confirming that the choice of $Z = 5.4$ is within an acceptable range.

This value of $Z$ leads to $\Gamma_L = \Delta\nu/Z = 1.9$ GHz. We note that given this small spectral bandwidth, the population initiated in a circular state will remain circular under such light stimulation, since the other possible transitions with $\Delta n = \pm 1$ and $\Delta m = +1, 0$ (e.g. $\nu_{(n-1,m-1)\to(n,m-1)}$) are detuned from the $\nu_{(n,m)\to(n-1,m-1)}$ transition by multiples of 14 GHz T$^{-1}$. Hence, in the zones where cooling is relevant to increase the trapped fraction (the first four zones), non-circular states cannot be populated. Figure 2 illustrates this point and highlights the deexcitation scheme.

The light sources' frequencies were simultaneously detuned linearly from the trap edge ($\Delta\nu = 10.5$ GHz) toward the trap center in $t_{\text{det}} = 2.9$ ms. The choice of a linear detuning function was made out of simplicity. We find that the results only marginally vary with this choice. More elaborate functions, for example $\Delta\nu(t) = k\sqrt{t_{\text{det}}-t}$ with a constant $k$, that better take into account the fact that $t_{\text{osc}} \propto \sqrt{r_{\text{max}}}$, where $r_{\text{max}}$ is the maximum oscillation radius in the trap, yielded similar results.

# 4. Results and discussion

Having chosen a given trap configuration and thus detuning parameters, we compare the performance of the stimulated decay process to that of the sole spontaneous emission process for an initial 40 K velocity distribution and $n = 30$ circular state, yielding an initial trap depth of 14.5 K.

## 4.1. Trapping fraction

Figure 3 shows the improvement in the trapped fraction as a function of $n'$ for different stimulated rates. The results show that, while for $n' = 9$ we recover similar results than with the proposed microwave solution (a few tens of percent improvement over spontaneous decay [25]), independently of the stimulated rate, above $n' = 9$ very little gain is observed with respect to spontaneous emission. The lifetime of the $n' = 10$ circular state ($\sim 9$ µs) is too long to significantly benefit from the induced cooling via stimulated deexcitation of the upper states. Indeed, if the spontaneous emission rate of the final $n'$ state is too long, the 20 wavelengths sent to drive all $\Delta n = -1$ transitions down to $n' = 10$ merely mix the $n$ levels without depopulating them, thus maintaining a high effective 'average' $n$ state which is prohibiting cooling. For lower targeted $n'$, the fraction of atoms trapped increases roughly linearly reaching a factor 3 improvement for the highest rate probed and $n' = 5$. A factor $\sim 50\%$ in the trapping fraction is gained for every doubling of the rate. Above a rate of $4 \times 10^7$ s$^{-1}$ we did not observe a significant improvement. Given the light intensity scaling with $n^4$, those rates correspond to high intensities for transitions in the THz region notoriously difficult to generate. As an example, a stimulated rate of $4 \times 10^6$ s$^{-1}$ down to $n' = 5$ which would lead to an improved trapped fraction of a factor 2 corresponds to a total intensity

of 13 mW cm$^{-2}$ which is close to the current state-of-the-art in THz generation using photomixers; a technology that might be suitable for the generation of multiple narrowband frequencies for the purpose of fast deexcitation [28, 36] in the low THz region.

Considering the currently available technology, an improvement over spontaneous decay limited to a factor of roughly 1.5 to 2 would be obtained. However, THz technology is a fast evolving field and it is not unreasonable to foresee significant improvements in the near future [37–39]. Additionally, the proposed THz deexcitation addresses the entire distribution of $n$ states below $n = 30$ which could lead to a significant improvement in the absolute number of atoms trapped. Finally, the results are highly dependent on the trap field configuration. Cesar et al [25] simulated an octupole trap which is the configuration used for trapping antihydrogen to minimize field asymmetries in the region where the charged particles are trapped. A linear trap is the best configuration for cooling via spontaneous emission [26], but if stimulated deexcitation is sufficiently fast, the gain over spontaneous emission could be enhanced in higher order traps. If the lifetime of the $n'$ state is the bottleneck, a larger diameter trap leading to a smaller gradient allowing more time for the stimulated decay to happen at a favourable position in the trap could also be beneficial. Figure 4 shows the enhancement factor over spontaneous decay for an octupole trap ($B \propto r^3$) of same dimensions and a three times larger (diameter of 2.7 cm) cusp trap, both with the same magnetic field depth as the originally simulated trap (labelled 'small cusp' in the figure). A large enhancement for both cases is observed although the slope of the improvement as a function of $n'$ is different as expected since the improved performances stem from different mechanisms. For the octupole trap, $n' = 5$ and a stimulated rate $\Gamma = 4 \times 10^7$ s$^{-1}$, an improvement factor close to 5 is achieved which represents a considerable gain. However the required intensity, of the order of 100 mW cm$^{-2}$, will be difficult to produce. Additionally, at these powers, operation in a cryogenic trap should be addressed with care. However, the same performance as the 'small cusp' trap with $n' = 5$ (factor roughly 3) is recovered for higher $n'$ ($n' = 7$) drastically decreasing the required intensity. We saw that the overall gain depends on the trap configuration. It also depends on the initial quantum state distribution and temperature. For example, for the 'small cusp' configuration with $\Gamma = 4 \times 10^7$ s$^{-1}$ and $n' = 5$, we found a linear scaling with initial temperature with a $\sim 50\%$ increase in the trapped fraction between 5 K and 40 K. Addressing higher Rydberg states is, as far as power is concerned, easy to achieve given the scaling discussed above. However, for higher initial $n$, the cooling by spontaneous decay gets also more efficient, potentially balancing the gain from the larger state distribution and the larger fraction of the initial velocity distribution addressed. Given the CPU-intensive nature of the simulations we did not investigate a larger parameter space, but by further optimization higher performance than those presented here could likely be obtained.
$(n = 30, m = 29)$ state ($\tau \sim 2.3$ ms) confirming that the choice of $Z = 5.4$ is within an acceptable range.

This value of $Z$ leads to $\Gamma_L = \Delta\nu/Z = 1.9$ GHz. We note that given this small spectral bandwidth, the population initiated in a circular state will remain circular under such light stimulation, since the other possible transitions with $\Delta n = \pm 1$ and $\Delta m = +1, 0$ (e.g. $\nu_{(n-1,m-1) \to (n,m-1)}$) are detuned from the $\nu_{(n,m) \to (n-1,m-1)}$ transition by multiples of 14 GHz T$^{-1}$. Hence, in the zones where cooling is relevant to increase the trapped fraction (the first four zones), non-circular states cannot be populated. Figure 2 illustrates this point and highlights the deexcitation scheme.

The light sources' frequencies were simultaneously detuned linearly from the trap edge ($\Delta\nu = 10.5$ GHz) toward the trap center in $t_{\text{det}} = 2.9$ ms. The choice of a linear detuning function was made out of simplicity. We find that the results only marginally vary with this choice. More elaborate functions, for example $\Delta\nu(t) = k\sqrt{t_{\text{det}}-t}$ with a constant $k$, that better take into account the fact that $t_{\text{osc}} \propto \sqrt{r_{\text{max}}}$, where $r_{\text{max}}$ is the maximum oscillation radius in the trap, yielded similar results.

# 4. Results and discussion

Having chosen a given trap configuration and thus detuning parameters, we compare the performance of the stimulated decay process to that of the sole spontaneous emission process for an initial 40 K velocity distribution and $n = 30$ circular state, yielding an initial trap depth of 14.5 K.

## 4.1. Trapping fraction

Figure 3 shows the improvement in the trapped fraction as a function of $n'$ for different stimulated rates. The results show that, while for $n' = 9$ we recover similar results than with the proposed microwave solution (a few tens of percent improvement over spontaneous decay [25]), independently of the stimulated rate, above $n' = 9$ very little gain is observed with respect to spontaneous emission. The lifetime of the $n' = 10$ circular state ($\sim 9$ µs) is too long to significantly benefit from the induced cooling via stimulated deexcitation of the upper states. Indeed, if the spontaneous emission rate of the final $n'$ state is too long, the 20 wavelengths sent to drive all $\Delta n = -1$ transitions down to $n' = 10$ merely mix the $n$ levels without depopulating them, thus maintaining a high effective 'average' $n$ state which is prohibiting cooling. For lower targeted $n'$, the fraction of atoms trapped increases roughly linearly reaching a factor 3 improvement for the highest rate probed and $n' = 5$. A factor $\sim 50\%$ in the trapping fraction is gained for every doubling of the rate. Above a rate of $4 \times 10^7$ s$^{-1}$ we did not observe a significant improvement. Given the light intensity scaling with $n^4$, those rates correspond to high intensities for transitions in the THz region notoriously difficult to generate. As an example, a stimulated rate of $4 \times 10^6$ s$^{-1}$ down to $n' = 5$ which would lead to an improved trapped fraction of a factor 2 corresponds to a total intensity

of 13 mW cm$^{-2}$ which is close to the current state-of-the-art in THz generation using photomixers; a technology that might be suitable for the generation of multiple narrowband frequencies for the purpose of fast deexcitation [28, 36] in the low THz region.

Considering the currently available technology, an improvement over spontaneous decay limited to a factor of roughly 1.5 to 2 would be obtained. However, THz technology is a fast evolving field and it is not unreasonable to foresee significant improvements in the near future [37–39]. Additionally, the proposed THz deexcitation addresses the entire distribution of $n$ states below $n = 30$ which could lead to a significant improvement in the absolute number of atoms trapped. Finally, the results are highly dependent on the trap field configuration. Cesar et al [25] simulated an octupole trap which is the configuration used for trapping antihydrogen to minimize field asymmetries in the region where the charged particles are trapped. A linear trap is the best configuration for cooling via spontaneous emission [26], but if stimulated deexcitation is sufficiently fast, the gain over spontaneous emission could be enhanced in higher order traps. If the lifetime of the $n'$ state is the bottleneck, a larger diameter trap leading to a smaller gradient allowing more time for the stimulated decay to happen at a favourable position in the trap could also be beneficial. Figure 4 shows the enhancement factor over spontaneous decay for an octupole trap ($B \propto r^3$) of same dimensions and a three times larger (diameter of 2.7 cm) cusp trap, both with the same magnetic field depth as the originally simulated trap (labelled 'small cusp' in the figure). A large enhancement for both cases is observed although the slope of the improvement as a function of $n'$ is different as expected since the improved performances stem from different mechanisms. For the octupole trap, $n' = 5$ and a stimulated rate $\Gamma = 4 \times 10^7$ s$^{-1}$, an improvement factor close to 5 is achieved which represents a considerable gain. However the required intensity, of the order of 100 mW cm$^{-2}$, will be difficult to produce. Additionally, at these powers, operation in a cryogenic trap should be addressed with care. However, the same performance as the 'small cusp' trap with $n' = 5$ (factor roughly 3) is recovered for higher $n'$ ($n' = 7$) drastically decreasing the required intensity. We saw that the overall gain depends on the trap configuration. It also depends on the initial quantum state distribution and temperature. For example, for the 'small cusp' configuration with $\Gamma = 4 \times 10^7$ s$^{-1}$ and $n' = 5$, we found a linear scaling with initial temperature with a $\sim 50\%$ increase in the trapped fraction between 5 K and 40 K. Addressing higher Rydberg states is, as far as power is concerned, easy to achieve given the scaling discussed above. However, for higher initial $n$, the cooling by spontaneous decay gets also more efficient, potentially balancing the gain from the larger state distribution and the larger fraction of the initial velocity distribution addressed. Given the CPU-intensive nature of the simulations we did not investigate a larger parameter space, but by further optimization higher performance than those presented here could likely be obtained.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Light characteristics used in the simulation for stimulating the deexcitation. Examples are given for the first two zones (b) and (c) in figure 1). The lowest lying transition, to a state labelled *n'* in the text, is varied in the simulation (in the figure, *n'* = 10 for illustration) to evaluate the performance of the scheme as a function of *n'*. Ground state is reached via a cascade of spontaneous emission from the *n'* state. The spectral bandwidth Γ<sub>L</sub> is the same for all lasers and kept constant. The detuning is varied linearly. Example of possible transitions to non-circular states within the *n* = 30 manifold are shown (dashed grey lines) to highlight the fact that they cannot be driven by the lasers used for the deexcitation.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** R<sub>st/sp</sub>: number of trapped atoms in the THz-stimulated decay scheme over number of trapped atoms with spontaneous emission only. The ratio is given as a function of the lowest stimulated level *n'* and for different stimulated rates. The error bars are statistical (each data point represents a simulation of typically 1150 atoms in the stimulated case normalized to the result obtained with 2150 atoms spontaneously decaying). The lines are a guide for the eye.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Ratio of trapped atoms with and without THz-stimulated decay (R<sub>st/sp</sub>) as a function of *n'* for different trap configurations. The stimulated rate is Γ = 4 × 10<sup>7</sup> s<sup>-1</sup>. The lines are a guide for the eye.</figcaption></figure>

## 4.2. Temperature distributions

The velocity distribution of the initial atomic cloud generated at 40 K, the initial velocity of the atoms eventually trapped and the final velocity distribution of the atoms trapped are plotted in figure 5, for the sole spontaneous emission decay and in figure 6 for stimulated decay.

For the stimulated case, the ‘small cusp’, ‘octupole trap’ and ‘large cusp’ are plotted for the *n'* = 5 and Γ = 4 × 10<sup>7</sup> s<sup>-1</sup> configuration. They are compared to the spontaneous case in the respective same traps. For the ‘small cusp’, we see that the stimulation of the decay enables to trap the entire velocity

distribution up to velocities about twice what is achieved with spontaneous decay. The final temperatures of the trapped cloud are similar in this trap configuration (we find 129 mK for the stimulated case and 118 mK for the spontaneous case).

The performance of the spontaneous decay induced cooling is very similar for the small and large cusp configuration. The improvement in the trapped fraction for the large cusp compared to the small one comes almost entirely from the enhanced performance of the stimulated scheme. On the contrary, for the octupole configuration, the cooling by spontaneous decay is less efficient, in line with the findings by Pohl *et al* [26]. The *onset* velocity (velocity above which the distribution is only partially trapped) is similar for the spontaneous and stimulated cases, but the stimulation allows to hold onto initially hotter atoms. The absolute number of trapped atoms is smaller than for the linear trap cases, the larger gain
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Light characteristics used in the simulation for stimulating the deexcitation. Examples are given for the first two zones (b) and (c) in figure 1). The lowest lying transition, to a state labelled *n'* in the text, is varied in the simulation (in the figure, *n'* = 10 for illustration) to evaluate the performance of the scheme as a function of *n'*. Ground state is reached via a cascade of spontaneous emission from the *n'* state. The spectral bandwidth Γ<sub>L</sub> is the same for all lasers and kept constant. The detuning is varied linearly. Example of possible transitions to non-circular states within the *n* = 30 manifold are shown (dashed grey lines) to highlight the fact that they cannot be driven by the lasers used for the deexcitation.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** R<sub>st/sp</sub>: number of trapped atoms in the THz-stimulated decay scheme over number of trapped atoms with spontaneous emission only. The ratio is given as a function of the lowest stimulated level *n'* and for different stimulated rates. The error bars are statistical (each data point represents a simulation of typically 1150 atoms in the stimulated case normalized to the result obtained with 2150 atoms spontaneously decaying). The lines are a guide for the eye.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Ratio of trapped atoms with and without THz-stimulated decay (R<sub>st/sp</sub>) as a function of *n'* for different trap configurations. The stimulated rate is Γ = 4 × 10<sup>7</sup> s<sup>-1</sup>. The lines are a guide for the eye.</figcaption></figure>

## 4.2. Temperature distributions

The velocity distribution of the initial atomic cloud generated at 40 K, the initial velocity of the atoms eventually trapped and the final velocity distribution of the atoms trapped are plotted in figure 5, for the sole spontaneous emission decay and in figure 6 for stimulated decay.

For the stimulated case, the ‘small cusp’, ‘octupole trap’ and ‘large cusp’ are plotted for the *n'* = 5 and Γ = 4 × 10<sup>7</sup> s<sup>-1</sup> configuration. They are compared to the spontaneous case in the respective same traps. For the ‘small cusp’, we see that the stimulation of the decay enables to trap the entire velocity

distribution up to velocities about twice what is achieved with spontaneous decay. The final temperatures of the trapped cloud are similar in this trap configuration (we find 129 mK for the stimulated case and 118 mK for the spontaneous case).

The performance of the spontaneous decay induced cooling is very similar for the small and large cusp configuration. The improvement in the trapped fraction for the large cusp compared to the small one comes almost entirely from the enhanced performance of the stimulated scheme. On the contrary, for the octupole configuration, the cooling by spontaneous decay is less efficient, in line with the findings by Pohl *et al* [26]. The onset velocity (velocity above which the distribution is only partially trapped) is similar for the spontaneous and stimulated cases, but the stimulation allows to hold onto initially hotter atoms. The absolute number of trapped atoms is smaller than for the linear trap cases, the larger gain
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Initial ($v_{ini}$) and final ($v_{fin}$) velocity distributions of atoms spontaneously decaying in a small cusp (black), large cusp (light gray) and octupole (dark gray) trap. In all cases, a total of 2000 atoms is initialized (black squares) with a 40 K Maxwell-Boltzmann (MB) velocity distribution (black continuous line showing only the portion of the distribution which was simulated—same for all traps). The velocity distribution of the finally trapped ground state atoms (circles) is fitted with a MB distribution (dashed lines). The fitted temperatures amount to 118 mK (black), 121 mK (light gray) and 176 mK (dark gray). The initial distribution of velocities of the trapped atoms in ground state is also plotted (triangles).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Initial ($v_{ini}$) and final ($v_{fin}$) velocity distributions of atoms when stimulating their decay ($n = 30 \rightarrow n' = 5$ and $\Gamma = 4 \times 10^7 \text{ s}^{-1}$) toward ground state. Similar to figure 5, the results are shown for a small cusp (black), large cusp (light gray) and octupole (dark gray) trap. For each scenario, a total of 2000 atoms is initialized (squares) with a 40 K MB velocity distribution (continuous line). The velocity distribution of the finally trapped ground state atoms (circles) is fitted with a MB distribution (dashed lines). The obtained temperatures amount to 129 mK (black), 124 mK (light gray) and 219 mK (dark gray). In addition, the initial distribution of velocities of the finally trapped atoms in ground state is plotted (triangles).</figcaption></figure>

coming from the under-performance of the spontaneous decay. The final, three-dimensional, temperature achieved is about 75% higher for the octupole configuration than both cusp traps for which the temperature reached is a factor ~2 higher than the state-of-the-art with Lyman-α laser-cooling in one dimension.

**5. Conclusion**

We have presented a cooling scheme for Rydberg atoms in a magnetic trap using THz radiation resonant at the turning point of the atoms in the field gradient. A time-dependent detuning enables efficiently addressing atoms of different velocities.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Initial ($v_{ini}$) and final ($v_{fin}$) velocity distributions of atoms spontaneously decaying in a small cusp (black), large cusp (light gray) and octupole (dark gray) trap. In all cases, a total of 2000 atoms is initialized (black squares) with a 40 K Maxwell-Boltzmann (MB) velocity distribution (black continuous line showing only the portion of the distribution which was simulated—same for all traps). The velocity distribution of the finally trapped ground state atoms (circles) is fitted with a MB distribution (dashed lines). The fitted temperatures amount to 118 mK (black), 121 mK (light gray) and 176 mK (dark gray). The initial distribution of velocities of the trapped atoms in ground state is also plotted (triangles).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Initial ($v_{ini}$) and final ($v_{fin}$) velocity distributions of atoms when stimulating their decay ($n = 30 \rightarrow n' = 5$ and $\Gamma = 4 \times 10^7 \text{ s}^{-1}$) toward ground state. Similar to figure 5, the results are shown for a small cusp (black), large cusp (light gray) and octupole (dark gray) trap. For each scenario, a total of 2000 atoms is initialized (squares) with a 40 K MB velocity distribution (continuous line). The velocity distribution of the finally trapped ground state atoms (circles) is fitted with a MB distribution (dashed lines). The obtained temperatures amount to 129 mK (black), 124 mK (light gray) and 219 mK (dark gray). In addition, the initial distribution of velocities of the finally trapped atoms in ground state is plotted (triangles).</figcaption></figure>

coming from the under-performance of the spontaneous decay. The final, three-dimensional, temperature achieved is about 75% higher for the octupole configuration than both cusp traps for which the temperature reached is a factor ~2 higher than the state-of-the-art with Lyman-α laser-cooling in one dimension.

**5. Conclusion**

We have presented a cooling scheme for Rydberg atoms in a magnetic trap using THz radiation resonant at the turning point of the atoms in the field gradient. A time-dependent detuning enables efficiently addressing atoms of different velocities.
Results showed several-fold improvements over spontaneous decay which represent up to an order of magnitude improvement over previous studies investigating the use of microwaves (MW). This is however at the expense of a more challenging technical implementation in particular with respect to available powers in the high THz region. This can be mitigated by lowering the targeted end state leading to slightly reduced performance. The bottleneck, for the trap configuration studied, lies around $n' = 10$, where the state mixing effect is counter-balanced by the too long spontaneous lifetime. This increased performance over MW mixing is a combination of several effects (i) the MW scheme couples together all levels in a given $n$ manifold ($n^2 \sim 1000$ states, for $n \sim 30$), in contrast to maximum 25 states in the THz case. The deexcitation time being proportional to the number of states addressed, the MW scheme alone can become quickly under-performing (ii) the THz scheme maintains the trapping potential maximal during the deexcitation by keeping the atoms in circular states. Additionally, it addresses the full distribution of $n$-states together and can be extended to higher $n$ states with very little additional power needed.

The results strongly depend on the initial state distribution but also on the initial temperature and trap configurations. We found the highest gain over spontaneous decay in the number of atoms trapped for the octupole case, at the expense of a larger final temperature. Lower final temperatures and improved performance are obtained for the larger cusp trap, indicating that a combination of larger dimensions and octupole order (as is the case in the ALPHA experiment) could yield even better results. Further optimization of the scheme could lead to settings improving the performance of the scheme beyond the maximum factor of $\sim 5$ found here.

Discussions on the technical implementations are beyond the scope of this paper, but we note that the access for the THz light could be axial, covering the entire trap or ideally hollow to avoid the center of the trap. In our work the turning point is close to the walls in the first zone, but is gradually moving to the center as colder atoms are addressed. We implemented spatial selection via spectral detuning, but one could imagine using spatial selectors to minimize the exposure of the trap walls to the THz light. In any case, total THz powers of roughly $20 \text{ mW cm}^{-2}$ should not constitute a major bottleneck in cryogenic trap configurations.

This cooling scheme, which also leads to an enhanced phase-space density, is not limited to the antihydrogen case and can be applied to any high magnetic moment atoms or molecules in a magnetic trap and could thus find applications in the cold molecular community [40]. Finally, we note that more involved schemes could be built upon this simple mechanism. For example, a Sisyphus cycle (similar to [41], but taking advantage of Rydberg states and fast stimulated deexcitation techniques) involving reexcitation to circular states at the center of the trap could lead to much colder temperatures.

## Acknowledgments

This work received funding by the Wolfgang Gentner Program of the German Federal Ministry of Education and Research (Grant No. 05E15CHA, university supervision by Norbert Pietralla) and the Austrian Doctoral Student Program at CERN. It was supported by the German Academic Scholarship Foundation. In addition, the research presented in this article was supported by the LASANTI grant (from Domaine d'Intérêt Majeur Science et Ingénierie en Région Île-de-France pour les Technologies Quantiques-DIM SIRTEQ-) and from the Initiative de Recherche stratégique of the Université Paris-Saclay IQUPS Project Number 156541.

## Data availability statement

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

## ORCID iDs

C Malbrunot https://orcid.org/0000-0001-6193-6601

T Wolz https://orcid.org/0000-0002-6810-2404

D Comparat https://orcid.org/0000-0001-5884-2882

## References

[1] Fried D G, Killian T C, Willmann L, Landhuis D, Moss S C, Kleppner D and Greytak T J 1998 *Phys. Rev. Lett.* **81** 3811–4

[2] Kleppner D 1989 Trapped atomic hydrogen *The Hydrogen Atom* (Berlin: Springer) pp 103–11

[3] Cesar C L, Fried D G, Killian T C, Polcyn A D, Sandberg J C, Yu I A, Greytak T J, Kleppner D and Doyle J M 1996 *Phys. Rev. Lett.* **77** 255–8

[4] Parthey C G et al 2011 *Phys. Rev. Lett.* **107** 203001

[5] GRASIAN 2021 Grasian homepage https://grasian.eu

[6] Ahmadi M et al 2018 *Nature* **557** 71–5

[7] Ahmadi M et al 2018 *Nature* **561** 211–5

[8] Ahmadi M et al 2020 *Nature* **578** 375–80

[9] Ahmadi M et al 2017 *Nature* **548** 66–9

[10] Bertsche W A 2018 *Phil. Trans. R. Soc. A* **376** 20170265

[11] Kuroda N et al 2014 *Nat. Commun.* **5** 3089

[12] Drobychev G Y et al 2007 Proposal for the AEGIS experiment at the CERN antiproton decelerator (antimatter experiment: gravity, interferometry, spectroscopy) *Technical Report Geneva CERN*

[13] Chardin G et al 2011 Proposal to measure the gravitational behaviour of antihydrogen at rest *Technical Report Geneva CERN*

[14] Ahmadi M et al 2017 *Nat. Commun.* **8** 681

[15] Gabrielse G et al 2011 *Phys. Rev. Lett.* **106** 073002

[16] Baker C J et al 2021 *Nat. Commun.* **12** 6139

[17] Cerchiari G, Yzombard P and Kellerbauer A 2019 *Phys. Rev. Lett.* **123** 103201

[18] Yzombard P, Hamamda M, Gerber S, Doser M and Comparat D 2015 *Phys. Rev. Lett.* **114** 213001

[19] Baker C J et al 2021 *Nature* **592** 35–42

[20] Michan J M, Polovy G, Madison K W, Fujiwara M C and Momose T 2015 *Hyperfine Interact.* **235** 29–36

[21] Gabrielse G et al 2018 *Opt. Lett.* **43** 2905–8
Results showed several-fold improvements over spontaneous decay which represent up to an order of magnitude improvement over previous studies investigating the use of microwaves (MW). This is however at the expense of a more challenging technical implementation in particular with respect to available powers in the high THz region. This can be mitigated by lowering the targeted end state leading to slightly reduced performance. The bottleneck, for the trap configuration studied, lies around $n' = 10$, where the state mixing effect is counter-balanced by the too long spontaneous lifetime. This increased performance over MW mixing is a combination of several effects (i) the MW scheme couples together all levels in a given $n$ manifold ($n^2 \sim 1000$ states, for $n \sim 30$), in contrast to maximum 25 states in the THz case. The deexcitation time being proportional to the number of states addressed, the MW scheme alone can become quickly under-performing (ii) the THz scheme maintains the trapping potential maximal during the deexcitation by keeping the atoms in circular states. Additionally, it addresses the full distribution of $n$-states together and can be extended to higher $n$ states with very little additional power needed.

The results strongly depend on the initial state distribution but also on the initial temperature and trap configurations. We found the highest gain over spontaneous decay in the number of atoms trapped for the octupole case, at the expense of a larger final temperature. Lower final temperatures and improved performance are obtained for the larger cusp trap, indicating that a combination of larger dimensions and octupole order (as is the case in the ALPHA experiment) could yield even better results. Further optimization of the scheme could lead to settings improving the performance of the scheme beyond the maximum factor of $\sim 5$ found here.

Discussions on the technical implementations are beyond the scope of this paper, but we note that the access for the THz light could be axial, covering the entire trap or ideally hollow to avoid the center of the trap. In our work the turning point is close to the walls in the first zone, but is gradually moving to the center as colder atoms are addressed. We implemented spatial selection via spectral detuning, but one could imagine using spatial selectors to minimize the exposure of the trap walls to the THz light. In any case, total THz powers of roughly $20 \text{ mW cm}^{-2}$ should not constitute a major bottleneck in cryogenic trap configurations.

This cooling scheme, which also leads to an enhanced phase-space density, is not limited to the antihydrogen case and can be applied to any high magnetic moment atoms or molecules in a magnetic trap and could thus find applications in the cold molecular community [40]. Finally, we note that more involved schemes could be built upon this simple mechanism. For example, a Sisyphus cycle (similar to [41], but taking advantage of Rydberg states and fast stimulated deexcitation techniques) involving reexcitation to circular states at the center of the trap could lead to much colder temperatures.

## Acknowledgments

This work received funding by the Wolfgang Gentner Program of the German Federal Ministry of Education and Research (Grant No. 05E15CHA, university supervision by Norbert Pietralla) and the Austrian Doctoral Student Program at CERN. It was supported by the German Academic Scholarship Foundation. In addition, the research presented in this article was supported by the LASANTI grant (from Domaine d'Intérêt Majeur Science et Ingénierie en Région Île-de-France pour les Technologies Quantiques-DIM SIRTEQ-) and from the Initiative de Recherche stratégique of the Université Paris-Saclay IQUPS Project Number 156541.

## Data availability statement

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

## ORCID iDs

C Malbrunot https://orcid.org/0000-0001-6193-6601

T Wolz https://orcid.org/0000-0002-6810-2404

D Comparat https://orcid.org/0000-0001-5884-2882

## References

[1] Fried D G, Killian T C, Willmann L, Landhuis D, Moss S C, Kleppner D and Greytak T J 1998 *Phys. Rev. Lett.* **81** 3811–4

[2] Kleppner D 1989 Trapped atomic hydrogen *The Hydrogen Atom* (Berlin: Springer) pp 103–11

[3] Cesar C L, Fried D G, Killian T C, Polcyn A D, Sandberg J C, Yu I A, Greytak T J, Kleppner D and Doyle J M 1996 *Phys. Rev. Lett.* **77** 255–8

[4] Parthey C G et al 2011 *Phys. Rev. Lett.* **107** 203001

[5] GRASIAN 2021 Grasian homepage https://grasian.eu

[6] Ahmadi M et al 2018 *Nature* **557** 71–5

[7] Ahmadi M et al 2018 *Nature* **561** 211–5

[8] Ahmadi M et al 2020 *Nature* **578** 375–80

[9] Ahmadi M et al 2017 *Nature* **548** 66–9

[10] Bertsche W A 2018 *Phil. Trans. R. Soc. A* **376** 20170265

[11] Kuroda N et al 2014 *Nat. Commun.* **5** 3089

[12] Drobychev G Y et al 2007 Proposal for the AEGIS experiment at the CERN antiproton decelerator (antimatter experiment: gravity, interferometry, spectroscopy) *Technical Report Geneva CERN*

[13] Chardin G et al 2011 Proposal to measure the gravitational behaviour of antihydrogen at rest *Technical Report Geneva CERN*

[14] Ahmadi M et al 2017 *Nat. Commun.* **8** 681

[15] Gabrielse G et al 2011 *Phys. Rev. Lett.* **106** 073002

[16] Baker C J et al 2021 *Nat. Commun.* **12** 6139

[17] Cerchiari G, Yzombard P and Kellerbauer A 2019 *Phys. Rev. Lett.* **123** 103201

[18] Yzombard P, Hamamda M, Gerber S, Doser M and Comparat D 2015 *Phys. Rev. Lett.* **114** 213001

[19] Baker C J et al 2021 *Nature* **592** 35–42

[20] Michan J M, Polovy G, Madison K W, Fujiwara M C and Momose T 2015 *Hyperfine Interact.* **235** 29–36

[21] Gabrielse G et al 2018 *Opt. Lett.* **43** 2905–8
[22] Saito N, Oishi Y, Miyazaki K, Okamura K, Nakamura J, Louchev O A, Iwasaki M and Wada S 2016 *Opt. Express* **24** 7566–74

[23] Gabrielse G 2021 Homepage of G Gabrielse group: new frontiers of hydrogen spectroscopy https://cfp.physics.northwestern.edu/gabrielse-group/lasercoolinghydrogenbeam.html

[24] Pérez P *et al* 2015 *Hyperfine Interact.* **233** 21–7

[25] Cesar C L, Robicheaux F and Zagury N 2009 *Phys. Rev. A* **80** 041404

[26] Pohl T, Sadeghpour H R, Nagata Y and Yamazaki Y 2006 *Phys. Rev. Lett.* **97** 213001

[27] Taylor C L, Zhang J and Robicheaux F 2006 *J. Phys. B: At. Mol. Opt. Phys.* **39** 4945–59

[28] Wolz T, Malbrunot C, Vieille-Grosjean M and Comparat D 2020 *Phys. Rev. A* **101** 043412

[29] Comparat D 2014 *Phys. Rev. A* **89** 043410

[30] Gabrielse G *et al* 2002 *Phys. Rev. Lett.* **89** 233401

[31] Malbrunot C *et al* 2018 *Phil. Trans. R. Soc. A* **376** 20170273

[32] Robicheaux F 2008 *J. Phys. B: At. Mol. Opt. Phys.* **41** 192001

[33] Radics B, Murtagh D, Yamazaki Y and Robicheaux F 2014 *Phys. Rev. A* **90** 032704

[34] Jonsell S and Charlton M 2019 *New J. Phys.* **21** 073020

[35] Flannery M R and Vrinceanu D 2003 *Phys. Rev. A* **68** 030502

[36] Vieille-Grosjean M, Mazzotta Z, Comparat D, Dimova E, Wolz T and Malbrunot C 2021 *Eur. Phys. J. D* **75** 27

[37] Mittleman D M 2017 *J. Appl. Phys.* **122** 230901

[38] Sarieddeen H, Alouini M-S and Al-Naffouri T Y 2021 *Proc. IEEE* **109** 1628–65

[39] Rahaman M H, Bandyopadhyay A, Pal S and Ray K P 2021 *IETE Tech. Rev.* **38** 465–78

[40] Heazlewood B R and Softley T P 2021 *Nat. Rev. Chem.* **5** 125–40

[41] Pritchard D E 1983 *Phys. Rev. Lett.* **51** 1336–9
[22] Saito N, Oishi Y, Miyazaki K, Okamura K, Nakamura J, Louchev O A, Iwasaki M and Wada S 2016 *Opt. Express* **24** 7566–74

[23] Gabrielse G 2021 Homepage of G Gabrielse group: new frontiers of hydrogen spectroscopy https://cfp.physics.northwestern.edu/gabrielse-group/lasercoolinghydrogenbeam.html

[24] Pérez P *et al* 2015 *Hyperfine Interact.* **233** 21–7

[25] Cesar C L, Robicheaux F and Zagury N 2009 *Phys. Rev. A* **80** 041404

[26] Pohl T, Sadeghpour H R, Nagata Y and Yamazaki Y 2006 *Phys. Rev. Lett.* **97** 213001

[27] Taylor C L, Zhang J and Robicheaux F 2006 *J. Phys. B: At. Mol. Opt. Phys.* **39** 4945–59

[28] Wolz T, Malbrunot C, Vieille-Grosjean M and Comparat D 2020 *Phys. Rev. A* **101** 043412

[29] Comparat D 2014 *Phys. Rev. A* **89** 043410

[30] Gabrielse G *et al* 2002 *Phys. Rev. Lett.* **89** 233401

[31] Malbrunot C *et al* 2018 *Phil. Trans. R. Soc. A* **376** 20170273

[32] Robicheaux F 2008 *J. Phys. B: At. Mol. Opt. Phys.* **41** 192001

[33] Radics B, Murtagh D, Yamazaki Y and Robicheaux F 2014 *Phys. Rev. A* **90** 032704

[34] Jonsell S and Charlton M 2019 *New J. Phys.* **21** 073020

[35] Flannery M R and Vrinceanu D 2003 *Phys. Rev. A* **68** 030502

[36] Vieille-Grosjean M, Mazzotta Z, Comparat D, Dimova E, Wolz T and Malbrunot C 2021 *Eur. Phys. J. D* **75** 27

[37] Mittleman D M 2017 *J. Appl. Phys.* **122** 230901

[38] Sarieddeen H, Alouini M-S and Al-Naffouri T Y 2021 *Proc. IEEE* **109** 1628–65

[39] Rahaman M H, Bandyopadhyay A, Pal S and Ray K P 2021 *IETE Tech. Rev.* **38** 465–78

[40] Heazlewood B R and Softley T P 2021 *Nat. Rev. Chem.* **5** 125–40

[41] Pritchard D E 1983 *Phys. Rev. Lett.* **51** 1336–9