Rydberg renaissance

Almost 130 years ago, Johannes Robert Rydberg wrote down a formula that contained the observed series of spectral lines of hydrogen and predicted the existence of energy states near the series limit. Atoms in such highly excited states are now known as Rydberg atoms. In this article we will briefly review their properties and history and explain why they have attracted renewed interest in recent years as building blocks for simulators of classical and quantum many-body systems.

1 Introduction

“To try to make a model of an atom by studying its spectrum is like trying to make a model of a grand piano by listening to the noise it makes when thrown downstairs.” This famous soundbite from the early days of atomic physics nicely sums up what towards the end of the 19th century seemed to be an insurmountable task: to understand the inner workings of atoms (whose existence would only be confirmed in 1909 by Jean Perrin) by looking at the light they emitted or absorbed. More than a hundred years later, one still stands in awe before the accomplishments of the scientists who managed, through sheer intuition and perseverance, to crack nature’s code. Extending the metaphor of the grand piano, one scientist stands out as the man with a piano tuner’s ear: Johannes Robert Rydberg (1854-1919, picture on this page). By analyzing vast amounts of (not always high-quality) spectroscopic data, he was able to come up with a simple mathematical formula that contained all the known hydrogen lines and predicted the wavelengths of many other lines that had not yet been observed. Niels Bohr, the father of the atomic model, called it a happy intuition that Rydberg “from the beginning [looked] for relations not between the directly measured wavelengths of the spectral lines, but between the reciprocal figures expressing the number of waves per unit length, now known as wave numbers”. Where Johann Jakob Balmer had tried to find a relationship between the wavelengths of the hydrogen spectrum of the form

$\lambda = \frac{bn^{2}}{n^{2}-4}$

(where b was an experimentally determined constant), Rydberg realized that relating the inverses of the wavelengths to each other worked much better. In 1889, this led him to propose a formula of the type

$\sigma = \frac{R}{2^{2}} - \frac{R}{n^{2}}$

for the wavenumber of the observed lines. The parameter R appearing in that formula now bears his name and is known as the Rydberg constant (today often written as R_∞ , or as Ry when expressed in energy units). As already mentioned, not only did the formula fit the observed spectral lines, it also predicted that there should be ever more closely spaced lines as the parameter n – which we now recognize as the principal quantum number – goes to infinity (the so-called series limit). Atoms in high-energy states giving rise to such lines are today known as Rydberg atoms. In the present article we will look at their characteristics, their importance during the early days of quantum mechanics and atomic physics, and at the surprising comeback they are now making in experiments and applications that make use of what have been described as the “exaggerated” properties of Rydberg atoms.

2 Rydberg atoms: properties and early history

Some twenty years after Rydberg’s spectacular success in finding a simple mathematical relationship between the spectral lines of hydrogen, Niels Bohr in Copenhagen tried to find an explanation for the observed wavelengths based on physical considerations. A few years earlier Ernest Rutherford had experimentally proved the existence of the atomic nucleus, and it was evident that a model of the atom had to allow for the fact that most of the atom was actually empty. Bohr, therefore, proposed a simple planetary orbit-type model, but with a twist. In addition to the balance between the centrifugal force of the orbiting electron and the attractive Coulomb force due to the nucleus, he also assumed that the angular momentum of the electron had to be quantized in units of Planck’s constant ħ. Putting together these two requirements immediately led to a simple expression for the energy difference between two atomic levels W1 and W2 characterized by quantum numbers $n_{1}$ and $n_{2}$:

$ W_{2} -W_{2} = \frac{k^{2}Z^{2}e^{4}m}{2\hbar^{2}} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})$

The factor in front of the parentheses (divided by hc to turn it into a wavenumber, where c is the velocity of light) turned out to be precisely the numerical constant Johannes Rydberg had found by crunching numbers, and so the value of Rydberg’s constant could now be explained in terms of the electronic charge e, the electron’s mass m, the nuclear charge Z (Z=1 for hydrogen) and Planck’s constant (in the formula, k = 1/(4 πε₀) is the Coulomb force constant related to the electric permittivity of the vacuum).
From Bohr’s model one can easily deduce some of the properties that make Rydberg atoms so special (see fig. 1). The most obvious of those is the radius of the electron’s orbit:

$ r = \frac{m^{2}\hbar^{2}}{2e^{2}mk} $

This radius scales as n², which means that for a Rydberg atom – typically defined as an excited atom with a principal quantum number larger than 20 – that radius, and hence the “size” of the Rydberg atom, can be quite large. To see just how large, let us take an example: a rubidium atom (an alkali with one electron in the outer shell, and hence very similar to hydrogen in its spectral properties1) in its 5S ground state has a size of about a third of a nanometer (or 5.6 Bohr radii, where the Bohr radius is defined as a₀ = 5.29×10^-11 m). A 100S Rydberg state (similar to the ones we will describe in our experiments further down), by contrast, has an electronic wave function (as it is called in modern quantum theory) that measures almost a micrometer. In other words, it should be possible to fit around 10 billion ground state rubidium atoms into a single 100S Rydberg atom. In fact, some of the early work on Rydberg atoms by Edoardo Amaldi and Emilio Segrè (two of the ragazzi di via Panisperna) tried to explain the line shifts of the Rydberg states of potassium in the presence of a high-pressure rare gas (e.g., argon or helium) by assuming that a large number of those atoms filled the empty space between the nucleus and the orbiting electron of the Rydberg atom. The rare-gas atoms thus modified the electric permittivity in that space compared to a vacuum. A few years later, however, Enrico Fermi showed that that assumption did not lead to the right result and found an alternative explanation in terms of scattering between the Rydberg electron and the rare gas atoms.
The size of a Rydberg atom is not the only quantity that can take on values that differ by orders of magnitude from those of ground-state atoms (or, indeed, of atoms in low-lying excited states, which one encounters in most experiments). The fact that the electron orbits the nucleus at a large distance also means that it can be easily disturbed by electric fields. A simple model describing the sum of the nuclear and the applied electric potentials shows that the field needed to ionize a Rydberg atom scales as $n^{-2}$. The electric field needed to ionize a 100S Rydberg state, for instance, is on the order of a few volts per centimeter, around 2000 times smaller than the one for a 5S ground state. This result implies a simple and powerful method for detecting Rydberg atoms: one just needs to apply a modest electric field and detect the resulting charged particles (we will use this method in the experiments reported in sect. 4).
Another consequence of the extraordinary size of Rydberg atoms is their radiative lifetime. Typically, once an atom is excited above its ground state, it quickly returns to it through the coupling of its electric dipole moment to vacuum fluctuations. For instance, the first excited state of rubidium – the 5P state – has a lifetime of just 26 nanoseconds. The 100S Rydberg state, by contrast, lives for almost 3 milliseconds (neglecting other decay mechanisms due to black-body radiation). Generally, the lifetime of an excited state scales as n³, which means that the lifetimes of Rydberg states exceed those of low-lying excited states by several orders of magnitude.
Finally, the high sensitivity of Rydberg atoms to electric fields also means that Rydberg atoms are highly polarizable and can interact strongly through induced electric dipoles. In fact, a small fluctuation in the relative position between electron and nucleus (still speaking in semi-classical terms, for simplicity) will result in a momentary dipole moment, which in turn will create an electric field that can induce an electric dipole in a nearby atom at distance r. The two dipoles then interact, and that so-called van der Waals interaction depends on the distance as $\frac{1}{r^{6}}$ – but more importantly, the prefactor of the interaction (called the C₆ coefficient) scales as n¹¹. For the two atomic states we looked at above – 5S and 100S in rubidium – this scaling predicts that two atoms in the 100S state at a certain distance from each other interact around 1000 quadrillion (10¹⁸) times more strongly than two 5S atoms at the same distance! It is because of this extreme scaling that in the literature on Rydberg atoms their properties are often described as “exaggerated” – an adjective that does not appear too often in scientific papers.
Despite their intriguing properties, for the first few decades after their implicit discovery by Rydberg and Bohr, Rydberg atoms were not studied very extensively. There was some interest in the astrophysics community in the 1970’s when it emerged that transitions between Rydberg states around $n=100 $ in interstellar matter could be detected as microwave signals at 2.4 GHz. Back on the ground, however, the physics of Rydberg atoms only made slow progress. One reason for this was that there were no experimental techniques available to controllably and reproducibly excite well-defined Rydberg states. The methods at hand were rather brute-force approaches: bombarding atoms with electrons or ions in order to kick an atom’s outer electron into a Rydberg orbit. Typically, such methods resulted in a mixture of many different Rydberg states. The detailed study of single Rydberg states, and indeed single Rydberg atoms, had to await more sophisticated tools.

3 The Rydberg renaissance: lasers and cold atoms

When Theodore Maiman built the first optical laser in 1960, the new device was famously called a “solution in search of a problem”. We now know that, in fact, it was a solution to a thousand problems, and not just in scientific research. For atomic physicists, to be sure, the laser was a godsend. In particular, the advent of the tunable dye laser in the 1970’s made it possible to shine light of (relatively) freely selectable wavelength and unprecedented spectral purity on atoms and precisely measure their response. Before long, researchers realized that such lasers were also an ideal tool for studying Rydberg atoms. Serge Haroche, the 2012 Nobel laureate, recalls that when bringing back the dye laser technology to Paris after a few years in Stanford working with Arthur Schawlow (one of the inventors of the laser), he immediately thought of applying this new and wonderful tool to Rydberg atoms: “Close to the atomic ionization limit, there was a very large number of levels forming a spectroscopic terra incognita of so-called Rydberg states with huge electron orbits. The lasers offered us the opportunity to prepare and study those states […]”. Besides those spectroscopic studies, which shed new light on the exact structure of the Rydberg levels and enabled the controlled excitation of single Rydberg states, Haroche was particularly interested in the huge polarizability of Rydberg atoms and their resulting strong coupling to microwaves. After initial successful experiments with atomic beams that were sent through a microwave cavity formed by two copper mirrors, it became clear that it should be possible to increase the quality factor of the cavity sufficiently so that one could observe the coupling of a single Rydberg atom to a single microwave photon. Thus, the research field of cavity quantum electrodynamics was born. Haroche and his colleagues then went on to use that technique for a careful study of the quantum states of the photons inside the cavity, effectively employing Rydberg atoms as a highly sensitive probe to count the cavity photons “one by one” (fig. 2) without destroying them. Once the Rydberg atoms had interacted with the photons in the cavity, all one had to do to obtain information about that interaction was to field-ionize the atoms (as we discussed above) in order to find out whether they were in a Rydberg state (by chirping the value of the ionizing field and recording at what value of the field an atom was ionized, it was possible to determine the quantum number of the Rydberg state).
While Haroche was performing his cavity QED experiments, the newly developed laser technology paved the way for another groundbreaking technology that would lead to renewed interest in Rydberg atoms. Shortly after the first lasers had been built, in 1975 Theoder Hänsch and Arthur Schwalow and, independently of them, David Wineland and Hans Dehmelt (all of whom would later win Nobel prizes, albeit for different discoveries) proposed a method for cooling atoms by shining laser light on them. If the frequency of the laser was slightly below an atomic resonance, they argued, then the Doppler effect would shift the frequencies “seen” by atoms moving towards or away from the laser source in opposite directions. Atoms moving towards the laser would be more likely to absorb a photon (as they saw the light blue-shifted and hence closer to resonance), whilst those moving away from it would very rarely absorb photons. Since each absorption event resulted in a recoil of the atom as the photon’s momentum was transferred to it, the net effect was to slow the atoms down. A simple extension of this principle to three dimensions led to the concept of “optical molasses”, in which atoms were slowed down whichever direction they moved in, much like in a viscous medium.
After initial experiments with trapped ions, in the early 1980’s it was experimentally demonstrated that optical molasses could actually cool a gas of neutral atoms down to incredibly low temperatures – in a few milliseconds they went from room temperature to less than a thousandth of a degree above absolute zero. Adding a magnetic field gradient gave spatial confinement on top of the cooling, and so the magneto-optical trap (or MOT) was born.
In the past three decades, laser cooling of atoms has gone from strength to strength, enabling, amongst other things, the observation of Bose-Einstein condensation in a dilute atomic gas in 1995, as well as ultra-precise atomic fountain clocks. As far as the study of Rydberg atoms is concerned, laser cooling ushered in an entirely new era. One of the direct consequences of cooling atoms to extremely low temperatures (or rather, another way of saying the same thing) is that their random motion slows down. At typical temperatures of magneto-optical traps of rubidium atoms around 100 micro-kelvin, the mean atomic velocity is on the order of a few centimeters per second, as opposed to 170 meters per second at room temperature. Since typical experiments are carried out on timescales of microseconds or milliseconds, the spatial motion of the atoms can be safely neglected and one effectively deals with a completely “frozen gas” (we will use this fact in sect. 4, where this approximation holds well for the experiments reported there). The frozen gas approximation makes it possible to perform experiments in which only the internal degrees of freedom of the atoms (represented by a fictitious spin ½) play a role. Laser cooling has certainly opened completely new avenues for studying Rydberg atoms. By the same token, Rydberg atoms are now starting to enrich the field of cold atoms. In particular, the strong van der Waals and dipole interactions make it possible to create highly correlated samples of cold atoms at modest densities. In cold atom experiments using ground state atoms, interactions are essentially limited to the contact interaction mediated by s-wave scattering (which, essentially, is a van der Waals interaction between ground-state atoms, and hence very small). Only when atoms are very close together, on the order of nanometers, do they appreciably feel each other’s presence. In typical cold atom experiments, however, the mean distance between atoms is on the order of a micrometer and they will collide only occasionally, which means that interaction energies (in frequency units) are on the order of tens to hundreds of hertz. In order to achieve stronger interactions, experimentalists have resorted to a number of tricks over the years. One of them is the use of so-called Feshbach resonances. By applying appropriate magnetic fields to the atoms, it is possible to shift a bound molecular state into resonance with the entrance and exit channels of the collision event , leading to s-wave interaction strengths that can be orders of magnitude larger than at zero magnetic field. Other approaches consists in creating ultracold molecules with a large electric dipole moment, or using atomic species with a large magnetic dipole moment. While all these techniques have been successfully demonstrated and are used in a number of experiments, Rydberg atoms still beat them all in terms of interaction strength. To give a concrete example: two ground state rubidium atoms at a distance of 10 micrometers essentially do not interact. Two 100S Rydberg atoms, on the other hand, have an interaction energy of 80 MHz at 10 micrometers (in frequency terms, compared to around 1 MHz of thermal energy at MOT temperatures), and at a separation of 5 micrometers that value shoots up to more than 5 GHz. Moreover, the strong interaction between Rydberg atoms can be switched on and off by exciting or de-exciting the atoms. The large value and controllability of their interactions make cold Rydberg atoms excellent candidates for creating highly controllable many-body systems. In the following we will present two examples, taken from our recent work in Pisa, of the type of physics that can be studied in such systems.

4 Artificial many-body systems with Rydberg atoms: kinetic constraints and dynamical phase transitions

Using the elements introduced above – Rydberg atoms and laser cooling – we can easily construct an artificial and controllable many-body system made from interacting, effective spin- ½ particles (see fig. 3). The two spin states correspond to the ground and Rydberg states of the atoms, and the interactions are the van der Waals interactions introduced at the beginning of this article. In particular, in the experiments described in the following we are interested in simulating a (quasi-)classical spin system, i.e., we want to ignore the coherences between the two spin states. To that end, we choose the parameters of our system such that the naturally occurring decoherence due to the finite linewidths of the excitation lasers and the residual motion of the atoms is faster than the excitation timescale, and hence the dynamics can be described by a classical rate equation in which the “spin flip” between the ground and Rydberg state occurs at a rate Γ.
In our experiments, we prepare cold samples of rubidium atoms in a MOT and excite 70S Rydberg states using a two-photon transition with a near-resonant intermediate state. This leads to an effective coherent coupling between the ground and the 70S state with a Rabi frequency Ω, whose value can be set through the laser intensities. Taking into account the decoherence rate γ, the resulting (resonant) spin-flip rate for the incoherent dynamics of a single atom is

$ \Gamma_{0} = \frac{\Omega^{2}}{2\gamma}$

After the excitation phase, atoms in the Rydberg state are field-ionized and detected by a charge multiplier.
The single atom spin-flip rate defined above only describes the dynamics if the driving laser is resonant with the transition and if there are no other Rydberg atoms nearby. Taking into account a possible detuning of the laser from resonance and the van der Waals interaction between atoms in the Rydberg state, the spin-flip rate for each atom is modified and depends on all the excited atoms in its vicinity as well as on the laser detuning (fig. 3 (a)). Two distinct effects are now possible, depending on the detuning of the excitation laser. If the laser is in resonance with a ground state-Rydberg transition of a single atom, then another excited atom in the vicinity will shift the Rydberg state we are trying to excite out of resonance. If that shift is larger than the linewidth of the laser, the probability of exciting the atom will be small. This is called the “dipole blockade” (fig. 3 (b)), and the distance below which the interaction between two Rydberg atoms is larger than the laser linewidth is called the “blockade radius”.
Going from two atoms to large samples of, say, hundreds of thousands of atoms in a MOT (which is a typical number for our experiments), it is easy to see that if one shines a resonant laser on a cloud of cold ground-state atoms, initially some Rydberg excitations will be created, as long as they are farther apart than the blockade radius. Beyond that point, subsequent excitations will be much less likely and hence the number of excitations eventually saturates (as seen in fig. 4 (a)). This also means that if we look at the fluctuations around the mean number of Rydberg excitations, we will find them suppressed beyond the value expected for a Poissonian random process. To quantify this, one can use the Mandel Q parameter Q = 〈ΔN²〉 /〈N〉 - 1 (originally conceived by Leonard Mandel for quantum optics experiments), where 〈ΔN²〉 is the variance and 〈N〉 the mean, that is zero for a Poissonian process, negative for sub-Poissonian processes and positive for super-Poissonian ones. In fig. 4 (b) one clearly sees that effect: the Q parameter is zero for short excitation times and goes towards its theoretical minimum of –1 as the number of excitations saturates.
A very different effect is expected if the laser driving the ground-state–Rydberg transition is detuned from resonance. For a single atom this simply implies that the rate of excitation Γ becomes small. If there is an already excited Rydberg atom nearby, however, the excitation rate can be drastically modified. In particular, if the sign of the van der Waals interaction matches that of the detuning (for repulsively interacting atoms as used in our experiments, the detuning has to be positive, i.e., on the blue side of the transition), for a particular distance between the two atoms the interaction exactly matches the detuning. In that case, the detuned laser is now resonant with the interaction-shifted Rydberg state, thus “facilitating” its excitation (fig. 3 (c)). The distance at which this facilitation occurs is called the facilitation radius.
Considering again a collection of ground-state atoms, this time excited by a detuned laser, we can imagine that initially not much will happen: excitation of single atoms is, as we have seen, unlikely in that case. Nevertheless, sooner or later one of the atoms in the cloud will be off-resonantly excited to a Rydberg state – and suddenly everything changes. Now there just has to be one atom among the many atoms in the cloud that is at the right distance – the facilitation radius – from the first excited atom. That atom will then be shifted into resonance and hence be excited much faster than the first one. Now there are two Rydberg excitations in the cloud, and of course those two can now shift further atoms into resonance, and those again will facilitate excitations, and so forth. In short, a chain reaction or avalanche ensues (visible as an acceleration of the dynamics in fig. 4 (a)). From this, we can see that for off-resonant excitation the fluctuations in the number of excitations will be larger than in the resonant case. In particular, the facilitation process amplifies small fluctuations in the early dynamics of the cloud, i.e., the time and position at which the first Rydberg excitation, also called the “seed”, occurs. Whenever a seed is created, subsequent facilitated excitations will happen fast (as long as there are atoms available to be excited at the facilitation radius) and result in a large total number of Rydberg excitations. In those cases where no seed was created, however, no facilitated excitations could take place, either, and so the total number of excitations will be small. Owing to the large, correlated fluctuations thus induced, the Q parameter in this case will be greater than 0. Figure 4 (b) illustrates this effect, with the Q parameter reaching a maximum value of 14, indicating strongly super-Poissonian fluctuations.
If we look at the experiments described above in terms of controllable simulators for (semi-classical) many-body dynamics, we notice that by simply changing the detuning of the excitation laser we can control the type of correlations that govern the dynamics of our system: anti-correlations for resonant excitation (meaning that existing excitations make subsequent ones less likely), and (positive) correlations for off-resonant excitation (i.e., existing excitations make subsequent ones more likely). Similar mechanisms also occur in other physical systems, for instance in glassy systems, colloids, or supercooled liquids. In all of those one encounters complex relaxation towards equilibrium, characterized by regions of activity and inactivity, as local re-arrangements of particles are strongly influenced by their surroundings. In some regions, the so-called steric hindrances will prevent particles from relaxing into a new position, whereas in other regions vacancies in the system will facilitate particle motion. It is easy to see that our Rydberg gases can be viewed as a model system for such complex many-body dynamics.
In glassy systems, steric hindrances and facilitation events have, so far, not been directly observed. However, as the rate equations describing those mechanisms are identical to those describing our model system, experiments of the type just described could help in understanding the microscopic behavior of real-world soft matter systems. In general, therefore, we conclude that cold Rydberg gases are a versatile model system for studying complex collective relaxation in many-body systems and associated non-ergodic phenomena.
The many-body model just described can be enriched by adding one more ingredient: dissipation. In the above experiments we implicitly neglected spontaneous decay of the Rydberg excitations back to the ground state, which was a good approximation as the longest experimental times (around 80 microseconds) were still shorter than the lifetime of the 70S Rydberg state (150 microseconds). By letting the system evolve for much longer times, spontaneous decay will become important and open up a dissipative channel that allows random down-flips of the spins. In “real-world” systems, this could be the effect of coupling the spins to a (low temperature) thermal bath .
When dissipation enters the picture there is a competition between the active driving of the atoms, with all the mechanisms described earlier, and passive decay from Rydberg to ground states. Such dissipative spin models have been extensively studied in recent years, and one of the salient properties that one expects to see is a dynamical phase transition between a passive phase – i.e., one in which the atoms remain in the ground state most of the time – and an active phase characterized by a large number of continuous excitation and decay events. In the crossover region between those two phases large fluctuations should occur, as the two phases coexist and the system randomly switches back and forth between them.
In our experiments we realized such a dissipative spin model by increasing the excitation time to almost a millisecond (six times the Rydberg lifetime). Since we will interpret our results in terms of a phase transition, it makes sense to relate our experimental parameters to those of a dissipative Ising spin model. In fact, it is easy to show that the Rabi frequency Ω (and, in the incoherent limit, the transition rate Γ) corresponds to the transverse field in the Ising model, while the detuning determines the coupling strength between the spins (which follows from the fact that the facilitation mechanism relies on the Rydberg-Rydberg interaction matching the detuning). The phase transition should then be visible in a plot of the mean number versus detuning and Rabi frequency as a division between a region of low mean number, close to zero, and one of high mean number. Also, in the crossover between the two regions enhanced fluctuations characterized by a large Q factor should be observed, as the system can coexist in the passive and active phases. Both features are clearly observed in our experiments, as seen in fig. 5.
In future experiments it will be interesting to investigate the dynamical phase transition more closely, by analyzing, for instance, the higher moments of the counting distributions such as the Binder cumulant, and by performing finite-size scaling through varying dimensions of the cold atom cloud. In the long run, of course, the aim will be to take our system into the regime of coherent dynamics, i.e., to turn it into a controllable quantum many-body system. The initial motivation for studying such systems was given more than 30 years ago.

5 Towards quantum simulators with Rydberg atoms

In 1981, Richard Feynman gave a keynote speech at the 1st Conference on Physics and Computation at MIT in Boston (attended by distinguished members of both the physics and computer science communities, among them Freeman Dyson and Konrad Zuse). At the beginning of his speech, Feynman asked a characteristically simple, but at the same time very profound question: How can we simulate a physical system on a computer? The problem, he explained, was that, on the one hand, we know that nature, at its most fundamental level, follows the rules of quantum mechanics; on the other hand, computers process numbers in the form of 0’s and 1’s – or “bits” – following classical rules such as AND, OR, NOT, and so forth. The ultimate question is, therefore, whether a quantum-mechanical system can be simulated by a classical system. A bit of reasoning along those lines quickly leads us into a dilemma. One can easily show that to store sufficient information about the full quantum state (i.e., without any approximations) of just 40 interacting spin-½ particles requires around one terabyte of memory.
For 80 spins the number of bits needed for storage exceeds the number of protons in the universe. It is this exponential scaling of resources (deriving from the exponential scaling of the system’s Hilbert space) needed for exactly computing a many-body quantum state that led Feynman to conclude that there were only two possible ways out of that dilemma: one either had to build computers that were governed by quantum mechanics (now known as quantum computers), or else one needed controllable laboratory models of a physical system of interest – a ferromagnet, say, or a high-temperature superconductor – and obtain the desired answers by performing measurements on the model system.
Interestingly, the latter solution proposed by Feynman precisely mimicked an approach that had already been successful some 60 years earlier, before the advent of digital computers, in a completely different context. Around the turn of the century, ever growing electrical networks using alternating current posed a serious problem to investors and engineers. While it was easy to write down the mathematical equations governing the behavior of the networks – a set of coupled differential equations of the type that James Clerk Maxwell had proposed thirty years earlier –, it was impossible to solve them analytically. Computers had not been invented yet, so scientists resorted to a trick: they built a scale model of the network in the laboratory, using inductances, capacitors and resistors, in such a way that the equations describing the model were the same as those for the actual electrical network. Now they could change voltages and phases in the model, or provoke an overload or short circuit, and read the results right off the model. For all intents and purposes, the scale model (also called a “network analyzer”) was a simulator of the real network – or, to put it differently, an analogue computer that “solved” the underlying differential equations.
What Feynman proposed in 1981, then, was effectively a return to analogue computers – but this time based on quantum rather than classical systems. If one could devise laboratory systems that had the same Hamiltonian as a real-world system of interest, then it was possible to get the laboratory system to effectively “calculate” quantities by simply letting it evolve naturally and making appropriate measurements. As we saw above, even interacting spin systems of modest size are impossible to calculate exactly on a classical computer. If one can create a model spin system in the laboratory, however, one can then use it as a "quantum simulator". Rydberg atoms, it turns out, lend themselves quite naturally to such a task.
We have already seen that a model system containing a hundred effective “spins” can easily be realized using Rydberg excitations in a cloud of cold atoms. In the experiments described in section 4, the dynamics of the spins was, to a good approximation, incoherent and therefore classical. All that is needed, in principle, to turn such a system into a quantum simulator is to make the excitations and the entire dynamics of the system coherent (by narrowing the linewidths of the excitation lasers, for instance, and by eliminating other sources of decoherence). Also, it might be desirable to place the atoms at well-known positions, for instance in periodic configurations that mimick solid-state crystals. The technology for achieving this – optical lattices – is well known and has been used in our lab (and dozens of others around the world) for many years. Finally, detecting not only the total number of Rydberg atoms, but also their positions, yields further information on the system’s properties and dynamics. Techniques that are capable of such a feat already exist and are being constantly refined and extended (see fig. 6).
Taking all of the above ingredients and putting them together should make it possible to perform quantum simulations of arbitrary spin systems – Ising models describing ferromagnetism, for instance, or coherent excitation transport in spin chains. With such simulators, one might one day test mathematical models for high-T_c superconductors or light harvesting complexes in plants, or even design new quantum materials with tailor-made properties. Moreover, Rydberg atoms have also been proposed as candidates for realizing quantum gates, which are the building blocks of quantum computers (the other solution proposed by Richard Feynman to solve the problem of the computability of quantum systems). All of this will, no doubt, require many more years of research. In any case, Rydberg atoms have come a long way in the past 130 years – from details in a formula to building blocks of tomorrow’s quantum technologies.

Acknowledgments

The author would like to thank C. Simonelli, M. Valado, N. Malossi, S. Scotto, M. Hoogerland, E. Arimondo and D. Ciampini, who contributed to the work described in section 4, as well as R. Mannella, P. Pillet, P. Huillery, I. Lesanovsky and J. Garrahan for providing theoretical support.