Having good neighbours can be very valuable – even in the atomic world. A team of Amsterdam physicists was able to determine an important property of strontium atoms, a highly useful element for modern applications in atomic clocks and quantum computers, to unprecedented precision. To achieve this, they made clever use of a nearby cloud of rubidium atoms. The results were published in the journal Physical Review Letters this week.

Strontium. It is perhaps not the most popularly known chemical element, but among a group of physicists it has a much better reputation – and rightfully so.

Strontium is one of six so-called alkaline earth metals, meaning that it shares properties with better-known cousins like magnesium, calcium and radium. Strontium atoms have 38 protons in their nucleus, and a varying number of neutrons – for the variations (or isotopes) of strontium that can be found in nature, either 46, 48, 49 or 50.

A mathematically inclined person might remark that only one of these numbers, 49, is odd. While this may seem like just a curious observation, it is actually that particular isotope of strontium, the one with 87 particles total in its nucleus, that has the most special properties. The odd number turns the nucleus into a type of object known as a fermion, whereas all other strontium nuclei are bosons. Importantly, the odd number also turns the nucleus into a tiny bar magnet, through a property called spin. While all the subatomic particles have spin, the two spins of a pair of identical particles can cancel each other completely, leading to total spin zero. This is exactly what happens in the bosonic isotopes of strontium, where the total spin of each of the (even-numbered) subatomic species (protons, neutrons and electrons) cancels to zero, and as a result the total spin of the atom is zero. It is the fermionic strontium atom that, through its nonzero nuclear spin, has the special properties that have inspired physicists for years now.

Atomic clocks and quantum computers

To begin with, ⁸⁷Sr, as the isotope is commonly abbreviated, is one of the prime candidate isotopes to be used in the next generation of atomic clocks – so-called optical clocks. Such clocks draw their extreme timing precision from very precisely determined frequencies of light that atoms emit or absorb. The emission or absorption arises when the atom transitions from one state to another. In the strontium atom, the most precise optical frequency corresponds to clear, red-colored light, with a wavelength of 698 nanometres. However, the problem with the bosonic versions of the atom is that the rules related to (zero) spin strictly forbid emission or absorption at what otherwise would be the ideal transition. Here the nuclear spin of the fermionic isotope comes to the rescue. The nuclear magnet resulting from the spin of ⁸⁷Sr allows “breaking” the spin rules just enough – yet not too much – for the optical clock transition to be viable for absorption and emission while remaining at a very well-defined and stable frequency. Furthermore, the strength of the nuclear magnet in ⁸⁷Sr is an important parameter for the operation of the clock.

This brings us to an effect discovered by Dutch Nobel Prize winner Pieter Zeeman in 1896, nowadays known as the Zeeman effect. In its original form, it describes the splitting of the different energy levels in which the electrons of an atom can be, and as a result outgoing photons – particles of light – can have many different but precisely determinable frequencies. At a more technical level, the same Zeeman effect occurs in an atomic nucleus that has spin, and the strength of the nuclear magnet determines the amount of splitting and the specific radio frequency (a much-lower-frequency form of light) with which one can flip the nuclear magnet from one state to another. As mentioned, this magnetic moment plays a key role in the operation of the optical clock, and its accurate determination can help researchers improve the calibration of such a clock.

Another special property, of the ⁸⁷Sr nucleus is that, through the Zeeman effect, its energy level splits into no less than ten (equally spaced) energy levels when a magnetic field is applied. These ten different states in which the nucleus can be, can be used as building blocks for a quantum computer. Where a classical computer uses bits that can be in two states – usually interpreted as “0” and “1” – a quantum computer uses qubits that can also be in a combined state – roughly “a little bit 0 and a little bit 1”. This notion of a qubit (and especially the controlled combination of many of these qubits) makes quantum computers much more powerful than classical computers when performing certain computations. However, the ten-fold degeneracy (and ten-fold splitting in a magnetic field) of the ⁸⁷Sr states opens up the possibility of also using qudits – analogues of qubits that can be in combinations of states “0”, “1”, “2”… all the way up to “9”. Such qudits would form even more versatile building blocks for quantum computers and quantum simulators.

The g-factor

In all of these applications, the Zeeman effect plays a central role, and so to get the most out of the potential uses of strontium, physicists want to know as accurately as possible by how much the energy levels of ⁸⁷Sr split. In other words: what is the strength of the nuclear magnet originating from its spin? The amount of splitting is determined by a quantity known as the g-factor of ⁸⁷Sr, and so the task is to determine this g-factor as precisely as possible. The small print: the value of the g-factor not only depends on the magnetic properties of the nucleus, but also on the small amount of magnetic shielding by the electron cloud surrounding the nucleus, constituting the total neutral atom. Calculating this to the desired level of precision is a seemingly insurmountable challenge, and hence precision measurement is crucial.

Very precise measurements of the g-factor had already been made more than fifty years ago, and so far these measurements had stood the test of time; no improvements were made afterwards. Now, however, a team of five physicists from the University of Amsterdam and quantum software research centre QuSoft have managed to achieve a hundredfold improvement of the previously known value.

As is so often the case, the breakthrough came from a rather unexpected direction. First author Premjith Thekkeppatt, who worked on the research as a PhD student in the Amsterdam group and is now a postdoctoral researcher at the Niels Bohr Institute in Copenhagen, explains: “The work grew out of our efforts to merge strontium atoms with another element, rubidium, to create rubidium-strontium molecules. This turned out to be extremely challenging, prompting us to investigate what we could achieve by having both species in one another’s vicinity, while avoiding overlap. Using a technique called optical trapping, we could achieve such a configuration.”

It turned out that trapping ⁸⁷Sr and very nearby also trapping rubidium atoms did not immediately help in building the desired molecules, but when employing a measuring technique known as nuclear magnetic resonance (in essence measuring the frequency corresponding to the energy splitting), it did help in very accurately determining the g-factor and therefore in determining the precise size of the Zeeman effect. The reason was that the properties of rubidium have been established extremely accurately, and so these could be used to precisely calibrate the magnetic field strength in the vacuum where both types of atom were trapped. This in turn enabled the strongly improved determination of the ⁸⁷Sr g-factor.

A challenging benchmark

The newly achieved precision will not only open the door for more precise strontium-based applications in atomic clocks and quantum computing, but may also lead to other steps forward. Thekkeppatt: “Our results form a new challenging benchmark for atomic structure calculations. We have shown that this new method works very well for precision measurements, and the demonstrated methods will inspire extension to further atomic species and states relevant for all sorts of applications.”