Four members of the NTU team that built a photonic lattice containing non-Hermitian Dirac particles. From left: Dr Letian Yu, Assoc. Prof. Yidong Chong, Prof. Baile Zhang, and Dr Ruixiang Guo. Photo credit: Nicholas Ang.
Mass is an intrinsic property of particles that determines their inertia, meaning how strongly they resist changes to their motion. But why do particles have mass? This turns out to be a fraught question, and over the years physicists have presented explanations at different levels of sophistication. For example, at the most fundamental level we know of, particles such as electrons, quarks and gluons acquire mass by interacting with another particle called the Higgs boson, whose discovery was honoured with the 2013 Nobel Prize in Physics.
One of the theories physicists use for describing massive particles is Dirac’s theory of the electron, formulated in 1928 by Paul Dirac. Dirac’s theory treats mass as a constant that determines the minimum energy an electron must have, according to the relation E=mc2 imposed by Einstein’s theory of relativity. As it turns out, the theory applies not just to electrons, but also to other relativistic particles such as quarks. It even describes non-fundamental particles, such as ‘quasiparticles’ appearing in certain exotic materials.
Paul Dirac (1902–1984) developed the relativistic theory of the electron in 1928.
A key assumption in Dirac’s theory is that energy is conserved. When an electron moves in an electric field, it can gain energy from the field or lose energy to it, but the total energy always remains the same. This is an ironclad feature of any ‘Dirac particle’ — or so physicists have long believed.
Now, a chink in the armour of Dirac’s theory has been found by a team of physicists at Nanyang Technological University (NTU) in Singapore and the Chinese University of Hong Kong (CUHK). These researchers, led by Professor Cesare Soci, Professor Baile Zhang and Associate Professor Yidong Chong of NTU Physics, and Assistant Professor Haoran Xue at CUHK, have shown theoretically and experimentally that many behaviors associated with massive Dirac particles can occur even if energy is not conserved.
These findings were published in the journal Nature in August 2024.
Laws (of physics) are made to be broken
Energy conservation is a bedrock principle of physics. It is commonly expressed by physicists via a property called ‘Hermitian symmetry’, named after the 19th century mathematician Charles Hermite. Theories in which energy is always conserved, like Dirac’s theory, are called ‘Hermitian’, while those that allow energy not to be conserved are ‘non-Hermitian’.
Non-Hermitian theories are not as pointless as one might expect. While energy is conserved at a fundamental level, many actual physical systems behave as though they are non-Hermitian. For instance, a ball rolling on a table slows down due to friction, and thus gradually loses kinetic energy. The lost energy is actually converted into heat, in the form of microscopic motions of other particles in the environment. As these extra particles usually fall outside the scope of the physics model, the ball in effect behaves as though friction is energy non-conserving.
Physicists refer to any friction-like process, which draws energy out of a system, as ‘loss’. The opposite process, which adds energy, is called ‘gain’. Systems with gain and loss are known to exhibit many important phenomena; lasers, for example, are electromagnetic resonators containing both gain and loss. However, no non-Hermitian system had ever been known to act like the particles in Dirac’s theory.
“Dirac particles, such as electrons, have many distinctive properties,” explains Associate Professor Yidong Chong, who co-led the NTU team. “For instance, they exhibit so-called ‘Klein tunneling’, whereby they pass right through any potential barrier, no matter how strong, when moving close to the speed of light. The question is, do these behaviours require energy conservation, as Dirac assumed? Or can they manifest in a non-Hermitian system?”
A strange symmetry
In 2020, a possible solution to this question was raised in a theoretical paper published in Physical Review Letters by several NTU researchers including Prof. Chong, Prof. Zhang, and Prof. Xue.
In place of Hermitian symmetry, the researchers postulated that a non-Hermitian system can obey an alternative symmetry that does not imply energy conservation.
“We call this ‘semi-Hermiticity’, as it is not the familiar Hermitian symmetry but gives similar outcomes,” says Prof. Haoran Xue, one of the joint first authors on the latest Nature paper. Formerly a graduate student and postdoctoral researcher at NTU, Prof. Xue joined CUHK in 2023 as an Assistant Professor in the Department of Physics.
“The next challenge we faced was that non-Hermitian systems are also usually not semi-Hermitian,” he adds. “So we had to create a system from scratch to obey this new symmetry.”
To test the theory of semi-Hermitian particles, an optical fibre experiment was set up by the researchers at NTU, including Dr Letian Yu and Dr Ruixiang Guo (pictured). Photo credit: Nicholas Ang.
To accomplish this, the researchers teamed up with Prof. Cesare Soci, an NTU Physics faculty member who is an expert in optics and photonics experiments. In Prof. Soci’s laboratory, physics graduate student Letian Yu worked with a pair of fibre optics experts, Dr Ruixiang Guo and Dr Eng Aik Chan, to assemble an apparatus of interconnected optical fibres. Their goal was to create a synthetic physical system, called a ‘photonic lattice’, that could realise semi-Hermitian symmetry.
“Dirac’s theory originally dealt with electrons. But in a photonic lattice, the same sorts of equations apply to light pulses traveling along optical fibres,” explains Prof. Baile Zhang. “Importantly, we can use devices called optical modulators to control how much gain and loss is experienced by the light pulses. And we can arrange things so that the pulses don’t obey Dirac’s equations, but a new set of equations invented by us.”
Mass effects
To test their setup, the researchers inserted packets of light and recorded how they moved through the photonic lattice. Unlike Dirac’s original theory, where mass is a constant, the particle mass in the semi-Hermitian photonic lattice is set via the gain and loss. With the mass set to a nonzero value, the team observed the packets of light spreading across the photonic lattice, with a range of speeds below a certain maximum speed (a nominal ‘speed of light’).
This behaviour is highly similar to the predictions of Dirac’s theory, which validated the team’s theoretical ideas, as non-Hermitian photonic lattices do not normally act this way. As a cross-check, they reconfigured the apparatus without semi-Hermitian symmetry, and saw that the light packets underwent runaway amplification instead, caused by the gain in the photonic lattice.
Next, the researchers undertook a series of more in-depth investigations.
“We have fine control over the photonic lattice’s gain, loss, and other parameters,” notes Dr Letian Yu, who is a joint first author on the Nature paper. After working on the experiment as a graduate student, he has subsequently received his PhD and is now a postdoctoral researcher at NTU. “We can set different masses at different positions and times, even switching between positive and negative mass. So we can study phenomena that are almost impossible to access by other means.”
Figure 1. Experimental results showing Klein tunneling in the photonic lattice. A wave-packet is incident from the left, and moves into a potential barrier (the region to the right of the red dashes) with almost zero reflection. Depending on the layout of the photonic lattice (top diagrams), the transmitted wave-packet can carry the same flux as before (left), less flux (middle), or more flux (right), due to the absence of energy conservation. Figure excerpted from Yu et al., Nature (2024).
One of the experiments Letian performed was a twist on Klein tunneling, a well-known phenomenon whereby Dirac particles pass, or ‘tunnel’, across a strong potential barrier. In the same way, light packets in the semi-Hermitian photonic lattice can be transmitted with nearly 100% efficiency across a barrier (Fig. 1, left panel).
However, the researchers also discovered that some features of Klein tunneling can be overturned in the semi-Hermitian system. In conventional Klein tunneling, a Dirac particle that has tunneled across a barrier retains the same ‘flux’, or rate of energy flow. But in the semi-Hermitian photonic lattice, the flux can decrease (Fig. 1, middle panel) or increase (Fig. 1, right panel) after tunneling.
The researchers also investigated the effects of changing the mass at a certain instant of time. This scenario, called ‘quenching’, is usually consigned to the realm of theoretical speculation, but could be accessed with the highly-controllable semi-Hermitian photonic lattice.
The team discovered that abruptly flipping the sign of the mass affects the motion of the particle in a velocity-dependent way. When the particle moves much slower than the speed of light, the mass-flip cases it to reverse direction, a phenomenon known as ‘time reflection’. But when the particle is moving close to the speed of light, it continues moving unimpeded.
“Physicists have been very interested in time reflection, and other phenomena caused by abrupt changes,” notes Prof. Zhang. “Our observation of velocity-dependent time reflection shows that many more possibilities can be opened up, once you include semi-Hermitian systems and other non-Hermitian systems.”
In the future, the researchers aim to use their photonic lattice to study how semi-Hermiticity alters other behaviours of Dirac particles. For example, Dirac particles in a magnetic field form special discrete energy levels called Landau levels, and it is currently unclear how exactly this might change if energy is not conserved.
“Dirac’s theory has been around for nearly a century, and we physicists have gained a lot of knowledge about how these particles behave,” says Prof. Chong. “Now we have an altenative theory that shares many characteristics, but not energy conservation. It’s exciting to figure out what carries over, and what might turn out to be completely different.”