Phase transitions, or, more generally, critical transitions, reveal themselves through identifiable signatures that mark a change in a system's behaviour. This is also typically accompanied by a change in the characteristics of excitations possible in the system. In certain physical systems with a high degree of control, such as cold atoms, it is possible to measure and characterise the required energy or spectra of possible excitations as the system crosses a phase transition.

In a recent work with our experimental collaborators from the University of Hamburg, published as a Letter in Physical Review Research, we demonstrated, using an atom-cavity system, a fascinating scenario. The system, consisting of a Bose-Einstein condensate interacting with a quantised light field inside an optical cavity, exhibits a quasiparticle called a "polariton". While this polariton is expected to remain stable only above a critical point, we showed that it can be excited with zero energy cost at parameter regimes below the critical point. The system remains stable despite this zero-energy excitation, making the polariton observable but short-lived.

Using a combination of numerical simulations and analytical calculations, we identify that the existence of this zero-energy excitation mode can be attributed to dissipation from photons leaking out of the cavity. However, it is crucial that the dissipation strength is not too large for this phenomenon to be observable in a wide range of parameters. 

Physical Review Research (Letter)

In both traditional (classical) and quantum computation, bit-flip operations, also known as NOT gates, are fundamental. They act as the basic switch that flips “0” to “1” or vice-versa, and such a simple operation is essential to building more complex logic gates.

Since any state that can represent a "1" or a "0" can be used to store information, we started wondering if the pair of "symmetry-broken" states found in parametric oscillators (e.g., LC circuit with capacitor plates that periodically move) and discrete time crystals could also encode these bits. If so, could we then design protocols to perform a bit-flip, or NOT gate, on them?

In our paper, we have theoretically tested how well a specific bit-flip protocol can withstand the disruptive behaviour of noise from thermal and quantum processes. Our simulations showed some exciting predictions: a protocol based on the sudden increase in the driving frequency for bits encoded in the resonant states of a parametric oscillator and discrete time crystals in the open Dicke model are robust against thermal and quantum noise, respectively. The discrete time crystal has an extra benefit: in the Dicke model, its superradiant state can “protect” the system from quantum disruptions. Our work represents an important step towards “taming” time crystals, which we hope will open doors for potential future applications

arXiv:2504.04900

Physical Review B

Critical transitions, such as a stock market crash and an abrupt climate change, can be understood using a mathematical framework called bifurcation theory. Typical examples of bifurcation include the pitchfork bifurcation, wherein the system goes from one fixed point to another, and the Hopf bifurcation, wherein a stable fixed point turns into an oscillating solution known as a limit cycle. An extremely rare type of transition is the Neimark-Sacker or torus bifurcation. 

Using a quantum gas experiment, we demonstrate, for the first time, a torus bifurcation as a continuous time crystal becomes unstable and starts "beating". That is, the continuous time crystal, characterised by a single well-defined frequency of oscillations, turns into a quasiperiodic dynamical state with an additional frequency incommensurate with respect to the original one. In contrast to the appearance of a limit cycle as a two-dimensional closed loop in phase, this quasiperiodic or beating dynamics manifests as a torus, a limit torus. We show in this work that the nature of transitions involving time crystals at the thermodynamic or mean-field level can be understood using bifurcation theory, commonly used in complex dynamical systems. Our work also provides an elementary platform for observing quasiperiodic dynamical phases.

arXiv:2411.00155

Physical Review Letters

Press release

In recent years, interest in phase transitions involving static and dynamical states has increased. An example of this is a transition involving a stationary superradiant phase and a time crystalline phase, which can be understood as a limit cycle solution, in an atom-cavity system (see our related work published in Science). For transitions involving static or equilibrium phases, it has been known that quantum noise smoothens the transition making it appear crossover-like. In our work, we predicted a similar smoothening of the transition but now between stationary and dynamical phases. This is the first time that the fundamental effects of quantum noise, inherent in microscopic and mesoscopic systems, are investigated and revealed in the context of phase transitions involving dynamical phases, particularly time crystals and limit cycles. A consequence of this quantum-induced smoothing behaviour is the unexpected emergence of limit cycles in parameter regimes, in which mean-field or thermodynamic predictions suggest otherwise. We also found that quantum fluctuations cause the oscillation frequencies of these limit cycles to spread out. This has important ramifications in synchronisation phenomena, such as entrainment, as the driving frequency must match an integer multiple of the limit cycle frequency for these to occur.

arXiv:2407.21390

Physical Review A

The Kibble-Zurek mechanism applies to various critical phenomena, wherein a system undergoes a continuous phase transition. To realise a phase transition, a tuning parameter that controls the phase of the system is typically varied, and according to this mechanism, the number of defects that will form in the system is related to the speed at which the critical point is crossed - the faster the transition is crossed, the larger the number of defects. While the idea was originally formulated for cosmological phase transition in the early universe, it eventually found applications in condensed matter. 

In our work published in Physical Review B (Editors' Suggestion), we demonstrate that the Kibble-Zurek mechanism also applies to open systems - systems that interact with their environment. Our key result is that for intermediate traversal speeds of a phase transition, dissipation may lead to an apparent delay in the Kibble-Zurek mechanism. This results in a lag between the actual time the system has entered a new phase and the time inferred from a threshold-based criterion for observable quantities, as done in most experiments. Our predictions apply to a general class of systems, including the Dicke model and its lattice version.

arXiv:2407.13424

Physical Review B (Editors' Suggestion)

Press release