NON-HERMITIAN PHYSICS

Exceptional quantum behaviour

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Non-Hermitian systems with gain and loss give rise to exceptional points with exceptional properties. An experiment with superconducting qubits now offers a first step towards studying these singularities in the quantum domain.

Non-conservative physical systems subject to dissipation (loss) and amplification (gain) exhibit degeneracies known as exceptional points (EPs)1. As a system approaches such a point in phase space, two eigenstates merge into one, giving rise to a plethora of interesting phenomena that recently caught the attention of the physics community. Exceptional points also emerge in the quantum domain, but the practical exploration of the related physical effects is a challenging task — loss and gain, in fact, lead to noise and decoherence too. Writing in Nature Physics, Mahdi Naghiloo and colleagues have now reported an experiment that provides access to EPs in a quantum system by means of modern qubit technology and a clever measurement scheme2.

The authors’ approach is rooted in the concepts of dissipation and decoherence, which are fundamental in quantum mechanics. Consider, for example, the iconic problem of the spontaneous emission of photons by an atom in free space, which happens as a result of the atom’s dissipative coupling with the surrounding electromagnetic modes. The time evolution of the atom in this process is typically described by a master equation, in which the central object is a density matrix of the atom rather than its wave function. This is because the atom decoheres while interacting with its electromagnetic bath and thus cannot be described by a pure state anymore.

Theoretical work demonstrated3, however, that a wave function description of the atom in this context is still possible at the expense of introducing randomly generated ‘quantum jumps’. For the case at hand, a jump would correspond to the decay of an excited state of the atom accompanied by the emission of a photon. Averaging over many such stochastic wave functions with random jumps can then be shown to be equivalent to a solution of the more involved master equation. One interesting aspect of this stochastic approach is the fact that for times until the first jump occurs, the time evolution of the atom is particularly simple and described by a conventional Schrödinger equation. The interaction with the environment enters this equation only as a dissipative term that makes the atomic Hamiltonian non-Hermitian. This is exactly what Naghiloo and colleagues exploited to study non-Hermitian physics with quantum states.

To start with, the authors didn’t use a real atom, but a highly tunable artificial atom known as a transmon qubit — a superconducting circuit developed about 12 years ago at Yale University in the US that can exist in well-defined charge states. Out of the three lowest levels of this qubit, the ground state is treated as an environment to which the higher two levels can decay. This creates an effective two-level atom with a dissipative lower level (see Fig. 1b in ref. 2).

To study the time evolution of this quantum system, the authors first initialized their effective two-level atom in its upper state. Following the stochastic wave function approach, this initial state can now either follow a non-Hermitian but coherent time evolution within the sub-manifold of the two-level system or execute a quantum jump to the decay channel outside the sub-manifold. The essential idea that allows the exclusive exploration of the non-Hermitian evolution, while masking out the jumps, is to restrict the analysis to those experimental data points for which no jump outside the sub-manifold occurred.

Although this post-selection of data can only be applied for no more than about 2 μs — as almost no states without a jump are left after that — very interesting physics takes place within this time frame. In particular, when the two levels of this non-Hermitian quantum system are coupled with each other by a coherent resonant drive, the authors observed a transition from a purely decaying behaviour for weak coupling to Rabi oscillations between the two levels for strong coupling. The transition point between these two regimes is then the EP that the authors are after.

The presence of an EP is verified as the parameters of the transmon circuit are tuned around it. Sweeping the coupling strength across the EP, for instance, revealed a square-root behaviour in the change of the oscillation frequency — a symptomatic feature of an EP (see Fig. 2d in ref. 2). Moreover, the experimental data nicely display how the two eigenstates of the non-Hermitian Hamiltonian transition from being orthogonal away from the EP to being parallel right at it. The two eigenstates merge at the EP into a unique state that is well identified by quantum state tomography carried out in the parametric vicinity of the EP. The work by Naghiloo and colleagues provides convincing evidence for the presence of an EP in a widely used experimental setting and, together with other recent advances in this field4, sets the stage for further explorations of EP-related phenomena in the quantum regime.

A pressing challenge arises from the question of whether the ubiquitous noise in microscopic systems5 will undermine the potential advantages of EPs for enhanced sensing6,7. Although in the current work the two-level atom was designed to have an effective parity–time (PT) symmetry with loss only, it will also be of interest to implement quantum systems with a genuine PT symmetry for which the balanced loss and gain lead to new dynamical regimes and unconventional lasing states5.

Also, many other features related to EPs and PT symmetry in the quantum domain now seem to be within reach. Examples are topologically protected states8, chiral dynamics9 and quantum correlations10. Studies in all of these directions will also eventually reveal which known exceptional features of EPs will survive the transition to the quantum realm. Even more interestingly, however, will be to identify which currently unknown and truly exceptional quantum behaviours EPs may give rise to in their new quantum habitat.

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Correspondence to Stefan Rotter.

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