Abstract
Open classical and quantum systems with effective parity-time ($\mathcal{PT}$) symmetry, over the past five years, have shown tremendous promise for advances in lasers, sensing, and non-reciprocal devices. And yet, how such effective $\mathcal{PT}$-symmetric non-Hermitian models emerge out of Hermitian quantum mechanics is not well understood. Here, starting from a fully Hermitian microscopic Hamiltonian description, we show that a non-Hermitian Hamiltonian emerges naturally in a double-quantum-dot-circuit-QED (DQD-circuit QED) set-up, which can be controllably tuned to the $\mathcal{PT}$-symmetric point. This effective Hamiltonian governs the dynamics of two coupled circuit-QED cavities with a voltage-biased DQD in one of them. Our analysis also reveals the effect of quantum fluctuations on the $\mathcal{PT}$ symmetric system. The $\mathcal{PT}$-transition is, then, observed both in the dynamics of cavity observables as well as via an input-output experiment. As a simple application of the $\mathcal{PT}$-transition in this set-up, we show that loss-induced enhancement of amplification and lasing can be observed in the coupled cavities. By comparing our results with two conventional local Lindblad equations, we demonstrate the utility and limitations of the latter. Our results pave the way for an on-chip realization of a potentially scalable non-Hermitian system with a gain medium in quantum regime, as well as its potential applications for quantum technology.
Highlights
For an isolated system, the Hamiltonian is the generator of its time evolution
We have shown that a setup of two coupled cQED cavities with a DQD in one of them can be used to explore the physics of the PT -symmetry breaking transition, including the effect of quantum fluctuations
Under a reasonable approximation on various energy scales consistent with state-of-the-art experiments, we have microscopically derived an effective non-Hermitian Hamiltonian, along with the quantum noise term, which governs the dynamics of bosonic complex quadratures that describe the cavity
Summary
For an isolated (quantum) system, the Hamiltonian is the generator of its time evolution. Our proposal is very different from a recent proposal for the realization of a PT -symmetric system in the cQED system by Quijandría et al [83] The latter considered two cQED cavities, each coupled to a qubit whose frequency is modulated via a coherent drive, and showed that the expectation values of the cavity field operators, in an appropriate parameter regime, are governed by an effective PT -symmetric Hamiltonian. The DQD voltage bias in our model acts as an incoherent drive, and our analysis extends to equations of motion for the cavity field operators and their bilinear combinations This allows us to study the effect of quantum fluctuations on the cavities, which was not explored in Ref. Certain details of the analytical calculations are delegated to the appendices
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