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arXiv:2212.06031 · Physical review. D/Physical review. D.

Quantum coherence of critical unstable two-level systems

Dimitrios Karamitros, Thomas McKelvey, Apostolos Pilaftsis A

Published Jul 13, 2023• 7 citations
arXivHTMLPDF

Abstract

We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-sphere formalism of qubits. By employing the Bloch-vector representation for such unstable qubit systems, we identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, $\mathbf{E}$ and $\mathbf{\ensuremath{\Gamma}}$, are orthogonal to one another, and the parameter $r=|\mathbf{\ensuremath{\Gamma}}|/(2|\mathbf{E}|)$ is less than 1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviors like coherence-decoherence oscillations when analyzed in an appropriately defined codecaying frame of the system. In the same frame, a unit Bloch vector $\mathbf{b}$ describing a pure critical qubit will sweep out unequal areas during equal intervals of time, while rotating about the vector $\mathbf{E}$. These phenomena emerge beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system. Interestingly enough, we observe that these new features will persist even for quasicritical scenarios, in which the vectors $\mathbf{E}$ and $\mathbf{\ensuremath{\Gamma}}$ are not perfectly orthogonal to each other. Applications of our results to quantum information and to unstable meson-antimeson and other systems are discussed.

DOI
10.1103/physrevd.108.016006

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We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-sphere formalism of qubits. By employing the Bloch-vector representation for such unstable qubit systems, we identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E\mathbf{E}E and \ensuremathΓ\mathbf{\ensuremath{\Gamma}}\ensuremathΓ, are orthogonal to one another, and the parameter r=∣\ensuremathΓ∣/(2∣E∣)r=|\mathbf{\ensuremath{\Gamma}}|/(2|\mathbf{E}|)r=∣\ensuremathΓ∣/(2∣E∣) is less than 1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviors like coherence-decoherence oscillations when analyzed in an appropriately defined codecaying frame of the system. In the same frame, a unit Bloch vector b\mathbf{b}b describing a pure critical qubit will sweep out unequal areas during equal intervals of time, while rotating about the vector E\mathbf{E}E. These phenomena emerge beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system. Interestingly enough, we observe that these new features will persist even for quasicritical scenarios, in which the vectors E\mathbf{E}E and \ensuremathΓ\mathbf{\ensuremath{\Gamma}}\ensuremathΓ are not perfectly orthogonal to each other. Applications of our results to quantum information and to unstable meson-antimeson and other systems are discussed.
We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-vector representation. We identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E{\bf E}E and Γ{\bf\Gamma}Γ, are orthogonal to one another, and the parameter r=∣Γ∣/(2∣E∣)r = |{\bf \Gamma}|/(2|{\bf E}|)r=∣Γ∣/(2∣E∣) is less than 1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviours like coherence--decoherence oscillations when analysed in an appropriately defined co-decaying frame of the system. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of non-sinusoidal oscillation of a CUQ. We apply the results of our formalism to the neutral-meson systems and derive generic upper limits on these new observables. In particular, we provide a compilation table of all well-explored meson--antimeson two-level systems in terms of Bloch-sphere parameters.
We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-vector representation. We identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E{\bf E}E and Γ{\bfΓ}Γ, are orthogonal to one another, and the parameter r=∣Γ∣/(2∣E∣)r = |{\bf Γ}|/(2|{\bf E}|)r=∣Γ∣/(2∣E∣) is less than~1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviours like coherence--decoherence oscillations when analysed in an appropriately defined co-decaying frame of the system. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of non-sinusoidal oscillation of a CUQ. We apply the results of our formalism to the neutral-meson systems and derive generic upper limits on these new observables. In particular, we provide a compilation table of all well-explored meson--antimeson two-level systems in terms of Bloch-sphere parameters.
We extend our previous work on a novel class of unstable qubits which we have identified recently and called them critical unstable qubits (CUQs). The characteristic property of CUQs is that the energy-level and decay-width vectors, E and Γ , are orthogonal to one another, and the key parameter r = | Γ | / | 2 E | is less than 1. Most remarkably, CUQs exhibit two atypical behaviors: (i) they display coherence-decoherence oscillations in a co-decaying frame of the system described by a unit Bloch vector b , and (ii) the unit Bloch vector b describing a pure CUQ sweeps out unequal areas during equal intervals of time, while rotating about the vector E . The latter phenomenon emerges beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system, which governs an ordinary qubit. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of oscillation of a CUQ. We apply the results of our formalism to the B 0 B ¯ 0 -meson system and derive, for the first time, generic upper limits on these new observables.
We extend previous studies on a novel class of unstable two-level systems which were called Critical Unstable Qubits (CUQs). In an appropriately defined co-decaying frame, the CUQs exhibit striking phenomena of indefinite anharmonic oscillations between two states and coherence-decoherence oscillations of mixed states. These features are distinct from the usual Rabi oscillations observed in the Hermitian counterpart of two-level systems, which are harmonic and preserve the coherence of the quantum state. We employ the density matrix formalism to study these phenomena for mixed states and delve into the nature of the trajectory traversed by these states in the Bloch sphere by studying the time evolution of the Bloch vector that describes the quantum state of the unstable qubit. In particular, we provide for the first time explicit geometric constructions to obtain trajectories of both pure and mixed CUQs in and on the Bloch sphere. This enables us to identify the stationary points of CUQs, at which the states do not evolve in time in the co-decaying frame. The potential implications of our findings for particle cosmology and quantum simulations of non-Hermitian Hamiltonians are discussed.
Dynamical decoupling protocols provide a versatile toolbox for robust control of quantum systems, suppressing noise that would otherwise limit their performance. This tutorial aims to provide an intuitive interpretation of these protocols. Written from an experimentalist's perspective, we use the Bloch sphere picture to visualise quantum dynamics for a range of sequences. Here each dynamical decoupling operation can be viewed as an "effective field vector", whilst the phase and population of the two-level system is encoded into a "Bloch vector". The two-level dynamics are calculated by the cross product between these two vectors, such that the quantum state represented by the Bloch vector rotates around the effective field vector. Several prominent control schemes are simulated using this picture, starting with a Rabi oscillation, followed by the Ramsey sequence and a selection of pulsed and continuous dynamical decoupling techniques. These sequences are engineered to preserve control of the quantum system for as long as possible, with their efficacy often quantified by a coherence time. We clarify the definition of the longitudinal T1T_1T1​, transverse T2T_2T2​ and inhomogeneous T2∗T_2^*T2∗​ coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.