Critical Unstable Qubits in Particle Physics

Abstract

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, ${\bf E}$ and ${\bfΓ}$, are orthogonal to one another, and the parameter $r = |{\bf Γ}|/(2|{\bf E}|)$ 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.

Figures (5)

• Figure 1: The maximum value of the magnitude of the co-decaying Bloch vector $\mathbf{b}(\tau )$ as a function of the parameter $r$, for a critical unstable qubit. A set of different initial conditions for $\mathbf{b}(0)\parallel \boldsymbol{\gamma}$ was assumed.
• Figure 2: Fourier series approximations for a CUQ with $r=0.85$ and $\mathbf{b}(0)=\mathbf{e}\times\boldsymbol{\gamma}$. The black line is the exact evolution of ${\bf b}(\tau)$, the blue dashed line is the Fourier series including terms up to and including $n=2$, and the red dashed line is the Fourier series including terms up to and including $n=6$. The left panel ($a$) shows the projection onto $\boldsymbol{\gamma}$, and the right panel ($b$) shows the projection onto $\mathbf{e}\times\boldsymbol{\gamma}$.
• Figure 3: Fourier series fit for the data given in LHCb:2016gsk, for the year 2011: panels ($a$) and ($b$), and the year 2012: panels ($c$) and ($d$). Panels ($a$) and ($c$) show the flavour asymmetry for the channel $B^0_{\rm d}\to D^-\mu^+\nu_\mu X$ and panels ($b$) and ($d$) for the channel $B^0_{\rm d}\to D^{-*}\mu^+\nu_\mu X$. The black data points and their associated error bars have been rewritten from their original form into projections of along the $\mathbf{e}\times\boldsymbol{\gamma}$ direction. For this fit, we make use of the first two harmonics of the Fourier series, as well as the constant term.
• Figure 4: The evolution of $\mathbf{b}$ as a function of the angle between $\mathbf{b}$ and $\boldsymbol{\gamma}$, $\varphi = \theta + {\pi\over 2}$. The solid red line shows values of $\mathbf{b}(\theta)$ in the range $[-\pi, 0]$. Given the oscillation period $\widehat{\rm P}$, we define $\tau_1 = \frac{1}{4}\widehat{\rm P}$, $\tau_2 = \frac{1}{2}\widehat{\rm P}$, and $\tau_1 = \frac{3}{4}\widehat{\rm P}$.
• Figure 5: Elliptical trajectories for the mixed state CUQ. The exact solution of the master evolution equation given the initial condition $\mathbf{b}(0)=\mathbf{0}$ are plotted for $r=0.2$, $r=0.4$, $r=0.6$ and $r=0.9$.