Qubit state tomography in superconducting circuit via weak measurements

The standard method of "measuring" quantum wavefunction is the technique of {\it indirect} quantum state tomography. Owing to conceptual novelty and possible advantages, an alternative {\it direct} scheme was proposed and demonstrated recently in quantum optics system. In this work we present a study on the direct scheme of measuring qubit state in the circuit QED system, based on weak measurement and weak value concepts. To be applied to generic parameter conditions, our formulation and analysis are carried out for finite strength weak measurement, and in particular beyond the bad-cavity and weak-response limits. The proposed study is accessible to the present state-of-the-art circuit-QED experiments.Comment: 7 pages,5figure

Direct measurement of the quantum state of photons in a cavity

We propose a scheme to measure the quantum state of photons in a cavity. The proposal is based on the concept of quantum weak values and applies equally well to both the solid-state circuit and atomic cavity quantum electrodynamics (QED) systems. The proposed scheme allows us to access directly the superposition components in Fock state basis, rather than the Wigner function as usual in phase space. Moreover, the separate access feature held in the direct scheme does not require a global reconstruction for the quantum state, which provides a particular advantage beyond the conventional method of quantum state tomography.Comment: 6 pages, 2 figure

Simple understanding of quantum weak values

In this work we revisit the important and controversial concept of quantum weak values, aiming to provide a simplified understanding to its associated physics and the origin of anomaly. Taking the Stern-Gerlach setup as a working system, we base our analysis on an exact treatment in terms of quantum Bayesian approach. We also make particular connection with a very recent work, where the anomaly of the weak values was claimed from the pure statistics in association with "disturbance" and "post-selection", rather than the unique quantum nature. Our analysis resolves the related controversies through a clear and quantitative way.Comment: 8 pages,2 figure

Weak-value-amplification analysis beyond the AAV limit of weak measurements

The weak-value (WV) measurement proposed by Aharonov, Albert and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this work, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain non-perturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV's prediction, we find that the SNR asymptotically gets worse when the AAV's WV $A_w$ becomes large, i.e., in the case $g|A_w|^2>>1$, where $g$ is the measurement strength. (ii) With the increase of $g$ (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach -- actually the former can slightly exceed -- the SNR of the standard measurement. However, along a further increase of $g$, the WV technique will become less efficient than the standard measurement, despite that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet their difference will become more prominent with the increase of $g$. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The non-perturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.Comment: 10 pages, 6figure

Enhanced super-Heisenberg scaling precision by nonlinear coupling and postselection

In quantum precision metrology, the famous result of Heisenberg limit scaling as $1/N$ (with $N$ the number of probes) can be surpassed by considering nonlinear coupling measurement. In this work, we consider the most practice-relevant quadratic nonlinear coupling and show that the metrological precision can be enhanced from the $1/N^{\frac{3}{2}}$ super-Heisenberg scaling to $1/N^2$, by simply employing a pre- and post-selection (PPS) technique, but not using any expensive quantum resources such as quantum entangled state of probes.Comment: 6 pages, 4 figure

Validating the Bogoliubov-de Gennes Treatment for Majorana Conductances in Three-Terminal Transports

In order to demonstrate the nature of nonlocality of the Majorana zero modes (MZMs),we consider a two-lead (three-terminal) setup of transport and construct a Majorana master equation (which is also valid for small bias voltage).We first carry out a representative result of current then show that only a modified Bogoliubov-de Gennes (BdG) treatment can consistently recover the same result. Based on the interplay of the two approaches, in contrast to the conventional BdG treatment, we predict the existence of nonvanishing channels of teleportation and crossed Andreev reflections at the limit $\epsilon_M \to 0$ (zero coupling energy of the MZMs),which consequently leads to specific predictions of different heights of the zero-bias-peak of the local conductance and different $\epsilon_M$-scaling behaviors of the "teleportation" conductance. Verification of these two predictions by experiments will further clarify the proposal of the present research to modify the conventional BdG treatment (in quantum transport), which is broadly employed in literature.Comment: 6 pages, 2 figure

Transport Signatures of a Majorana Qubit and Read-out-induced Dephasing

Motivated by recent proposals of Majorana qubits and the read-out of their quantum state we investigate a qubit setup formed by two parallel topological wires shunted by a superconducting bridge. The wires are further coupled to two quantum dots, which are also linked directly, thus creating an interference loop. The transport current through this system shows an interference pattern which distinguishes two basis states of the qubit in a QND measurement. We analyze various properties of the interference current and the read-out process, including the resulting dephasing and relaxation. We also analyze the effects of varying control parameters such as gate voltages on the current. The characteristic dependencies could serve as a signature of Majorana bound states.Comment: 10 pages,12 figure