Entanglement with a mode observable via a tunable interaction with a qubit

Abstract

We study the possibility of detection of ``spin-boson'' entanglement by qubit only measurements. Such entanglement is impossible to detect by previously proposed schemes that involve a fixed system-environment interaction, because of inherent symmetries within the coupling and the initial state of the environment. We take advantage of the possibility of tuning of qubit-environment coupling, that is available in some qubit realizations. As an example we study a superconducting transmon qubit interacting with a microwave cavity, which is one of such systems and is, furthermore, essential in the context of quantum information processing. We propose suitable Hamiltonian parameters for the preparation and measurement phases of the detection scheme that allow for an experimental test, and verify that the reported signal is nonnegligibly large still at finite temperatures.

Figures (3)

• Figure 1: Circuit representing a single run of the QEE detection scheme. Initially the qubit is in pointer state $|i\rangle$ and environment is in state $\hat{R}(0)$. They interact for time $t$ preparing the environmental state $\hat{R}_{ii}(t)$, while the qubit is not affected. Then the Hadamard gate is used to prepare a superposition qubit state and the interaction is changed to the probe settings, followed by qubit coherence measurements (e. g. measurements of observables defined by the Pauli operators $\hat{\sigma}_x$ and $\hat{\sigma}_y$).
• Figure 2: Evolution of the real and imaginary parts of qubit coherence [given by Eq. (\ref{['deta']}) and corrected for the minus sign stemming from the Hadamard gate] for the measurement part of the QEE detection scheme, Fig. \ref{['g1']}, at zero temperature. The preparation time is (a) $\beta t/\hbar=0$ (separable state at time $t$) and (b) $\beta t/\hbar=2$ (entangled state at time $t$). Real part of coherence (upper curves): $i=0$ - dotted, red lines; $i=1$ - dashed green lines. Imaginary part of coherence (lower curves): $i=0$ - solid blue lines; $i=1$ - dotted-dashed black lines. The Hamiltonian parameters are set to $\alpha/\beta=(1+i)/2$ in the preparation part, and $\alpha/\beta=1/\sqrt{2}$ in the measurement part. Any discrepancy between the two upper curves or the two lower curves signifies entanglement, thus entanglement is witnessed in panel (b).
• Figure 3: Evolution of the difference of qubit coherence in the measurement stage of the QEE detection scheme; real part is on the left and imaginary part on the right. Different curves correspond to different preparation times: dashed red lines for $\beta t/\hbar =\pi/6$, solid blue lines for $\beta t/\hbar=2$, and dotted green lines $\beta t/\hbar=3\pi/2$. Different panels correspond to different temperatures: (a),(b) $k_BT/\Gamma =0$, (c),(d) $k_BT/\Gamma =0.5$, (e),(f) $k_BT/\Gamma =1$, (f),(h) $k_BT/\Gamma =2$. The Hamiltonian parameters are set to $\alpha/\beta=(1+i)/2$ in the preparation part, and $\alpha/\beta=1/\sqrt{2}$ in the measurement part.