Continuous Quantum Nondemolition Measurement of the Transverse Component of a Qubit

Quantum jumps of a qubit are usually observed between its energy eigenstates, also known as its longitudinal pseudospin component. Is it possible, instead, to observe quantum jumps between the transverse superpositions of these eigenstates? We answer positively by presenting the first continuous quantum nondemolition measurement of the transverse component of an individual qubit. In a circuit QED system irradiated by two pump tones, we engineer an effective Hamiltonian whose eigenstates are the transverse qubit states, and a dispersive measurement of the corresponding operator. Such transverse component measurements are a useful tool in the driven-dissipative operation engineering toolbox, which is central to quantum simulation and quantum error correction.

Figure 1

(a) Schematic of the experiment setup. A 3D transmon qubit-cavity system is continuously driven by three tones termed as sideband (red), readout (blue), and Rabi (green). The readout tone is transmitted through the output port, amplified by a Josephson parametric converter amplifier and demodulated at room temperature to give $I$, $Q$ signals. (b) Frequency landscape. Our system consists of a qubit at frequency ${\omega }_q$ and cavity with qubit-state-dependent frequency ${\omega }_c\pm \chi /2$ and linewidth $\kappa$. The qubit frequency correspondingly depends on the number of photons in the cavity, changing by $\chi$ for every photon. We apply a strong sideband tone (red) detuned from the cavity frequency by ${\mathrm{\Delta }}_c$ and a strong Rabi tone (green) at the qubit frequency. Readout is performed by applying a weak readout tone (blue) at the cavity resonance frequency ${\omega }_c$ to read out the system.

Figure 2

Histograms of $I$, $Q$ measurements in the presence of the readout, sideband, and Rabi drives for various indicated values of ${\mathrm{\Omega }}_{\mathrm{R}}$. For each ${\mathrm{\Omega }}_{\mathrm{R}}$, a 1 s continuous trace was recorded and integrated in 400 ns chunks to give an $I$, $Q$ value, in units of $\sqrt{\text{photon}}$ in the integrated chunk. The sideband and readout drive powers resulted in a steady state population of ${\overline{n}}_{sb}=12$ and $\overline{n}=0.9$ photons in the cavity respectively. The two Gaussian distributions corresponding to the eigenstates of ${\mathit{\sigma }}_{\mathit{z}}$ are visible in (a) and disappear as ${\mathrm{\Omega }}_{\mathrm{R}}$ is increased. For large values of ${\mathrm{\Omega }}_{\mathrm{R}}$ two new distributions appear, corresponding to the eigenstates of ${\mathit{\sigma }}_{\mathit{x}}$. The faint distribution near the center in (e) corresponds to the second excited state of the qubit ($|f⟩$).

Figure 3

The qubit is prepared in $|-⟩$ (a), $|+⟩$ (b), $|g⟩$ (c) and $|i⟩$ (d) as is shown by the Bloch spheres. We then measure $I$, $Q$ histograms in the presence of the readout, sideband, and Rabi drives (${\mathrm{\Omega }}_{\mathrm{R}}/2\pi =70\mathrm{MHz}$). The results for the initializations $|-⟩$ and $|+⟩$ show two Gaussian distributions separated by $5.4\sigma$. The initializations $|g⟩$ and $|i⟩$ show nearly $50:50$ weight in both distributions as expected for $⟨{\mathit{\sigma }}_{\mathit{x}}⟩$. Outcomes above (below) the separatrix (dashed white line) are identified as $|+⟩$ ($|-⟩$). The expectation value $⟨{\mathit{\sigma }}_{\mathit{x}}⟩$ is plotted in blue for states prepared on the Bloch sphere surface in the ${\mathit{\sigma }}_{\mathit{x}}-{\mathit{\sigma }}_{\mathit{y}}$ plane (e), ${\mathit{\sigma }}_{\mathit{x}}-{\mathit{\sigma }}_{\mathit{z}}$ plane (f), and ${\mathit{\sigma }}_{\mathit{y}}-{\mathit{\sigma }}_{\mathit{z}}$ plane (g). The dashed red lines in (e)–(g) show the ideal expectation value $⟨{\mathit{\sigma }}_{\mathit{x}}⟩$, while the solid red lines show a theoretical prediction based on a simulation of the cavity-qubit system, including measurement imperfection.

Figure 4

A cut of a continuous trace of quantum jumps taken from the measurement histogrammed in Fig. 2. The blue line shows the angle in phase space around the point shown in the inset, and we observe jumps between the eigenstates of ${\mathit{\sigma }}_{\mathit{x}}$. The dashed orange line is a two-point filter estimate of the state of the qubit. The qubit is sometimes in its second excited state ($|f⟩$).