Deterministic Entanglement of Photons in Two Superconducting Microwave Resonators

Quantum entanglement, one of the defining features of quantum mechanics, has been demonstrated in a variety of nonlinear spinlike systems. Quantum entanglement in linear systems has proven significantly more challenging, as the intrinsic energy level degeneracy associated with linearity makes quantum control more difficult. Here we demonstrate the quantum entanglement of photon states in two independent linear microwave resonators, creating $N$-photon NOON states (entangled states $|N0⟩+|0N⟩$) as a benchmark demonstration. We use a superconducting quantum circuit that includes Josephson qubits to control and measure the two resonators, and we completely characterize the entangled states with bipartite Wigner tomography. These results demonstrate a significant advance in the quantum control of linear resonators in superconducting circuits.

Figure 1

(a) Device circuit schematic. Coupling resonator $C$ is connected to $q_0$ and $q_1$ using $\sim 1.9\mathrm{fF}$ coupling capacitors that yield 20 MHz coupling strengths, while storage resonator $A$ ($B$) is connected to $q_0$ ($q_1$) through a $\sim 1.9\mathrm{fF}$ coupling capacitor with 17 MHz coupling strength. (b) Qubit spectroscopy, showing probability $P_e$ (color bar) vs microwave frequency and flux bias for each qubit. Avoided-level crossings near 6.8 GHz (dash-dotted lines) are due to the coupling resonator $C$ and near 6.3 GHz (dashed lines) due to each qubit’s storage resonator. Lower right panel shows magnified view of circled area, upper right panel shows three qubit levels.

Figure 2

NOON state preparation sequence. Resonators are represented by dashed lines and the qubit $|g⟩\leftrightarrow |e⟩$ and $|e⟩\leftrightarrow |f⟩$ transitions by dark and light (color) solid lines, respectively. (1) $q_0$ is excited to $|e⟩$ and half-swapped to $C$, generating the Bell state $|e0⟩+|g1⟩$. (2) Coupling resonator swapped to $q_1$, generating $|eg⟩+|ge⟩$. (3a) $N=1$ NOON state $|10⟩+|01⟩$ generated by fully swapping each qubit to its storage resonator. For higher $N$ states: (4) Qubits excited to $|fg⟩+|gf⟩$. (5) One photon swapped into storage resonators, generating $|eg10⟩+|ge01⟩$. Steps (4) and (5) are repeated $N-1$ times, generating $|eg(N-1)0⟩+|ge0(N-1)⟩$. (6) Final photon transfer generates (7) $N=2$ (or higher $N$) NOON state.

Figure 3

Qubit coincidence probability measurements for $N=1$, 2, and 3 NOON states, and for a mixed state. $P_{ge}$ (blue) and $P_{eg}$ (red) oscillate with interaction time $\tau$ at a rate $\propto \sqrt{N}$, while $P_{ee}$ (brown) always remains small. For the mixed state, behavior is nearly identical to the $N=1$ NOON state. Lines are fits to the data. Statistical errors, from the measured probability spread of $\sim 2\%–3\%$, are shown only for $P_{gg}$ [26]. Horizontal and vertical axes are same for all plots.

Figure 4

NOON and mixed state density matrix amplitudes, reconstructed in the photon-number basis from bipartite Wigner tomography; states are labeled $|mn⟩$ where $m$ is the photon number in resonator $A$ and $n$ that in $B$. Bar heights and colors represent matrix element amplitudes. The dominant amplitudes for all three NOON states are in the expected locations, although the off-diagonal elements decrease with $N$, due to the finite qubit dephasing time. Errors for the density matrix elements are not shown but are small [26].

Figure 5

Phase sensitivity of NOON states up to $N=3$. (a) Density matrix for $N=1$ NOON state at different times after qubit entanglement. Each element is represented by an arrow, with orientation determined by the phase angle (scale on bottom left). Off-diagonal elements rotate with delay time (black: 0 ns, blue: 12 ns, red: 24 ns), due to frequency difference between the qubit operating point and resonator. Dephasing causes a decrease in amplitude of off-diagonal elements. (b) Phase angle of the upper left nonzero off-diagonal element in (a) versus delay time. Line is fit to a rotation rate of $\Delta f=21.6\pm 0.8\mathrm{MHz}$, which corresponds well to set frequency difference. Error bars indicate maximum phase uncertainty. (c) Rotation angle of off-diagonal element versus controlled, additional phase angle used in coherent state displacement. Lines are fits (slopes indicated on plot), consistent with expected phase sensitivity. Confidence bounds are as in (b), with uncertainty increasing with $N$ due to increased phase sensitivity.