Deterministic entanglement of photons in two superconducting microwave resonators

References

I Materials and Methods

The device fabrication is similar to that published previously S (1). The half-wavelength superconducting coplanar waveguide resonators are made of rhenium deposited on a $c$-axis single-crystal sapphire substrate, with a $5$ $\mu$m-wide center signal trace and 10 $\mu$m gaps to the ground plane metallization on either side of the center trace. We place a single lithographed shorting strap connecting the two ground planes at the midpoint of each resonator to improve the quality of the grounding. This point is a voltage node for the fundamental half-wave resonant mode, so that there is minimal additional dielectric loss from the shorting strap’s underlying amorphous Si insulating film S (2).

In the circuit layout, the coupling resonator $C$ is designed to have a higher resonance frequency than the two state storage resonators $A$ and $B$. This prevents the qubit frequencies from having to cross the $C$ resonator frequency during NOON state amplification. The two storage resonators $A$ and $B$ are designed with slightly different resonance frequencies, to avoid possible interference between the resonators. The full frequency span in the design was chosen to be about $550$ MHz, within the dynamic range of our custom microwave electronics. The two superconducting phase qubits and coplanar waveguide resonators are fabricated together, using our standard multi-layer process S (3). We use interdigitated coupling capacitors between the qubits and the resonators, calculated to each have a coupling capacitance of 1.9 fF. The actual coupling strengths vary slightly with resonator frequency; the detailed component parameters are listed in Table S1.

II Generation Sequence Tune-Up

The time required for each qubit-resonator $i$-SWAP is calibrated separately. The swap times obtained from these calibrations scale correctly as $\sqrt{n}$ with the number of photons $n$ in the resonator S (5) and also depend on the state of the qubit. Examples of the swap calibrations for a one-photon swap between qubit $q_{0}$ and resonator $A$ are shown in Fig. S1, for both the $|g\rangle\leftrightarrow|e\rangle$ and $|e\rangle\leftrightarrow|f\rangle$ transitions. The swap time for the $|e\rangle\leftrightarrow|f\rangle$ transition is approximately $1/\sqrt{2}$ times that for the $|g\rangle\leftrightarrow|e\rangle$ transition. This scaling is as expected, as the multi-level phase qubit can be well-approximated as a weakly nonlinear harmonic oscillator for the energy levels confined by the qubit’s metastable potential well. The scaling confirms that we can use harmonic-oscillator-like raising and lowering operators for the three-level qubit, as was done in the numerical simulations (see below).

Figure S2 shows the detailed pulse sequence used to generate and measure the $N=2$ NOON state. Sequence steps in general are calibrated and checked separately to maximize preparation fidelity, when possible. For example, we first optimize the qubit Bell state preparation at the end of step I in the preparation sequence detailed in Fig. S2. The fidelity for the Bell state is above $0.80$, with entanglement of formation 0.59, which agrees well with numerical simulations, performed using a pure dephasing time $T_{\phi}=300$ ns for the qubits (see below). State tomography of the qubits is also as expected: Fig. S2(b) shows the density matrices extracted from coupled qubit tomography, measured at different times during the $N=2$ NOON state preparation. We note that at the end of the sequence, both qubits should return to their ground states. Experimentally we observe small populations in the excited states $|e\rangle$ due to decoherence and pulse imperfections. The exact qubit state after the NOON state generation is measured and is used as the initial state for the qubits when performing Wigner tomography on the storage resonators (see below).

III Numerical Simulations

Numerical simulations were performed using the model Hamiltonian

where $H_{q_{i}}$ is the Hamiltonian of the qubit $q_{i}$, $a_{q_{i}}^{+}$ and $a_{q_{i}}$ ($b_{j}^{+}$ and $b_{j}$) are the raising and lowering operators for the 3-level qubit $q_{i}$ (resonator $j$), $g_{q_{i}\leftrightarrow j}$ is the coupling strength between qubit $q_{i}$ and resonator $j$, with a sum over all possible qubit-resonator combinations, such that $(q_{i},j)\in\left\{(q_{0},A),(q_{0},C),(q_{1},B),(q_{1},C)\right\}$ and $\Omega_{q_{i}}(t)$ is the time-dependent, two-tone ($f_{|g\rangle\leftrightarrow|e\rangle}$ and $f_{|e\rangle\leftrightarrow|f\rangle}$) microwave drive on qubit $q_{i}$.

The 3-level qubit Hamiltonian $H_{q_{i}}$ was approximated as

where for simplicity we assumed a constant nonlinearity $f_{|g\rangle\leftrightarrow|e\rangle}-f_{|e\rangle\leftrightarrow|f\rangle}=200$ MHz, so that $f_{|g\rangle\leftrightarrow|f\rangle}\simeq 2f_{|g\rangle\leftrightarrow|e\rangle}-200$ MHz. We approximated the multi-level qubit $a_{q_{i}}^{+}$ and $a_{q_{i}}$ by the raising and lower operators for the lowest three levels of a harmonic oscillator, as discussed above.

Decoherence was approximated using the Lindblad master equation taking into account the Markovian environment S (6), where two characteristic decay times, the energy relaxation time $T_{1}$ and the pure dephasing time $T_{\phi}$, were used for each resonator and qubit.

The simulations do not directly account for the non-Markovian $1/f$ character of the phase noise in the qubits. To account for this, we used a sequence-time dependent $T_{\phi}$ for each qubit, as obtained from Ramsey interference measurements. We used $T_{\phi}=300$ ns for $\sim$50 ns-long sequences and $T_{\phi}=200$ ns for $\sim$100 ns-long sequences. The resulting simulations agree reasonably well with the experimental measurements.

IV Bipartite Wigner Tomography

We use every possible combination of values of $\alpha$ and $\beta$ distributed over the circles of the same radius for tomography, i.e., for each value of $\alpha$, we use all $\beta$ values with the same amplitude as $\alpha$. The total number of displacement pulse combinations is thus quite large and increases with the photon number $N$ in the NOON state, typically involving of order a few hundred pulses. The displacement pulses can be expressed as

IV.2 Photon Populations

By bringing both qubits (initially in their $|g\rangle$ states) on resonance with the resonators, the joint number states contained in two resonators, i.e., the diagonal elements of $\rho$, can be read out, as each diagonal element swaps with the qubits at a different rate S (5), resulting in a distinct time-dependence for the probabilities $P_{gg}$, $P_{ge}$, $P_{eg}$, and $P_{ee}$. These can be numerically simulated using the device parameters from Table S1.

As displayed in Fig. S2(b), there is a small non-zero occupation of the excited state of each qubit after the state generation sequence, due to decoherence and pulse imperfections. We use the measured qubit state after the state generation sequence as the qubit initial condition when numerically simulating the tomography. Using these simulations, we obtain the time dependence for the each of the probabilities $P_{gg}$, $P_{ge}$, $P_{eg}$, and $P_{ee}$ corresponding to different combinations of photon number (Fock) states in the two storage resonators. Examples of these probability traces are shown in Fig. S3(a) for some selected initial states. The time-dependent traces for these probabilities, for the set of Fock states $\left\{\left|m\right\rangle_{A}\left|n\right\rangle_{B},m=0,1,2,\ldots,n=0,1,2,\ldots\right\}$, are then used to decompose the experimentally-measured time traces, which yields the probability distribution for the Fock number states contained in the storage resonators. This thus yields the diagonal elements of the experimentally-measured displaced density matrix $\rho$.

We obtain the diagonal elements of $\rho$ by doing a least-squares fit of the time-dependent probabilities, corrected for measurement fidelity. We use the MATLAB packages YALMIP and SeDuMi for the fitting. The number of fitting parameters is the number of diagonal elements, directly determined by the maximum photon number state contained in the resonators, plus the number of photon quanta added by the displacement pulses $|\alpha\rangle$ and $|\beta\rangle$. Fits are done with constraints $P_{mn}\geq 0$ and $\sum_{m,n}{P_{mn}}=1$ to return meaningful probability values. Examples of these fits are shown in Fig. S3, for the $N=1$ NOON state.

IV.3 Joint-Resonator Density Matrices

V NOON State Decay Dynamics

The Wigner tomography allows us to study the decoherence dynamics of the bipartite system S (4). The experimental results, compared with numerical simulations, are shown in Fig. S4, with relevant elements in Table S2. We note that the time evolution of the off-diagonal elements in the two-resonator density matrix, which represent inter-resonator coherence, is different from the evolution of the corresponding off-diagonal elements for a single resonator, which represent intra-resonator coherence S (4).

Statistical errors in the qubit probability measurements as well as uncertainty in the amplitude calibration for the displacement pulses $\alpha_{j}$ and $\beta_{k}$ are used to estimate the uncertainty in the amplitude and phase of each density matrix element. These errors are found to be small, in part because the constraints on the analysis filters unrealistic values. Instead we find that slow phase drifts in the electronics, perhaps dominated by ambient temperature fluctuations, give the main phase uncertainties, especially during long measurements. Evaluating a single density matrix usually takes a relatively short time during which these drifts are minimal. However, measuring a series of density matrices such as Fig. 5 in the main paper, takes a much longer time, typically $12$ to $20$ hours, allowing for more significant drifts. These mostly affect the phases of the density matrix elements, rather than the amplitudes.

V.2 Validation

The bipartite Wigner tomography was validated by several consistency checks. (1) The density matrix is as expected for a range of different states, including the highly entangled NOON states, the energy eigenstates $|2\rangle_{A}|0\rangle_{B}$ (Fig. S5(a)), the separable (product) state $(|0\rangle-|1\rangle)_{A}(|0\rangle-|1\rangle)_{B}$ (Fig. S5(b)) and the un-entangled mixed states (see below). (2) The NOON states display the expected phase sensitivity as a function of photon number $N$, as shown in Fig. 5 in the main paper and Table S3. (3) The NOON state fidelity and entanglement of formation agree reasonably well with numerical simulations. (4) The time-dependence of the density matrix elements, showing uniform exponential decay of all elements, is as expected and agrees with numerical simulations, as shown in Fig. S4 and Table S2. (5) The calculated negativities are significantly above zero for the NOON states and precisely zero (within the measurement error) for the unentangled mixed state (see below).

We note that bipartite Wigner tomography can measure any matrix element with a relatively high accuracy, as we can displace the system by an arbitrary amount in phase space. Even for relatively small off-diagonal elements, such as the desired off-diagonal term in the $N=3$ NOON state, the tomography can unambiguously evaluate this element and measure its sensitivity to external phase perturbations.

V.3 Ensemble of Mixed States

We use a synthetic ensemble of mixed states to illustrate the hazards involved in relying purely on coincidence measurements for demonstrating NOON-state entanglement. The ensemble comprises a mixture of 50% $|1\rangle_{A}|0\rangle_{B}$ and 50% $|0\rangle_{A}|1\rangle_{B}$ states, i.e., an ensemble with the same probability of being measured in either of the states forming the $N=1$ NOON state, but without any entanglement. This is done by generating the pure state $|1\rangle_{A}|0\rangle_{B}$ and measuring the time-dependent joint probabilities $P_{gg}(\tau)$, $P_{ge}(\tau)$, $P_{eg}(\tau)$, and $P_{ee}(\tau)$ for this state. We then generate the other component of the ensemble, the pure state $|0\rangle_{A}|1\rangle_{B}$, and repeat the generation and measurement procedure. Each value of $\tau$ involves 300 repeats of the preparation and measurement sequence for each of the pure states. We then combine the measurement results with equal weights, creating the joint probabilities for the synthetic ensemble; these data are shown in the main paper. The tomographic analysis yielding the density matrices is done in the usual way. The outcome of the ensemble measurements are shown in the main text, with the joint probabilities evolving in a way indistinguishable from the entangled NOON states, but the density matrix for the ensemble revealing a complete lack of entanglement, as witnessed by the negligible values for the off-diagonal elements.

VI MOON State

The NOON-state generation protocol can be simply generalized to generate MOON states, with different photon numbers in the two entangled resonators. The generation is similar to the NOON state sequence shown in Fig. S2. We assume $M>N$: After generating the Bell entanglement between two qubits at the end of step I (Fig. S2(a)), we repeat step II $N-1$ times, yielding the four-fold entangled state $|eg(N-1)0\rangle+|ge0(N-1)\rangle$. The photon amplification and transfer process (step II) is then applied $M-N$ times, but only to qubit $q_{0}$ and resonator $A$, yielding the state $|eg(M-1)0\rangle+|ge0(N-1)\rangle$. The final qubit excitations are then transferred in step III, resulting in the MOON state $|gg\rangle\otimes\left(|M0\rangle+|0N\rangle\right)$, with the qubits disentangled from the resonators. A MOON state generated in this fashion, with $M=2$ and $N=1$, is shown in Fig. S6.

References