Engineering direct nonlinear coupling between microwave photons

Technical Summary: Engineering Direct Nonlinear Coupling Between Microwave Photons

Problem Statement
Quantum harmonic oscillators, particularly those realized in superconducting cavities, offer a promising platform for quantum information processing due to their large Hilbert spaces and hardware-efficient encoding capabilities. However, a critical bottleneck remains the implementation of strong, direct nonlinear coupling between bosonic modes. Existing strategies for entangling oscillators typically rely on parametrically driven interactions mediated by a nonlinear ancillary element (such as a transmon). These methods necessitate populating the excited states of the coupler, which introduces leakage outside the protected bosonic code space and creates additional decoherence channels. This leakage limits gate fidelity and hinders the realization of fault-tolerant bosonic quantum computation.

Methodology
The authors propose and experimentally demonstrate a fundamentally different approach: a Raman-assisted cross-Kerr interaction between microwave photons in two superconducting cavities (Alice and Bob) that does not populate the mediating coupler.

• System Architecture: The device consists of two 3D aluminum superconducting cavities dispersively coupled to a common, flux-tunable transmon coupler. The coupler frequency ($\omega_c$) is tunable from 4.3 GHz to 5.7 GHz.
• Interaction Mechanism: Instead of resonantly driving the coupler, the authors off-resonantly drive a coherent exchange interaction between the states $|1\rangle_a|0\rangle_b|g\rangle_c$ and $|0\rangle_a|1\rangle_b|e\rangle_c$ (where $g$ and $e$ denote the ground and excited states of the coupler).
• Effective Hamiltonian: By operating in the Raman regime where the drive detuning $\Delta$ is much larger than the exchange rate $g_1$, the system undergoes a second-order process. Through a Schrieffer–Wolff transformation, this virtual transition generates an effective cross-Kerr Hamiltonian:
  $$\hat{H}_{eff}/\hbar = g_{ab} \hat{a}^\dagger \hat{a} \hat{b}^\dagger \hat{b} |g\rangle\langle g|$$
  where the coupling strength is $g_{ab} \approx g_1^2 / \Delta$. Crucially, the coupler remains in its ground state $|g\rangle$ throughout the operation, preserving the bosonic code space.
• Calibration: The exchange rate $g_1$ is calibrated via coherent population oscillations. The detuning $\Delta$ is optimized to balance gate speed against residual coupler excitations ($P_e$), selecting an operating point of $\Delta/2\pi = -6$ MHz to achieve a coupling strength of $g_{ab}/2\pi \approx 90$ kHz while keeping $P_e < 1\%$.

Key Contributions and Results

1. Direct CPHASE Gates: The engineered cross-Kerr interaction functions as a direct controlled-phase (CPHASE) gate. The authors demonstrated this in two encodings:

   • Single-photon subspace (0/1): A controlled-Z (CZ) gate was implemented with a duration of approximately 5.2 $\mu$s. The gate generated a maximally entangled Bell state from $|+\rangle|+\rangle$ inputs.
   • Two-photon subspace (0/2): The interaction was applied to a biased-erasure encoding using the $|0\rangle$ and $|2\rangle$ Fock states. Due to bosonic enhancement, the CZ gate time was reduced to 1.94 $\mu$s.

2. Gate Performance:

   • The average infidelity per gate was measured to be below 5% (specifically $4.2 \pm 0.3\%$ for the $|+\rangle|+\rangle$ state in the 0/1 subspace).
   • The dominant source of error was identified as cavity decoherence (photon loss) and drive-induced decay, rather than leakage out of the code space.
   • The measured coupler population remained below 1%, confirming the virtual nature of the interaction.

3. All-Bosonic Syndrome Extraction:

   • The authors utilized the entangling operation to perform a parity check on a storage cavity (Alice) using an ancillary cavity (Bob) instead of a nonlinear ancilla.
   • By mapping the photon-number parity of Alice onto the phase of Bob, they demonstrated a method to detect photon loss without entangling the storage mode with a nonlinear element.
   • Lifetime Extension: Post-selecting on the bosonic parity check (where the ancillary cavity indicates even parity) extended the effective lifetime of the storage oscillator's $|2\rangle$ state. The decay constant increased from $763 \pm 11$ $\mu$s (bare) to $1173 \pm 41$ $\mu$s (post-selected), effectively suppressing errors caused by single-photon loss events.

Significance
The paper establishes a qualitatively different strategy for bosonic control that avoids the fundamental limitations of coupler-population-based schemes. By engineering a direct photon-photon interaction that preserves the bosonic code space and keeps the nonlinear coupler in its ground state, the work provides a pathway toward:

• Fault-tolerant quantum information processing: The method is inherently compatible with rotationally symmetric bosonic codes and enables syndrome extraction without exposing the storage mode to nonlinear-ancilla-induced decoherence.
• Nonlinear Quantum Optics: It offers a new route to explore strongly interacting photonic many-body phenomena, such as photon blockade and Bose-Hubbard physics, in the microwave regime.

The authors note that while current gate fidelities are limited by drive-induced decoherence (a technical challenge common to four-wave mixing architectures), the virtual-process approach is not fundamentally constrained. Future improvements could be achieved through active cancellation techniques, multi-tone drives, or the use of alternative nonlinear couplers (e.g., SNAIL) to further suppress spurious dynamics.