Amplification of bosonic interactions through squeezing in the presence of decoherence

1. Problem Statement

In quantum information processing, weak interactions between bosonic modes (such as beamsplitter or cross-Kerr couplings) are often too slow to compete with environmental noise and decoherence. While Hamiltonian amplification via parametric squeezing can speed up desired interactions, a critical challenge exists: squeezing typically amplifies both the desired dynamics and the detrimental noise processes simultaneously. If noise is amplified at the same or a higher rate than the interaction, the protocol fails to improve state preparation fidelity. The paper addresses the question: Under what conditions can squeezing protocols amplify desired bosonic interactions faster than specific noise and decoherence mechanisms, thereby enabling high-fidelity state preparation in open quantum systems?

2. Methodology

The authors investigate this problem using two primary bosonic interaction models and two distinct noise models, employing both analytical derivations and numerical simulations.

• Target Interactions:

  1. Beamsplitter Interaction (Quadratic): $H_{bs} = g(ab^\dagger + a^\dagger b)$. Used to generate entangled states like $|\Phi\rangle = \frac{1}{\sqrt{2}}(|01\rangle + i|10\rangle)$.
  2. Cross-Kerr Interaction (Quartic): $H_{cK} = \chi a^\dagger a b^\dagger b$. Used to generate states like $|\Psi\rangle = \frac{1}{\sqrt{2}}(|0+\rangle + |1-\rangle)$ and implement controlled-phase gates.

• Amplification Protocol:
  The authors utilize Hamiltonian amplification via bang-bang control (rapidly alternating squeezing operations along orthogonal quadratures) and discuss continuous parametric control.

  • Sequence: The protocol alternates between free evolution under the target Hamiltonian and squeezing operations $S_{\theta}(r) = \exp[\frac{r}{2}(a^2 e^{-i\theta} - a^{\dagger 2} e^{i\theta})]$.
  • Amplification Factor ($\lambda$): In the limit of infinite steps ($N \to \infty$), the effective Hamiltonian becomes $H_{\lambda} = \lambda H$.
    • For Beamsplitter: $\lambda_2 = \cosh(2r)$.
    • For Cross-Kerr: $\lambda_2 = \cosh^2(2r)$ (using mode-specific squeezing).

• Noise Models:

  1. Random Displacements: Modeled as stochastic field fluctuations (generalized forces) added to the Hamiltonian, representing static disorder or dynamic noise.
  2. Decoherence (Lindblad): Modeled via a master equation including:
     • Excitation Losses: Jump operators $L \in \{a, b\}$ (coupling to a vacuum bath).
     • Dephasing: Jump operators $L \in \{a^\dagger a, b^\dagger b\}$.

• Analysis Framework:
  The authors analyze the scaling of the desired interaction strength versus the noise strength as a function of the squeezing parameter $r$. They calculate the infidelity ($1-F$) of the target Bell-type states under various noise strengths and squeezing levels.

3. Key Contributions & Results

A. Differential Scaling of Noise vs. Interaction

The core finding is that the amplification factor for the desired interaction often scales differently (and more favorably) than the amplification factor for specific noise types.

• Random Displacements:

  • The desired interaction scales as $\lambda_2 \approx e^{2r}$ (Beamsplitter) or $e^{4r}$ (Cross-Kerr).
  • The displacement noise scales as $\cosh(r) \approx e^r$.
  • Result: The ratio of noise-to-signal decreases exponentially with $r$. For Cross-Kerr, the suppression is even stronger ($\propto e^{-3r}$). The protocol successfully outperforms displacement noise, reducing infidelity by an order of magnitude for moderate noise strengths.

• Excitation Losses (Lindblad):

  • The amplification protocol modifies the Lindblad superoperator. It enhances the original loss term by $\cosh^2(r)$ but introduces a new "heating" term (excitation from a thermal bath) proportional to $\sinh^2(r)$.
  • Beamsplitter Case: The effective loss rate scales similarly to the interaction rate ($\cosh^2(r)/\cosh(2r) \approx \text{const}$). Result: Hamiltonian amplification fails to overcome losses for beamsplitter interactions; infidelity remains high.
  • Cross-Kerr Case: The interaction scales as $\cosh^2(2r)$, while the effective loss/heating rates scale as $\cosh^2(r)$. The ratio scales as $\sim e^{-2r}$. Result: The protocol succeeds in overcoming losses for Cross-Kerr interactions, significantly improving fidelity for large $r$.

B. Numerical Validation

• Figures 2 & 3: Simulations confirm that for Cross-Kerr interactions, increasing the squeezing strength ($r$) and the number of Trotter steps ($N$) drastically reduces infidelity in the presence of both random displacements and excitation losses.
• Trotter Error Trade-off: The authors note that for very weak noise, high squeezing can initially worsen performance due to Trotter errors (discretization errors from finite $N$). However, increasing $N$ mitigates this, allowing the noise-suppression benefits to dominate.

C. Analytical Derivation of Effective Dynamics

The paper provides rigorous derivations (Appendices B and C) showing how the squeezing sequences transform the Lindblad operators.

• For losses, the effective bath temperature becomes $T(r) \propto \ln[\coth^2(r)]$, effectively heating the system.
• The authors demonstrate that for Cross-Kerr, the "heating" rate is suppressed relative to the interaction speed, whereas for Beamsplitter, it is not.

4. Significance and Implications

• New Control Paradigm: The work challenges the conventional view that squeezing is merely a resource for sensing or that it inevitably amplifies all noise. It establishes a pathway where fast parametric control can be tailored to make desired dynamics faster than detrimental processes, even in open systems.
• Robust Quantum Gates: The findings are particularly relevant for photonic quantum computing and trapped-ion systems. Specifically, the ability to amplify Cross-Kerr interactions while suppressing loss-induced decoherence enables faster, high-fidelity implementation of two-qubit entangling gates (controlled-phase gates), which are notoriously difficult due to weak non-linearities and photon loss.
• Hybrid Systems: The authors suggest that combining Hamiltonian amplification with dynamical decoupling could offer powerful control strategies for hybrid quantum systems (coupling bosonic modes to spins), selectively enhancing specific interactions while suppressing others.
• Design Guidelines: The paper provides a "rule of thumb": Amplification is advantageous if the desired interaction involves bosonic operators of a higher order (e.g., quartic) than the detrimental noise processes (e.g., linear losses or quadratic displacements).

Conclusion

Tiwari et al. demonstrate that while Hamiltonian amplification generally amplifies all dynamics, careful selection of the interaction type (specifically Cross-Kerr) and noise model allows for a regime where the desired interaction is accelerated exponentially faster than the noise. This enables the preparation of complex entangled states with high fidelity in noisy environments, offering a viable strategy for scaling up bosonic quantum technologies.