Coherent Generation and Protection of Anticoherent Spin States

Technical Summary: Coherent Generation and Protection of Anticoherent Spin States

Problem Statement
Anticoherent (AC) spin states represent a class of quantum states characterized by the vanishing of multipole moments up to a specific order $t$. Unlike coherent spin states, which resemble classical angular momentum, AC states maximize quantum uncertainty and exhibit rotational invariance, making them ideal for tasks such as Cartesian reference frame alignment and multi-parameter estimation in quantum metrology. While first-order AC states (e.g., Schrödinger cat states) have been realized on various platforms, higher-order AC states have so far been generated exclusively in multiphotonic systems. There is a significant gap in protocols capable of generating these states on atomic or solid-state platforms using standard spin control techniques. Furthermore, the high degree of symmetry and quantum sensitivity of AC states renders them particularly fragile against decoherence sources such as local disorder and dipole-dipole interactions.

Methodology
The authors propose a two-pronged approach: a pulse-based protocol for the deterministic generation of AC states and a dynamical decoupling strategy to protect these states from decoherence and finite-duration errors.

1. Generation Protocol:
   The generation scheme relies on a Hamiltonian comprising a rotation term and a one-axis twisting (squeezing) term: $H(t) = \Omega(t)J_y + \chi(t)J_z^2$. The protocol consists of $n_C$ cycles, where each cycle (except the first) applies a rotation about the $y$-axis followed by a squeezing operation along the $z$-axis.

   • Mechanism: The authors leverage the distinct coupling behaviors of these operations on the multipole expansion of the density matrix. Squeezing ($J_z^2$) couples multipoles of order $L$ to $L \pm 1$, while rotation ($J_y$) couples multipoles of the same $L$ but different $M$. By strategically rotating the population into specific $M$ states (specifically $M=0$ and $M=\pm L$) that are decoupled from lower-order multipoles during squeezing, the protocol prevents the backflow of population to lower orders, effectively suppressing multipole moments up to order $t$.
   • Optimization: The control parameters (rotation angles $\theta_i$ and squeezing strengths $\eta_i$) are optimized to maximize a figure of merit based on the Bures distance ($A_t$), which quantifies the degree of anticoherence. The authors employ both numerical optimization (Nelder-Mead algorithm) and analytical derivations for specific cases (e.g., $t=2$).

2. Protection Strategy (Dynamical Decoupling):
   To mitigate dephasing caused by disorder ($H_{dis}$) and dipole-dipole interactions ($H_{dd}$), the authors integrate Dynamically Corrected Gates (DCGs) into the generation protocol.

   • Sequence Design: They utilize specific dynamical decoupling (DD) sequences, TEDD and TEDDY, which are constructed on the Cayley graph of a decoupling group. These sequences are modified to include "balanced pairs" of pulses (a stretched target pulse and an identity pulse) that cancel finite-duration errors to first order.
   • Implementation: Two protection strategies are evaluated: (i) protecting each individual pulse (rotation or squeezing) with a DCG, and (ii) protecting each full cycle (rotation + squeezing) as a single block. The effectiveness is analyzed using a distance metric between the noisy and ideal propagators.

Key Results

• Generation Efficiency: The pulse-based protocol successfully generates pure AC states of various orders. Numerical optimization demonstrates the ability to generate states up to order $t=9$ for spin $j=24$ and order $t=2$ for collective spins up to $j=5000$.
• Analytical Solutions: For $t=2$, the authors derived analytical expressions for the control parameters. For integer spins, the protocol requires $n_C = j$ cycles to achieve high-fidelity AC states. Specific analytical values for squeezing parameters were found to follow power laws ($\eta_2 \propto j^{-1/2}$, $\eta_3 \propto j^{-1}$), allowing for scalable generation.
• Robustness: The introduction of DCGs significantly reduces the infidelity of the state preparation in the presence of noise.
  • In the disorder-dominated regime ($\delta > \Delta$), protecting individual pulses yields slightly better performance than protecting full cycles.
  • In the interaction-dominated regime ($\Delta > \delta$), both strategies perform similarly, with cycle protection being slightly advantageous in some parameter ranges.
  • The DCGs are effective when the noise strength relative to the control amplitude ($\|H_{err}\|/\chi$) is below a critical threshold (approximately $10^{-1.6}$).
• Control Errors: The study analyzes the impact of systematic control errors (flip-angle errors). It is found that errors in the DD pulses are self-corrected to first order by the Eulerian design of the sequence. However, errors in the balanced pair (specifically type II errors where identity pulses introduce new errors) can degrade performance. The authors identify parameter regimes where DCGs still outperform unprotected protocols despite these errors.

Significance and Claims
The paper claims to present the first protocol specifically designed to generate anticoherent spin states of different orders on platforms utilizing spin rotation and squeezing, filling a gap left by previous work restricted to multiphotonic systems. The authors emphasize that their method is experimentally accessible with current technology, citing feasibility in systems like cesium and dysprosium atoms, and solid-state spin ensembles (e.g., NV centers).

The significance of the work lies in:

1. Scalability: The ability to generate high-order AC states for large spin numbers ($j$) using a systematic, pulse-based approach.
2. Robustness: The development of a protection scheme that preserves the fragile coherence of AC states against realistic noise sources (disorder and dipolar interactions) and finite-duration pulse errors.
3. Applications: The authors posit that the generated states are valuable for quantum sensing (due to their sensitivity to external perturbations) and quantum metrology (specifically for simultaneous multi-parameter estimation and alignment of reference frames). They also note the potential relevance for quantum entanglement studies, as AC states correspond to maximally entangled symmetric multiqubit states, and suggest their utility in fault-tolerant quantum computing schemes where logical qubits are encoded in spin-$j$ AC states.

The authors remain modest regarding future directions, suggesting that while their deterministic protocol is efficient, probabilistic generation via post-selection or dissipative methods, as well as the use of more complex Hamiltonians (e.g., two-axis countertwisting), represent viable avenues for future research to further speed up generation or enhance robustness.