[Paper Review] Measurement of Translational Diffusion Constant using Noon State

Abhishek Shukla, Manvendra Sharma+1 more|arXiv (Cornell University)|Oct 7, 2011

Spectroscopy and Quantum Chemical Studies7 references3 citations

TL;DR

This paper proposes using NOON states in NMR to measure translational diffusion constants in liquids with higher sensitivity to weak pulsed-field gradients and shorter diffusion delays. By preparing a 10-qubit NOON state in an AM₉ spin system (trimethylphosphite), the method achieves identical diffusion constants to standard single-quantum methods but with reduced time and gradient strength, enabling study of slow diffusion and systems with limited gradient strength.

ABSTRACT

A method for measuring translational diffusion constant in liquids via NOON state is described using a quantum circuit and is experimentally demonstrated using a model system. When compared with the standard single quantum method, the NOON state method requirs shorter diffusion delays and weaker gradients. These improvements depend on the on the size of the NOON state. Due to the high sensitivity of the NOON state for the changes in the local magnetic fields, this method enables studying slow diffusion and studying diffusion with limited strengths of pulsed-field-gradients.

Motivation & Objective

• To develop a quantum-enhanced method for measuring translational diffusion constants in liquids using NOON states.
• To reduce the required diffusion delay and pulsed-field-gradient strength compared to conventional single-quantum NMR methods.
• To enable the study of slow diffusion processes that are difficult to access with standard PFG-NMR due to relaxation or gradient limitations.
• To experimentally demonstrate the feasibility of NOON-state-based diffusion measurement in a liquid-state NMR system.
• To compare the NOON-state method with standard single-quantum coherence methods in terms of accuracy, sensitivity, and experimental complexity.

Proposed method

• Prepare a pseudopure NOON state in a 10-spin AM₉ system (31P and nine equivalent 1H spins) using a sequence of single-qubit Hadamard and multi-qubit CNOT gates.
• Utilize J-coupling between the 31P nucleus and the nine equivalent 1H spins to implement the CNOT operation via a J-evolution sequence.
• Implement a gradient selection protocol with G₃/G₂ = 23.23 to selectively excite the 10-quantum coherence pathway.
• Apply a two-step phase cycle involving X̄ₘ pulses and receiver phase control to suppress artifact signals.
• Measure echo intensity as a function of gradient strength G₁ using 1H NMR, fitting to a Gaussian decay model: I(G₁) = I₀ exp(−γₑff²δ²G₁²Δ/6).
• Use the effective gyromagnetic ratio γₑff = lγ_H, where l = 9.4 is the lopsidedness parameter, to extract the diffusion constant D from the decay rate.

Experimental results

Research questions

• RQ1 Can NOON states enhance the sensitivity of PFG-NMR measurements for translational diffusion in liquids?
• RQ2 Does the NOON-state method allow for shorter diffusion delays and weaker pulsed-field gradients compared to standard single-quantum methods?
• RQ3 What is the accuracy and precision of the diffusion constant measured using the NOON-state method compared to conventional PFG-NMR?
• RQ4 Can the NOON-state approach be used to study slow diffusion processes that are inaccessible with standard techniques?
• RQ5 How do experimental imperfections, such as relaxation and coherence pathway selection errors, affect the performance of the NOON-state method?

Key findings

• The NOON-state method achieved a diffusion constant of D = (6.17 ± 0.25) × 10⁻¹⁰ m²/s in trimethylphosphite, in excellent agreement with the standard single-quantum method (D = 6.24 ± 0.06 × 10⁻¹⁰ m²/s).
• Despite similar accuracy, the NOON-state experiment required an order of magnitude shorter time between the PFG pulses (Δ) and weaker gradient strengths compared to the single-quantum method.
• The lopsidedness parameter l = 9.4 in the AM₉ system enabled effective encoding of z-coordinate changes via the 10-quantum coherence pathway, enhancing sensitivity to diffusion.
• The increased error bar in the NOON-state measurement (±0.25 × 10⁻¹⁰ m²/s) was attributed to experimental complexity, coherence pathway imperfections, and faster relaxation of the NOON state.
• The method demonstrates potential for studying slow diffusion and time-varying diffusion processes due to its ability to capture diffusion effects in shorter time windows.
• The results suggest that combining NOON states with single-scan PFG techniques could enable ultra-fast, real-time diffusion measurements in dynamic systems.