[Paper Review] Generating optical nonlinearity using trapped atoms

Alexei Gilchrist, G. J. Milburn+2 more|ArXiv.org|May 28, 2003

Quantum optics and atomic interactions2 references3 citations

TL;DR

This paper proposes a near-deterministic optical nonlinearity using trapped atoms via high-efficiency fluorescence shelving measurements, enabling universal Hamiltonian evolution on optical Fock-state qudits. By conditioning on atomic state measurements after a Raman-mediated interaction, the scheme generates a conditional square-root-number operator that enables nonlinear phase shifts and cat-state preparation with high fidelity.

ABSTRACT

We describe a scheme for producing an optical nonlinearity using an interaction with one or more ancilla two-level atomic systems. The nonlinearity, which can be implemented using high efficiency fluorescence shelving measurements, together with general linear transformations is sufficient for simulating arbitrary Hamiltonian evolution on a Fock state qudit. We give two examples of the application of this nonlinearity, one for the creation of nonlinear phase shifts on optical fields as required in single photon quantum computation schemes, and the other for the preparation of optical Schrodinger cat states.

Motivation & Objective

• To overcome the lack of intrinsic optical nonlinearities in materials for quantum optics.
• To develop a deterministic method for generating optical nonlinearities using trapped atoms and high-efficiency measurements.
• To demonstrate that conditional atomic measurements combined with linear optics enable universal Hamiltonian evolution on qudit systems.
• To show feasibility for implementing nonlinear gates and Schrödinger cat states in photonic quantum computing.

Proposed method

• Uses a Raman-coupled two-level atomic transition in a cavity to mediate an effective interaction between a quantized optical field and the atom.
• Employs a Hamiltonian $ H = \kappa(a^\dagger\sigma^- + a\sigma^+) $, where $ a^\dagger, a $ are field operators and $ \sigma^-, \sigma^+ $ are atomic raising/lowering operators.
• Applies a conditional measurement on the atomic state after interaction, yielding a nonlinear transformation $ \Upsilon_g(\tau) = \cos(\tau\sqrt{a^\dagger a}) $ on the field state.
• Relies on high-efficiency atomic readout (e.g., >99%) via cycling transitions to achieve near-deterministic operation.
• Uses the conditional field evolution to simulate arbitrary Hamiltonian dynamics on truncated Fock-state qudits.
• Demonstrates that this nonlinearity, combined with linear optics, is universal for quantum computation in the Fock basis.

Experimental results

Research questions

• RQ1 Can a deterministic optical nonlinearity be generated using trapped atoms and high-efficiency measurements?
• RQ2 Is the conditional transformation $ \cos(\tau\sqrt{a^\dagger a}) $ sufficient for universal quantum computation on qudits?
• RQ3 Can this scheme generate nonclassical states such as optical Schrödinger cat states?
• RQ4 What are the required interaction times and parameters for practical implementation in current experimental setups?
• RQ5 How does the nonlinearity compare in efficiency and fidelity to measurement-based schemes relying on photo-detection?

Key findings

• The scheme achieves near-deterministic nonlinear phase shifts with probability approaching one via high-efficiency atomic measurements.
• The conditional transformation $ \cos(\theta\sqrt{a^\dagger a}) $ enables universal simulation of arbitrary Hamiltonian evolution on qudits.
• For coherent input states, the conditional field state evolves into a superposition localized symmetrically on the imaginary axis in phase space, resembling a Schrödinger cat state.
• Numerical analysis shows that for $ |\alpha| \gg 1 $, the Q-function amplitude is peaked at $ \phi = \theta/(2|\alpha|) $, enabling controlled state localization.
• With typical experimental parameters ($ g = 4.5 $ MHz, $ \Omega = 30 $ MHz, $ \Delta = 6 $ MHz), coupling strengths of ~70 MHz are achievable, requiring interaction times of 0.1–5 $\mu$s.
• The method is robust against cavity decay if the interaction time is kept short, making it feasible with current cavity QED technology.