[Paper Review] Continuous Measurement Quantum State Tomography of Atomic Ensembles

Riofrio Almeida, A R Carlos|arXiv (Cornell University)|Jan 1, 2012

Quantum Information and Cryptography65 references4 citations

TL;DR

This paper presents a continuous measurement quantum state tomography protocol for atomic ensembles, using time-dependent control and collective probing to generate an informationally complete measurement record. Applied to a 16-dimensional Hilbert space in cold 133Cs atoms, the method achieves >95% fidelity for low-complexity states and >92% for arbitrary random states using maximum likelihood and compressed sensing estimation.

ABSTRACT

Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, ${}^{133}$Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace ($F=3 \oplus F=4$), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.

Motivation & Objective

• To develop an efficient quantum state tomography protocol for atomic ensembles that avoids repeated preparation by enabling continuous measurement.
• To address the inefficiency of traditional tomography requiring many independent preparations and measurements.
• To demonstrate full control and reconstruction of quantum states in a 16-dimensional Hilbert space (F=3 ⊕ F=4) of 133Cs atoms.
• To validate the protocol experimentally with high-fidelity state reconstruction using both maximum likelihood and compressed sensing methods.

Proposed method

• Utilizes continuous collective probing of an atomic ensemble via external fields (microwaves and RF magnetic fields) during time-evolving control.
• Employs time-dependent control fields to steer the ensemble through a sequence of unitary operations that generate an informationally complete measurement record.
• Applies maximum likelihood and compressed sensing estimation techniques to invert the continuous measurement record and reconstruct the initial quantum state.
• Relies on a general formalism for continuous measurement tomography applicable to systems with collective observables and time-continuous data streams.
• Uses the full 16-dimensional electronic ground state manifold (F=3 and F=4 hyperfine levels) of 133Cs as the Hilbert space for state encoding and control.
• Employs a continuous measurement record derived from the collective response of the atomic ensemble to external fields, enabling real-time state estimation.

Experimental results

Research questions

• RQ1 Can continuous measurement protocols achieve informationally complete quantum state tomography in large-dimensional Hilbert spaces?
• RQ2 How does the fidelity of state reconstruction scale with state complexity in a 16-dimensional atomic ensemble?
• RQ3 To what extent can maximum likelihood and compressed sensing estimation improve reconstruction fidelity in continuous measurement schemes?
• RQ4 Can full control of a 16-dimensional Hilbert space be experimentally demonstrated using microwave and RF fields in a cold atomic ensemble?
• RQ5 What is the experimental feasibility and accuracy of reconstructing arbitrary quantum states via continuous measurement tomography?

Key findings

• The continuous measurement quantum tomography protocol achieves state reconstruction fidelities exceeding 95% for low-complexity quantum states in the 16-dimensional Hilbert space of 133Cs.
• For arbitrary random quantum states, the protocol maintains a fidelity above 92%, demonstrating robustness across diverse state types.
• The experimental results show excellent agreement with theoretical predictions, validating the proposed formalism and estimation techniques.
• The method enables efficient state reconstruction without requiring repeated state preparation, significantly reducing measurement time compared to standard tomography.
• The protocol demonstrates full control over the 16-dimensional electronic ground state manifold (F=3 ⊕ F=4) of 133Cs using microwave and RF magnetic fields.
• Both maximum likelihood and compressed sensing estimation methods successfully reconstruct the initial state from the continuous measurement record, confirming the method's versatility and accuracy.