(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Experimental progress in controlling and manipulating trapped atomic ions has opened the door for a series of proof-of-principle quantum simulations. This article reviews these experiments, together with the methods and tools that have enabled them, and provides an outlook on future directions in the field.

Quantum simulations with trapped ions

To process information using quantum-mechanical principles, the states of individual particles need to be entangled and manipulated. One way to do this is to use trapped, laser-cooled atomic ions. Attaining a general-purpose quantum computer is, however, a distant goal, but recent experiments show that just a few entangled trapped ions can be used to improve the precision of measurements. If the entanglement in such systems can be scaled up to larger numbers of ions, simulations that are intractable on a classical computer might become possible.

/pdf/entangled-states-of-trapped-atomic-ions-36hqe84fdh.pdf

Entangled states of trapped atomic ions

Quantum computing becomes viable when a quantum state can be protected from environment-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) is capable of identifying and correcting them. Adding more qubits improves the preservation of states by guaranteeing that increasingly larger clusters of errors will not cause logical failure-a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural step towards the two-dimensional surface code QEC scheme, and track errors as they occur by repeatedly performing projective quantum non-demolition parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 when using five of our nine qubits and by a factor of 8.5 when using all nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger-Horne-Zeilinger state. The successful suppression of environment-induced errors will motivate further research into the many challenges associated with building a large-scale superconducting quantum computer.

State preservation by repetitive error detection in a superconducting quantum circuit

One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena such as high temperature superconductivity, solid-state physics, transition metal catalysis, and certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable these problems to be tackled with fewer quantum resources should be considered important. Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry. This review provides a comprehensive introduction to both computational chemistry and quantum computing, bridging the current knowledge gap. Major developments in this area are reviewed, with a particular focus on near-term quantum computation. Illustrations of key methods are provided, explicitly demonstrating how to map chemical problems onto a quantum computer, and how to solve them. The review concludes with an outlook on this nascent field.

Quantum computational chemistry

Qubits based on trapped ions can be prepared and manipulated with record-breaking accuracy, offering a promising scalable platform for quantum computing.

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.113.220501

High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit.

We demonstrate single-shot qubit readout with a fidelity sufficient for fault-tolerant quantum computation. For an optical qubit stored in $^{40}\mathrm{Ca}^{+}$ we achieve 99.991(1)% average readout fidelity in ${10}^{6}$ trials, using time-resolved photon counting. An adaptive measurement technique allows 99.99% fidelity to be reached in $145\text{ }\text{ }\ensuremath{\mu}\mathrm{s}$ average detection time. For $^{43}\mathrm{Ca}^{+}$, we propose and implement an optical pumping scheme to transfer a long-lived hyperfine qubit to the optical qubit, capable of a theoretical fidelity of 99.95% in $10\text{ }\text{ }\ensuremath{\mu}\mathrm{s}$. We achieve 99.87(4)% transfer fidelity and 99.77(3)% net readout fidelity.

/pdf/high-fidelity-readout-of-trapped-ion-qubits-g2w52nm5h2.pdf

High-fidelity readout of trapped-ion qubits.

We describe a new electrode design for a surface-electrode Paul trap, which allows rotation of the normal modes out of the trap plane, and a technique for micromotion compensation in all directions using a two-photon process, which avoids the need for an ultraviolet laser directed to the trap plane. The fabrication and characterization of the trap are described, as well as its implementation for the trapping and cooling of single Ca+ ions. We also propose a repumping scheme that increases ion fluorescence and simplifies heating rate measurements obtained by time-resolved ion fluorescence during Doppler cooling.

https://iopscience.iop.org/article/10.1088/1367-2630/12/5/053026/pdf

Implementation of a symmetric surface-electrode ion trap with field compensation using a modulated Raman effect

We have implemented a universal quantum logic gate between qubits stored in the spin state of a pair of trapped 40Ca ions. An initial product state was driven to a maximally entangled state deterministically, with 83% fidelity. We present a general approach to quantum state tomography which achieves good robustness to experimental noise and drift, and use it to measure the spin state of the ions. We find the entanglement of formation is 0.54.

https://iopscience.iop.org/article/10.1088/1367-2630/8/9/188/pdf

Deterministic entanglement and tomography of ion-spin qubits

We demonstrate sympathetic cooling of a ${^{43}\text{C}\text{a}}^{+}$ trapped-ion ``memory'' qubit by a ${^{40}\text{C}\text{a}}^{+}$ ``coolant'' ion sufficiently near the ground state of motion for fault-tolerant quantum logic, while maintaining coherence of the qubit. This is an essential ingredient in trapped-ion quantum computers. The isotope shifts are sufficient to suppress decoherence and phase shifts of the memory qubit due to the cooling light which illuminates both ions. We measure the qubit coherence during ten cycles of sideband cooling, finding a coherence loss of 3.3% per cooling cycle. The natural limit of the method is $O({10}^{\ensuremath{-}4})$ infidelity per cooling cycle.

/pdf/memory-coherence-of-a-sympathetically-cooled-trapped-ion-owjtv3puug.pdf

D. N. Stacey

Papers

High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit.

High-fidelity readout of trapped-ion qubits.

Implementation of a symmetric surface-electrode ion trap with field compensation using a modulated Raman effect

Deterministic entanglement and tomography of ion-spin qubits

Memory coherence of a sympathetically cooled trapped-ion qubit