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Single trapped atoms are a natural candidate for the nodes of a quantum network. We’ve already seen how long-lived qubits can be implemented with the electronic configuration of atoms in Chap. 3 and we will discuss our particular implementation with trapped Barium ions in Chap. 6. Atoms of the same isotope are naturally identical in all respects and, if the atoms are well insulated from their environment, photons scattered from atoms can be made to interfere with high visibility. Experimental techniques for isolating, manipulating and addressing single trapped atoms are well developed. Furthermore, although scaling trapped-ion quantum computers is challenging, they are currently the most advanced platform for universal quantum computation [42, 155]. Atom-light interfaces with trapped atoms are a means of networking such systems, and perhaps a feasible path to scaling trapped-atom processors by networking small quantum processors [57]. Apart from their advantages as photon sources for quantum networks, this provides an additional motivation for trapped-ion based atom-light couplers in particular.

However, the interaction cross section of atoms and photons in free space is small. To couple atomic and photonic qubits deterministically requires an efficient atom-light interface. Enormous progress has been made towards this goal in the last decade, par- ticularly in the field of cavity QED. In the sections below we will consider the relative advantages of cavity and free-space based atom-light couplers, and review recent results with rudimentary optical networks of trapped atoms.

(a) (b)



g V

= 1 ⌘

Figure 5.1: (a) Schematic of a resonator-coupled atom showing the important cavity parameters. (b) Free space coupling by a parabolic mirror.

5.4.1 Coupling with optical resonators

Resonators can be used to couple atoms to either optical or microwave photons, but we will restrict this review to atom-light couplers for optical photonic networks, which have longer coherence times. When an atom is positioned at an anti-node of a cavity standing wave, the atom-photon interaction strength can be much larger than in free space. Resonators e↵ectively increase the interaction strength in two ways: by confining the photon to a small resonator mode and by increasing the interaction time. We call the regimes dominated by one or the other of these e↵ects the ‘Purcell’ and ‘strong’ coupling regimes respectively [156].

The important cavity parameters are the coherent atom-field coupling constant for the cavity field g, the free-space atomic emission rate and the cavity decay rate , which is inversely related to the cavity finesse F shown in Fig. 5.1(a). Together these give the cooperativity C = g2_{/( ) and Purcell emission rate enhancement P = 1 + 2C. The} collection efficiency in such a cavity is the product of three probabilities [157]

⌘=T ✓ 2C 1 + 2C ◆ ✓ 2 2+ ◆ , (5.2)

whereT is the transmission of the cavity, the second term is the cavity capture proportion and the third term is the cavity loss proportion of the atom-cavity system. Written in this way, it becomes clear that efficient atom-cavity coupling requires both largeg and.

There are two typical strategies for designing resonator-based atom-light couplers. First, improving the cavity finesse allows photons to interact with the atom over many cavity round-trips or, equivalently, to interact with many mirror images of the atom at the same time. This is the strong coupling regime,g >, andC 1, typified by oscillatory exchange between the atom and cavity field. In this regime the coupling strength g has been increased so that it exceeds all the dissipative processes in the system, but at the cost of lower cavity linewidth.

In practice many implementations fall short of strong coupling. However, high-finesse cavity sources can be operated efficiently with⇤-atoms and Raman coupling schemes, even when the free-space spontaneous emission rate is faster than the cavity dynamics. This is achieved by taking advantage of the di↵erent detuning dependence between the coherent and incoherent Raman processes and is sometimes called the ‘intermediate’ regime [158].

§5.4 Trapped atoms 69

The e↵ective Raman atom-cavity coupling and spontaneous decay rates are

ge↵ =g ⌦e 2 , (5.3) e↵ = ✓ ⌦e 2 ◆2 . (5.4)

By choosing the excitation power⌦eand detuning such that> ge↵ > e↵, the atomic

excitation may be efficiently transferred to the field. However, the repetition rate of the source is still restricted by the cavity decay rate . There is an implicit trade-o↵in the design of both strong and intermediate resonator coupled photon sources: the better the cavity finesse, the lower the linewidth and the lower the speed of the network.

The second route to resonator-based networks is reducing the cavity mode volume. In quantizing the electric field we saw that the field strength per photon depends on the mode volumeV, Eqn. 2.4. As the mode volume of a single photon decreases, the strength of the field increases. Confining photons to a smaller mode therefore improves the interaction strength according tog_/1/pV. With sufficiently small mode volume even leaky cavities can be efficient couplers. This is the Purcell regime of small, leaky cavities with g

andC 1. In this regime the cavity increases the atom-light coupling strength by shaping the photonic spatial mode density in the vicinity of the atom. In Sec. 10.7.2 we consider a single-pass approach to atom-light coupling which uses the same technique, and realizes an extreme of the Purcell resonator regime.

As we described in Sec. 3.4.2, photons can be generated reversibly from ⇤-atoms in cavities by vacuum StiRAP with the cavity mode. This has been performed with cold neutral atoms falling through cavities [159, 160], neutral atoms in optical traps [161, 162] and trapped ions [163]. Compared to electromagnetically trapped ions, optically trapped atoms have limited trapping time and are less well localized, which compromises atom- cavity coupling and is an additional source of decoherence. On the other hand, trapped ions are restricted by the geometries of large ion traps (a problem that we will also face in Chap. 11) and coupling ions to small cavities is difficult because charged particles interact strongly with nearby surfaces. To date no trapped-ion photon-source has been operated in the strong or Purcell cavity QED regimes.

Here we will highlight recent experiments with resonator-based atom-light couplers. Refs. [156, 164] contain a historical overview and further detail for readers interested in the landmark results in this field. StiRAP has been performed with trapped ions in Fabry- P´erot cavities with collection efficiency ⌘I = 0.88 and total efficiency ⌘ = 0.045 at a rate

of 2.4 kHz [165]. The source is essentially dark count limited, withA= 0.015. The photon source may be operated as a unidirectional atom-light qubit interface with ⌘ = 0.01 and fidelity F = 0.66[166] and to demonstrate atom-photon entanglement [167].

StiRAP has also been demonstrated with neutral atoms in large cavities with⌘I = 0.56

and ⌘ = 0.15 at a rate of 100 kHz[168]. Once again, the number-purity of the source is essentially dark-count limited, A = 0.02. Operated as a unidirectional atom-light qubit interface the fidelity is F = 0.93[169]. The trapped atom source has been operated as a deterministic quantum memory with efficiency ⌘ = 0.17 and fidelity F = 0.93 [169, 170] as well as a heralded quantum memory with⌘= 0.39 and F = 0.86[171].

To this point, a state of the art atom-light quantum network consists of two nodes universally coupled (read and write) to a single network link. Teleportation has been performed with neutral trapped atoms in Fabry-P´erot cavities in this configuration with

fidelity F = 0.88 and success probability P = 0.001 at an attempt rate of 10 Hz [172]. The interference visibility in that experiment was limited by inhomogeneous transition broadening toV = 0.64. Ground state cooling has since been demonstrated with trapped neutral atoms in optical cavities [173] which would improve the source indistinguishability considerably.

Although these elementary atom-light networks have been constructed from relatively large cavities, miniaturization is the route to fast, high efficiency single photons from atom-resonator systems. To this end atoms have been coupled to fibre-based Fabry-P´erot cavities [174, 175], microtoroidal resonators [176, 177] and photonic crystal cavities [178]. Fibre cavities are a promising path towards coupling trapped ions in particular because the dielectric surface required can be small, and shielded using metal fibre sleeves [179, 180].