Remote-Entanglement Protocols for Stationary Qubits with Photonic Interfaces

The generation of entanglement between distant quantum systems is at the core of quantum networking. In recent years, numerous theoretical protocols for remote-entanglement generation have been proposed, many of which have been experimentally realized. Here, we provide a modular theoretical framework to elucidate the general mechanisms of photon-mediated entanglement generation between single spins in atomic or solid-state systems. Our framework categorizes existing protocols at various levels of abstraction and allows for combining the elements of different schemes in new ways. These abstraction layers make it possible to readily compare protocols for different quantum hardware. To enable the practical evaluation of protocols tailored to specific experimental parameters, we have devised numerical simulations based on the framework with our codes available online.

Popular Summary

Entanglement between distant quantum systems is an essential building block for quantum networks, enabling applications that range from secure quantum communication to distributed quantum computation. Photons can bridge the distance between stationary qubits to realize entanglement remotely. Several qubit systems have a natural interaction with light, including cold atoms, trapped ions, color centers in solid state, and quantum dots. In recent years many different protocols have been developed and experimentally realized.

However, there still is no user manual for remote-entanglement generation. On the contrary, after years of advances in theory and experiments, the proliferation of entanglement protocols may at first glance warrant an encyclopedia.

To tame this “zoo” of protocols, we present a two-part solution: a taxonomy to classify protocols at varying levels of abstraction and a “user's guide” to compose protocols by modular subcomponents. This greatly simplifies the understanding and numerical analysis of various entanglement protocols, as shown by illustrative use cases with open-source code.

Figure 1

An overview of the remote-entanglement-protocol modular framework. With regard to the entanglement link, the objective of entanglement protocols is to entangle two remote spins (purple), where the entanglement is created by photons (red). (a) Logical building blocks and topology (Sec. 3). The topology for remote-entanglement generation is deconstructed in logical building blocks (LBBs). From left to right, we consider three main REP topologies: the detection-in-midpoint, sender-receiver, and source-in-midpoint topologies. (b) Encoding (Sec. 4). The encoding layer chooses the basis for the photonic qubits and the levels for the spin qubit. We illustrate the time-bin encoding of sender-receiver topology with spin-photon gates. (c) Physical building blocks (Sec. 5). The physical building blocks (PBBs) are the quantum channels describing the operations of the physical systems, including imperfections. This layer shows the REP at the physical level. The first panel describes the construction of spin-photon gates with an overcoupled cavity-QED system for state conditional amplitude reflection. The second panel shows the implementation of a measurement of the time-bin qubit in the $X$ basis, using a Mach-Zehnder interferometer (see Figs. 3 and 4). (d) Quantum optical modeling (Sec. 6). This layer models the hardware used in the PBB. In this example, the relevant parameters for the PBBs (e.g., the state-dependent reflection coefficient, $r_{|0⟩,|1⟩}$) can be calculated from physical variables (e.g., the spin-cavity coupling rate, $g$) using a detailed quantum optical model. We implemented these descriptions in a software package publicly available on GitHub (Sec. 7). Section 8 shows the simulation and benchmarking results with our framework.

Figure 2

The remote-entanglement-protocol topologies and logical building blocks (LBBs). (a) The three available topologies for REP—detection-in-midpoint, sender-receiver topology, and source-in-midpoint—and a depiction of their operation with circuit diagrams. Here, we use the notation $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. (b) The LBBs available to construct the REP. The circuit description outlines their idealized operations, which the user can match with the diagrams in (a). For the spin-photon projector and the spin-photon absorption, an approximate gate diagram is given. This can be used to see how they can be used in the quantum circuits in (a). The exact gate description for these LBBs would be a projector operation and a swap gate. (c) Three different implementations of the midpoint detector, as an example. In this case, the spin-photon emission, spin-photon gate, and spin-photon projector are used as the spin-photon interface, respectively.

Figure 3

The encoding of photonic qubits. (a) The basis states of photonic encoding. (b) In every basis choice, the basis can be implemented with single photons or those can be approximated with a weak coherent state.

Figure 4

The photonic PBBs for the different photon encodings. (a) Each basis has its own implementation of the quantum operations but some do not have natural implementation without converting them to a different basis (usually to dual-rail encoding). The measurement in the computational basis is performed with a single-photon detector. For time-bin encoding, it needs to be time resolved. For polarization and frequency, the qubit is usually converted to and detected in dual-rail encoding. Two-qubit gates are implemented using mode mixing with a beam splitter. The mode mixing lets two photonic qubits interfere with each other. For a $Z$ rotation, the phase between the two basis states needs to be altered. This is implemented in Fock-state encoding with a delay line, in time-bin encoding with an electric optic modulator (EOM) shifting one time bin, in polarization encoding with a wave plate, in dual-rail encoding with delaying one of the lines, and in frequency encoding with a dispersive medium. The $X$ rotation requires changing the photon between the two eigenstates, which is not feasible for Fock-state encoding, goes via dual-rail encoding for time-bin encoding, can be easily implemented with a wave plate in polarization encoding, and uses a Mach-Zehnder interferometer for dual-rail and nonlinear processes for frequency encoding. (b) The conversion between the basis that can be used for the easier physical implementation of quantum operations. Fock-state encoding cannot easily be converted. All other encodings can be converted to and from dual-rail encoding. For this, one element adds a new encoding (switch, PBS, or cavity or grating), and another removes the old encoding (in the white dashed box, a delay, wave plate, or frequency-shifting EOM). The latter can be omitted if the photon is detected afterward, as no indistinguishability is required. (c) The LBB of Bell-state measurement can be constructed using mode mixing and single-photon detection in each of the encodings.

Figure 5

Spin-photon-interface PBBs. (a) State-selective optical transitions as spin-photon interfaces. For the spontaneous-emission spin-photon interface, the excitation pulse is much shorter than the optical lifetime and excites the $|1⟩$ transition: it will emit a single photon by spontaneous emission. The coherent-scattering spin-photon interface scatters a weak pulse from the $|1⟩$ optical transition, resulting in a photon with a weak coherent state. The Raman-scattering spin-photon interface is excited with a pulse but as the optical excitation ends up in a different spin state, only a single photon is emitted. In the conditional-phase-reflection spin-photon interface, a photon is reflected from the cavity with a coupled spin. For $|1⟩$, the phase is flipped and therefore entangled with the spin state. An overcoupled cavity is used for this purpose. In the conditional-amplitude-reflection spin-photon interface, the photon is transmitted or reflected depending on the spin state. A critically coupled cavity is used for this case. (b) A schematic example of how a spin-photon-interface LBB can be implemented in the different photonic encodings, starting from a spin-photon PBB. Below are examples of experimental realizations of the emission [22], projection [35], and gate [32] LBBs in different encodings.

Figure 6

The quantum modeling layer: spin-photon interface. (a) The inefficiencies and noises of the optical transition in the spin-photon interface. Left: zero-phonon-line (ZPL) decay (${\gamma }_r$), phonon-sideband (PSB) decay, and nonradiative decay combined give a total decay rate $\gamma$, resulting in the natural broadening (Lorentzian). The ratios of the transitions are expressed with an internal quantum efficiency (${\eta }_{\mathrm{QE}}$) and Debye-Waller factor (${\eta }_{\mathrm{DW}}$). Right: additional slow noise adds Gaussian spectral diffusion with a standard deviation of ${\sigma }_{\omega }$, while the fast noise gives Lorentzian pure dephasing of line width ${\gamma }^{\ast }$. (b) Left: the cavity-QED system parameters. $g$ is the vacuum Rabi frequency, $\gamma$ is the spontaneous-emission rate, and ${\kappa }_r,{\kappa }_t,$, and ${\kappa }_l$ are the cavity decay rates to the reflection, transmission, and loss ports, respectively. Right: the level diagram of the laser, cavity, and spin. We assume that the $|1⟩$ state of spins is desirably interacting with photons; interaction with the $|0⟩$ state is negligible when $\gamma \ll {\delta }_{01},g$.

Figure 7

A comparison of different protocols for entanglement generation using the $\mathrm{Si}$$V$ center in diamond coupled to an optical cavity as a spin-photon interface. (b) The circuit diagram and LBB descriptions of the protocols under investigation. (c) The photonic encodings for each protocol. (d) The protocol after the LBBs are compiled to hardware-aware PBBs and including imperfections such as loss. The photon states are labeled by node of interaction ($A$ and $B$ for Alice and Bob), by time-bin encoding ($E$ and $L$ for the early and late modes), and by whether the photon is resulting from incoherent emission ($\text{Incoh.}$). The initial photon state is defined as $|+⟩=1/\sqrt{2}(|E⟩+|L⟩)$. (e) Simulated success probability-infidelity curves for the different protocols, using the parameters of Table 3. Protocol C is also simulated for a weak-coherent-state (WCS) input instead of single photons. (f)–(h) The performance of protocol C when the cooperativity is changed by improving either the cavity decay rate $\kappa$ or the emitter-cavity coupling $g$, following Eq. (2). (f) The fidelity for two different success probabilities $\eta$ when varying $\kappa$ or $g$ (left and right graphs, respectively). (g),(h) The success-probability–infidelity curves for some of the cooperativities. The dashed lines mark the values of $\eta$ shown in (f).

Figure 8

The construction of loss, transmission, and reflection with beam-splitters and phase shifters.