Cavity-based quantum networks with single atoms and optical photons

Distributed quantum networks will allow users to perform tasks and to interact in ways which are not possible with present-day technology. Their implementation is a key challenge for quantum science and requires the development of stationary quantum nodes that can send and receive as well as store and process quantum information locally. The nodes are connected by quantum channels for flying information carriers, i.e., photons. These channels serve both to directly exchange quantum information between nodes and to distribute entanglement over the whole network. In order to scale such networks to many particles and long distances, an efficient interface between the nodes and the channels is required. This article describes the cavity-based approach to this goal, with an emphasis on experimental systems in which single atoms are trapped in and coupled to optical resonators. Besides being conceptually appealing, this approach is promising for quantum networks on larger scales, as it gives access to long qubit coherence times and high light-matter coupling efficiencies. Thus, it allows one to generate entangled photons on the push of a button, to reversibly map the quantum state of a photon onto an atom, to transfer and teleport quantum states between remote atoms, to entangle distant atoms, to detect optical photons nondestructively, to perform entangling quantum gates between an atom and one or several photons, and even provides a route toward efficient heralded quantum memories for future repeaters. The presented general protocols and the identification of key parameters are applicable to other experimental systems.

Figure 1

Basic physical situation considered in CQED. A three-level emitter, e.g., a single atom, is located in an optical cavity, schematically represented by two facing mirrors. The atom exchanges energy with the optical field at a rate $2g$. The decay rate of photons out of the resonator is $2\kappa =2{\kappa }_{\text{out}}+2{\kappa }_{\text{loss}}$. The atomic polarization decay rate is $\gamma$. (b) Atomic energy level scheme. The transition frequency between the coupled ground state $|c⟩$ and the excited state $|e⟩$ is ${\omega }_a$. The cavity resonance frequency is ${\omega }_c$. Atom and cavity are detuned by ${\mathrm{\Delta }}_{ac}={\omega }_a-{\omega }_c$. The third, uncoupled level $|u⟩$ is detuned from $|c⟩$ by ${\mathrm{\Delta }}_u$.

Figure 2

The Jaynes-Cummings ladder of the dressed energy eigenstates $|N,\pm ⟩$ of a strongly coupled atom-cavity system, where the cavity frequency is kept fixed and the emitter frequency is scanned over the resonance. Compared to the uncoupled situation (gray), the coupled states (black) exhibit pronounced avoided level crossings. At zero detuning, the levels are separated by $2g\sqrt{N}$.

Figure 3

(a) Vacuum-Rabi oscillations in the time domain. A short laser pulse excites a single atom to the excited state $|e⟩$, from which it decays back to the ground state while emitting a single photon. In the cavity output mode, damped vacuum-Rabi oscillations are observed. With increasing detuning $|{\mathrm{\Delta }}_{ac}|$ (front to back), the vacuum-Rabi frequency increases. The solid curves are fits to the data using a theoretical model. Adapted from [25]. (b) Vacuum-Rabi splitting in the spectral domain, recorded with exactly one trapped atom. The detuning of the atom with respect to the cavity is varied by changing the power of the intracavity dipole-trap laser from 250 nW (top) to 340 nW (bottom). The data points (gray) clearly show a vacuum-Rabi splitting and are in good agreement with Monte Carlo simulations of the system (black lines). Adapted from [154].

Figure 4

Purcell-enhanced atomic emission. On resonance (upper fit curve), the presence of a photonic-crystal cavity with $\kappa =2\pi \times 12.5\mathrm{GHz}$ guides the fluorescence of a single atom into the cavity mode and shortens the excited state lifetime from 26 to 3 ns. Both effects are less pronounced at a detuning of $-41\mathrm{GHz}$ (lower lying fit curve). The inset shows the dependence of the decay rate on the atom-cavity detuning. Adapted from [241].

Figure 5

Visualization of the field annihilation operators describing the input and output of the cavity field.

Figure 6

(a) Transmission and (b) reflection spectra of a single-sided, overcoupled cavity in the strong-coupling regime. Without the presence of an atom (black circles and black fit curves), the cavity exhibits a Lorentzian transmission resonance, which is also observed as a drop in reflection. The sum of transmission and reflection is less than 100% due to mirror absorption and scattering. With an atom trapped in the mode (gray squares and gray theory curves), the vacuum-Rabi peaks are clearly resolved. The discrepancy of the data points from the theory curve is explained by inhomogeneous broadening of the atomic transition frequency. (c), (d) Phase response in the coupled (gray) and uncoupled (black) cased in (c) transmission and (d) reflection. On resonance, a phase shift of $\pi$ is observed in reflection for the case of an empty cavity.

Figure 7

Electromagnetically induced transparency, observed with about 15 atoms trapped in a cavity. When the cavity is empty or the atoms are prepared in the uncoupled state (black), a Lorentzian transmission peak is obtained. When the atoms are prepared in the coupled state, the vacuum-Rabi splitting is observed (dashed theory curve). When an additional control laser is switched on (data with error bars and theory curve), the transmission peaks are shifted outward, and the spectrum exhibits a narrow transparency window, which indicates the presence of a coherent dark state. Adapted from [165].

Figure 8

Optical resonator designs. (a) Top: Schematic of a Fabry-Perot cavity. The resonator is formed by two facing mirrors with high reflectivity. A small residual mirror transmission allows for incoupling and outcoupling of light. Bottom: Photograph of a fiber-based Fabry-Perot resonator. The mirrors are the coated end facets of two optical fibers, which allows for improved optical access. The fibers are surrounded by a copper sleeve to facilitate ion trapping in the $\sim 200\mu \mathrm{m}$ small gap between the mirrors. Adapted from [32]. (b) Microtoroid (left) and bottle (right) resonators. Atoms are coupled to the evanescent field of a whispering-gallery mode that circulates in the glass structure. The mode diameter is typically on the order of $100\mu \mathrm{m}$. The resonator is coupled to a nanofiber with a rate that can be adjusted via the fiber distance. Bottom: Scanning electron microscope image of a microtoroid with a diameter of $120\mu \mathrm{m}$. Adapted from [249] and [108]. (c) Photonic-crystal cavity. Top: Sketch of the experiment. Atoms are trapped in a standing-wave optical tweezer $\sim 0.2\mu \mathrm{m}$ from the surface of the resonator. The electric field of the resonator mode is confined to a volume of less than ${\lambda }^3$ using a periodic modulation of the refractive index. The cavity is coupled to a propagating mode via a fiber taper visible at the left side. Bottom: Scanning electron microscope image. Adapted from [241].

Figure 9

Observation of single atoms launched from an atomic fountain on their pass through a Fabry-Perot cavity. (a) Sketch of the used experimental setup. (b) Cavity transmission signal. When an atom arrives in the mode, the transmission of a weak probe laser drops. The depth and duration of the dip are determined by the cavity parameters and the specific atomic trajectory. Adapted from [169].

Figure 10

(a)–(d) Measurement of the spatial structure of the electromagnetic field within a high-finesse Fabry-Perot cavity. An ion is trapped in a radio-frequency potential. When the trap is shifted, the rate of photons scattered into free space out of a laser beam that drives the resonator mode reveals the spatial structure of the cavity field. The solid curve is a fit that includes saturation of the atomic transition. By changing the incoupling, different resonator modes can be mapped (a)–(d). The insets show the calculated mode structures, and arrows indicate the scan paths. Adapted from [85]. (e) State-insensitive trapping of single atoms in a Fabry-Perot cavity in the strong-coupling regime. The depth of transmission suppression of a faint resonant probe laser allows for continuous monitoring of the number of atoms $N$ trapped in the resonator (when $N<3$). In both depicted traces, the initial drop is caused by several atoms that are cooled into the cavity mode. Subsequently, the transmission increases in steps whenever one atom is lost from the trap. Adapted from [158].

Figure 11

Full control over the atomic motion. Atoms are trapped in a three-dimensional optical lattice with different trap frequencies along the three orthogonal axes. When a Raman sideband spectrum is recorded after intracavity Sisyphus cooling, three peaks are resolved on either side of the carrier (dashed Lorentzian fit), at the positions of the three red (left side) and the three blue (right side) sidebands. After Raman sideband cooling, the atom is prepared in the three-dimensional ground state of the trapping potential and the red sideband peaks vanish. From [204].

Figure 12

Atomic-state detection (a) Fluorescence state detection. The cavity output photons are measured when the atom is driven on a closed transition with a laser beam applied from the side of the cavity. By setting a suitable threshold (dashed line), one can unambiguously discriminate the fluorescent “bright” state, where several photons are detected within a certain time interval, from the detuned “dark” state, where no photons are detected. Adapted from [24]. (b) Transmission state detection. A weak resonant laser drives the cavity along its axis. When the atom is in a detuned state $|u⟩$, it does not alter the transmission of the resonator. Within a certain time interval, one observes a Poissonian distribution of detected photons. If the atom is in the coupled state ($|c⟩$), the transmission is strongly suppressed. The two obtained distributions are clearly separated, which allows for unambiguous discrimination of the two states. Adapted from [27]. (c) Continuous detection of and feedback onto the combined state of two atoms trapped in the same resonator. Top: Experimentally obtained cavity transmission signal. The vertical lines indicate the instants when a feedback pulse is applied to transfer one of the atoms to the coupled or uncoupled ground state. Bottom: Number of atoms $\alpha$ in the coupled state according to the Bayesian state estimation. Because of the feedback, the probability for $\alpha =1$ is maximized. Right: Histograms of the resonator transmission with exactly (0,1,2) atoms prepared in the coupled state. When two atoms are loaded into the resonator and the feedback is applied, the histogram matches that observed with exactly one atom in the coupled state. Adapted from [31].

Figure 13

(a) Bloch-sphere representation of a two-level system. Any pure atomic superposition state $\alpha |u⟩+\beta \exp (i\phi )|c⟩$ is represented by a point on the surface of a sphere. The latitude represents the ratio of $\alpha$ and $\beta$, while the longitude gives $\phi$. Mixed quantum states can be represented by points within the sphere. (b) Rabi oscillations demonstrating controlled atomic-state rotations. The atom is prepared in the state $|c⟩$. Subsequently, a pair of Raman lasers drives transitions to the state $|u⟩$. The atomic population in $|c⟩$ is measured by fluorescence state detection. The visibility of the resulting Rabi oscillations (here 98%) is a good measure for the combined quality of the state preparation, rotation, and readout process. The observed decay of the oscillation can be caused by fluctuating laser beam intensities and fluctuating ground-state energies, e.g., due to changes in the magnetic field. Adapted from [202].

Figure 14

Cavity-based single-photon sources. Using a vacuum-stimulated Raman transfer from one of the atomic ground states to the other, single photons are emitted into the cavity mode. With alternating repump pulses, a single-photon stream is generated. (a) Photons are generated with atoms falling through a Fabry-Perot resonator. The cavity output is analyzed in a Hanbury Brown and Twiss setup to measure the correlation function. With respect to the independently measured background caused by detector dark counts (hatched area), the absence of coincidence counts at $\tau =0$ demonstrates the single-photon character of the emitted field. The temporal distance of the peaks is given by the repetition rate. The envelope of the correlation function is determined by the atom transit time. Adapted from [130]. (b) With trapped rather than falling atoms, a continuous stream of single photons can be generated, and the peaks in the correlation function remain at a constant height. Adapted from [157]. (c) Control of the photonic wave packet envelope. The intensity of the control laser (dashed line) that drives the Raman transfer is changed and the arrival time of the generated photons is measured. The data points are in good agreement with a numerical model (solid lines) of the photon generation process. Adapted from [113].

Figure 15

Time-resolved two-photon interference. Two photons are generated one after the other from an atom-cavity system. One of the photons is sent to an optical delay line, such that both arrive at the same time in the two input ports of a beam splitter. When the photons have the same frequencies, no coincidence counts are observed (not shown). When a frequency offset is externally applied to the photons, a quantum beat is observed (open circles), which proves the coherence of the photon generation process. The solid circle data show a reference run with distinguishable photons of orthogonal polarization. Adapted from [226].

Figure 16

(a) Atom-photon state transfer. The atomic qubit is encoded in the states $m_F=+1\equiv |\uparrow ⟩$ and $m_F=-1\equiv |\downarrow ⟩$. The atom is initially in the superposition state $\alpha |\uparrow ⟩+\beta |\downarrow ⟩$. A photon is generated using a $\pi$-polarized control laser (up arrows), which transfers the atom to $m_F=0$, independent of its initial state. The polarization of the emitted photons (curly arrows), however, depends on the initial state, such that $\alpha |\uparrow ⟩+\beta |\downarrow ⟩\to \alpha |\circlearrowleft ⟩+\beta |\circlearrowright ⟩$. (b) Generation of atom-photon entanglement. The atom is prepared in the $m_F=0$ state. Using a $\pi$-polarized control laser (up arrow), a photon can be generated on two different transitions ${\sigma }^{\pm }$ with $\mathrm{\Delta }{m}_F=\mp 1$. Depending on the polarization of the emitted photon (curly arrows), the atom ends up in a different state. Thus, the process generates atom-photon entanglement. (c) Density matrix of an atom-photon entangled state, reconstructed by mapping the atomic state onto a second photon and performing quantum-state tomography on the two-photon polarization state in the horizontal (H)—vertical (V) qubit basis. The resulting state has a fidelity of $\mathcal{F}=86\%$ with the $|{\mathrm{\Psi }}^{-}⟩$ state. Adapted from [263].

Figure 17

An elementary cell of a quantum network that consists of single atoms in optical cavities. An optical fiber (1) connects two independent setups that are separated by 21 m. In both setups, a single atom (2) is trapped in a Fabry-Perot resonator (3). The insets show typical fluorescence images. Quantum states are exchanged between the atoms in the form of a single photon (4). To this end, an atomic coherent dark state is controlled with two external laser fields (5). Increasing the field intensity at node $A$ transfers the atomic state to the state of the photon, while decreasing it performs the reverse operation at node $B$. Adapted from [210].

Figure 18

Single-atom quantum memory. (a) Atomic level scheme for photon storage using the STIRAP technique. The atom is prepared in the ground state with $m_F=0$ (ball). An impinging photon (curly arrows) drives ${\sigma }^{\pm }$ transitions, depending on its polarization. A $\pi$-polarized control laser (down arrows) thus transfers the photonic polarization onto the spin state of the atom. (b) Atomic level scheme for an alternative proposal, termed deterministic single-photon Raman passage, that uses an atom-photon SWAP gate to achieve photon storage in the Purcell regime. When the atom is initially in the state $m_F=-1\equiv |\downarrow ⟩$, a left-circularly polarized photon is not coupled to it and thus reflected with unchanged polarization. In contrast, a right-circularly polarized photon will be reflected with left-circular polarization (curly arrows) due to interference with the atomic emission, and in this process the atom is transferred to the state $m_F=+1\equiv |\uparrow ⟩$. In this way, the photonic polarization is transferred to the atomic state. (c) Quantum memory results, achieved with the STIRAP scheme. The qubit state after storage and readout is reconstructed for six unbiased input states using quantum-state tomography. The result is visualized as colored circles in the Bloch-sphere representation. Via quantum process tomography, the memory performance for any input state can be predicted, as visualized by the sphere. The average fidelity of the depicted measurement is $\overline{\mathcal{F}}=93\%$. From [227].

Figure 19

Remote atom-atom entanglement. In the experiment, atom-photon entanglement is created at node $A$ and the photon is sent via an optical fiber to node $B$, where it is coherently absorbed. The resulting atom-atom state is reconstructed using quantum-state tomography, which results in the depicted density matrix. The fidelity with the $|{\mathrm{\Psi }}^{-}⟩$-Bell state is 85%. From [210].

Figure 20

(a) Photonic Bell-state measurement. Two photons arrive simultaneously in the two input ports of a nonpolarizing beam splitter (NPBS), which is followed by two polarizing beam splitters (PBS). Single-photon counters in the individual output ports allow one to independently detect horizontal ($H$) and vertical ($V$) polarizations. Coincident detection events project the photonic state onto the $|{\mathrm{\Psi }}^{+}⟩$ or the $|{\mathrm{\Psi }}^{-}⟩$ Bell state. (b) Time-resolved two-photon quantum interference from two atoms trapped in independent setups. Photons of parallel (solid bars) or orthogonal (open bars) polarization impinge onto a nonpolarizing beam splitter. For indistinguishable photons, the ratio of parallel and orthogonal coincidence events (circles) is expected to be zero. The observed dip indicates photon indistinguishability with a contrast of 64%. Selecting a shorter coincidence time window allows one to obtain a better interference contrast at the price of a reduced number of events. (c) Teleportation between remote atoms. The sender atom is initialized in six unbiased input states. After the teleportation process, the resulting state of the receiver atom is reconstructed, giving the colored circles in the figure. The sphere is the result of quantum process tomography. The average fidelity in the depicted experiment was 88.0(1.5)%. Adapted from [177].

Figure 21

Entanglement of two ions, heralded by the detection of two photons in the cavity output. The fidelity of the entangled state is bounded by three measurements. First, the sum of population terms ${\rho }_{\downarrow \uparrow ,\downarrow \uparrow }$ and ${\rho }_{\uparrow \downarrow ,\uparrow \downarrow }$, indicated by the horizontal line. Second, the parity after two $\pi /2$ rotations with relative phase $\varphi$ on the $|c⟩\leftrightarrow |u⟩$ transition (circles and dashed sinusoidal fit curve). Third, the parity after one $\pi /2$ pulse with absolute phase $\varphi$ (triangles). From the obtained values, a lower bound to the state fidelity of 92% can be derived. Adapted from [43].

Figure 22

Nondestructive detection of an optical photon. (a)–(d) Ramsey sequences. (a) The atomic state, visualized on the Bloch sphere, is first prepared in the superposition state $(|u⟩+|c⟩)/\sqrt{2}$. (b) When a single photon is reflected from the resonator, the phase of the superposition state is flipped $(|u⟩+|c⟩)/\sqrt{2}\to (|u⟩-|c⟩)/\sqrt{2}$. (c) This state is transferred to $|c⟩$ in a subsequent Raman rotation. (d) State detection in fluorescence thus facilitates detecting whether a photon has been reflected or not. (e) Result of a nondestructive detection process. A photon is destructively detected by an SPCM after reflection from an atom-cavity system (line in the left trigger interval). But before being absorbed, the photon has left its trace in the atomic state. Therefore, after $\pi /2$ rotation, many fluorescence photons are observed in the $25\mu \mathrm{s}$ long readout interval (right), unambiguously signaling the atomic-state change induced by the reflected photon. (f) Arrival-time histogram of the photons detected with the SPCMs after reflection from the setup. The data taken during the nondestructive photon measurement (squares) do not show a significant deviation from the reference curve recorded without an atom (points). Adapted from [205].

Figure 23

(a) Truth table of the atom-photon cnot quantum gate, showing the normalized probability of obtaining a specific output state for a complete set of atom-photon input states. The experimentally obtained values (filled bars) are close to the ideally expected ones (open bars). (b) Atom-photon-photon entangled state, generated via subsequent interactions of two photons with the same atom. The depicted density matrix is obtained by quantum-state tomography. The fidelity with a maximally entangled GHZ state (open bars) is 67%, proving genuine three-particle entanglement. From [203].

Figure 24

A two-mode photon-blockade effect. A faint, linearly polarized probe laser, detuned from the empty cavity resonance by the coupling strength $g$, drives the cavity mode. After passing an orthogonal polarizer, the transmitted light is analyzed in a Hanbury Brown and Twiss setup. The obtained correlation function exhibits clear antibunching. This demonstrates the fact that because of its nonlinear eigenspectrum, photons can pass the atom-cavity system only one at a time. Adapted from [18].