Figure 1

Principle of quantum repeaters. In order to distribute entanglement over long distances, say between locations $A$ and $Z$, one proceeds step by step: (a) Entanglement is first created independently within short elementary links, say between the locations $A$ and $B$, $C$ and $D$, …, $W$ and $X$, $Y$ and $Z$. (b) Entanglement is then swapped between neighboring links such that the locations $A$ and $D$, …, $W$ and $Z$ share entanglement. (c) Entanglement swapping operations are performed successively in a hierarchical fashion until entanglement is distributed over the desired distance separating the locations $A$ and $Z$. Squares represent quantum memories. The dotted arrows connecting two remote memories indicate that they are entangled.Reuse & Permissions

Figure 2

Basic level scheme for the creation of collective atomic excitations in atomic ensembles via spontaneous Raman emission (write process) and for their readout (read process), as proposed in the DLCZ protocol. Write process: All atoms start out in $g_1$. A laser pulse off-resonantly excites the $g_1\mathrm{\text{−}}e$ transition, making it possible for a photon to be emitted on the $e\mathrm{\text{−}}{g}_2$ transition (with small probability). Read process: A resonant laser is applied on the $g_2\mathrm{\text{−}}e$ transition, promoting the single atomic excitation from $g_2$ back to $e$, followed by collective emission on the $e\mathrm{\text{−}}{g}_1$ transition of a Stokes photon in a well-defined direction.Reuse & Permissions

Figure 3

Entanglement creation between two remote ensembles located at $A$ and $B$ within the DLCZ protocol. The circles embedded in squares represent DLCZ-type atomic ensembles that probabilistically emit Stokes photons (dots). These photons are sent through long optical fibers (dotted line) to a central station. The detection of a single Stokes photon at the central station in mode $d$ or $\tilde{d}$, which could have come from either location $A$ or $B$, heralds the storage of a single excitation ($s_a$ or $s_b$) in one of the two ensembles. Half circles represent photon detectors. The vertical bar represents a beam splitter (BS).Reuse & Permissions

Figure 4

Entanglement connection between two links $A$-$B$ and $C$-$D$. The ensemble $A$ ($C$) is initially entangled with $B$ ($D$) as described by $|{\psi }_{ab}⟩$ ($|{\psi }_{cd}⟩$). The memories $B$ and $C$ are read out, and the resulting anti-Stokes photons are combined on a beam splitter. The detection of a single photon after the beam splitter, which could have come from either location $B$ or $C$, heralds the storage of a single excitation ($s_a$ or $s_d$) in the ensembles $A$ and $D$ and projects them into an entangled state $|{\psi }_{ad}⟩$.Reuse & Permissions

Figure 5

Postselection of two-photon entanglement. Entanglement has been distributed independently within two chains (labeled by the subscript 1 or 2) such that the ensembles $A_1$-$Z_1$ and $A_2$-$Z_2$ share an entanglement. The atomic excitations at the same location are read out and the emitted anti-Stokes photons are combined into a beam splitter and then counted. Measurements in arbitrary basis can be done by changing the beam-splitter transmission coefficients and phases.Reuse & Permissions

Figure 6

First-level entanglement swapping based on a two-photon detection. Long-distance entanglement between the ensembles located at $A$ and $B$ ($C$ and $D$) is created via single-Stokes-photon detections following the DLCZ protocol; see Fig. 3. The subscript $h$ or $v$ refers to horizontal or vertical polarization. The spin-wave excitation stored in atomic ensembles $B_h$,$B_v$,$C_h$,$C_v$ are read out, and the corresponding anti-Stokes photon in modes $b_h^{\prime }$,$b_v^{\prime }$,$c_h^{\prime }$,$c_v^{\prime }$ are combined at a central station using the setup shown. Vertical bars within squares label polarizing beam spitters (PBSs) that transmit (reflect) $h$- ($v$-) polarized photons. The central PBS with a circle performs the same action in the $\pm 45°$ basis. The coincident detection between modes $d_{+}$ and ${\tilde{d}}_{+}$ heralds the storage of two excitations ($s_{ah}$-$s_{dh}$ or $s_{av}$-$s_{dv}$), either in the ensembles $A_h$ and $D_h$ or in the ensembles $A_v$ and $D_v$; cf. Eq. (14).Reuse & Permissions

Figure 7

Higher-level entanglement swapping based on a two-photon detection. The ensembles located at $A$ and $D$ ($E$ and $H$) are entangled based on the principle shown in Fig. 6 and are described by the state $|{\Psi }_{ad}⟩$ ($|{\Psi }_{eh}⟩$) [see Eq. (14)]. The spin waves stored in ensembles $D_h$, $D_v$, $E_h$, and $E_v$ are converted back into anti-Stokes photons, which are combined using the set of linear optics shown. A twofold coincident detection between $d_h^{\prime }+e_v^{\prime }$ and $e_h^{\prime }+d_v^{\prime }$ nondestructively projects the ensembles $A$ and $H$ into the state $|{\Psi }_{ah}⟩$.Reuse & Permissions

Figure 8

Setup for entanglement creation based on two-photon detection as proposed by 34, Sec. III.B. The ensembles $A_h$ and $B_h$, as well as $A_v$ and $B_v$, where all four ensembles are located at the same node $AB$, are entangled two by two as in the DLCZ protocol; see Fig. 3. The ensembles $A_h$-$B_h$ ($A_v$-$B_v$) store a single delocalized photon with horizontal (vertical) polarization. In a similar way, the ensembles $C_h$ and $D_h$ ($C_v$ and $D_v$), located at a different node $CD$, have been entangled independently. The excitations stored in the ensembles $B_h$,$B_v$,$C_h$,$C_v$ are read out, and the resulting photonic modes are combined at a central station using the setup shown. Ideally, the coincident detection of two photons in $d_{+}$ and ${\tilde{d}}_{+}$ projects nondestructively the atomic cells $A$-$D$ into the entangled state $|{\Psi }_{\mathrm{ad}}⟩$ of Eq. (14).Reuse & Permissions

Figure 9

Separation between entanglement creation and storage using photon-pair sources and absorptive quantum memories. Circles represent sources emitting photon pairs in modes $a$-$a^{\prime }$ for location $A$ and in modes $b$-$b^{\prime }$ for location $B$. The prime modes $a^{\prime }$ and $b^{\prime }$ are stored in neighboring quantum memories (squares), whereas the modes $a$ and $b$ are combined on a beam splitter (vertical bar) at a central station such that the detection of a single photon in one of the output modes heralds the entanglement between the quantum memories in $A$ and $B$.Reuse & Permissions

Figure 10

Entanglement creation with temporal multimode memories. The source can be triggered a large number of times in every communication time interval $\frac{L_0}{c}$. One mode from each pair is sent to the central station, and the other one is stored in the multimode memory.Reuse & Permissions

Figure 11

Entanglement swapping with temporal multimode memories. One has to be capable of recombining at the beam splitter exactly those modes whose partners have participated in a successful entanglement generation at the lower level.Reuse & Permissions

Figure 12

Reduced memory time requirements due to spatial multiplexing. The horizontal axis corresponds to the memory storage time $\tau$, which is measured in units of $L_0/c$. The vertical axis shows the ratio of the repeater rate for storage time $\tau$, labeled $f_{\tau }$, over the ideal rate for infinite storage time, labeled $f_{\infty }$. $n$ is the number of multiplexed memories per site. Memory time requirements for strictly parallel operation always correspond to $n=1$. From 46.Reuse & Permissions

Figure 13

Entanglement creation between two remote ensembles located at $A$ and $B$ based on single-photon sources. Circles represent single-photon sources. A single photon is generated at each location and sent through an asymmetric beam splitter with small transmission and high reflectivity, leading to a superposition of modes $a$ and $a^{\prime }$ ($b$ and $b^{\prime }$) at location $A$ ($B$). The modes $a^{\prime }$ and $b^{\prime }$ are stored in local memories, whereas $a$ and $b$ are sent to a central station where they are combined on a $50:50$ beam splitter. The detection of a single photon at the central station heralds the storage of the second one within the memories with high probability, due to the asymmetry of the local beam splitters. This creates the entanglement of the two remote memories that share a single excitation.Reuse & Permissions

Figure 14

Partial readout of a single collective excitation. A single-photon source whose output is partially stored in a memory, as required in the protocol of 179 described in Sec. 3e, can be emulated with a DLCZ-type atomic ensemble, in which an atomic excitation is first created, accompanied by a Stokes-photon emission, and then partially read out through the application of a read pulse with an area that is smaller than $\pi$. The same principle is also used in the protocol of 180 described in Sec. 3f.Reuse & Permissions

Figure 15

Setup for generating high-fidelity entangled pairs of atomic excitations. Squares containing a circle represent atomic ensembles that probabilistically emit Stokes photons (dots). The conditional detection of a single Stokes photon heralds the storage of one atomic spin-wave excitation. In this way an atomic excitation is created and stored independently in each ensemble. Then all four ensembles are simultaneously read out partially, creating a probability amplitude to emit an anti-Stokes photon (dots). The coincident detection of two anti-Stokes photons in $d_{+}$ and ${\tilde{d}}_{+}$ projects nondestructively the atomic cells into the entangled state $|{\Psi }_{ab}⟩$ of Eq. (24).Reuse & Permissions

Figure 16

Entanglement creation based on two-photon detections using two locally prepared entangled pairs. The excitations stored in the ensembles $B_h$,$B_v$,$C_h$,$C_v$ are read out, and the resulting photonic modes are combined at a central station using the setup shown. Ideally, the coincident detection of two photons in $d_{+}$ and ${\tilde{d}}_{+}$ projects nondestructively the atomic cells $A$-$D$ into the entangled state $|{\Psi }_{ad}⟩$.Reuse & Permissions

Figure 17

Entanglement swapping based on two-photon detections. The spin waves stored in ensembles $D_h$,$D_v$,$E_h$, and $E_v$ are converted back into anti-Stokes photons, which are combined using the set of linear optics shown. A twofold coincident detection between $d_h^{\prime }+e_v^{\prime }$ and $e_h^{\prime }+d_v^{\prime }$ nondestructively projects the ensembles $A$ and $H$ into the state $|{\Psi }_{ah}⟩$.Reuse & Permissions

Figure 18

Comparison of quantum repeater protocols based on atomic ensembles and linear optics. The quantity shown is the average time needed to distribute a single entangled pair with a final fidelity $F=0.9$ for the given distance. We assume losses of $0.2\mathrm{dB}/\mathrm{km}$, corresponding to telecom fibers at a wavelength of $1.5\mu \mathrm{m}$. $A$: as a reference, the time required using direct transmission of photons through optical fibers with a single-photon generation rate of 10 GHz. $B$: original DLCZ protocol that uses single-photon detections for both entanglement generation and swapping (60). $C$: protocol of 106, which uses entanglement swapping based on two-photon detections. $D$: the protocol of Sec. III.B of 34, which first creates single-photon entanglement locally using single-photon detections, and then generates long-distance entanglement using two-photon detections. $E$: protocol of 190 that uses photon-pair sources (which can be realized with ensembles) and multimode memories to implement a temporally multiplexed version of the DLCZ protocol. The corresponding entanglement distribution time is that of the DLCZ protocol divided by the number of temporal modes $N$ that can be stored. Based on the analysis of 1, we have assumed a memory that can store $N=100$ modes. $F$: protocol of 179 that uses quasi-ideal single-photon sources (which can be implemented with atomic ensembles) plus single-photon detections for generation and swapping. $G$: protocol of 180, which creates high-fidelity entangled pairs locally and uses two-photon detections for entanglement generation and swapping, thus following the approach of Sec. III.C of 34, but using an improved method of generating the local entanglement. We have assumed a basic source repetition rate of 10 MHz, which is a limiting factor for this protocol. For all the curves we have assumed memory and detector efficiencies of 90%. We imposed a maximum number of 16 links, which is larger than or equal to the optimal link number for all protocols apart from curve $D$, for which the effect is also less than a factor of 2.Reuse & Permissions

Figure 19

Robustness of various protocols with respect to nonunit memory efficiency. The quantity exhibited is the average time for the distribution of an entangled pair for a distance $L=600\mathrm{km}$ as a function of the memory efficiency. The letters refer to the same protocols as in Fig. 18.Reuse & Permissions

Figure 20

Robustness of various protocols with respect to imperfect photon detector efficiency. The quantity shown is the average time for the distribution of an entangled pair for a distance $L=600\mathrm{km}$ as a function of the photon detector efficiency. The letters refer to the same protocols as in Fig. 18.Reuse & Permissions

Figure 21

Typical experimental configuration used to measure the single-photon character of the conditional anti-Stokes fields. A detection of the Stokes field in detector $D_s$ is used as a trigger, and the anti-Stokes field is split at a beam splitter and detected by detectors $D_1$ and $D_2$.Reuse & Permissions

Figure 22

Retrieval efficiency vs read optical depth $d$ (denoted here as $N\eta$) at a write-read delay of 120 ns with the atomic ensemble inserted into an optical cavity. The dashed line shows the predicted retrieval efficiency for a three-level system; the solid line is the prediction from a model including dephasing from additional excited states. From 195.Reuse & Permissions

Figure 23

(a)–(c) The cross-correlation $g_{A,AS}$ vs the storage time $\delta t$ for different angles $\theta$ between the write beam and the Stokes field. By reducing the angle, the lifetime is increased from 25 to $283\mu \mathrm{s}$, which implies that the decoherence is mainly caused by the dephasing induced by atomic random motion. (d) The measured lifetime ${\tau }_D$ as a function of detection angle $\theta$, where the horizontal error bars indicate measurement errors in the angles. The solid line is the theoretical curve with $T=100\mu \mathrm{K}$. The experimental results are in good agreement with the theoretical predictions. From 231.Reuse & Permissions

Figure 24

(a) Experiment demonstrating a DLCZ quantum memory with an atomic ensemble loaded into a one-dimensional optical lattice. (a) Schematic of the experiment. Between ${10}^5$ and ${10}^6$ sub-Doppler-cooled ${}^{87}\mathrm{Rb}$ atoms are loaded into an optical lattice, and detection of the signal field, generated by Raman scattering of the write laser pulse (red detuned by 20 MHz), heralds the presence of a write spin-wave excitation. A resonant read-control field converts the surviving atomic excitation into an idler field after a storage period $T$s. The inset shows the atomic level scheme of ${}^{87}\mathrm{Rb}$ with levels $a$ and $b$ being the hyperfine components of the ground ${}^{5}S_{1/2}$ level, and level $c$ being a hyperfine component of the excited ${}^{5}P_{1/2}$ level. (b) Retrieval efficiency (including detection) as a function of storage time for atoms optically pumped in clock states in the optical lattice for two different lattice depths $U_0$ (diamonds, $U_0$ $80\mu \mathrm{K}$; circles $U_0$ $40\mu \mathrm{K}$). The intrinsic retrieval efficiency at the output of the ensemble is roughly 4 times larger. From 233.Reuse & Permissions

Figure 25

An overview of the experiment to entangle two atomic ensembles . (a) Setup for generating entanglement between two pencil-shaped ensembles $L$ and $R$ located within spherical clouds of cold cesium atoms. The atomic level structure for the writing process consists of the initial ground state $|g⟩$ (${}^{6}S_{1/2}$, $F=4$ level of atomic cesium), the ground state $|s⟩$ for storing a collective spin flip (${}^{6}S_{1/2}$, $F=3$ level), and the excited level $|e⟩$ (${}^{6}P_{3/2}$, $F=4$). The transition $|g⟩\to |e⟩$ in each ensemble is initially coupled by a write pulse detuned from resonance to generate the forward-scattered anti-Stokes field 1 from the transition $|e⟩\to |s⟩$. The $L$ and $R$ ensembles are excited by synchronized writing pulses obtained from beam splitter ${\mathrm{BS}}_w$. After filtering, the anti-Stokes fields $1L$ and $1R$ are collected, coupled to fiber-optic channels, and interfere at beam splitter ${\mathrm{BS}}_1$, with outputs directed toward two single-photon detectors $D_{1a}$ and $D_{1b}$. (b) Schematic for verification of entanglement between the $L$ and $R$ ensembles by conversion of atomic to field excitation by way of simultaneous read pulses obtained from ${\mathrm{BS}}_r$. The read pulses reach the samples after a programmable delay from the write pulses and couple the transition $|s⟩\to |e⟩$ ($|e⟩$ being the ${}^{6}P_{1/2}$, $F=4$ level), leading to the emission of the forward-scattered Stokes fields $2_L$ and $2_R$ from the transition $|e⟩\to |g⟩$. The upper inset shows the configuration used to measure the diagonal elements of the density matrix from the photodetection events at $D_{2a}$, $D_{2b}$, and $D_{2c}$. The off-diagonal elements are measured by interfering the fields $2_L$ and $2_R$ at the beam splitter ${\mathrm{BS}}_2$ (see lower inset). The 12 m arms of both write and read interferometers are actively stabilized using an auxiliary Nd:YAG laser at 1.06 mm. From 38.Reuse & Permissions

Figure 26

Signature of the coherent superposition of a single excitation delocalized between two atomic ensembles located 3 m away. After the detection heralding entanglement, the stored excitations are converted into anti-Stokes photons and combined at a beam splitter, forming a Mach-Zehnder interferometer. (a) The detection of the anti-Stokes fields after the beam splitter as a function of the phase of the interferometer conditioned on the detection of a Stokes photon (heralding event), when the Stokes fields are combined with the same polarization. (b) The same measurement when the Stokes fields are combined with orthogonal polarizations. This highlights the importance of the indistinguishability of the Stokes photons in order to generate an entangled state. From 38.Reuse & Permissions

Figure 27

Heralded entanglement between two atomic ensembles. Concurrence $C$ (without correcting for propagation and detection losses) as a function of the normalized cross-correlation function $g_{S,AS}$ (denoted here $g_{12}$), for the two possible heralding events (detection at $D_{1a}$ or $D_{1b}$ after the beam splitter). Inset: Average visibility of the interference fringe between the two field-2 modes. From 122.Reuse & Permissions

Figure 28

Setup for the elementary link of a DLCZ quantum repeater between two quantum nodes ($L$, $R$) separated by 3 m. The inset at the bottom left shows the relevant atomic levels for the ${}^{6}S_{1/2}\to {}^{6}P_{3/2}$ transition in atomic cesium, as well as the associated light fields. With this setup, a photodetection event at either detector $D_{1a}$ or $D_{1b}$ indicates entanglement between the collective excitation in $LU$ and $RU$, and a photodetection event at either detector $D_{1c}$ or $D_{1d}$ indicates entanglement between the collective excitation in $LD$ and $RD$ (20). Two orthogonal polarizations in one fiber beam-splitter implement ${\mathrm{BS}}_U$ and ${\mathrm{BS}}_D$, yielding excellent relative path stability. A heralding detection event triggers the control logic to gate off the light pulses going to the corresponding ensemble pair ($U$ or $D$) by controlling the intensity modulators (I.M.). The atomic state is thus stored while waiting for the second ensemble pair to be prepared. After both pairs of ensembles $U$ and $D$ are entangled, the control logic releases strong read pulses to map the states of the atoms to photons that are combined with orthogonal polarizations on the polarizing beam splitters ${\mathrm{PBS}}_L$ and ${\mathrm{PBS}}_R$. If only coincidences between the fields at both nodes are registered, the state is effectively equivalent to a polarization maximally entangled state. From 39.Reuse & Permissions

Figure 29

The scheme of the experimental setup to generate entanglement between a photon and a stored atomic excitation using the two-modes approach. A weak horizontal polarized write pulse illuminates the cold Rb atom cloud. The spontaneous Raman scattered anti-Stokes fields $A{S}_L$ and $A{S}_R$ with vertical polarization are collected at $\pm 3°$ to the propagating direction of the write beam, defining the spatial mode of the atomic ensembles $L$ and $R$, respectively. (Note that in this experiment the storage state has a lower energy than the initial state. Hence, the field generated by the write beam is called the anti-Stokes field and the collective field generated by the read beam is called the Stokes field, contrary to the convention used in this review.) The $A{S}_R$ mode is rotated to be horizontally polarized, combined with $A{S}_L$ mode on a polarizing beam splitter PBS1, and sent to the polarization analyzer. This creates the entanglement between the polarization of the anti-Stokes field and the spatial modes of spin excitation in the atomic ensemble. From 32.Reuse & Permissions

Figure 30

Source of nondegenerate photon pairs using atomic cascade transitions. (a) The atomic structure for the proposed cascade emission scheme involving excitation by pumps I and II. Pump II and the signal photons lie in the telecommunication wavelength range when a suitable level of orbital angular momentum $L=0$ or $L=2$ is used as level $|d⟩$. (b) Schematic of experimental setup based on ultracold ${}^{85}\mathrm{Rb}$ atomic gas. From 28.Reuse & Permissions

Figure 31

Suppression of the two-photon component of the anti-Stokes field conditioned on the detection events in the Stokes field, measured by the autocorrelation parameter $\alpha$ (denoted here $w$) as a function of $g_{S,AS}$ (denoted here $g_{12}$). The lowest value obtained is $0.007\pm 0.003$. From 124.Reuse & Permissions

Figure 32

Storage and retrieval of single-photon fields using EIT in a cold Cs ensemble. The points around $\tau =0\mu \mathrm{s}$ represent “leakage” of the signal field due to the finite optical depth and length of the ensemble. The points beyond $\tau =1\mu \mathrm{s}$ show the retrieved signal field. The overall storage and retrieval efficiency is $(17\pm 1)\%$. The solid line is the estimated Rabi frequency of the control pulse. From 37.Reuse & Permissions

Figure 33

The principles of the atomic frequency comb (AFC) quantum memory. (a) An inhomogeneously broadened optical transition $|g⟩\mathrm{\text{−}}|e⟩$ is shaped into an AFC by frequency-selective optical pumping to the $|aux⟩$ level. The peaks in the AFC have width $\gamma$ (FWHM) and are separated by $\Delta$, where we define the comb finesse as $F=\Delta /\gamma$. (b) The input mode is completely absorbed and coherently excites the AFC modes, which will dephase and then rephase after a time $2\pi /\Delta$, resulting in a photon-echo–type coherent emission. A pair of control fields on $|e⟩\mathrm{\text{−}}|s⟩$ allow for long-time storage as a collective spin wave in $|s⟩$, and on-demand readout after a storage time $T_s$. From 1.Reuse & Permissions

Figure 34

Phase preservation during the storage of a time-bin qubit by an atomic frequency comb. Time-bin qubits with different phases $\Phi$ are stored and analyzed using the light-matter interface. The analysis is performed by projecting the time-bin qubit on a fixed superposition basis, which here is achieved by two partial readouts (see text for details). The inset shows the histogram of arrival times, where there is constructive interference for $\Phi =2\pi$ and destructive interference for $\Phi =\pi$ in the middle time bin. For this particular interference fringe, a raw visibility of 82% and a visibility of 95% when subtracting detector dark counts are obtained. From 50.Reuse & Permissions

Figure 35

The coefficients $A_n$ and $B_n$ as a function of $n$. One sees the scaling with $N^2=2^{2n}$ and $N^4=2^{4n}$, respectively.Reuse & Permissions