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Functional Quantum Nodes for Entanglement Distribution over Scalable Quantum Networks

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Science  01 Jun 2007:
Vol. 316, Issue 5829, pp. 1316-1320
DOI: 10.1126/science.1140300

Abstract

We demonstrated entanglement distribution between two remote quantum nodes located 3 meters apart. This distribution involves the asynchronous preparation of two pairs of atomic memories and the coherent mapping of stored atomic states into light fields in an effective state of near-maximum polarization entanglement. Entanglement is verified by way of the measured violation of a Bell inequality, and it can be used for communication protocols such as quantum cryptography. The demonstrated quantum nodes and channels can be used as segments of a quantum repeater, providing an essential tool for robust long-distance quantum communication.

In quantum information science (1), distribution of entanglement over quantum networks is a critical requirement for metrology (2), quantum computation (3, 4), and communication (3, 5). Quantum networks are composed of quantum nodes for processing and storing quantum states, and quantum channels that link the nodes. Substantial advances have been made with diverse systems toward the realization of such networks, including ions (6), single trapped atoms in free space (7, 8) and in cavities (9), and atomic ensembles in the regime of continuous variables (10).

An approach of particular importance has been the seminal work of Duan, Lukin, Cirac, and Zoller (DLCZ) for the realization of quantum networks based on entanglement between single photons and collective excitations in atomic ensembles (11). Critical experimental capabilities have been achieved, beginning with the generation of nonclassical fields (12, 13) with controlled waveforms (14) and extending to the creation and retrieval of single collective excitations (1517) with high efficiency (18, 19). Heralded entanglement with quantum memory, which is the cornerstone of networks with efficient scaling, was achieved between two ensembles (20). More recently, conditional control of the quantum states of a single ensemble (2123) and of two distant ensembles (24) has also been implemented; the quantum states are likewise required for the scalability of quantum networks based on probabilistic protocols.

Our goal is to develop the physical resources that enable quantum repeaters (5), thereby allowing entanglement-based quantum communication tasks over quantum networks on distance scales much larger than those set by the attenuation length of optical fibers, including quantum cryptography (25). For this purpose, heralded number-state entanglement (20) between two remote atomic ensembles is not directly applicable. Instead, DLCZ proposed the use of pairs of ensembles (Ui,Di) at each quantum node i, with the sets of ensembles {Ui}, {Di} separately linked in parallel chains across the network (11). Relative to the state of the art in our previous work (20), the DLCZ protocol requires the capability for the independent control of pairs of entangled ensembles between two nodes.

In our experiment, we created, addressed, and controlled pairs of atomic ensembles at each of two quantum nodes, thereby demonstrating entanglement distribution in a form suitable both for quantum network architectures and for entanglement-based quantum communication schemes (26). Specifically, two pairs of remote ensembles at two nodes were each prepared in an entangled state (20), in a heralded and asynchronous fashion (24), thanks to the conditional control of the quantum memories. After a signal indicating that the two chains are prepared in the desired state, the states of the ensembles were coherently transferred to propagating fields locally at the two nodes. The fields were arranged such that they effectively contained two photons, one at each node, whose polarizations were entangled. The entanglement between the two nodes was verified by the violation of a Bell inequality. The effective polarization-entangled state, created with favorable scaling behavior, was thereby compatible with entanglement-based quantum communication protocols (11).

The architecture for our experiment is shown in Fig. 1. Each quantum node, L (left) and R (right), consists of two atomic ensembles, U (up) and D (down), or four ensembles altogether, namely (LU, LD) and (RU, RD), respectively. We first prepared each pair in an entangled state, in which one excitation is shared coherently, by using a pair of coherent weak write pulses to induce spontaneous Raman transitions |g 〉→ |e 〉→ |s 〉 (bottom left, Fig. 1). The Raman fields (1LU, 1RU) from (LU, RU) were combined at the 50-50 beamsplitter BSU, and the resulting fields were directed to single-photon detectors. A photoelectric detection event in either detector indicated that the two ensembles were prepared. The remote pair of D ensembles, (LD, RD), was prepared in an analogous fashion.

Fig. 1.

Setup for distributing entanglement between two quantum nodes (L,R) separated by 3 m. The inset at the bottom left shows the relevant atomic levels for the 6S1/2→ 6P3/2 transition in atomic cesium, as well as the associated light fields. The ensembles are initially prepared in |g 〉. Weak write pulses then induce spontaneous Raman transitions |g 〉 (F = 4)→|e 〉 (F′ = 4)→ |s 〉 (F = 3), resulting in the emission of anti-Stokes fields (Field 1) near the |e 〉→|s 〉 transition along with the storage of collective excitations in the form of spin-flips shared among the atoms (11). With this setup, a photo-detection event at either detector D1a or D1b indicates entanglement between the collective excitation in LU and RU, and a photo-detection event at either detector D1c or D1d indicates entanglement between the collective excitation in LD and RD (20). Two orthogonal polarizations in one fiber beamsplitter implement BSU and BSD, 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,D are entangled, the control logic releases strong read pulses to map the states of the atoms to Stokes Field 2 fields through |s 〉→ |e 〉→ |g 〉. Fields 2LU and 2LD are combined with orthogonal polarizations on the polarizing beamsplitter PBSL to yield field 2L; fields 2RU and 2RD are combined with orthogonal polarizations on the polarizing beam splitter PBSR to yield field 2R. If only coincidences between fields 2L and 2R are registered, the state is effectively equivalent to a polarization maximally entangled state.

Conditioned upon the preparation of both ensemble pairs (LU, LD) and (RU, RD), a set of read pulses was triggered to map the stored atomic excitations into propagating Stokes fields in well-defined spatial modes through |s 〉→ |e 〉→ |g 〉 with the use of a collective enhancement (11) (bottom left, Fig. 1). This generated a set of four fields denoted by (2LU,2RU) for ensembles (LU, RU) and by (2LD, 2RD) for ensembles (LD, RD). In the ideal case and neglecting higher-order terms, this mapping results in a quantum state for the Field 2 fields given by Math(1) Math Math Here, |nx is the n-photon state for mode x, where x ∈ {2LU,2RU,2LD,2RD}, and ηU and ηD are the relative phases resulting from the writing and reading processes for the U and D pair of ensembles, respectively (20). The ± signs for the conditional states U,D result from the unitarity of the transformation by the beamsplitters (BSU, BSD). The extension of Eq. 1 to incorporate various nonidealities is given in the supporting online material (SOM) text.

Apart from an overall phase, the state jψ2LU,2RU,2LD,2RD 〉 can be rewritten as follows: Math(2) Math Math Math Math where |vac2i denotes |0 〉2iU |0 〉2iD. If only coincidences between both nodes L,R are registered, the first two terms (i.e., with eiηD, eiηU) do not contribute. Hence, as noted by DLCZ, excluding such cases leads to an effective density matrix equivalent to the one for a maximally entangled state of the form of the last term in Eq. 2. Notably, the absolute phases ηU and ηD do not need to be independently stabilized. Only the relative phase η = ηU – ηD must be kept constant, leading to 1/2 unit of entanglement for two quantum bits (i.e., 1/2 ebit).

The experimental demonstration of this architecture for implementing the DLCZ protocol relies critically on the ability to carry out efficient parallel preparation of the (LU, RU) and (LD, RD) ensemble pairs, as well as the ability to stabilize the relative phase η. The first requirement is achieved by the use of real-time control, as described in Felinto et al. (24) in a simpler case. As shown in Fig. 1, we implemented control logic that monitors the outputs of Field 1 detectors. A detection event at either pair triggers electro-optic intensity modulators (IM) that gate off all laser pulses traveling toward the corresponding pair of ensembles, thereby storing the associated state. Upon receipt of signals indicating that the two pairs of ensembles, (LU, RU) and (LD, RD), have both been independently prepared, the control logic triggers the retrieval of the stored states by simultaneously sending a strong read pulse into each of the four ensembles. Relative to the case in which no logic is implemented, this process resulted in a 19-fold enhancement in the probability of generating this overall state from the four ensembles (SOM text).

The second requirement—stability of the relative phase η—could be accomplished by active stabilization of each individual phase ηUD, as in (20). Instead of implementing this challenging technical task (which ultimately would have to be extended across longer chains of ensembles), our setup exploits the passive stability between two independent polarizations propagating in a single interferometer to prepare the two ensemble pairs (27). No active phase stabilization is thus required. In practice, we found that the passive stability of our system was sufficient for operation overnight without adjustment. Additionally, we implemented a procedure that deterministically sets the relative phase η to zero.

We also extended the original DLCZ protocol (Fig. 1) by combining fields (2LU,2LD) and (2RU, 2RD) with orthogonal polarizations on polarizing beamsplitters PBSL and PBSR to yield fields 2L and 2R, respectively. The polarization encoding opens the possibility of performing additional entanglement purification and thus superior scalability (28, 29). In the ideal case, the resulting state would now be effectively equivalent to a maximally entangled state for the polarization of two photons Math(3) where |H 〉 and |V 〉 stand for the state of a single photon with horizontal and vertical polarization, respectively. The sign of the superposition in Eq. 3 is inherited from Eq. 1 and is determined by the particular pair of heralding signals recorded by (D1a,D1b) and (D1c,D1d). The entanglement in the polarization basis is well suited for entanglement-based quantum cryptography (11, 25), including security verification by way of the violation of a Bell inequality, as well as for quantum teleportation (11).

As a first step to investigate the joint states of the atomic ensembles, we recorded photoelectric counting events for the ensemble pairs (LU,RU) and (LD,RD) by setting the angles for the half-wave plates (λ/2)L,R shown in Fig. 1 to 0°, such that photons reaching detectors D2b and D2d come only from the ensemble pair U, and photons reaching detectors D2a and D2c come only from the ensemble pair D. Conditioned upon detection events at D1a or D1b (or at D1c or D1d), we estimated the probability that each ensemble pair U,D contains only a single, shared excitation as compared with the probability for two excitations by way of the associated photoelectric statistics. In quantitative terms, we determined the ratio (20) Math(4) where pX,mn is the probability to register m photodetection events in mode 2LX and n events in mode 2RX (X ={U,D}), conditioned on a detection event at D1. A necessary condition for the two ensembles (LX, RX) to be entangled is that hX(2) <1, where hX(2) = 1 corresponds to the case of independent (unentangled) coherent states for the two fields (20). Figure 2 shows the measured hX(2) versus the duration τM (where M stands for memory) that the state is stored before retrieval. For both U and D pairs, h(2) remains well below unity for storage times τM < ∼10 μs. For the U pair, the solid line in Fig. 2 provides a fit by the simple expression h(2) = 1 – Aexp[–(τM/τ)2]. The fit gives A = 0.94 ± 0.01, where the error is SD, and τ = 22±2 μs, providing an estimate of a coherence time for our system. A principal cause for decoherence is an inhomogeneous broadening of the ground state levels by residual magnetic fields (30). The characterization of the time dependence of h(2) constitutes an important benchmark of our system (SOM text).

Fig. 2.

Suppression h(2) of the probabilities for each ensemble to emit two photons compared with the product of the probabilities that only one photon is emitted, as a function of the duration τM that the state is stored before retrieval. The solid line gives a fit for the U pair. Error bars indicate SD.

We next measured the correlation function ELR), defined by Math(5) Here, Cjk gives the rates of coincidences between detectors D2j and D2k for Field 2 fields, where j,k ∈ {a,b,c,d}, conditioned upon heralding events at detectors D1a,D1b and D1c,D1d from Field 1 fields. The angles of the two half-wave plates (λ/2)L and (λ/2)R are set at θL/2 and θR/2, respectively. As stated above, the capability to store the state heralded in one pair of ensembles and then to wait for the other pair to be prepared markedly improves the various coincidence rates Cjk by a factor that increases with the duration τM that a state can be preserved (24) (SOM text).

Figure 3 displays the correlation function E as a function of θR, for θL = 0° (Fig. 3A) and θL = 45° (Fig. 3B). Relative to Fig. 2, these data are taken with increased excitation probability (higher write power) to validate the phase stability of the system, which is evidently good. Moreover, these four-fold coincidence fringes in Fig. 3A provide further verification that predominantly one excitation is shared between a pair of ensembles. The analysis provided in the SOM text with the measured cumulative h(2) parameter for this set of data, h(2) = 0.12 ± 0.02, predicts a visibility of V = 78 ± 3% in good agreement with the experimentally determined V ≅ 75%. Finally, one of the fringes is inverted with respect to the other in Fig. 3B, which corresponds with the two possible signs in Eq. 3. As for θL = 45°, the measurement is sensitive to the square of the overlap ξ of photon wave-packets for fields 2U,D; we may infer ξU,D ≅ 0.85 from the reduced fringe visibility (V ≅ 55%) in Fig. 3B relative to Fig. 3A, if all the reduction is attributed to a nonideal overlap. An independent experiment for two-photon interference in this setup has shown an overlap ξ ≅ 0.90, which confirms that the reduction can be principally attributed to the nonideal overlap. Other possible causes include imperfect phase alignment η ≠ 0 and imbalance of the effective-state coefficients (SOM text).

Fig. 3.

Measured correlation function EL, θR) as a function of θR with θL fixed at (A) 0° and (B) 45°. The excitation probabilities for the ensembles are increased by ∼1.5 times relative to Fig. 2, with each point taken for 30 min at a typical coincidence rate of 400 per hour for each fringe. Error bars indicate SD.

With the measurements from Figs. 2 and 3 in hand, we verified entanglement unambiguously by way of the violation of a Bell inequality (31). For this purpose, we chose the canonical values, θL = {0°,45°} and θR = {22.5°,–22.5°}, and constructed the Clauser-Horne-Shimony-Holt (CHSH) parameters Math(6) Math Math(7) Math for the two effective states |ψ2L,2R±eff in Eq. 3. For local, realistic hidden-variable theories, S± ≤ 2 (31). Figure 4 shows the CHSH parameters S± as functions of the duration τM up to which one pair of ensembles holds the prepared state, in the excitation regime of Fig. 2. As shown in the SOM text, the requirements for minimization of higher-order terms are much more stringent in this experiment with four ensembles than with simpler configurations (21).

Fig. 4.

Measured CHSH parameters S±, for the two possible effective states in Eq. 3, as functions of duration τM for which the first ensemble pair holds the prepared state. The excitation probabilities are kept low for high correlation (as in Fig. 2). (A and B) Binned data. The horizontal solid lines indicate the size of the bins used. (C and D) Cumulative data. The coincidence rate for these measurements is about 150 per hour for each effective state. Error bars indicate SD.

Figure 4, A and B, gives the results for our measurements of S± with binned data. Each point corresponds to the violation obtained for states generated at τM ± ΔτM/2 (ΔτM is marked by the thick horizontal lines in Fig. 4). Strong violations are obtained for short memory times—for instance, S+ = 2.55 ± 0.14 > 2 and S = 2.61± 0.13 > 2 for the second bin—demonstrating the presence of entanglement between fields 2L and 2R. Therefore, these fields can be exploited to perform entanglement-based quantum communication protocols, such as quantum key distribution with, at minimum, security against individual attacks (11, 32).

As can be seen in Fig. 4, the violation decreases with increasing τM. The decay is largely due to the time-varying behavior of h(2) (Fig. 2 and SOM text). In addition to this decay, the S+ parameter exhibits modulation with τM. We explored different models for the time dependence of the CHSH parameters, but thus far have found no satisfactory agreement between model calculations and measurements. Nevertheless, the density matrix for the ensemble over the full memory time is potentially useful for tasks such as entanglement connection, as shown by Fig. 4, C and D, in which cumulative data are given. Each point at memory time τM gives the violation obtained by taking into account all the states generated from 0 to τM. Overall significant violations are obtained, namely S+ = 2.21 ± 0.04 > 2 and S = 2.24± 0.04 > 2 at τM ∼10 μs.

In our experiment, we were able to generate excitation-number entangled states between remote locations, which are well suited for scaling purposes, and, with real-time control, we were able to operate them as if they were effectively polarization-entangled states, which can be applied to quantum communications such as quantum cryptography. Measurements of the suppression h(2) of two-excitation components versus storage time explicitly demonstrates the major source that causes the extracted polarization entanglement to decay, emphasizing the critical role of multi-excitation events in the experiments aiming for a scalable quantum network. The present scheme, which constitutes a functional segment of a quantum repeater in terms of quantum state encoding and channel control, allows the distribution of entanglement between two quantum nodes. The extension of our work to longer chains involving many segments becomes more complicated and is out of reach for any current system. For long-distance communication, the first quantity to improve is the coherence time of the memory. Better cancellation of the residual magnetic fields and switching to new trap schemes should improve this parameter to ∼0.1 s by using an optical trap (30), thereby increasing the rate of preparing the ensembles in the state of Eq. 1 to ∼100 Hz. The second challenge that would immediately appear in an extended chain would be the increase of the multi-excitation probability with the connection stages. Recently, Jiang et al. (28) have theoretically demonstrated the prevention of such growth in a similar setup, but its full scalability still requires very high retrieval and detection efficiency, and photon-number resolving detectors. These two points clearly show that the quest of scalable quantum networks is still a theoretical and experimental challenge. The availability of our first functional segment opens the way for fruitful investigations.

Supporting Online Material

www.sciencemag.org/cgi/content/full/1140300/DC1

Materials and Methods

SOM Text

Figs. S1 to S3

Table S1

References

References and Notes

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