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Entangled States of More Than 40 Atoms in an Optical Fiber Cavity

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Science  11 Apr 2014:
Vol. 344, Issue 6180, pp. 180-183
DOI: 10.1126/science.1248905

All Together Now

In quantum entanglement, correlations between particles mean that the measurement of one determines the outcome of the other(s). Generally, when trying to exploit quantum entanglement, the larger the number of entangled particles, the better. However, the size of entangled systems has been limited. Haas et al. (p. 180, published online 27 March; see the Perspective by Widera) prepared a small ensemble of ultracold atoms into a collective entangled state. Starting from one internal quantum state, the system of cold atoms was excited with a weak microwave pulse leading to a small excitation probability. Because it is not known which atom is promoted into the excited state, the detection of one quantum of excitation projects the system into an entangled quantum state, called a W-state. A fast repeat-until-success scheme produced such W-states quasi-deterministically. Using such a technique was able to yield entangled states of more than 40 particles. The relatively large ensemble-entangled states could potentially in the future find use in quantum sensing or enhanced quantum metrology applications.

Abstract

Multiparticle entanglement enables quantum simulations, quantum computing, and quantum-enhanced metrology. Yet, there are few methods to produce and measure such entanglement while maintaining single-qubit resolution as the number of qubits is scaled up. Using atom chips and fiber-optical cavities, we have developed a method based on nondestructive collective measurement and conditional evolution to create symmetric entangled states and perform their tomography. We demonstrate creation and analysis of entangled states with mean atom numbers up to 41 and experimentally prove multiparticle entanglement. Our method is independent of atom number and should allow generalization to other entangled states and other physical implementations, including circuit quantum electrodynamics.

For entanglement-enabled technologies as well as from a fundamental point of view, an important goal is to scale up the number of entangled particles in many-qubit systems. In a bottom-up approach, individual addressing and universal quantum gates allow full control on the quantum state, and, in principle, any entangled state can be produced. However, because the number of gate operations scales up with particle number, experiments based on this method (such as ion traps) are currently limited to less than 20 entangled qubits (1).

Many important entangled states can be produced in a “top-down” approach with collective interactions or quantum nondemolition (QND) measurements, the complexity of which does not increase with particle number. In atomic ensembles, spin-squeezed (2, 3) and twin Fock states (4) have been produced by collisional interactions; collective QND measurement (58) and cavity-mediated interactions (9) have been used to produce spin squeezing and have been proposed for Schrödinger cats (10) and Dicke states, including twin Fock states (11). However, state-of-the-art QND measurements in ensembles are still far from the single-particle resolution that would be required, for example, to reach the Heisenberg limit of quantum metrology (12). Moreover, the full quantum state of a system cannot be experimentally determined nor efficiently analyzed beyond 10 to 20 qubits, in general (13), so that new methods that specifically identify and characterize relevant forms of entanglement in large systems are required.

Here, we use a cavity-based measurement that distinguishes one particular many-particle quantum state from the orthogonal subspace of all other states. This allows us to prepare many-body entangled states projectively and to directly measure their quasiprobability distribution with high resolution. Consider an ensemble of N atoms, all strongly coupled to a single mode of a high-finesse cavity (1418) (Fig. 1, A and B). The cavity and probe beam are tuned for detecting the hyperfine state, Embedded Image or Embedded Image (19). Transmission through the cavity is observed only when all N atoms are in Embedded Image. A single atom in Embedded Image makes the cavity fully reflecting (20), and no further substantial changes occur when more than one atom is in Embedded Image. Due to the very strong coupling of the atom-resonator system (21), this is true for all atomic positions, so that the atoms are indistinguishable when probed by the cavity mode. Furthermore, we set the probe power such that the total probability for a spontaneous emission event is much smaller than one (19), limiting the amount of atom-distinguishing information that leaks into the environment. Thus, to good approximation, measuring the cavity transmission is a projective measurement with two eigenvalues: high transmission corresponding to Embedded Image and low transmission to the orthogonal subspace containing all other states, where at least one atom is in Embedded Image. This measurement enables the generation of multiparticle entanglement as follows: Atoms are prepared in a premeasurement state Embedded Image, then are measured as described. Low transmission signals preparation of Embedded Image (meaning that the cycle must be repeated), whereas high transmission prepares the system in Embedded Image, where c is a normalization factor. For a suitable choice of Embedded Image, Embedded Image can have interesting nonclassical properties. Here, we prepare a Embedded Image that is a good approximation ofEmbedded Image(1)known as the W state or the first Dicke state. It represents a fundamental class of entangled states (22), which are robust against particle loss and enable some meteorological gain over nonentangled states (23).

Fig. 1 Cavity-assisted generation of entanglement.

(A) Relevant level scheme of 87Rb. A resonant 6.8 GHz microwave allows applying arbitrary rotations to the atomic qubit. The cavity and probe laser are resonant with the transition Embedded Image, where F′ and mF denote the hyperfine and magnetic sublevels of the excited state, respectively. (21). MW, microwave pulse. (B) Principle of the collective QND measurement based on the normal-mode splitting. R, reflection; T, transmission. (C) Preparation sequence of the W state. The asymmetry in 3b originates from the nonzero contribution of higher-order Dicke states, which is due to the finite value of p. (D) The spin states with norm J and z component –J + n, where 2Jn ≥ 0, form a basis for the total atomic pseudospin (with this notation, n is the number of atoms in Embedded Image). In the symmetric subspace (J = N/2), the atomic state is fully characterized by its Husimi Q distribution, measured in our experiment. Shown are calculated distributions for the Embedded Image, Embedded Image, and Embedded Image Dicke states.

To obtain the W state, we start from Embedded Image and apply a weak coherent microwave pulse on the qubit transition. If the excitation probability p is small, this prepares Embedded Image. Measuring Embedded Image as described either projects back onto Embedded Image or prepares Embedded Image. Low transmission heralds successful preparation, and the state is then available for further experiments. Figure 1C shows the expected evolution during the sequence. Note that the system is always in a known, pure state, as in a quantum feedback scheme, and in contrast to single-photon heralded schemes (24, 25).

To fully characterize the produced state, we have developed a tomography technique that measures the Husimi Q distribution of the total spin (26). In the ideal sequence (Fig. 1C), the state evolves inside the symmetric subspace, which contains the states with maximum pseudo-spin J = N/2 (Fig. 1D). In this subspace, which is spanned by the Dicke states and can be represented on a generalized Bloch sphere, the Q function is defined as Embedded Image (26), where Embedded Image is the probability that all atoms are in Embedded Image after a rotation Embedded Image of the state with density matrix ρ. This expression shows that a direct measurement of Q is obtained by combining a rotation Embedded Image (performed with a microwave pulse here) with the measurement described above. Indeed, the probability to obtain high transmission for a given state measures its overlap with Embedded Image (21, 27). This method strongly differs from marginal distribution measurements (3, 28) and is similar to one developed in quantum optics (29). Note that our binary individual measurement is sufficient to distinguish all symmetric states in the tomography when performed with a sufficient number of repetitions. For states outside the symmetric subspace (J < N/2), spin conservation under Embedded Image entails that they never transform into Embedded Image and, thus, have a zero contribution to the measured P0 for all Embedded Image (Fig. 1D) (21). Therefore, the norm of Q of a given state yields its probability of lying in the symmetric subspace, whereas the shape of Embedded Image fully characterizes the symmetric part of the state.

In our experiment, a small atom number is prepared from an ultracold ensemble close to quantum degeneracy and trapped in a single antinode of a one-dimensional intracavity optical lattice, where each atom is strongly coupled to the resonant cavity mode (15, 21). Figure 2A shows the tomography curve Embedded Image for Embedded Image, the state before the preparation sequence. The measured curve is well approximated by the expected cos2N(θ/2), from which we deduce the mean number of atoms N that contribute to the state. Because of our atom number preparation (21), we expect the prepared atom number to follow a binomial distribution with SD σ = 4.2, 3.6, and 2.8 for N = 41, 23, and 12.

Fig. 2 Tomography of coherent and W states.

(A) Tomography of a coherent state Embedded Image (the initial state before the W state preparation). From a fit (red line), we obtain the number of atoms, N = 41. (B to D) Tomography of W states with 41, 23, and 12 atoms, respectively. Each point corresponds to ~50 measurements. For each atom number, the red curve results from a maximum likelihood state reconstruction taking into account all known imperfections of the tomography technique (21). For comparison, the dashed green lines show the theoretical P0(θ) for an ideal W state, and the dashed orange lines indicate that of a statistical mixture in which one known atom among N is in Embedded Image. For each N, the populations of the Embedded Image state and the first and second Dicke states Embedded Image and Embedded Image are indicated, as deduced from the state reconstruction.

Applying the entanglement preparation method [see (21) for experimental details], the tomography results P0(θ) of the prepared W states for N = 41, 23, 12 are shown in Fig. 2, B to D. The curves feature the characteristic central dip expected for the nonclassical W state. The nonzero value for θ = 0 indicates some remnant population in Embedded Image, which is below 10% for all atom numbers. In addition, the curves have a high contrast, indicating a population in the symmetric subspace above 40% for all atom numbers. This clearly sets them apart from their classical counterparts, the statistical mixtures of all states with one localized excitation, Embedded Image (orange curves in Fig. 2, B to D). For those states, the symmetric subspace population is only 1/N, so that the maximum value of P0(θ) is small.

Because of the rotational symmetry of the state, a fair estimate of the W state fidelity is obtained from the single polar cut Embedded Image of the Q distribution. In fact, we expect that other cuts do not contain additional information because of a slow drift of the magnetic field during the long acquisition time of our data (several days), which randomizes the phases between the different Dicke states. This can be seen in the symmetry of the tomography data shown in Fig. 2 [P0(–θ) ≈ P0(θ)]. We also measured Embedded Image and verified that it is very similar to P0(θ,0). To partially reconstruct the density matrix ρ in the Dicke basis, we make the simplifying assumption that ρ is purely diagonal and deduce the populations ρii using a maximum likelihood algorithm. We have checked that this assumption does not overestimate the fidelity ρ11, nor does it underestimate ρ00 for our data (21). The red solid lines in Fig. 2 show the tomography curves corresponding to the reconstructed density matrices displayed as insets, where we truncated the basis to the first three Dicke states Embedded Image (21). A fidelity ρ11 of 0.42, 0.37, and 0.31 was obtained for N = 41, 23, and 12. The 1σ error is below 0.1 in all cases (21).

One important limit to the fidelity is false-positive detection during preparation (atoms detected in Embedded Image although they are in Embedded Image). Discarding runs with ambiguous measurement results reduces this error and increases fidelity, at the cost of reducing success probability. To decide whether the cavity is in the high- or low-transmission state, we analyze both the reflected and transmitted photon numbers, NR and NT, and compare to predefined thresholds. Figure 3A shows the count distribution of NR (21). All data was acquired using NT = 0, NR ≥ 4 as the criterion signaling successful preparation, leaving room for postselection using a higher threshold NR,min. Figure 3B shows the fidelity as a function of NR,min, confirming that this is an important contribution to the preparation error. The data shown in Fig. 2 correspond to values of NR,min [slightly different for each atom number (21)] such that the fraction of successful runs is ~10%.

Fig. 3 Fidelity control of a W state with 41 atoms.

(A) Distribution of counts in reflection during successful state preparation. The count rate in transmission is always zero, otherwise the preparation step is repeated. (B) Fidelity of the prepared W state and success probability as functions of the reflection threshold NR,min. The data shown in Fig. 2B corresponds to NR,min = 11, for which the preparation success is 9%. Error bars indicate 1 SD.

Knowing how much entanglement is present in a many-particle state is difficult, even when the full density matrix is known. We now establish criteria for entanglement in the vicinity of the W state by solely comparing the two populations ρ00 and ρ11. Bipartite entanglement being extremely difficult to prove with a W state (30), we rather look for multipartite entanglement and for the minimal number of entangled atoms. A state with density matrix ρ contains, at most, k entangled particles if ρ can be decomposed as a convex sum of density matrices Embedded Image, where each ρi corresponds to a density matrix containing, at most, k atoms, and MN (30). Starting from this expression, we first calculate the upper bound for ρ11 as a function of ρ00 for a fully separable state with k = 1 (21). The bound, which is tight, is shown as the solid thick blue line in Fig. 4. Any state corresponding to a point outside the blue shaded region necessarily contains at least two-particle entanglement. The calculation can be extended to larger k, and the green lines show the bounds for k = 8, 12, and 16. The experimental state with the highest fidelity of 0.42 lies ~1 SD above the k = 12 curve, indicating that it contains at least 13 entangled atoms.

Fig. 4 Multipartite entanglement for a W state with 41 atoms.

Fully separable states lie within the blue shaded area. From bottom to top, the green curves show the bound for k-separable states with k = 8, 12, and 16. A state above a bound contains at least k + 1 entangled particles. Shaded areas limit the bounds when varying the atom number from 37 to 45. The red points are the data from Fig. 3B, showing the increase of the minimal number of entangled particles with increasing NR,min. Error bars indicate 1 SD.

In our present setup, fidelity is limited by decoherence due to differential light shifts in the dipole trap (21) and a probability of ~0.2 for spontaneous emission from Embedded Image during the QND detection. For large atom numbers, spontaneous emission from Embedded Image will eventually become dominant. Simulations show that, with state-of-the-art optical cavities, the entanglement process can be scaled up to ensembles with N > 104. Because our method relies only on coherent evolution and collective QND measurement, it can be adapted to many systems and, in particular, to other forms of cavity quantum electrodynamics, including addressable qubits such as ions in optical cavities (31) or superconducting qubits in microwave cavities (32), as long as they are indistinguishable by cavity measurement. Furthermore, by including multiple rotations and QND detection intervals, or by combining it with other entanglement schemes such as cavity-induced spin squeezing (6), our scheme can be extended to a large range of entangled states. In combination with the inherent single-particle resolution, this makes it possible to investigate the fundamental limits of metrologically relevant forms of entanglement and could considerably enhance the precision of interferometric devices based on quantum metrology. In addition, our method is well suited to investigate quantum Zeno dynamics (33), where permanent QND observation of a degenerate eigenvalue limits the coherent evolution of the system to a given subspace, enabling preparation of a large variety of entangled states (34).

Supplementary Materials

www.sciencemag.org/content/344/6180/180/suppl/DC1

Materials and Methods

Figs. S1 to S6

References (3537)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We thank G. Semerjian for help with the calculations on the entanglement criterion and B. Huard for discussions. This work was supported by the European Union Information and Communication Technologies project QIBEC (Quantum Interferometry with Bose-Einstein Condensates) (GA 284584) and the Integrating Project AQUTE (Atomic Quantum Technologies) (GA 247687). F.H. acknowledges a scholarship by Institut Francilien de Recherche sur les Atomes Froids. Author contributions: F.H. and J.V. performed the experiment; R.G. made contributions in its early stages; and F.H., J.V., J.R., and J.E. contributed to data analysis and interpretation, as well as to the manuscript.
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