Abstract
Heralded entanglement between macroscopical samples is an important resource for present quantum technology protocols, allowing quantum communication over large distances. In such protocols, optical photons are typically used as information and entanglement carriers between macroscopic quantum memories placed in remote locations. Here we investigate theoretically a new implementation which employs more robust xray quanta to generate heralded entanglement between two crystalhosted macroscopical nuclear ensembles. Mössbauer nuclei in the two crystals interact collectively with an xray spontaneous parametric down conversion photon that generates heralded macroscopical entanglement with coherence times of approximately 100 ns at room temperature. The quantum phase between the entangled crystals can be conveniently manipulated by magnetic field rotations at the samples. The inherent long nuclear coherence times allow also for mechanical manipulations of the samples, for instance to check the stability of entanglement in the xray setup. Our results pave the way for first quantum communication protocols that use xray qubits.
Introduction
As a purely quantum mechanical property, quantum entanglement has been demonstrated and is nowadays routinely realized with photons, typically in the long wavelength, optical regime, or with quantum particles such as electrons, ions or atoms^{1,2,3,4,5}. However, it has been shown that also large, macroscopical collections of the latter may experience entanglement. While from a fundamental point of view this is interesting for the study of the boundary between the quantum realm and the classical world^{6}, practically it may be of paramount importance for quantum communication applications. The ability to create entanglement between quantum memories in a heralded manner^{5} can be advantageous for quantum communication developments, for instance for quantum repeaters^{1} and quantum networks^{7}. The search for scalable quantum repeaters has come up with solidstate resources, which require entanglement between quantum memories hosted in spatially separated macroscopical crystals^{5}. So far, such macroscopical entanglement has been typically limited in either time duration, working temperature or sample size/number of atoms involved, as shown in Table 1 that lists some key achievements. As generic feature, these experiments make use of optical photons as entanglement carriers^{2,3,4,5}.
The commissioning of the first xray free electron lasers (XFEL)^{8,9} and recent developments in xray optics^{10,11,12,13} and single xray quanta manipulation^{14,15,16} open new perspectives for quantum information and quantum communication protocols using xray qubits. The advantages of higher frequency photons would be better focusing, deeper penetration power, robustness and improved detection^{15}. The latter two are in particular pertinent for quantum technology applications, while the improved focusability might help shrinking future photonic devices^{14}. Xray quantum optics^{17} promises so far in theory exciting applications in metrology^{18,19} and information technology^{15,20}, as well as for generation of photon entanglement in the keV regime using nuclear rather than atomic transitions^{21}. Due to their more suitable transition energies, nuclei arise as natural candidates for xray quantum optics, especially Mössbauer nuclei which allow for a collective, delocalized excitation throughout macroscopic nuclear ensembles.
Here we investigate a scenario to create and manipulate heralded entanglement between two macroscopic solid objects, i.e., crystals containing Mössbauer nuclei, using xrays. An implementation in the xray regime and using for the first time nuclear systems instead of atoms would demonstrate the universality of quantum optics for a new energy regime and degree of complexity. From the practical point of view, such a setup presents several advantages compared to existing realizations, which are (i) room temperature handling, (ii) long coherence times, and (iii) solidstate macroscopical samples with a large number of constituents, particularly appealing for decoherence studies at the borderline between the classical and quantum worlds, (iv) comparing with other setups, we put forward the first scheme to directly manipulate macroscopic entangled objects while other proposals or experiments rather aim at manipulating the single photon that is used to entangle the macroscopic objects. This may provide a stronger evidence of nonlocality of macroscopic entanglement.
An xray parametric down conversion (XPDC) process xray → xray + extreme ultraviolet (X → X + EUV)^{22,23} in a diamond crystal provides an xray quantum that impinges on an xray interferometer^{24} as shown in Fig. 1. The detection of the EUV idler photon at the detector A heralds the presence of the xray photon in the setup. In a first step, a 50/50 beam splitter^{12} BS 1 transfers the signal photon into a twopath entanglement state , where states and refer to the onephoton Fock state and the vacuum state at the left (right) path, respectively. This singlexray entanglement state subsequently reaches two crystals containing ^{57}Fe nuclei and may drive the magnetic dipole transition from the nuclear ground to the first excited state at 14.4 keV. With recoilless absorption and reemission of the xray photon due to the Mössbauer effect, the nuclear excitation is coherently spread over the entire nuclear ensemble, and remains delocalized. This nuclear exciton state comprises the th nucleus at position excited by the incident xray with the wave number , whereas all other (N − 1) nuclei remain in their ground state^{14,25}. As a result of quantum interference between the emission from each crystal site, a directional remission of a single photon along the incident follows the decay of state . Also known under the name of singlephoton superradiance^{26}, this coherent, cooperative decay is strongly enhanced compared to the spontaneous decay channel^{25} and is routinely observed in nuclear forward scattering experiments with nuclear solidstate targets typically few μm thick and few mm in diameter^{25}. The specific speedup decay characteristics are determined mostly by the optical thickness of the sample. We note here that since at most only one resonant photon is present in the system at any given time, the superradiance experienced by the Mössbauer nuclei is different from traditional atomic superradiance which involves a strong excitation of the sample.
Due to the incident twopath entanglement state, the resonant scattering on the two remote crystal samples labeled by L and R leads to the formation of an entangled state between two distant parties
where and stand for the L(R) ensemble being in the excited state and in the ground state, respectively, and ϕ is the relative phase between two components. As additional control parameter, each nuclear sample is under the action of a hyperfine magnetic field, denoted by B_{L} and B_{R}, as illustrated in Fig. 1a. The nuclear response is recombined by a second beam splitter BS 2 and monitored by two detectors B and C.
The key for the setup is the arrangement of xray and EUV detectors such that without interacting with the nuclear sample crystals, either both detectors A and B, or both detectors A and C simultaneously register the two XPDC photons. Each successful creation of the entanglement state in Eq. (1) is heralded by the click at detector A while no photon is registered at detectors B, C (registering the coherent decay of the nuclear exciton) or any other detectors monitoring the 4π emission angle (for photon loss or incoherent, spontaneous decay of the nuclear exciton). Modern xray detectors are few up to 10 centimeters in size and much larger than the nuclear samples, thus facilitating a wide solidangle monitoring. The missing count of an xray signal photon is attributed to the absorption by the two remote crystals. The absorbed and rescattered signal photon reaches the detectors B or C and is recorded only later, with a time delay given by the nuclear excited state lifetime. We recall that the latter will be influenced by superradiant decay channels in the system. By choosing a moderate optical thickness , the xray emission is predominantly occurring in the forward direction, facilitating detection, while the speedup of the decay remains relatively small, leading to the signal delay of few tens up to 100 ns. In order to minimize the effects of false nondetection events, which are unavoidable despite high detection efficiency, a valid data record will require the timedelayed coincidence of two detectors, either detectors A and B, or detectors A and C. Thus, false nondetection events will not jeopardize the fidelity of the prepared state, but rather just reduce the rate with which heralded entanglement can be recorded.
For a detectable production rate of the heralded macroscopic entanglement, the key requirement is that the XPDC source produces downconverted xray signal photons which are broadband relatively to the nuclear resonance width and cover the hyperfinesplit nuclear absorption lines. Over this narrow range, the nuclear resonance absorption exceeds by orders of magnitude the atomic background processes^{25}. An estimate of the flux of generated xray parametric down conversion photons by an incoming XFEL pulse (see Methods) gives 3 × 10^{6} signal photons/s with a 1 eV bandwidth. This corresponds to a 0.1 Hz production rate of heralded macroscopic entanglement. We expect that the latter rate can be further increased by one or two orders of magnitude, i.e., R_{E} ~ 10 Hz, by a tighter focusing on a diamond crystal which would enhance the nonlinear efficiency of XPDC. We also note that the signal photon rate is low enough to allow sufficient potential recording time (several hundreds ns) between single shots. Further attention is required for avoiding losses by air absorption of the heralding EUV photon^{22} and also for the mechanical alignment of the setup, with XPDC source, beam splitters and mirrors all having angular acceptances of μrad^{11,22,24}.
To verify the entanglement between the two nuclear sample crystals, we invoke the method of quantum state tomography^{3,5,27} to determine the density matrix of of the coherently reemitted single xray photon from two targets (see Methods). The key quantity to be determined experimentally is the visibility V of the interference fringe at detectors B and C. The remitted single photon is allowed to interfere with itself on beam splitter BS 2 for different phase shifts while measuring the interference fringe. Typically, an additional Si phase shifter or a vibrating crystal are used to mechanically vary the phase between the two arms in an interferometer^{24} for the determination of V. Here we propose a novel magnetic, nonmechanical solution for phase modulation that directly and locally controls the nuclear dynamics in each ensemble and can provide an indication of entanglement between the two remote parties. The phase modulation can be achieved via a fast rotation of the hyperfine magnetic field at one of the crystals. We note that the magnetic field rotations provide direct control over the entangled macroscopic ensembles rather than over the scattered photons^{3,4}. In principle, this renders possible new decoherence tests by superimposing mechanical movement in parallel to monitoring the effects of the magnetic phase modulation. More practical aspects related to the mechanical stability requirements of the setup will be discussed in the following.
Due to the hyperfine magnetic field, each ^{57}Fe 14.4 keV nuclear transition is split into a sextet (Fig. 1b). The typical bandwidth of the downconverted photons of approx. 1 eV^{22} is much broader than the linewidth of the interacting nuclear transition such that the scattering photon can drive any of the six transitions between the hyperfine ground state and excited state levels. The setup geometry (Fig. 1a) is chosen such that linearly polarized xray photons will drive simultaneously the two Δm = 0 transitions, with Zeeman energy shifts ±ħΔ_{B}. The quantum coherences in the two crystals (see Methods) are simultaneously driven by the downconverted signal photon and rotate in time as shown by Fig. 1(c,d). Due to the Zeeman shifts ±ħΔ_{B}, the two pairs of coherences accumulate a phase for a constant Δ_{B}. If just one of the magnetic fields, for instance B_{R} is inverted at t = T_{ϕ}, the right mode turns into cos(ϕ − Δ_{B}t), whereas the left wavepacket is still proportional to cos(ϕ + Δ_{B}t), leading to a phase shift between the two samples. We thus can magnetically control, without need of a mechanical solution, the quantum phase between the two spatially separated entangled nuclear crystals. Rapid manipulations of the hyperfine magnetic field in iron samples have been demonstrated with antiferromagnetic ^{57}FeBO_{3} crystals^{28}.
The resulting interference fringe can be analyzed by investigating the output intensities at the two detectors B and C, and , respectively. Here, τ is the photon counting time after T_{ϕ}, θ is the phase shift experienced by the xray photon when transmitted by the beam splitters BS 1 and BS 2 and and are the photon number operators for the two output fields at detectors B and C, respectively. The output fields can be obtained by considering the action of the beam splitter BS 1, nuclear scattering in samples L and R, mirror and beam splitter BS 1, respectively, on the incident XPDC field. In matrix form, this can be written as^{29}
where and and θ and φ are the phase shift of the transmitted and reflected xrays, respectively, relative to the incident xrays. We assume in Eq. (2) that reflectivity = transmittance = s. As the field is in the vacuum state, the number operators or . Numerical results for the interference fringes Q_{B} and Q_{C} are presented in Fig. 1e for optical thickness α = 1, natural decay rate Γ = 1/141 GHz for ^{57}Fe and Δ_{B} = 30Γ. The coherence time of the entanglement between two crystals is approx. 60 ns in Fig. 1e, corresponding to the chosen optical thickness value. The lifetime of the entanglement state can be prolonged up to the natural mean lifetime of the ^{57}Fe excited state of 141 ns by either rotating^{28} the hyperfine magnetic field as has been demonstrated experimentally in ^{57}FeBO_{3} crystals^{28} or by switching it off^{14} to obtain a quantum memory. The fidelity under the action of the quantum memory could be defined as and . Both are the measure of the similarity between the retrieved wavepacket and the wavepacket ψ_{R(L)}(T_{s} + t) to be stored, where T_{s} is the instant of storage, and T_{r} is the instant of retrieval. Since storage via magnetic field rotation^{28} does not freeze the magnetic phase evolution, we expect that switching off the magnetic field^{14} to have the hyperfine splitting completely vanish may prepare the required state with higher fidelity. We note that 141 ns would be the longest coherence time achieved for macroscopic entanglement of solidstate samples^{4,5}, see Table 1.
The degree of entanglement may be influenced by the performance of the beam splitters BS 1 and BS 2. Considering imperfect beam splitters, we may write and for a α < 1 and an integer value n. The visibility fringe becomes in this case , showing that the maximum entanglement occurs at θ = nπ/2. A further dynamical decoherence mechanism may come into play with vibrations or displacements of the target. In order to simulate this effect theoretically and find out the tolerance vibration of the system, we envisage that the two entangled targets experience movement with random velocities u along the direction of photon propagation. Displacements in the plane orthogonal to the photon propagation would lead to misalignments in the interferometer setup and failure to recover the photon at detectors B and C. Let us assume that u_{R}(t) and u_{L}(t) are random numbers chosen within a certain range for each time instant t. Figure 2 demonstrates numerical results with random velocities in the ranges of a (−0.1, 0.1) mm/s, b (−0.2, 0.2) mm/s and c (−0.4, 0.4) mm/s showing that the interference fringes gradually become blurred when extending the maximum vibration speed to ku_{R} ~ Δ_{B} or ku_{L} ~ Δ_{B}. These values are much larger than the typical vibration fluctuation of a stabilized xray interferometer^{24}, confirming that macroscopical entanglement can be generated and sustained with the proposed setup.
Mechanical vibrations of the samples act as classical dephasing to the entanglement setup. In particular, random classical dephasing can be used to mimick quantum decoherence^{6}. Thus, an experimental observation of the behaviour shown in Fig. 2 may shed light on the nature of quantum decoherence occurring in the entanglement of two crystals with Mössbauer nuclei. This is the unique feature of our setup due to the long nuclear decoherence times, the solidstate nature of the samples and finally to the magneticfield phase control, which does not require any mechanical manipulation for entanglement checks in the first place. The theoretical treatment of specific decoherence models goes however beyond the scope of this paper.
In conclusion, we have demonstrated a scheme that employs xray quanta in a new parameter regime to create macroscopical entanglement between two crystals hosting Mössbauer nuclei. The use of stable and well isolated nuclear systems allows longer coherence times together with room temperature handling. Entanglement relies on a delocalized nuclear excitation which can be spread over a large number of nuclei, for typical sample and focus parameters of up to approx. 10^{14}. Quantum state tomography in conjunction with a novel magneticphase control technique can be employed to characterize the entanglement state. The entangled crystals can then be subjected to decoherence tests involving mechanical movement. We expect that heralded entanglement using xrays and nuclear transitions can thus open a new research avenue for both applied ideas related to quantum technology as well as more foundational studies of the boundary between the quantum and classical worlds.
Methods
For the calculation of the nuclear response we use the wellknown MaxwellBloch equations^{29}: , and . Here, ρ_{mn} is the coherence between states m〉 and n〉, with {m, n} ∈ {1, 2, 3, 4} as depicted in Fig. 1b, the ClebschGordan coefficient, c the speed of light, and L the crystal thickness. With the boundary condition ψ(t, 0) = δ(t) for a broadband incident xray, the coherently scattered xray wavepacket off the nuclear crystals reads^{25} , and . Here, J_{1} is the first order Bessel function of the first kind due to multiple scattering events in the sample, α = g^{2}β/2 the effective resonant thickness and Γ the spontaneous decay rate of the nuclear excited state. Furthermore, the trigonometric oscillation is caused by the quantum beat of the two split nuclear transitions, and the exponential decay term describes the incoherent spontaneous decay of the excited states.
Entanglement realization can be checked by means of quantum state tomography. In the photonnumber basis the density matrix of the coherently reemitted single xray photon reads^{3,27}
where p_{ij} is the probability of detecting i photons from the left crystal and j photons from the right one. Furthermore, d_{tpe} is the coherence between the two components of TPE〉 and . The concurrence from a measured then quantifies a lower bound for entanglement such that for maximal entanglement and for a pure quantum state^{3,27}. With the approximation p_{00} ≈ 1 − (p_{01} + p_{10} + p_{11}), the diagonal terms in Eq. (3) can be determined experimentally by conditional measurements that distinguish between photons scattered by the L or R samples, e.g., by removing the second beam splitter BS 2. What concerns the coherence term d_{tpe}, it has been shown that this can be approximated as^{3,27} V(p_{01} + p_{10})/2, where V is the visibility of the interference fringe at detectors B and C.
For an experimental implementation, we now estimate the possible production rate of heralded macroscopic entanglement. With a nuclear resonance cross section of σ = 2.5 Mbarn for the 14.4 keV transition of ^{57}Fe, already a sample of 20 μm thickness is likely to absorb all incoming resonant photons. Assuming 100% detection efficiency^{15}, the flux R_{E} of produced resonant photons equals the rate of heralded entanglement creation. The flux can be estimated as R_{E} = ξ_{s}ΔE_{n}/ΔE_{s}, where ΔE_{n} = 4.66 neV is the linewidth of the considered ^{57}Fe nuclear transition, and ΔE_{s} = 1 eV and ξ_{s} are the bandwidth and the flux, respectively, of the XPDC signal photons^{22}. According to ref. 22, , where I_{p} is the photon density of the pump field, and the 111 Fourier coefficient of the second order nonlinear susceptibility for a diamond (111) crystal^{30}. By introducing ω_{p} = ω_{s} + ω_{i} ^{22}, we obtain the susceptibility
where ω_{p}, ω_{s}, and ω_{i} are the angular frequencies of pump, signal and idler photons, respectively, N is the number density of unit cells, the linear structure factor of bound electrons^{22,30} and the 111 reciprocal lattice vector of a diamond crystal. Further parameters in Eq. (4) are m the electron mass, e the electron charge and ε_{0} the vacuum permittivity. Given ℏω_{s }= 14.4 keV and ℏω_{i} = 100 eV, C/N, having the same order of magnitude as for the case of ℏω_{s} = 10.9 keV reported in ref. 22. Since for the latter, SR pulses were used as pump field, the pump photon density can be enhanced by considering an XFEL pulse. Considering a train of XFEL pulses with 10^{12} photons/pulse and repetition rate^{8} f = 2.7 × 10^{4}, on a spot size^{22} of 5000 μm^{2}, by simple scaling, we then obtain ξ_{s} = 2.9 × 10^{6} signal photons/s.
Additional Information
How to cite this article: Liao, W.T. et al. Xraygenerated heralded macroscopical quantum entanglement of two nuclear ensembles. Sci. Rep. 6, 33361; doi: 10.1038/srep33361 (2016).
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Acknowledgements
We are grateful to K. Tamasaku for valuable discussions. W.T.L. is supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST 1052112M008001MY3). W.T.L. is also supported by the National Center for Theoretical Sciences, Taiwan.
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W.T.L. and A.P. conceived the project. W.T.L. developed the analytical model and carried out the calculations and A.P. supervised the project. W.T.L., C.H.K. and A.P. all contributed to the development of ideas, discussion of the results, and preparation of the manuscript.
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Liao, W., Keitel, C. & Pálffy, A. Xraygenerated heralded macroscopical quantum entanglement of two nuclear ensembles. Sci Rep 6, 33361 (2016). https://doi.org/10.1038/srep33361
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