Entanglement and non-locality between disparate solid-state quantum memories mediated by photons

TL;DR

This paper demonstrates entanglement and non-locality between two different solid-state quantum memories at disparate wavelengths mediated by photons, advancing quantum network development.

Contribution

It presents the first experimental evidence of entanglement preservation and non-locality between quantum memories operating at different wavelengths.

Findings

Non-classical correlations with $g^{(2)}_{12} = 53\pm8$

Input-output fidelity of approximately 93%

Violation of Bell inequality with $S= 2.6\\pm0.2$

Abstract

Entangling quantum systems with different characteristics through the exchange of photons is a prerequisite for building future quantum networks. Proving the presence of entanglement between quantum memories for light working at different wavelengths furthers this goal. Here, we report on a series of experiments with a thulium-doped crystal, serving as a quantum memory for 794 nm photons, an erbium-doped fibre, serving as a quantum memory for telecommunication-wavelength photons at 1535 nm, and a source of photon pairs created via spontaneous parametric down-conversion. Characterizing the photons after re-emission from the two memories, we find non-classical correlations with a cross-correlation coefficient of $g^{(2)}_{12} = 53\pm8$; entanglement preserving storage with input-output fidelity of $\mathcal{F}_{IO}\approx93\pm2\%$; and non-locality featuring a violation of the Clauser-Horne-Shimony-Holt Bell-inequality with $S= 2.6\pm0.2$. Our proof-of-principle experiment shows that entanglement persists while propagating through different solid-state quantum memories operating at different wavelengths.

Tables

Table 1. Table 1: Values were calculated using the density matrices in Fig.  3 , and uncertainties estimated using Monte-Carlo simulation.

	Entanglement of formation (%)	Purity (%)	Fidelity with $|{\varphi }^{+}⟩$ (%)	Input-output fidelity (%)
${\rho }_{in}$	81.10$\pm$2.23	84.57$\pm$1.47	91.68$\pm$0.83	93.77$\pm$2.18
${\rho }_{out}$	72.61$\pm$11.70	80.14$\pm$7.22	87.68$\pm$4.67

Table 2. Table 1: Efficiency versus wavelength for pairs of wavelengths available in the SPDC source.

${\lambda }_{\mathrm{signal}}$ [nm]	${\lambda }_{\mathrm{idler}}$ [nm]	${\eta }_{\text{Tm}}$	${\eta }_{\text{Er}}$	${\eta }_{\text{Tm}}\times {\eta }_{\text{Er}}$
795.15	1534.05	$0.93\%$	$0.50\%$	$0.0095\%$
794.85	1535.17	$1.87\%$	$0.48\%$	$0.0182\%$
794.68	1535.80	$2.00\%$	$0.50\%$	$0.0200\%$
794.55	1565.29	$1.49\%$	$0.49\%$	$0.0148\%$

Table 3. Table 2: Joint-detection probabilities used for the reconstruction of the density matrices before (in) and after (out) storage.

	${\sigma }_z\otimes {\sigma }_z$	${\sigma }_z\otimes {\sigma }_x$	${\sigma }_z\otimes {\sigma }_y$	${\sigma }_x\otimes {\sigma }_z$	${\sigma }_x\otimes \left({\sigma }_x+{\sigma }_y\right)$	${\sigma }_x\otimes \left({\sigma }_x-{\sigma }_y\right)$
P${}_{\text{in}}$ %	49.55$\pm$2.48	25.51$\pm$0.43	24.96$\pm$0.43	26.85$\pm$0.41	41.72$\pm$1.08	40.15$\pm$1.06
P${}_{\text{out}}$ %	48.21$\pm$2.48	25.00$\pm$5.89	24.49$\pm$7.07	25.00$\pm$5.10	40.91$\pm$6.82	42.50$\pm$10.31
	${\sigma }_x\otimes {\sigma }_y$	${\sigma }_y\otimes {\sigma }_z$	${\sigma }_y\otimes {\sigma }_x$	${\sigma }_y\otimes \left({\sigma }_x+{\sigma }_y\right)$	${\sigma }_y\otimes \left({\sigma }_x-{\sigma }_y\right)$
P${}_{\text{in}}$ %	23.82$\pm$0.57	27.15$\pm$0.42	25.22$\pm$0.59	8.66$\pm$0.58	40.39$\pm$1.05
P${}_{\text{out}}$ %	22.22$\pm$5.56	25.26$\pm$5.16	26.39$\pm$6.05	6.82$\pm$3.94	40.00$\pm$8.16
	${\sigma }_z\otimes {\sigma }_{-z}$	${\sigma }_z\otimes {\sigma }_{-x}$	${\sigma }_z\otimes {\sigma }_{-y}$	${\sigma }_x\otimes {\sigma }_{-z}$	${\sigma }_x\otimes \left({\sigma }_{-x}+{\sigma }_{-y}\right)$	${\sigma }_x\otimes \left({\sigma }_{-x}-{\sigma }_{-y}\right)$
P${}_{\text{in}}$ %	2.26$\pm$0.10	23.37$\pm$0.42	23.63$\pm$0.41	20.81$\pm$0.36	9.04$\pm$0.50	9.85$\pm$0.53
P${}_{\text{out}}$ %	2.30$\pm$0.54	26.39$\pm$6.05	26.53$\pm$7.36	27.08$\pm$5.31	9.09$\pm$3.21	7.50$\pm$4.33
	${\sigma }_x\otimes {\sigma }_{-y}$	${\sigma }_y\otimes {\sigma }_{-z}$	${\sigma }_y\otimes {\sigma }_{-x}$	${\sigma }_y\otimes \left({\sigma }_{-x}+{\sigma }_{-y}\right)$	${\sigma }_y\otimes \left({\sigma }_{-x}-{\sigma }_{-y}\right)$
P${}_{\text{in}}$ %	26.18$\pm$0.60	20.48$\pm$0.36	24.78$\pm$0.59	41.34$\pm$1.26	9.61$\pm$0.51
P${}_{\text{out}}$ %	27.78$\pm$6.21	24.21$\pm$5.05	23.61$\pm$5.73	43.18$\pm$9.91	10.00$\pm$4.08
	${\sigma }_{-z}\otimes {\sigma }_z$	${\sigma }_{-z}\otimes {\sigma }_x$	${\sigma }_{-z}\otimes {\sigma }_y$	${\sigma }_{-x}\otimes {\sigma }_z$	${\sigma }_{-y}\otimes {\sigma }_z$
P${}_{\text{in}}$ %	2.15$\pm$0.11	24.45$\pm$0.43	24.74$\pm$0.42	28.16$\pm$0.42	28.54$\pm$0.43
P${}_{\text{out}}$ %	2.17$\pm$0.53	23.61$\pm$5.73	24.49$\pm$7.07	26.04$\pm$5.21	25.26$\pm$5.16
	${\sigma }_{-z}\otimes {\sigma }_{-z}$	${\sigma }_{-z}\otimes {\sigma }_{-x}$	${\sigma }_{-z}\otimes {\sigma }_{-y}$	${\sigma }_{-x}\otimes {\sigma }_{-z}$	${\sigma }_{-y}\otimes {\sigma }_{-z}$
P${}_{\text{in}}$ %	46.03$\pm$0.41	26.66$\pm$0.44	26.67$\pm$0.44	24.18$\pm$0.39	23.83$\pm$0.39
P${}_{\text{out}}$ %	47.31$\pm$2.46	25.00$\pm$5.89	24.49$\pm$7.07	21.88$\pm$4.77	25.26$\pm$5.16

Table 4. Table 3: Correlation coefficients used to test the CHSH-Bell inequality.

	a $\otimes$ b	a $\otimes$ $b^{\prime }$	$a^{\prime }$ $\otimes$ b	$a^{\prime }$ $\otimes$ $b^{\prime }$
Ein %	60.59$\pm$1.34	64.39$\pm$1.27	-61.56$\pm$1.31	65.40$\pm$1.48
Eout %	68.73$\pm$11.29	63.03$\pm$8.22	-59.28$\pm$10.32	68.07$\pm$10.34

Equations

$$|\phi^{+}\rangle=\big{(}|ee\rangle+|\ell\ell\rangle\big{)}/\sqrt{2},$$

$$S=|E({\alpha }_1,{\beta }_1)+E({\alpha }_1,{\beta }_2)+E({\alpha }_2,{\beta }_1)-E({\alpha }_2,{\beta }_2)|\le 2.$$

$$|\psi {⟩}_A=\frac{1}{\sqrt{N}}\sum _{j=1}^N{c}_j{e}^{-i2\pi n_j\Delta \cdot t}{e}^{ik\cdot z}|{\mathsf{g}}_1,...,{\mathsf{e}}_j,...,{\mathsf{g}}_N⟩.$$

$${\eta }_{device}=(d_1/F{)}^2{e}^{-d_1/F}{e}^{-7/F_2}{e}^{-d_0},$$

$$g_{12}^{(2)}(\delta t)=\text{R}(\delta t)/⟨\text{R}(\delta t+12.5\text{ns}){⟩}_{\text{n}}.$$

$$E_{F}(\rho)=H\Big{(}0.5+0.5\sqrt{1-C^{2}(\rho)}\Big{)},$$

$$C(\rho )=\max \{0,{\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4\}.$$

$$F(\rho,\sigma)=\Bigg{(}\textrm{Tr}\Big{(}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\Big{)}\Bigg{)}^{2}$$

$$P=\text{Tr}({\rho }^2).$$

$$E\left(a,b\right)=\frac{C\left(a,b\right)-C\left(a,-b\right)}{C\left(a,b\right)+C\left(a,-b\right)}$$

Full text

Entanglement is central to applications of quantum mechanics horodecki2009Entanglementreview . In particular, photon-mediated distribution of entanglement over different and widely spaced quantum systems underpins the creation of a future quantum network wehner2018quantum . Various materials, devices and protocols are currently being studied towards this end lvovsky2009optical ; eisaman2011invited ; somaschi2016near , but, so far, there is no certainty about which elements will constitute its fundamental building blocks. However, it appears likely that they will operate within different wavelength regions, ranging from visible hensen2015loophole ; hedges2010efficient via near-infrared 2013entanglementRb ; yu2019entanglement ; saglamyurek2011broadband ; clausen2011Entangled ; hanschke2018quantum ; krutyanskiy2019light , to telecommunication wavelengths saglamyurek2015FiberENt ; kim2016two . This will allow leveraging the best properties of each device, and thereby offer heightened capabilities compared to a network consisting of identical quantum systems.

The development of such networks therefore creates a need, immediately and in the future, for hybridization experiments to bridge existing frequency and bandwidth mismatches. One example is linking quantum memories for light operating at different wavelengths through the exchange of photons. However, so far, only a few investigations have been reported Rempe2011BEC800nm ; 2017Nv13C ; Riedmatten2017Nature , and none of them has included a quantum memory functioning at telecommunication wavelength. In this work, we demonstrate entanglement between two atomic frequency comb (AFC)-based quantum memories afzelius2009AFC in ensembles of cryogenically-cooled rare-earth ions, one for 794 nm and one for 1535 nm wavelength photons. The first memory employs a thulium-doped lithium-niobate (Tm3+:LiNbO3) crystal, the second an erbium-doped fibre (Er3+:SiO2). Entanglement is created through the interaction with entangled photons created by spontaneous parametric down-conversion. Both memories allow buffering and re-emitting multiplexed quantum data in feed-forward-controlled spectral or temporal modes, either of which makes them suitable for quantum repeaters sinclair2014spectral ; simon2007rate . We note that our experiment can also be interpreted as a quantum frequency conversion interface mediated by the spontaneous parametric down-conversion(SPDC) source that connects two quantum memories operating at different wavelengths Riedmatten2017Nature .

Experimental setup. Our experimental setup, outlined in Fig. 1 and further detailed in the Supplemental Material, consisted of four parts: a source of entangled photon pairs; two solid-state memories for light (one doped with thulium, and one with erbium); and a detection system comprising analyzers (including detectors), coincidence electronics and data processing software. Appropriate configuration of the source and detection system allowed measuring the cross-correlation function, reconstructing density matrices, and testing Bell inequalities with photon pairs before and after storage.

To create time-bin entangled pairs of photons at 794 nm and 1535 nm wavelengths, short laser pulses at 523 nm and 80 MHz repetition rate (reference clock) were split by an unbalanced Mach-Zehnder (MZ) interferometer into early, $e$, and late, $\ell$, temporal modes, and used to pump a non-linear crystal that is phase-matched for frequency non-degenerate spontaneous parametric down-conversion (SPDC). Assuming the annihilation of exactly one pump photon, the quantum state of the resulting photon pair is described by the so-called $|\phi^{+}\rangle$ Bell state

[TABLE]

where $|xy\rangle\equiv|x\rangle_{\mathrm{794}}\otimes|y\rangle_{\mathrm{1535}}$, with subscripts denoting each photon’s wavelengthbrendel1999SPDC . The remaining pump light was removed, and the photons of different wavelengths were separated by a dichroic mirror and sent to separate rare-earth-ion-doped memories. In addition, their spectra were filtered to 8 GHz using an air-spaced etalon (for the 1535 nm photons) and 10 GHz using the inhomogeneously-broadened absorption line of Tm (for the 794 nm photons; see the Supplemental Material for details). We adjusted the paths of the photons such that their storage in the memories was temporally overlapped.

The Tm3+:LiNbO3 crystal, used for storing 794 nm photons, and the Er3+:SiO2 fibre, used for storing 1535 nm photons, were cooled to $\sim$0.6 K and exposed to magnetic fields of 125 G and 1500 G, respectively. Under these conditions it is possible to tailor the inhomogeneously broadened absorption lines of each ensemble of rare-earth ions into a large number of spectrally equidistant absorption peaks separated by frequency $\Delta$. This feature, known as an atomic frequency comb (AFC), was first described as a quantum memory in afzelius2009AFC . After a storage time $\tau=1/\Delta$ the photon is re-emitted from the memory (for a short theoretical description of the storage protocol see the Supplemental Material). In our experiment, we prepared storage times of 32 ns and 6 ns, and memory bandwidths of 10 GHz and 8 GHz for the 794 nm and 1535 nm photons. Note that the short storage times were partly determined by the impossibility to independently optimize the magnetic fields to which the ensembles were exposed in a single cryostat (described in the Supplemental Material).

After re-emission, the photons were directed to analyzers consisting either of a short fibre or an interferometer featuring the same path-length difference as that acting on the pump pulses, and detected using superconducting nanowire single-photon detectors (SNSPDs) cooled to 0.8 K in a second cryostat. Finally, the resulting electronic signals were processed in a time-to-digital converter (TDC), and single-detector count rates as well as time-resolved coincidence count rates were recorded by a computer.

Results. Initially, we characterized the source and the two memories individually. For this, we blocked the long arm of the pump interferometer such that the source produced pairs of photons in a single temporal mode. We measured the system storage efficiencies for heralded single photons for each memory, finding 0.1% for the Er-doped fibre and 0.4% for the Tm-doped crystal. Taking independently characterized input and output coupling efficiencies and transmission loss into account, the corresponding device efficiencies were 0.5% and 2%. See Fig. 1 and Supplemental Material for more information.

Then, with the long arm of the pump interferometer still blocked, we measured the cross correlation function $g^{(2)}_{12}(\delta t)$ of the photon pairs before storage in the memories (see the Supplemental Material for more information). Creating photons pairs using the highest pump power available, we found $g^{(2)}_{12}(0)$= 61$\pm$9, which exceeds the maximum value of 2 that is consistent with the assumption of classical fields kuzmich2003generation by 4.5 standard deviations (see Supplemental Material for additional measurements). This verifies the non-classical nature of the photon-pairs. Repeating the measurement with memories in place, we found $g^{(2)}_{12}(\delta t\!=\!32~{}\textrm{ns}\!-\!6~{}\textrm{ns}=26~{}\textrm{ns})$=53$\pm$8 (shown in Fig. 2), where 32 ns and 6 ns denote the pre-set storage time of the thulium and erbium memories, respectively. This result demonstrates that non-classical correlations between the members of photon pairs were preserved during storage.

Next, we opened both arms of the pump interferometer, thereby creating the entangled state described in Eq. 1. We used two approaches to establish that the photons remain entangled after storage, and hence that we preserved the entanglement during storage in the distinct quantum memories.

First, we reconstructed the density matrices of the photon pairs before and after storage. This requires measuring a set of joint (bipartite) projectors (see Supplemental Material). Using a maximum likelihood estimation, the density matrices that best fit the measured projection probabilities were reconstructed; the results are shown in Fig 3. In turn, this allowed us to calculate the entanglement of formation, the purity, the fidelity of the measured states with the target state $|\phi^{+}\rangle$ in Eq. 1, and the fidelity between input and output states (see Supplemental Material for more information). These figures of merit, summarized in Table 1, confirm that the photons remained entangled.

Finally, we also performed a test of the CHSH Bell inequality clauser1969CHSH , which states that the sum of four correlation coefficients $E(\alpha,\beta)$ is upper-bounded by two if the measurement results can be described by local theories:

[TABLE]

Here, $\alpha_{i}$ and $\beta_{i}$ denote different measurement settings – in our case phases set by analyzing interferometers. Using the data given in the Supplemental Material, we found Sin=2.52$\pm$0.02 before, and Sout=2.6$\pm$0.2 after storage, both of which significantly exceed 2.

Violating the CHSH Bell inequality (or any other Bell inequality) proves non-locality rather than entanglement. However, Bell tests allow certifying entanglement within a device-independent framework, that is without making assumptions about, e.g., the dimensions of the Hilbert spaces describing the individual quantum systems scarani2012device . (Note that such certification requires the Bell test to be loophole free, while we made the common assumptions of fair sampling and no signaling.) As such, a Bell-inequality violation is a more stringent test of entanglement than finding positive values for the entanglement of formation.

Conclusion and discussion. Our results show that entanglement persists while propagating through different solid-state quantum memories operating at different wavelengths. However, for future use in a quantum network, several factors must be improved. To increase the device efficiency and storage time of the memory for the 794 nm photons, the Tm3+:LiNbO3 crystal could be replaced by Tm3+:Y3Ga5O12 (Tm:YGG)thiel2014Tm:YGG , which features better coherence properties, inside an impedance-matched cavity afzelius2010impedancecavity . Similarly, the erbium-doped fibre could be replaced by a 167Er-doped Y2SiO5 crystal (Er:YSO) Morgan2018coherence , again inside a cavity. For more information see the Supplemental Material.

Acknowledgements

The authors thank Erhan Saglamyurek and Neil Sinclair for discussions, and acknowledge funding through Alberta Innovates Technology Futures (AITF), the National Science and Engineering Research Council of Canada (NSERC), and the Netherlands Organization for Scientific Research (NWO). Furthermore, WT acknowledges funding as a Senior Fellow of the Canadian Institute for Advanced Research (CIFAR), and VBV and SWN partial funding for detector development from the Defense Advanced Research Projects Agency (DARPA) Information in a Photon (InPho) program. Part of the detector research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA).