Experimental demonstration of switching entangled photons based on the Rydberg blockade effect
Yi-Chen Yu, Ming-Xin Dong, Ying-Hao Ye, Guang-Can Guo, Dong-Sheng Ding, Bao-Sen Shi
EExperimental demonstration of switching entangled photons based on the Rydbergblockade effect
Yi-Chen Yu,
1, 2
Ming-Xin Dong,
1, 2
Ying-Hao Ye,
1, 2
Guang-Can Guo,
1, 2
Dong-Sheng Ding,
1, 2, ∗ and Bao-Sen Shi
1, 2, † Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China. Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China. (Dated: November 3, 2020)The long-range interaction between Rydberg-excited atoms endows a medium with large opticalnonlinearity. Here, we demonstrate an optical switch to operate on a single photon from an entangledphoton pair under a Rydberg electromagnetically induced transparency configuration. With thepresence of the Rydberg blockade effect, we switch on a gate field to make the atomic mediumnontransparent thereby absorbing the single photon emitted from another atomic ensemble via thespontaneous four-wave mixing process. In contrast to the case without a gate field, more than50% of the photons sent to the switch are blocked, and finally achieve an effective single-photonswitch. There are on average 1 ∼ Keywords: Rydberg blockade, entangled photon,quantum switchPacs numbers: 32.80.Rm, 42.50.Gy, 42.50.Ct,42.50.Nn1. Introduction
In analogy to classical electronic counterparts, quan-tum switches are regarded as basic building blocks forquantum circuits and networks [1–4]. Switching statesin the full-quantum regime where single particles controla quantum qubit or entanglement from another systemmay enable further applications in quantum informationscience, such as in quantum computing [5], distributedquantum information processing [6–8], and metrology [9].Many efforts have been done towards constructing a pro-totype; examples include a micro-resonator coupled witha single atom [10], cold atoms trapped in a microscopichollow fiber [11], cold atoms coupled to a cavity [12],strongly coupled quantum cavity-dots [13], and single dyemolecules [14].The strong interaction offered by Rydberg-excitedatoms shifts the energy levels of the surrounding atomsdramatically and suppresses all further excitation of theseneighboring atoms. This interaction between cold atomsgives rise to excitation blockade [15–21], multipartite en-tanglement [22–25], spatial correlations [26–28], strongoptical nonlinearities [29–35], plasma formation [36], andphoton-photon gate [37]. The single-photon nonlinear-ity arising from the strong interaction between Ryd-berg atoms shows great potential in constructing single-photon transistors [38–40]. The fundamental aspects ofquantum nonlinearity based on Rydberg atoms have beenstudied before; a type of " photonic hourglass " single-photon device is constructed [31, 41]. Such " photonic hourglass " is a kind of medium exhibiting strong absorp-tion of photon pairs while remaining transparent to sin-gle photons. However, all the relative experiments onswitching were demonstrated with a weak coherent field,thus there are no reports on switching of a true single-photon. Operating on true single photons is more chal-lenging than attenuated coherent pulses.Here, we demonstrate an experiment of switching truesingle-photons from entangle photon pairs. The entan-gled photon pairs are prepared in one atomic cloud andpropagate through another atomic cloud for switchingoperation. The switching operation is using a single-photon level gate field to turn on or turn off the accep-tance of that true single-photon. We perpare a gate pulsewith 1 ∼ . ± .
5% and 80 . ± .
3% in the absence and pres-ence of a gate field, respectively, with a switch contrastlarger than 50%. By increasing the principal quantumnumber n , the switching effect becomes stronger andthe required photon number of the gate field decreases.Implementing a Rydberg-mediated switch device undernon-classical fields could enable the implementation ofquantum computation and information processing withthe interaction between Rydberg atoms and entangledphotons [5], such as building a Toffoli gate [43] and quan-tum computation [44–47] with Rydberg ensembles, andswitching a distributed quantum node. a r X i v : . [ qu a n t - ph ] O c t
2. Results Rb ) trapped in different magneto-optic traps (MOTs), labeled MOT 1 and MOT 2.Schematics of the energy levels, time sequence, and ex-perimental setup are shown in figure 1 (a)–(c). A cigar-shaped Rb atomic ensemble is first prepared in MOT1 and then cooled down to about 100 µ K via the opticalmolasses technique; the atomic cloud has dimensions of10 × × . We prepare non-classical photon pairsby spontaneous four-wave mixing (SFWM) in this atomicensemble. The energy levels involved here correspond tothe double-Λ system, consisting of both the D1 and D2lines of Rubidium 85. The two pump fields couple theatomic transition 5 S / ( F = 3) → P / ( F = 3) witha detuning of − π ×
110 MHz and the atomic transition5 S / ( F = 2) → P / ( F = 3) under resonance. Thegenerated signal photons (labeled signal 1 and signal 2)are correlated in the time domain. The signal-2 photonpassing through an acousto-optic modulator (AOM) isfrequency shifted by +2 π ×
120 MHz, after which it isexactly resonant with the atomic transition 5 S / ( F =3) → P / ( F = 4). Then, the signal-2 photon propa-gates through the three-dimensional Rb atomic cloudin MOT 2 for the demonstration of the switching process.Finally, we perform quantum state tomography for thephotonic entanglement before two signals are detectedby two single-photon detectors. The coils of MOT 2are switched off during the switching measurement. Thespherical atomic cloud of MOT 2 has a size of 500 µ mwith a temperature ∼ µ K and an average densityof 3 . × cm − . The coupling field is resonant withthe atomic transition 5 P / ( F = 4) → (cid:12)(cid:12) nD / (cid:11) , that is∆ c = 0. Rabi frequency of coupling light is Ω c = 2 π × µ m in thecenter of MOT 2 which is obtained by using a short-focuslens. With a pulsed coupling beam (TA-SHG, Toptica),we demonstrate Rydberg-EIT in the ladder-type atomicconfiguration, consisting of a ground state | g i , an ex-cited state | e i , and a highly-excited state | nD i ; here, n = 50. The gate field has a beam waist of 18 µ m inthe center of MOT 2 and couples the atomic transition5 S / ( F = 3) → P / ( F = 4). The coupling field witha beam waist of 30 µ m covers both the gate and signal-2 beams. The smaller the beam waist, the stronger theblockade effect [31]. However the beam size is limited tothe size of the transmission window of our MOT. Analyz-ing the van der Waals interactions between the Rydbergatoms with an effective coefficient C = 2 π ×
32 GHz · µ m for the rubidium 50 D / by considering weighted aver-age of the interaction effects of all Zeeman sublevels, wecan calculate an average blockade radius ∼ µ m with r b = | C √ (2∆ c ) +Ω c | / [48] [49]. Since the coupling Rabifrequency is larger than the bandwidth of the signal pho-ton, we use the strong coupling configuration to calculatethe blockade radius. Figure 1(d) describes the switchingeffect on a weak coherent pulse, the red and blue linesrepresent the Rydberg-EIT spectra with and without acoherent gate field.2.2. Bandwidth matchingIn order to switch single photons, we need to connecttwo physical systems. One is to generate entangled pho-ton; the other is to operate on that photon. We matchthe frequency and the bandwidth between the signal-2photon and the absorption window of the atomic ensem-ble in MOT 2. This can be realized by changing thefrequency and Rabi frequency Ω p of the pump 2 fieldas explained above. The switching effect obviously de-creases when the bandwidth of signal-2 photon increases.Due to the narrow transparency window in the spectrumof Rydberg-EIT, the optical response on two-photon res-onance is strongly affected by the level shifts induced byRydberg atoms interaction and the linewidths of the in-put lasers [50]. Compared with S state, D state has widertransparency window resulting from the larger dipole ma-trix element to D state. Thus, we use Rydberg- D stateto get larger bandwidth and higher transmission rate ofRydberg-EIT window.As a result, the mismatching between the signal-2 pho-ton and the absorbtion bandwidth of the atomic ensem-ble in MOT 2 decreases the switch contrast. Becausethe high-frequency component of the signal-2 photon isunable to fall within the Rydberg-EIT window, the reab-sorption of the signal-2 photon weakens although the gatefield is present. The switch contrast decreases with in-creasing Rabi frequency Ω p of pump 2 field (see figure 2).The bandwidth of the signal-2 photon depends signifi-cantly on Ω p [51, 52], because the profile of the wavepacket of the signal-2 photon can be modulated by tun-ing the Λ-EIT transparency window. The single-photonbandwidth becomes narrower with the Ω p decreasing.Only when the bandwidth of the single photon is nar-rower than the Rydberg-EIT window, can we get a higherabsorption rate and switch contrast.This data in figure 2hints that the switching effect becomes more obvious witha smaller Ω p . For the optimized case Ω p ∼ π × ∼ π × ∼ π ×
13 MHz. That is, the signal-2 photoncan completely fall within the Rydberg-EIT window. Ifdecreasing Ω p further, the signal to noise ratio of two-photon coincidence becomes worse.2.3. Switch entangled photons ( ) × () Without gate fieldWith gate field |2>|3>
S 1 |1>
P 1 |4> |3>
P 2 S 2 M O T |e>| n D>|g>
CS 2 G |e′> M O T ∆T P 1P 2G P B S BD λ/2λ/2 M O T BD λ/2λ/4 P B S D M L en s L en s M O T D M Signal 2 Pump 1Pump 2 Gate CouplingSignal 1 ab c
AOM d C λ/2λ/2 λ/2 P B S L en s L en s L en s L en s L en s L en s P B SP B S λ/2λ/4 Figure 1. (a) Energy diagram of entanglement generation and the switching processes. The double- Λ atomic configurationcorresponds to the Rb states 5 S / ( F = 2) ( | i ), 5 P / ( F = 3) ( | i ), 5 S / ( F = 3) ( | i ), and 5 P / ( F = 3) ( | i ), respectively.The pump fields are P1 and P2 and the signal fields are S1 and S2. The right-side energy diagram is a ladder-type energylevel with ground state 5 S / ( F = 3) ( | g i ), excited state 5 P / ( F = 4) ( | e i ), and highly-excited state (cid:12)(cid:12) nD / (cid:11) ( | nD i ). Labels:P-pump, S-signal. (b) Time sequence for the preparation and switching on single photons. ∆ T represents the experimental timewindow. (c) Schematic overview of the experimental setup. Labels: PBS-polarizing beam splitter, DM-dichroic mirror, λ/ λ/ n = 50. The solid lines are fitted by the function e − w s /c (1+ χ/ L with OD = 8, Ω c = 2 π × . δ ∆ = 2 π × γ rg = 2 π × .
03 MHz, Γ de = 2 π × .
07 MHz and OD = 8, Ω c = 2 π × δ ∆ = 2 π × . γ rg = 2 π × . To demonstrate the switching effect under quantumregime, we firstly prepared non-classical photon pairs viaSFWM process [51, 52] in MOT 1. The generated pho-ton pairs are correlated in time domain. We constructtwo optical paths L and R by using two beam displac-ers (BDs) to build a passive-locking interferometer [53–55] where the perturbations between L and R opticalpaths can be mutually eliminated. The signal photonsin each path are collinear, as the phase matching con- S w it c h C on t r a s t ! Figure 2. The measured switch contrast against with Ω p , thesolid red curve is guided for eye which is fitted by a function y = A ∗ Exp[ − x/t ] + y with parameters of A = 0 . y =0 . t = 2 . dition k p1 − k s1 = k p2 − k s2 should be satisfied in theSFWM process. With two half-wave plates inserted inthe R optical path, the signal photons along these twooptical paths can be coherently combined by BDs. Theform of the entanglement is | ψ i = ( | H s i | V s i + e iθ | V s i | H s i ) / √ θ , the relative phase between L and R optical paths,setting to zero in our experiment; | H s ,s i and | V s ,s i represent the horizontal and vertical polarized states ofthe signal photons. The details about generating entan- a b with atomsno atoms EIT with gateEIT with no gate Figure 3. Coincidence counts (CC) under different situationscorresponding to: (a) with (blue) and without atoms (red),and (b) under the Rydberg-EIT configuration with (blue) andwithout (red) gate field. All of these CC were detected in4000 s. gled photon pair are in our supplementary materials.In order to demonstrate the switching effect of entan-gled photons, we input entangled photons into MOT 2.In this situation, we use a 50-m fibre to introduce a timedelay in the path of the signal-1 photons. This guaran-tees that the entanglement does not collapse before theswitching process has finished. The results are shownin figure 3(a) and (b); the former shows the coincidencecounts of photon pairs when the atoms in MOT 2 are ab-sent (red) and present (blue), whereas the latter showsthe results under Rydberg-EIT without (red) and with(blue) gate field. Obviously, the coincidence counts de-crease when the gate field is applied as the signal-2 pho-ton is significantly absorbed when compared with the no-gate situation. The central physics behind the operationof a single-photon switch is that the long-range interac-tion between Rydberg atoms endows the Rydberg-EITmedium with a large optical nonlinearity [31, 56], andthe resulting dipole blockade effect makes the mediumnon-transparent. We define a switch contrast to charac-terize the switching effect,C switch = CC
EIT − CC gate CC EIT , (2)where CC EIT and CC gate represent the total coincidencecounts between the signal-1 and signal-2 photons withoutand with a gate field. From the data [figure 3 (a) and (b)],we obtain a switch contrast of C switch = 77 . ± . ∼ π ×
13 MHz.Although the bandwidth of the signal-2 photon wave-packet may be tuned by decreasing the power of pump2 field [52], there is always a high-frequency componentin the wave-packet of the signal-2 photon. And the com-ponent falls outside of the bandwidth of the absorption,which induces an optical precursor [57, 58]. The switchcontrast is also limited by the broadening effect of theRydberg-EIT window, which is maybe caused by the de-phasing of the distribution of Rabi frequencies with theunpolarized atoms in MOT 2. Our experiment is demon-strated with no bias magnetic fields. The atoms can betreated as averagely distributed in all sublevels.We change the detected state ϕ s2 of the signal-2 pho-ton and recorded the coincidence counts under differentsignal-1 states ϕ s1 of | H i , | V i , | H i− i | V i , and | H i + | V i .To obtain the differences with and without the gate field,we recorded these coincidence counts under these situa-tions [figure 4 (a)–(d)]. The coincidence counts with-out (semi-transparent blue)/with (blue) the gate fieldare obviously different. We obtain switch contrasts withC V H switch = 81 . C HV switch = 64 . C RR switch = 52 . C DD switch = 79 .
9% under the four situations ϕ s1 = | V i , ϕ s2 = | H i ; ϕ s1 = | V i , ϕ s2 = | H i ; ϕ s1 = ϕ s2 = | H i− i | V i and ϕ s1 = ϕ s2 = | H i + | V i . The switching operationis effective for any polarization state with treating thesignal-2 photon as if it were in a mixed state. The ob-tained switching contrasts are different depending on thedetected states due to the non-perfect balance of the pho-ton generation rate in the two optical paths and the noiseof each path. In addition, we measure two-photon inter-ference without and with the gate field under the signal-2basis of | H i and | H i + | V i [figure 4 (e) and (f)]. From fig-ure 4 (e), we find the visibility without the gate field ex-ceeding the threshold 70.7%, which means that the entan-glement can be preserved after the transmission throughthe EIT window. It is easy to observe that the propa-gation of the signal-2 photon through the Rydberg-EITmedium doesn’t destroy the entanglement, because theRydberg-EIT is independent on the polarization of thesignal-2 photon.We also perform quantum state tomography [59] forthe photonic entanglement to compare the entanglementproperties before and after the switching process. Signal-1 and signal-2 are polarization entangled, their entangledstate being | ψ i = ( | H i s1 | V i s2 + | V i s1 | H i s2 ) / √
2. Us-ing the polarizing beam splitter, half-wave plate, andquarter-wave plate, we project the two photon statesonto the four polarization states | φ ∼ i ( | H i , | V i ,( | H i − i | V i ) / √
2, ( | H i + | V i ) / √ ρ ideal using the formula F i = Tr( p √ ρρ ideal √ ρ ) ,we obtain the fidelity to be 87 . ± .
5% correspond-ing to the fidelity of input photons. By comparingwith the input density matrix using the formula of F = Tr( p √ ρ output ρ input √ ρ output ) , the fidelity for theoutput state without gate field is 85 . ± . . ± . n to change the interaction strength to mea-sure both the Rydberg-EIT transmission contrast andthe switch contrast. Here the Rydberg-EIT contrastis defined as C EIT = (CC no − atom − CC EIT ) / CC no − atom , with Gateno Gate φ =| H > with Gateno Gate φ =| H >+| V > Signal 1 state ( φ )| V > | H >- i | V > | H >+| V >| H > | V > | H >- i | V > | H >+| V >| H > Signal 1 state ( φ ) φ =| H >with Gateno Gate φ =| V >with Gateno Gate φ =| H >- i | V >with Gateno Gate φ =| H >+| V >with Gateno Gate a c e b d f Figure 4. Switching with different bases: (a)–(d) are the coincidence counts without a gate field (semi-transparent blue column)and with a gate field (blue column) under different signal-1 states of | H i , | V i , | H i − i | V i , and | H i + | V i . (e) and (f) arethe recorded two-photon interference curves with signal-2 states of | H i and | H i + | V i without gate field (red) and with gatefield (blue). Their interference visibilities are 87 . ± .
8% ((e), red), 82 . ± .
7% ((e), blue), 72 . ± .
1% ((f), red), and57 . ± .
6% ((f), blue). The solid lines are fitted curves to the measured data. All these coincidence counts were recordedover a interval of 1000 s. Error bars are ± CC no − atom representing the total coincidence counts be-tween the signal-1 and signal-2 photons without atoms.We change the principal quantum number n by chang-ing the wavelength of the coupling laser. Each time wechange the wavelength, we adjust the experimental op-tical system to keep the coupling Rabi frequency a con-stant. The results (figure 5) show that the Rydberg-EIT contrast decreases with the increase of n ; this is be-cause the transition amplitude for | e i → | nD i decreases.In contrast, because the dipole interaction strength in-creases, the switch contrast of the signal-2 photon ob-viously increases by comparing two situations, n = 60( r b = 5 . µ m) and n = 40 ( r b = 1 . µ m). The switchcontrast is larger than 50% when n >
45, revealing aneffective switching operation. In this way, the interactionbetween Rydberg atoms becomes stronger with the prin-cipal quantum number increasing. Although the gatefield has hundreds of photons because of the relativelylarge size of the atomic cloud in our experiment, we cal-culate that there are on average 1 ∼ ~ ∗ ω . Eventually, the single-photon switchwith a single gate photon can be realized by trapping theatoms into the scale of the blockade radius and increasingthe principal quantum number n .
3. Discussion
In addition, in order to avoid saturation of our switchsystem where the gate and coupling field would depletethe atoms after a certain duration, we set the experi-mental time window to 25 µ s. The lifetime of Rydberg-excited atom is estimated to be ∼
800 ns by measuring
Figure 5. Dependence of the switch contrast (red) andRydberg-EIT contrast (blue) on the principal quantum num-ber. As visual guides, the data are fitted with function y = A ∗ Exp[ − x/t ] + y (dotted lines) with parameter set-tings A = − . y = 0 . t = − .
77 and A = − . y = 0 . t = 7 .
0. In this process, the gate field intensity isset to average ∼ ± the Rydberg spinwave through storage process [53, 60],which guarantees an adequate interaction time for eachswitch operation during near 200 ns arrival time of thesignal-2 photon given in figure 3. In our experiment,the dephasing of Rydberg D state [33] is not obviousdue to the small gate photon numbers used here. Withsmall photon numbers as input, we have not investi-gated an obvious time dependence of the transmissionon Rydberg-EIT resonance, but with an obvious timedependence of the transmission for large photon num-bers, more details are shown in supplementary materi-als figure 2. Thus, the nonlinearity behind the switchexperiment is offered by Rydberg blockade effect (seemore details in supplementary materials figure 3). Be-sides, there are some challenges to improve the switchcontrast: 1. Decreasing the bandwidth mismatch be-tween signal-photon and the Rydberg-EIT transparencywindow. 2. Increasing the principle quantum number n to achieve large dipole-dipole interaction strength, asshown in figure 5. 3. Trapping the atoms into the scaleof the blockade radius. 4. Using a single-photon withultra-narrow bandwidth. The bandwidth of transparencywindow of Rydberg-EIT with large principle quantumnumber n would become narrower due to smaller naturalline width for high- n Rydberg state. This means thatit needs narrower bandwidth of single-photon, such assubnatural-linewidth single-photon source [52].In summary, we have demonstrated an optical switchon entangled photons based on Rydberg nonlinearitywith two atomic ensembles. The emitted signal-2 pho-ton correlated with the signal-1 photon is blocked by an-other gate field under the Rydberg-EIT configuration.Switching effect depends on the principal quantum num-ber, the bandwidth of the emitted single photons, andthe average photon number of the gate field. We havesuccessfully realized optical switch on one of the entan-gled photons of the pair, with more than 50% of pairsbeing blocked. These results on switching single pho-tons using the strong dipole interaction hold promises indemonstrating quantum information processing betweenRydberg atoms and entangled photons.
4. Acknowledgements
The authors thank Prof. Lin Li from huazhong univer-sity of science and technology and Prof. Yuan Sun fromnational university of defense technology for valued dis-cussions and critical reading our manuscript. This workwas supported by National Key Research and Devel-opment Program of China (2017YFA0304800), the Na-tional Natural Science Foundation of China (Grant Nos.61525504, 61722510, 61435011, 11174271, 61275115,11604322), Anhui Initiative in Quantum InformationTechnologies (AHY020200), and the Youth InnovationPro motion Association of Chinese Academy of Sciencesunder Grant No. 2018490. ∗ [email protected] † [email protected][1] Juan Ignacio Cirac, Peter Zoller, H Jeff Kimble, andHideo Mabuchi, “Quantum state transfer and entangle-ment distribution among distant nodes in a quantum net-work,” Phys. Rev. Lett. , 3221 (1997).[2] Jeremy L O’brien, Akira Furusawa, and Jelena Vuˇckovi´c,“Photonic quantum technologies,” Nat. 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1, 2
Ming-Xin Dong,
1, 2
Ying-Hao Ye,
1, 2
Guang-Can Guo,
1, 2
Dong-Sheng Ding,
1, 2, ∗ and Bao-Sen Shi
1, 2, † Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China. Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China. (Dated: November 3, 2020)
Experimental time sequence.
The repetition rateof our experiment is 200 Hz, and the MOT trapping timeis 4.71 ms. The experimental time window is 25 µ s. Themagnetic field is switched off during the switch process.The fields of pumps 1 and 2 are controlled by two AOMs,and therefore the frequencies of signals 1 and 2 photonscan be tuned. The optical depth in MOT 1 is about 20.Each of the two lenses with a focal length of 300 mm, isused to couple the signal fields into the atomic ensem-ble in MOT 1. The short-focus lens (the blue lens infigure 1) with a focal length of 30 mm is used to cou-ple the signal-2 field into the atomic ensemble in MOT2. The fields of P1 and P2 are collinear, and the sig-nal fields S1 and S2 are collinear. The phase match-ing condition k p − k s = k p − k s is satisfied in thespontaneous four-wave mixing process. The two signalphotons are collected into their respective single-modefibers and are detected by two single-photon detectors(avalanche diode, PerkinElmer SPCM-AQR-16-FC, 60%efficiency, maximum dark count rate of 25/s). The twodetectors are gated by an arbitrary function generator.The gated signals from the two detectors are then sent toa time-correlated single-photon counting system (Time-Harp 260) to measure their time-correlated function. Generating entangled photons . We generate en-tangled photon pairs in MOT 1 which is presented inthe upper half of fig.1 (c). We generate the entan-gled photons via SFWM process by using the counter-propagating pump fields 1 and 2. The SFWM process isbased on double- Λ atomic configuration with energy lev-els of ground state | i , metastable state | i , excited states | i and | i , which correspond to 5S (F=2), 5S (F=3),5P (F’=3) and 5P (F’=3). Pump 1 at 795 nm withhorizonal polarization is from an external-cavity diodelaser (DL100, Toptica). Pump 2 at 780 nm with verti-cal polarization into the atomic medium is from anoterexternal-cavity diode laser (DL100, Toptica). These twopump lasers with different polarizations are propagat-ing in a strictly collinear geometry, with an angle of 2 ◦ away from the direction of collected photons [1]. Signal1 and signal 2 photons are non-classically correlated [2].Afterwards, we form a phase self-stabilized multiplexingstructure where the relative phase between different sig-nal paths can be eliminated because of the symmetricalstructure, by using the beam displacer 1, beam displacer2 and two half-wave plates. Then, signal 1 and signal 2 photons are not only non-classical correleted but alsopolarization entangled. The form of the entanglement is | ψ i = ( | H s i | V s i + e iθ | V s i | H s i ) / √
2. In Ref [3], wehave already checked the self-stabilization of our struc-ture and the high fidelity of polarization entanglement.We can produce all four Bell states by adjusting the twoBDs.
Theoretical analysis.
We use the Lindblad masterequation: dρ/dt = − i [ H, ρ ] / ~ + L/ ~ in the absence ofRydberg-mediated interactions to characterize the inter-action of the EIT light fields with an ensemble of threelevel atoms, where ρ is the the atomic ensemble’s densitymatrix and H is the atom-light interaction Hamiltoniansummed over all the single-atom Hamiltonians under ro-tating wave approximation with H = P k H [ ρ ( k ) ]. H [ ρ ( k ) ] = − ~ p p − p Ω c Ω c − p + ∆ c ) The Lindblad superoperator L = P k L [ ρ ( k ) ] is com-prised of the single-atom superoperators. The Lindbladmaster equation includes both spontaneous emissions anddephasing. Ω p and Ω c are the Rabi frequency of theprobe and coupling light. ∆ p and ∆ c are the detuning ofprobe and coupling light, respectively. Considering thedephasing of Rydberg state, we introduce γ d to charac-terize the decoherence. Since the dephasing of Rydbergstate doesn’t include population transfer, it ought to beincluded only as a decay of the coherence, that is in theoff-diagonal terms of the Lindblad operator. The single-atom Lindblad superoperator is :2 L [ ρ ( k ) ] / ~ = Γ e ρ ( k ) ee − Γ e ρ ( k ) ge − Γ r ρ ( k ) gr − Γ e ρ ( k ) eg − Γ e ρ ( k ) ee + 2 Γ r ρ ( k ) rr − ( Γ e + Γ r ) ρ ( k ) er − Γ r ρ ( k ) rg − ( Γ e + Γ r ) ρ ( k ) re − Γ r ρ ( k ) rr + − γ de ρ ( k ) ge − γ de ρ ( k ) eg − γ de ρ ( k ) er − γ de ρ ( k ) re a r X i v : . [ qu a n t - ph ] O c t + − γ dr ρ ( k ) gr − γ dr ρ ( k ) er − γ dr ρ ( k ) rg − γ dr ρ ( k ) re where Γ e and Γ r are the natural population decayrates of excited state | e i and Rydberg state | r i , respec-tively. γ de and γ dr are the decay of the coherence of ex-cited state | e i and Rydberg state | r i caused by collisionsand stray fields. We solve the Lindblad master equa-tion under the condition of steady state corresponding to t → ∞ , dρ/dt = 0 to get ρ eg . The complex susceptibilityof the EIT medium is χ = ( N | µ ge | /(cid:15) ~ ) ρ eg [4], with N being the atom number density in the medium, and µ ge being the electric dipole moment of the state | g i → | e i .Under the plane-wave approximation, the transmissionof the signal-2 photon through the EIT medium can beobtained from the susceptibility via e − w s /c (1+ χ/ L ,where L is the length of the atomic medium, w s the fre-quency of the signal-2 photon, c the speed of light in avacuum. We calculate the complex linear susceptibility[5] : χ = α k w s + δ ∆ + iγ rg ) γ eg Ω c − w s + δ ∆ + iγ rg )( w s + δ ∆ + iγ eg ) (1)where α = OD/L is the absorption coefficient when thecoupling field is not present, OD is the optical depth ofatomic ensemble. k is the wave vector. w s is thedetuning of the signal-2 photon. δ ∆ is the frequencyshift used for fitting EIT spectrum. γ eg = Γ e + γ de isthe decay rate of atomic transition | e i → | g i correspond-ing to the natural linewidths of | e i . γ rg = Γ r + γ dr isthe decay rate of atomic transition | nD i → | g i [6]. Ω c represents the Rabi frequency of the coupling field. Weuse this equation 1 to simulate the results given in fig-ure 1 (a) and (b). The strong dipole interaction couplesthe nearby Rydberg atoms so that the evolution of theseatoms are fundamentally linked, thereby modifying theindividual atomic energy levels [7]. As a result of theRydberg dipole interactions, the behaviour of an ensem-ble of N -atoms cannot simply be described by summingthe response of a single atom N times. To describe thebehavior for Rydberg-EIT with a gate field, we introduce γ d to characterize the decoherence to fit the data fromour experimental result. If there are no Rydberg dipoleinteractions, the transmission of the N -atom system canbe traced to a summation of N single-atom contributions. γ d would increase when the Rydberg atoms interact, andthe response of each atom is modified significantly. Be-cause of this process, the response of Rydberg atomswould exhibit non-transparency behavior for signal pho-tons when γ dr is large enough. Thus, we can control thesingle-photon transmission behavior of Rydberg-EIT de-pending on whether nearby Rydberg atoms are excited. In our experiment, the coupling Rabi frequency is higherthan the polarization decay rate of the excited state butnot much higher enough to absolutely separate the twoabsorption spectra [8]. Furthermore, we have checked thelinear susceptibility under the condition of either EIT orAT effect, confirming that the result can be used in allvalue of Ω c . Our scheme of switching an entangled pho-ton could work well under both EIT and AT conditions. a b × × Figure 1. Transmission of the signal-2 photon under differentconditions. (a) Absorption spectra of the signal-2 photon withatoms present in MOT 2. The data are fitted by function e − w s /c (1+ χ/ L + a (solid lines) with parameter valuesof Ω c = 0, γ eg = 2 π × γ rg = 2 π × OD = 20,and a = 0 .
04. (b) Rydberg-EIT effect of the signal-2 photonwith coupling field present. The data are fitted by the samefunction above with parameter values of Ω c = 2 π ×
11 MHz, γ eg = 2 π × γ rg = 2 π × . OD = 20, a = 0 . δ ∆ = − π × . Frequency matching between signal 2 andatoms in MOT 2 . We need to match the frequencywindows to connect two different physical systems. Forthis point, the emitted signal-2 photon from MOT 1 maynot be matched with the working window of Rydberg-EIT in MOT 2. The detuning of the signal-2 photonis performed by changing the frequency of the pump-2 field, which is controlled by an AOM. The pump 2field passes through the AOM with a frequency shiftfrom − π × ∼ π ×
17 MHz. There is another AOMadded in the optical path to give the signal-2 photon(see figure 1 (c) in main text) a frequency shift with+2 π ×
120 MHz. By this method, the frequency of thesignal-2 photon can be tuned from the atomic transi-tion 5 S / ( F = 3) → P / ( F = 3) to the atomictransition 5 S / ( F = 3) → P / ( F = 4). To checkwhether the signal-2 photon falls into the atomic tran-sition window 5 S / ( F = 3) → P / ( F = 4) in MOT2, we measure its absorption and EIT transmission spec-tra (figure 3) by changing the frequency of the emittedsignal-2 photon. To check this process, we added a cou-pling field, which is resonant with the atomic transition5 P / ( F = 3) → D / to demonstrate the Rydberg-EIT. Time dependence of the transmission spectrumof Rydberg- D state. The Ref. [9] has reported thedephasing of Rydberg- D state polaritons, namely, timedependence of the transmission on EIT resonance. As Time (µs) T r a n s m i ss i on ( no r m a li ze d ) T r a n s m i ss i on ( a . u . ) Time (µs) a b
Figure 2. (a) The normalized transmission of probe fieldon Rydberg-EIT resonance. The red line represents 1.5photons/ µs , the blue line corresponds to 38.7 photons/ µs . (b)Time dependence of probe transmission with different pho-tons (not normalized, only see the profile of the transmissionline). The red line represents 38.7 photons/ µs , the blue linecorresponds to 601.4 photons/ µs . stated in this reference, the Rydberg- D state dephas-ing effect is caused by the interaction-induced couplingto degenerate Zeeman sublevels, preventing other pho-tons propagating through the cloud. We also checkedthe D -state dephasing in our experiment to demonstrateprobe field transmission under the EIT resonance con-dition. We use a probe field with 1.5 photons/ µs as aninput field, the transmission is higher than using 38.7photons/ µs , (see the results given in Fig. 2 (a)). Wewant to show that the gate field with 1.5 photon/ µs isnot enough to reach a strong quantum nonlinearity toobviously destroy the transparency. Thereby, the trans-mission rate is almost 1 (actually lower than 1). But thestrong interaction between Rydberg atoms could easilytunes the transition out of resonance on the conditionof the gate field with 38.7 photons/ µs . The compari-son is consistent with the result of our experiment. Andthe gate field is almost 15.5 photons/ µs during our ex-periment. We use the normalized transmission in theleft panel to distinguish the transmission rate under twodifferent conditions. The Rydberg- D state dephasing isnot obvious for small photon numbers, but with an obvi-ous time dependence with much more photon numbers,given in Fig. 2 (b). So, the D -state dephasing caused bythe gate field is not obvious in our experiment, for thegate photon number we used in our experiment is 15.5photons/ µs , which is smaller than 38.7 photons/ µs . Inaddition, we also compared the Rydberg-EIT effects for D state and S state by tuning the coupling laser wave-length, the only difference between them in our configura-tion is the height of the transparent peak with the trans-parent peak of D state higher than that of S state. In or-der to obviously observe the blockade effect in Rydberg-EIT, we choose Rydberg- D state to demonstrate the ex-periment. Nonlinear response of Rydberg blockaded en-semble.
We provide the visualized evidence of nonlin-ear response of Rydberg blockaded ensemble. Before theswitch operation in our experiment, we change the input
30 µm N input N output Figure 3. We change the input photon numbers N input from aweek coherent light beam. Then we record the output photonnumbers N output after the propagation through the MOT 2proving that N output nonlinearly increase as N input increase. photon numbers (N input ) of signal-2 photons which arefrom a week coherent light beam. Then we record theoutput photon numbers (N output ) after the propagationthrough the MOT 2 Rydberg-EIT transparent windowby a fast and high-efficient camera (PI-MAX 4, prince-ton instruments) shown in Fig. 3. As we increase N input ,N output nolinearly increase or even decrease because ofthe Rydberg-blockade effect. Thus, there is a thresholdin our Rydberg ensemble. If the input photon numbersexceed that threshold, our ensemble exists large nolin-earity to block the input photons. In fact, the gate fieldis applied to our ensemble to approach that threshold. ∗ [email protected] † [email protected][1] Kaiyu Liao, Hui Yan, Junyu He, Shengwang Du, Zhi-Ming Zhang, and Shi-Liang Zhu, “Subnatural-linewidthpolarization-entangled photon pairs with controllable tem-poral length,” Phys. Rev. Lett. , 243602 (2014).[2] Dong-Sheng Ding, Zhi-Yuan Zhou, Bao-Sen Shi, Xu-BoZou, and Guang-Can Guo, “Generation of non-classicalcorrelated photon pairs via a ladder-type atomic configu-ration: theory and experiment,” Optics express , 11433–11444 (2012).[3] Yi-Chen Yu, Dong-Sheng Ding, Ming-Xin Dong, ShuaiShi, Wei Zhang, and Bao-Sen Shi, “Self-stabilized narrow-bandwidth and high-fidelity entangled photons generatedfrom cold atoms,” Phys. Rev. A , 043809 (2018).[4] Stephen E Harris, JE Field, and A Imamo˘glu, “Nonlinearoptical processes using electromagnetically induced trans-parency,” Physical Review Letters , 1107 (1990).[5] Shanchao Zhang, JF Chen, Chang Liu, Shuyu Zhou, MMTLoy, George Ke Lun Wong, and Shengwang Du, “A dark-line two-dimensional magneto-optical trap of 85rb atomswith high optical depth,” Review of Scientific Instruments , 073102 (2012).[6] U. Raitzsch, R. Heidemann, H. Weimer, V. Bendkowsky,and T. Pfau, “Investigation of dephasing rates in an in-teracting rydberg gas,” New Journal of Physics , 6456–6460 (2008).[7] James Keaveney, Collective Atom–Light Interactions inDense Atomic Vapours (Springer, 2014).[8] Tony Y. Abi-Salloum, “Interference between competing pathways in the interaction of three-level ladder atomsand radiation,” Journal of Modern Optics (2007),10.1080/09500341003658147.[9] Christoph Tresp, Przemyslaw Bienias, Sebastian We- ber, Hannes Gorniaczyk, Ivan Mirgorodskiy, Hans PeterB¨uchler, and Sebastian Hofferberth, “Dipolar dephas-ing of rydberg d-state polaritons,” Phys. Rev. Lett.115