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Photon-Photon Interactions via Rydberg Blockade

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Photon-Photon Interactions via Rydberg Blockade Alexey V. Gorshkov, 1 Johannes Otterbach, 2 Michael Fleischhauer, 2 Thomas Pohl, 3 and Mikhail D. Lukin 4 1 Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA 2 Department of Physics and Research Center OPTIMAS, Technische Universit¨at Kaiserslautern, 67663, Kaiserslautern, Germany 3 Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 4 Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA (Dated: March 22, 2011) We develop the theory of light propagation under the conditions of electromagnetically induced transparency (EIT) in systems involving strongly interacting Rydberg states. Taking into account the quantum nature and the spatial propagation of light, we analyze interactions involving few- photon pulses. We demonstrate that this system can be used for the generation of nonclassical states of light including trains of single photons with an avoided volume between them, for imple- menting photon-photon quantum gates, as well as for studying many-body phenomena with strongly correlated photons. PACS numbers: 42.50.Nn, 32.80.Ee, 42.50.Gy, 34.20.Cf The phenomenon of electromagnetically induced trans- parency (EIT) [1] in systems involving Rydberg states [2] has recently attracted significant experimental [3–10] and theoretical [11–21] attention. While EIT allows for strong atom-light interactions without absorption, Ryd- berg states provide strong long-range atom-atom inter- actions. Therefore, the resulting combination of EIT with Rydberg atoms is ideal for implementing mesoscopic quantum gates [2, 16] and for inducing strong photon- photon interactions, with applications to photonic quan- tum information processing [2, 11–14, 19–22] and to the realization of many-body phenomena with strongly inter- acting photons [23]. At the same time, the many-body theoretical description of EIT with arbitrarily strongly interacting Rydberg atoms, taking into account the full quantum dynamics and the spatial propagation of light, has not been reported previously. In this Letter, we develop such a theory by analyzing the problem for at most two incident photons, which, in turn, provides intuition for understanding the full multi- photon problem. We show that Rydberg atom inter- actions give rise to photon-photon interactions, which, below a critical inter-photonic distance, turn the EIT medium into an effective two-level medium. This can be used to implement photon-atom and photon-photon phase gates and to enable deterministic single-photon sources. The basic physics is illustrated by considering a simple case [Fig. 1(b)], in which a single-photon wavepacket E propagates in an EIT medium [level scheme in Fig. 1(a)] with a central control atom at z = 0 prepared in a Ry- dberg state |r 0 i. Atoms in another Rydberg state |ri, coupled by the EIT control laser [Fig. 1(a)], experience a van der Waals potential V (z)= C 6 /z 6 due to the in- teraction with the control atom, which is decoupled from the applied fields. Alternatively, one could apply an elec- tric field to induce dipole moments in states |ri and |r 0 i, resulting in V 1/z 3 [2]. Sufficiently far away from z = 0, the incident photon propagates in a standard EIT medium featuring a two- photon-resonant control field with Rabi frequency Ω. In the vicinity of the control atom, however, the state |ri is shifted so strongly out of resonance that the photon sees only a two-level (|gi, |ei) medium with transition linewidth 2γ . The critical z, at which the interaction is equal to the EIT linewidth, separates these two regimes and corresponds to the Rydberg blockade radius [11, 24]. When the single-photon detuning Δ = 0, the resonant blockade radius z b is thus defined by V (z b )=Ω 2 (~ = 1), while for Δ γ , we define the off-resonant blockade radius z B via V (z B ) = Ω 2 /Δ (we assumed Δ/C 6 > 0). Since the blockade region extends over 2z b(B) [Fig. 1(b)], the presence of the control atom locally cre- ates an absorbing or refractive medium with an effective optical depth d b(B) =2dz b(B) /L, where d is the resonant optical depth of the |gi-|ei medium (Ω = 0) of length L. Interesting effects occur at large blockaded optical depths d b(B) . In the resonant case, assuming d b 1, the pres- ence of the |r 0 i excitation causes complete scattering, i.e. (a) (b) (c) (d) FIG. 1. (a) EIT level scheme, in which a ground state |gi, an excited state |ei, and a strongly interacting Rydberg state |ri are coupled by a quantum probe field E and a classical con- trol field with Rabi frequency Ω and single-photon detuning Δ. (b) Interaction of one photon with a Rydberg excitation stored at z = 0, which modifies the propagation within the blockade region |z| <z b(B) . (c,d) Interaction of two counter- propagating (c) or co-propagating (d) photons. arXiv:1103.3700v1 [quant-ph] 18 Mar 2011
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Page 1: Photon-Photon Interactions via Rydberg Blockade

Photon-Photon Interactions via Rydberg Blockade

Alexey V. Gorshkov,1 Johannes Otterbach,2 Michael Fleischhauer,2 Thomas Pohl,3 and Mikhail D. Lukin4

1Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA2Department of Physics and Research Center OPTIMAS,

Technische Universitat Kaiserslautern, 67663, Kaiserslautern, Germany3Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

4Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA(Dated: March 22, 2011)

We develop the theory of light propagation under the conditions of electromagnetically inducedtransparency (EIT) in systems involving strongly interacting Rydberg states. Taking into accountthe quantum nature and the spatial propagation of light, we analyze interactions involving few-photon pulses. We demonstrate that this system can be used for the generation of nonclassicalstates of light including trains of single photons with an avoided volume between them, for imple-menting photon-photon quantum gates, as well as for studying many-body phenomena with stronglycorrelated photons.

PACS numbers: 42.50.Nn, 32.80.Ee, 42.50.Gy, 34.20.Cf

The phenomenon of electromagnetically induced trans-parency (EIT) [1] in systems involving Rydberg states[2] has recently attracted significant experimental [3–10]and theoretical [11–21] attention. While EIT allows forstrong atom-light interactions without absorption, Ryd-berg states provide strong long-range atom-atom inter-actions. Therefore, the resulting combination of EITwith Rydberg atoms is ideal for implementing mesoscopicquantum gates [2, 16] and for inducing strong photon-photon interactions, with applications to photonic quan-tum information processing [2, 11–14, 19–22] and to therealization of many-body phenomena with strongly inter-acting photons [23]. At the same time, the many-bodytheoretical description of EIT with arbitrarily stronglyinteracting Rydberg atoms, taking into account the fullquantum dynamics and the spatial propagation of light,has not been reported previously.

In this Letter, we develop such a theory by analyzingthe problem for at most two incident photons, which, inturn, provides intuition for understanding the full multi-photon problem. We show that Rydberg atom inter-actions give rise to photon-photon interactions, which,below a critical inter-photonic distance, turn the EITmedium into an effective two-level medium. This canbe used to implement photon-atom and photon-photonphase gates and to enable deterministic single-photonsources.

The basic physics is illustrated by considering a simplecase [Fig. 1(b)], in which a single-photon wavepacket Epropagates in an EIT medium [level scheme in Fig. 1(a)]with a central control atom at z = 0 prepared in a Ry-dberg state |r′〉. Atoms in another Rydberg state |r〉,coupled by the EIT control laser [Fig. 1(a)], experiencea van der Waals potential V (z) = C6/z

6 due to the in-teraction with the control atom, which is decoupled fromthe applied fields. Alternatively, one could apply an elec-tric field to induce dipole moments in states |r〉 and |r′〉,

resulting in V ∝ 1/z3 [2].

Sufficiently far away from z = 0, the incident photonpropagates in a standard EIT medium featuring a two-photon-resonant control field with Rabi frequency Ω. Inthe vicinity of the control atom, however, the state |r〉is shifted so strongly out of resonance that the photonsees only a two-level (|g〉, |e〉) medium with transitionlinewidth 2γ. The critical z, at which the interaction isequal to the EIT linewidth, separates these two regimesand corresponds to the Rydberg blockade radius [11, 24].When the single-photon detuning ∆ = 0, the resonantblockade radius zb is thus defined by V (zb) = Ω2/γ(~ = 1), while for ∆ γ, we define the off-resonantblockade radius zB via V (zB) = Ω2/∆ (we assumed∆/C6 > 0). Since the blockade region extends over 2zb(B)

[Fig. 1(b)], the presence of the control atom locally cre-ates an absorbing or refractive medium with an effectiveoptical depth db(B) = 2dzb(B)/L, where d is the resonantoptical depth of the |g〉−|e〉 medium (Ω = 0) of length L.Interesting effects occur at large blockaded optical depthsdb(B). In the resonant case, assuming db 1, the pres-ence of the |r′〉 excitation causes complete scattering, i.e.

(a) (b)

(c)(d)

FIG. 1. (a) EIT level scheme, in which a ground state |g〉, anexcited state |e〉, and a strongly interacting Rydberg state |r〉are coupled by a quantum probe field E and a classical con-trol field with Rabi frequency Ω and single-photon detuning∆. (b) Interaction of one photon with a Rydberg excitationstored at z = 0, which modifies the propagation within theblockade region |z| < zb(B). (c,d) Interaction of two counter-propagating (c) or co-propagating (d) photons.

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Page 2: Photon-Photon Interactions via Rydberg Blockade

2

loss, of the incoming photon. Off resonance, for dB 1and dB(γ/∆)2 1, the interaction with the |r′〉-atomimprints a phase ∼ dBγ/∆ on the probe photon and re-duces its group delay by ∼ dBγ/Ω

2, as its group velocityis increased to the speed of light, c, within the blockaderegion.

In the off-resonant case, this simple system has di-rect practical applications. First, by encoding a qubitin the ground and |r′〉 states of the central control atom,one can implement a phase gate between the probe pho-ton and the atom. Second, the protocol of Ref. [25] al-lows to implement a phase gate between two photonsby successively sending them past the control atom thatis appropriately prepared and manipulated between thepasses. Selective manipulation of the control atom can beachieved particularly simply if it is of a different speciesor isotope. Third, a phase gate between two photons canalso be achieved by storing one of them in the |r′〉 stateof the control atom and sending the other one throughthe medium. While storing a single photon in a singleatom is difficult, the same effect can be achieved by stor-ing [26, 27] the photon in a collective |r′〉 excitation, aswe will discuss below.

The results of this simple problem can be extended tothe case of multi-photon EIT propagation in Rydbergmedia. First, off-resonance, two counter-propagatingphotons [Fig. 1(c)] can pick up a phase ∼ dBγ/∆, en-abling the implementation of a two-photon phase gate[12, 14]. Second, a pulse of co-propagating photons [Fig.1(d)] will evolve into a non-classical state correspondingto a train of single photons [19] and exhibiting correla-tions similar to those of hard-sphere particles with ra-dius zb(B)/2. These correlations arise from scattering ofphoton pairs within the blockade region. Third, in theregime where zb is larger than the EIT-compressed pulselength, σ, both co- and counter-propagating resonant se-tups might be usable as single-photon sources since allbut one excitation will be extinguished. In the follow-ing, we present a detailed theoretical analysis of thesephenomena.

Interaction of a photon with a stationary excitation.—We begin by detailing the solution of the problem of a sin-gle photon propagating in a medium where state |r〉 expe-riences a potential V (z) [Fig. 1(b)]. Treating the mediumin a one-dimensional continuum approximation, workingin the dipole and rotating-wave approximations, and adi-abatically eliminating the polarization on the |g〉 − |e〉transition, the slowly varying electric field amplitude Eof the single-photon wavepacket and the polarization Son the |g〉 − |r〉 transition obey [26, 27]

(∂t + c∂z)E(z, t) = − g2nΓ E(z, t)− g√nΩ

Γ S(z, t), (1)

∂tS(z, t) = −iU(z, t)− Ω2

Γ S(z, t)− g√nΩ

Γ E(z, t). (2)

Here Γ = γ − i∆, U(z, t) = V (z)S(z, t), g is the atom-field coupling constant, and n is the atomic density. We

have neglected the depletion of state |g〉 and the finitelifetime of the Rydberg state |r〉, which is typically muchlonger than the propagation times considered here [2].Assuming that all atoms are in state |g〉 before the arrivalof the photon, Eqs. (1,2) can be solved to give

E(L2 , t)

=

∫ ∞−∞

dωe−iω(t−L

c )+ik2

∫ L2

−L2

dzχ(z,ω)E(−L2 , ω

),(3)

where

kχ(z, ω) =1

L

dγ[ω − V (z)]

Ω2 − (∆ + iγ)[ω − V (z)](4)

and E (−L/2, ω) is the Fourier transform of thewavepacket incident at z = −L/2.

For narrowband pulses, we expand χ in ω and, assum-ing ∆ γ and L 2zB, reduce Eq. (3) to

E (L/2, t) ≈ E (−L/2, t− L′/vg) eiϕ−η, (5)

where vg ≈ cΩ2/(g2n) = 2Ω2L/(dγ) is the EIT groupveclocity. In order to avoid the Raman resonance atV + Ω2/∆ = 0, we assumed ∆/C6 > 0. Since the photontravels at c within the blockade region, the group delaycomes from a reduced medium length L′ = L − 7

9πzB ≈L − 2zB. Additionally, the emergence of a two-levelmedium within |z| < zB gives an intensity attenuation ofe−2η with 2η = 5π

18 dB(γ/∆)2 ≈ dB(γ/∆)2 and a picked-up phase of ϕ = −π6 dB(γ/∆) ≈ − 1

2dB(γ/∆). Thus, withdB 1 and a properly chosen ∆ γ, one can get aconsiderable phase and/or change in group delay with-out significant absorption. For the same derivation onresonance (∆ = 0), the main effect is an intensity atten-uation of≈ exp(−db), as expected for a two-level mediumof length 2zb.

It is straightforward to extend our analysis to a delo-calized |r′〉 excitation, i.e. a spin wave, that is spread overmany atoms. Far off resonance, the effect of the controlatom is independent of its position, such that a singlecontrol atom and a corresponding spin wave affect theincident photon identically. On resonance, with db 1,the |r′〉 spin wave causes complete scattering of the in-coming photon. At the same time, after tracing out thescattered photon, which carries information about the lo-cation of the scattering, the spin wave itself disentanglesinto a classical mixture of pieces of length ∼ zb.Interaction of propagating photons.—We now consider

the problem of propagating photons interacting with eachother. Regarding E and S in Eqs. (1,2) as operatorswith same-time commutation relations [E(z), E†(z′)] =[S(z), S†(z′)] = δ(z − z′) [27] and taking U(z) =∫dz′V (z−z′)S†(z′)S(z′)S(z), Eqs. (1,2) become Heisen-

berg operator equations [28] for the case of photons co-propagating in a Rydberg EIT medium [Fig. 1(d)]. Al-ternatively, for the case of two counter-propagating pho-tons [Fig. 1(c)], we define operators E1(2) and S1(2) for

Page 3: Photon-Photon Interactions via Rydberg Blockade

3

the right- (left-)moving photon. For S1, the interaction

is U(z) =∫dz′V (z−z′)S†2(z′)S2(z′)S1(z), and vice versa

for S2.Since the physics of two counter-propagating pho-

tons is similar to the spin-wave problem above,we begin our analysis with this case [Fig. 1(c)].Letting |ψ(t)〉 be the two-excitation wavefunction[29], we define ee(z1, z2, t) = 〈0|E1(z1)E2(z2)|ψ(t)〉,es(z1, z2, t) = 〈0|E1(z1)S2(z2)|ψ(t)〉, se(z1, z2, t) =〈0|S1(z1)E2(z2)|ψ(t)〉, and ss(z1, z2, t) =〈0|S1(z1)S2(z2)|ψ(t)〉. Eqs. (1,2) then yield a sys-tem of equations for these four variables. Defininges± = (es ± se)/2, one finds that es− is small anddoes not significantly affect the dynamics. Droppinges−, defining center-of-mass and relative coordinatesR = (z1 + z2)/2 and r = z1 − z2, and taking a Fouriertransform in time, one obtains c∂rv = M(r, ω)v, wherev = ee(R, r, ω), es+(R, r, ω) and

M(r, ω) =

[iw2 −

g2nΓ − g

√nΩ

Γ

− g√nΩ

Γ iω − Ω2

Γ + ig2n[ω−V (r)]2Ω2+iV (r)Γ−iωΓ

].(6)

R enters only through boundary conditions and is, thus,not important in the present case. For narrowbandpulses, we can expand M(r, ω) ≈M0(r) +ωM1(r), with

M0 = − 1

Γ

[g2n g

√nΩ

g√nΩ Ω2 + g2nV

], (7a)

M1 = i

[ 12 0

0 1− 2g2nΩ2V2

Γ2V 2

]. (7b)

Here we defined the effective potential V = ΓV/(ΓV −i2Ω2). Outside (inside) the blockade region, V ≈iΓV/(2Ω2) (V ≈ 1). For |r| zb(B), the two pho-tons propagate as dark-state polaritons [26], i.e. we havees+/ee = −g√n/Ω, which is an eigenstate of M0 witheigenvalue 0. Since g

√n Ω, the group velocity can be

read out from the last entry of M1, which gives twice theEIT group velocity vg since the two polaritons propagatetowards each other. Within the blockade radius, whereV ≈ 1 and V/V ≈ 0, the polariton solution ceases to bean eigenstate of M0, and Eq. (7b) predicts a speed up to∼ c. Since the time ∼ zb(B)/c it takes to cross the block-ade region is much less than the inverse width of the EITwindow, the dynamics is highly non-adiabatic such thatthe main result of the interactions is a picked-up factorof exp

[−∫drg2nV(r)/(cΓ)

]= exp(iϕ − η). This is a

generalization of the result of Refs. [12, 14] beyond theperturbative regime.

On resonance, 2η ≈ db. Thus, analogously to the spin-wave problem above, the entire EIT-compressed two-particle wavefunction decays provided it fits inside themedium and db 1. The resulting state is a statisticalmixture of right- and left-moving excitations.

Off resonance, es+ picks up ϕ ≈ − π21/66

γ∆dB ≈ − γ

2∆dB

and η ≈ 5π21/636

γ2

∆2 dB ≈ γ2

2∆2 dB. Additionally, the off-

(a) (b) (c)

0

0.5

1

1.5

2

2.5

3(d)

0 5 10 15 20

-1

-0.5

0

0.5

1

0 5 10 15 200

0.2

0.4

0.6

0.8

1(e)

t = 2Tt = 1.8Tt = 1.2Tt = T-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4

z1/Lz1/Lz1/Lz1/L

z 2/L

cos(ϕ)

exp(−

η)

ϕ

dBγ/∆

FIG. 2. (a)-(d) Two-photon counter-propagation for ∆ =20γ, Ω = 2∆, g

√n = 20∆ and zB = 0.055σ, where σ is the

compressed pulse length inside the medium. The color codingshows the local phase of ee, while the opacity reflects the two-photon density |ee|2. The dashed lines are |z1−z2| = zB. Thefull movie is provided in the supplementary material [30]. (e)Numerically obtained phase shift ϕ [we plot cosϕ] and atten-uation e−η as a function of dB compared to the analytical pre-dictions (lines). The numerical data corresponds to two dif-ferent parameter scans g

√n = 400∆, zB = 0.0025σ, ..., 0.03σ

(dots) and zB = 0.03σ, g√n = 80, ..., 390 (squares).

diagonal terms in M0 result in a small admixture of thebright-state polariton [26], which decays after the wave-function exits the blockade region.

To verify these conclusions, we show in Fig. 2 and inthe supplementary movie [30] the results of numerical so-lutions of the full equations for ee, es, se, and ee in theoff-resonant case. Despite the bright-polariton-inducedoscillations of ee inside and near the blockade region [30],the final phase of the outgoing two-photon pulse perfectlyagrees with our analytical prediction [Fig. 2(e)]. Whilealso showing good agreement with the analytical result,the obtained loss is slightly larger due to the bright-state polariton admixture, which was neglected withinthe above approximate treatment.

Provided the EIT-compressed two-particle wavefunc-tion fits inside the medium, this process, thus, allows forthe implementation of a nearly lossless phase gate be-tween two photons. Taking a specific example of coldRb atoms with |e〉 = 52P1/2 and |r〉 = 702S1/2 and us-ing Ω/2π = 2MHz and ∆ = 20γ, we find zB = 15µm,which, for a dense cloud with n = 1012 cm−3, givesdB = 3

2πλ2(2zB)n ≈ 9. This yields a significant phase

of ϕ ≈ −0.2 and a very small attenuation 2η ≈ 0.02.One can increase dB further by using photonic waveg-uides [31–34] and working with a BEC [34].

In the co-propagating case, we define ee(z1, z2, t) =〈0|E(z1)E(z2)|ψ(t)〉, es(z1, z2, t) = 〈0|E(z1)S(z2)|ψ(t)〉,and ss(z1, z2, t) = 〈0|S(z1)S(z2)|ψ(t)〉 [Fig. 1(d)]. Defin-ing es±(z1, z2) = [es(z1, z2)±es(z2, z1)]/2, dropping es−,and taking the Fourier transform in time, we obtainc∂Rv = 2M(r, ω)v. That is, the only difference fromthe counter-propagating case is the replacement of ∂rwith (1/2)∂R. The resulting equations can be solved

Page 4: Photon-Photon Interactions via Rydberg Blockade

4

-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

t = T/2 t = 3T/4 t = 5T/4

(a) (b) (c)

z1/L

z 2/L

z1/L z1/L

|ee| 2[arb.u.]

FIG. 3. Time evolution of |ee|2 for two co-propagating pho-tons for Ω = γ, g

√n = 100γ, and zb = 0.08σ. The dashed

lines are |z1 − z2| = zb, in agreement with the numerical re-sults, which show the decay of ee within the dashed lines. Thefull movie is provided in the supplementary material [30].

separately at each r. As before, outside the block-ade radius, the two-photon dark-state polariton propa-gates with group velocity vg. Inside the blockade ra-dius, M0 results in fast attenuation on a lengthscale∼ L

d (γ2 + ∆2)/γ2. This is confirmed by our numericalcalculations, shown in Fig. 3 and in the supplementarymovie [30]. Therefore, the two-excitation wavefunctionevolves into a statistical mixture of a single excitationand a correlated train of two photons separated by zb(B).On resonance, if the photon is scattered when the EIT-compressed pulse of length less than zb is fully inside themedium, we expect the remaining excitation to propa-gate in its original spatiotemporal mode. For a coher-ent input pulse, one similarly expects the wavepacket toevolve with some probability into a correlated train ofblockade-radius-separated photons. Furthermore, if theblockade radius is larger than the EIT-compressed pulselength, there may be a regime in which such a systemcould function as a deterministic single-photon source.

In summary, we have shown that Rydberg blockade inEIT media can be harnessed for inducing strong photon-photon interactions, with applications to generating non-classical states of light, implementing nonlinear photonicgates, and studying many-body phenomena with stronglycorrelated light. This work opens several promising av-enues of research. With an eye towards single-photongeneration, one can extend the presented wavefunctiontreatment to a density matrix approach and explicitlyanalyze the propagation of the remaining excitation af-ter the interaction-induced decay of multi-photon states.In addition, a gas of bosons (Rydberg polaritons) witha hard-sphere core (of radius zb(B)/2) can be investi-gated both theoretically and experimentally in the co-propagating case. In particular, the previously neglectedeffects of es− endow these bosons with an effective mass∝ −idγ/(LvgΓ), which plays a significant role for prop-agation distances larger than those considered in thepresent Letter. By including the effects of the coordi-nates transverse to the propagation axis, one can ex-tend this problem to higher dimensions. Furthermore, for

∆/C6 < 0, the effective potential shows a resonant fea-ture, which can give rise to two-polariton bound states.

We thank S. Hofferberth, M. Bajcsy, T. Peyronel, M.Hafezi, and N. Yao for discussions. This work was sup-ported by NSF, the Lee A. DuBridge Fellowship, DFGthrough the GRK 792, CUA, DARPA, and the PackardFoundation.

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propagating [Fig. 3] photons are provided in the supple-mentary material (download the source files to see them).

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