Engineering and harnessing giant atoms in high-dimensional baths: a cold atoms' implementation
EEngineering and harnessing giant atoms in high-dimensional baths: a cold atoms’ implementation
A. Gonz´alez-Tudela, ∗ C. S´anchez Mu˜noz, and J. I. Cirac Instituto de F´ısica Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain. Clarendon Laboratory, University of Oxford, Oxford OX13PU, UK Max-Planck-Institut f¨ur Quantenoptik Hans-Kopfermann-Str. 1. 85748 Garching, Germany
Emitters coupled simultaneously to distant positions of a photonic bath, the so-called giant atoms, representa new paradigm in quantum optics. When coupled to one-dimensional baths, as recently implemented withtransmission lines or SAW waveguides, they lead to striking effects such as chiral emission or decoherence-freeatomic interactions. Here, we show how to create giant atoms in dynamical state-dependent optical lattices,which offers the possibility of coupling them to structured baths in arbitrary dimensions. This opens up newavenues to a variety of phenomena and opportunities for quantum simulation. In particular, we show how to en-gineer unconventional radiation patterns, like multi-directional chiral emission, as well as collective interactionsthat can be used to simulate non-equilibrium many-body dynamics with no analogue in other setups. Besides,the recipes we provide to harness giant atoms in high dimensions can be exported to other platforms where suchnon-local couplings can be engineered.
The design and exploration of novel forms of light-matterinteraction have been a driving force in quantum optics trig-gering both fundamental and technological advances. Aparadigmatic example of this was the observation that atomiclifetimes renormalize within cavities [1], which opened thefield of cavity QED [2, 3]. This seemingly simple light-matter coupling leaded to many other fundamental discover-ies, such as the creation of mixed light-matter particles (po-laritons), and applications, e.g., in quantum information [4].Another timely example is the interaction of (natural or ar-tificial) emitters with the structured propagating photons (ormatter-waves) which appear in nanophotonics structures [5–14], circuits [15–17], or state-dependent optical lattices [18–22]. In these systems, the bath displays structured energydispersions, leading to a plethora of effects absent in otherenvironments. On the fundamental level, they generate non-exponential relaxations [23–29], whereas in the more appliedperspective they lead to the emergence of bound states out-side [30–35] or in the continuum [36–45], which can be har-nessed for (out-of) equilibrium quantum simulation [46–49].In all these setups the emitters are typically much smallerthan their associated wavelength, leading to inherently lo-cal light-matter couplings. This picture, however, has beenrecently challenged with the design of the so-called ”giant-atoms”, which are emitters coupled to several points of SAWwaveguides [50–54] or transmission lines [55] separated be-yond their characteristic wavelength. These giant atoms rep-resent another paradigm change in quantum optics since thecoupling to different bath positions induces strong interfer-ence effects which can be exploited for applications [20, 21,56–58]. For instance, when coupled to one-dimensional bathsthey lead to decoherence-free atomic interactions [58], orto chiral emission [20, 21] without exploiting polarization,something impossible to realize with ”small” emitters. Ex-porting this paradigm to higher dimensional baths, where, forexample, quantum simulation will show its full power, is a de-sirable, but challenging, goal. On the one hand, to our knowl-edge there is still no implementation to do so, since wiring uphigh-dimensional circuits becomes complicated. On the other (a) (b) T i m e Effective giant atom
Figure 1. (a) State-dependent optical lattice scheme to simulate quan-tum optical phenomena: one deep lattice V b ( R b ) (blue) traps theatomic state that mimics the QE behaviour, whereas a shallower one, V a ( R a ) , lets matter-wave propagation at rate J . The two atomic statecan be connected through a local laser(s) or microwave field (green)with strength Ω n j . The relative position between the lattices, and ofthe local laser can be dynamically tuned R b ( t ) . (b) Pictorial repre-sentation on how the effective giant atom couplings emerge from thestroboscopic movement between the lattices. hand, even if achieved, it is not obvious how to harness giantatoms when coupled to high dimensional baths. The reason isthat the resonant photons mediating the interactions, definedby the isofrequencies of ω ( k ) at the emitters frequency, arecontours (or surfaces) in the k -space, instead of points, mak-ing perfect interference more difficult.In this manuscript, we address both issues showing: i) Aproposal to engineer effective giant atoms coupling to bathswith high dimensions. We use ultra-cold atoms in dynami-cal state-dependent optical lattices [18, 19, 22] (see Fig. 1),such that by moving the relative position between the poten-tials [59–61] fast enough, the giant emitter couples effectivelyto several bath positions. ii) A way to harness them to observephenomena with no analogue in other setups by coupling themto structured photonic reservoirs with a Van-Hove singular-ity [37–39, 49]. In particular, we show how giant quantumemitters (QEs) can modify the non-Markovian nature of thedynamics, and lead to unconventional emission patterns, e.g.,chiral emission in one or several directions, which translateinto unconventional collective QE interactions when severalof them couple to the bath. Even though we make the discus- a r X i v : . [ qu a n t - ph ] M a r (a) (b)(c) (d) Figure 2. Bath population at a time tJ = N / ( , ) , ( , ) , such that g n [ n ] = g cos ( ω t / )[ sin ( ω t / )] , with g = . J and ω as depicted in the legend. Bath linear size is N = sion of (i-ii) together along the manuscript, the recipes thatwe provide for (ii) can be exported to other implementationswhere such couplings can be engineered.Let us first recall how to obtain the standard quantumoptical Hamiltonian with ultra-cold atoms [18, 19, 22], seeFig. 1(a): one needs an atom with two states a / b subject todifferent potentials V a / b ( R ) , whose dimensionality can be op-tically controlled [62]. The b -atoms are trapped in a deeppotential such that they mostly localize within a lattice site,and in the strongly interacting regime, which means that therewill be at most one b -excitation per lattice site such that theirexcitations can be represented by spin operators σ n j ge , with σ n αβ = | α (cid:105) n (cid:104) β | . On the contrary, when the atoms are in the a -state, they can hop to their nearest neighbours at a rate J without interactions, mimicking photon propagation. Besides,one needs an extra field that transfers the b excitations into a ones (and viceversa), which can be obtained via a Raman ormicrowave transition [22, 63] (or a direct one in the case ofAlkaline-Earth atoms [64–66]). Let us denote as Ω n j the a - b coupling at site n j , which can be controlled in both magnitudeand phase though the lasers. As derived in Refs. [18, 19], theHamiltonian describing the dynamics of the excitations of the a and b atoms mimics the one standard light-matter interac-tions, that is, H = H S + H B + H int , where: H S = ω e σ ee , H B = ∑ k ω ( k ) a † k a k , (1) H int = (cid:0) Ω n e a † n e σ ge + H . c . (cid:1) , (2)where for illustration we restrict to a single QE, droppingthe superindex in σ ge . The a k ( a † k ) are the annihilation (cre-ation) operator of a matter-wave excitation with momentum k , whose energy dispersion ω ( k ) is controlled by the geome-try of V a ( R ) . The QE is in the strong confinement limit such that its coupling will be local like with optical photons [67].To effectively transform this local coupling into a non-localone among { n α } N p α = positions, one can dynamically move therelative position between the V a / b ( R ) potentials in a periodicfashion, e.g., changing the relative phase between the laserscreating the potentials [62]. If the movement is adiabatic, thatis | ˙ R ( t ) | (cid:28) d ω t for all t , where d ground state size, and ω t thetrap frequency [59–61], the atoms remain in their motionalground state and can still be described by a Hamiltonian asin Eqs. 1-2 but with time-dependent parameters. For example,assuming that the simulated QE probes the { n α } N p α = positionsand that the laser parameter change as needed in each position,the Hamiltonian will now read: H int → H int , mov ( t ) = N p ∑ α = (cid:0) Ω n α ( t ) a † n α σ ge + H . c . (cid:1) . (3)Now, to formally derive how the desired non-local cou-plings emerge using Floquet analysis, we consider that QEmoves periodically along N p positions with period T (and fre-quency ω = π / T ), probing each position during a constanttime interval T / N p with coupling strength g n α [68]. Withthat assumption, we can apply Floquet theory [69] to obtainan effective Hamiltonian description in the high-frequencylimit. To the lowest order, it corresponds to the non-locallight-matter couplings that we want to obtain (see Sup. Ma-terial [70]): H int , eff ≈ N p ∑ α = (cid:18) g n α N p a † n α σ ge + H . c . (cid:19) , (4)where g n α / N p is the time average of Ω n α ( t ) . We can alsocalculate the next-order term contribution which is of order ∼ | g max | N p ζ [ ] / ( π ω ) (cid:28) max | g n α | for our situations of in-terest. Summing up, to obtain the desired behaviour the peri-odic movement has to be slow enough to stay within the low-est band of the tight-binding Hamiltonians of Eqs. 1-2, butfast compared to the induced QE timescales, such that it ef-fectively couples to several positions, i.e., ω t (cid:29) ω ( L / d ) (cid:29) max | g n α | (assuming a constant speed over the distance L thatwe displace the potentials). Since the couplings are tune-able and they can always be made small, the lower bound ofthese inequalities will be ultimately provided by the decoher-ence rate Γ ∗ of the setup, which should be smaller than thesimulated parameters. To provide some estimation, we canfirst take the recent realization of our proposed setup [22],where two hyperfine Rb levels were used to engineer theoptical potentials, | a / b (cid:105) = | F = / , m F = − / (cid:105) , with trap-depths of the order ω t ∼ π ×
10 kHz, and typical decoher-ence rates ∼ −
100 Hz. Another possibility is to use theground/excited metastable state in Alkaline-Earth atoms (seeRef. [64] for a concrete proposal with Strontium). This plat-form shows similar ω t , but decoherence can be substantiallydecreased since it will be mostly determined by the excitedstate lifetime which can be Γ ∗ / ( π ) (cid:46) .
01 Hz, thus leavingseveral orders of magnitude to adiabaticaly move the lattice. (a)
Local (b)
Figure 3. (a) Bath probability amplitude at a time tJ = N / G pur ( k ) , g = . J and bath linear sizeis N = k dependence of G pur ( k ) , i.e., D eff ( E ) = ∑ k | G pur ( k ) | δ ( E − ω ( k )) . Let us now show how to exploit giant QEs coupled tohigher dimensional baths to obtain phenomena with no ana-logue in other setups. In particular, we illustrate it by studyingthe spontaneous decay of an excited QE coupled to a two-dimensional bath with ω ( k ) = ω a − J [ cos ( k x ) + cos ( k y )] .When the QE interacts locally in space with frequency ω e = ω a , it couples equally to all the resonant k ’s defined by: k x ± k y = ± ( ∓ ) π . This contour, which includes points withzero group velocity ( v g ( , ± π ) = v g ( ± π , ) = ( , ) ) respon-sible of a Van-Hove singularity in the density of states [71],leads to two remarkable effects in the QE spontaneous de-cay [37–39]: i) its emission pattern is highly anisotropic,as shown in Fig. 2(a), emitting mostly in four directionswith some diffraction due to the inhomogeneous group ve-locity of the wavepacket; ii) its dynamics is intrinsically non-Markovian due to divergence of the density of states at thisfrequency [38, 39]. Now, we will show how building up onthis behaviour, giant QEs can lead to very flexible and unusualemission patterns and interactions. Quasi-1D emission.
First, we show how to cancel theemission in one of the diagonals of Fig. 2(a) by coupling totwo lattice sites n / = ( , ) / ( , ) . To numerically showhow the Floquet averaged Hamiltonian H int , eff emerges, weassume that the movement between the lattices is such that Ω n ( t ) = g cos ( ω t / ) , Ω n ( t ) = g sin ( ω t / ) , and solve thedynamics using H int , mov ( t ) . In Fig. 2(b-d) we plot the bathpopulation in real space after a time tJ = N / g = . J ,and for several ω ’s. As expected, for ω (cid:28) g , the emission oc-curs in four directions as if the QE was locally coupled. How-ever, as ω increases, the interference between the bath emis-sion in two different points occurs, until it cancels the emis-sion in one of the diagonals. This behaviour can be understoodfrom the asymptotic bath state in the perturbative limit [39]: C k ( t → ∞ ) ∝ G ( k ) e − i ω ( k ) t ω ( k ) − ω e + i Γ M / , (5)where Γ M is the Markovian decay rate, and G ( k ) is the ef- fective light-matter coupling between the emitter and the k -modes, H int , eff = ∑ k (cid:16) G ( k ) a † k σ gs + H . c . (cid:17) , which reads: G ( k ) = N p N p ∑ α = g n α e − i k · n α . (6)In this case G ( k ) ∝ + e − i ( k x + k y ) , which satisfies G ( k x , ± π − k x ) ≡
0. Thus, the giant QE is effec-tively uncoupled from the k -modes responsible of the for-ward/backward direction in the diagonal where the giant QEis coupled to, and does not decay into them. After havingnumerically seen how H int , eff emerges from H int , mov ( t ) for thisexample, from now on we use H int , eff to analyze the dynamics. Trapped emission.
Let us now consider that the QE movesaround four positions, i.e., ( ± , ) , ( , ± ) . The effective k -coupling will be: G trap ( k ) = g (cid:0) e ik x + e − ik x + e ik y + e − ik y (cid:1) / k -lines.Thus, the giant QE will not decay, while keeping some thephoton population trapped between the four positions (notshown). As in the 1D counterpart [58], these confinedphotons will mediate coherent interactions between thesedecoherence-free QEs. Filtering non-Markovian emission.
Another feature thatcan be achieved by coupling to few lattice sites is the effec-tive decoupling from zero-group velocity terms occurring at k = ( , ± π ) and ( ± π , ) . For that, we can couple the QEto the positions ( ± , ± ) , ( ± , ∓ ) , with an alternating ± G pur ( k ) = g sin ( k ) sin ( k ) . This has two con-sequences: first, the QE shows a more homogeneous direc-tional emission, as observed in Fig. 3(a). Second, it smoothensthe effective spectral density probed by the QE, as plotted inFig. 3(b), making its dynamics more Markovian. Thus, giantQEs provide a way of decoupling directional emission fromnon-Markovian dynamics in Van-Hove singularities. Reverse design: chiral and V-type emission.
In the previ-ous examples it was possible to guess the spatial couplingsrequired to obtain the desired behaviour. An alternative ap-proach consists of first guessing the G ( k ) required to obtaina given behaviour, and then Fourier transforming it to get thespatial dependent couplings, that is G ( n ) = N ∑ k G ( k ) e − i k · n . (7)For example, let us imagine we want to obtain perfect chiralemission in one or two orthogonal directions out of the fourappearing with local couplings. It is easy to see that: G chi ( k ) ∝ cos (cid:18) k − k (cid:19) (cid:20) + sin (cid:18) k + k (cid:19)(cid:21) , (8) G V ( k ) ∝ (cid:20) − sin (cid:18) k − k (cid:19)(cid:21) (cid:20) − sin (cid:18) k + k (cid:19)(cid:21) . (9)cancels the coupling to the light emitted in three (or two) ofthe four directions, respectively. Then, using Eq. 7 we ob-tain the spatial profile of the couplings whose absolute value (a) (b) Figure 4. (a-b) Bath probability amplitude at a time tJ = N / G trunc ( n ; n tr ) , respectively, g = . J and bath linear size is N = G ( n ) using Eq. 7. In red the truncation we use to plotthe figure. (c-d) 1 − F , where F is the fraction of the emission intothe desired directions for the parameters of panel (a-b), as a functionof the number of terms in the sum G trunc ( n ; n tr ) . | G ( n ) | is plotted in the inset of Figs. 4(a-b). The couplingspatial pattern is more intricate than in the previous situationsbecause it requires adding complex phases (not shown), andinvolve the coupling to many lattice sites. Since this will beexperimentally challenging, one needs to adopt a truncationstrategy in which one approximates the sum by a finite num-ber n tr of terms, G ( n ) ≈ G trunc ( n ; n tr ) . This is what we do inFigs. 4(a-b), where we observe that even for a small n tr , theQE emits approximately with the desired behaviour. Finally,in Fig. 4(c-d) we show how increasing n tr , the light collimatedin the desired directions can go close to 100 %. Interactions.
Let us finally point how these unconven-tional emission patterns will translate into exotic QE inter-actions when N e QEs are coupled to the bath. For sim-plicity, let us assume that each QE has a k -dependent cou-pling G j ( k ) = G ( k ) e − i k · n j , where e − i k · n j is a global phasefactor which indicates the giant QE central position ( n j ),and G ( k ) is a common k dependent coupling defined bythe non-local couplings around the position n j . Then, if wetrace out the bath degrees of freedom under the Born-Markovapproximation, the QE reduced density matrix ( ρ ) dynam-ics is governed by [72]: ∂ t ρ = i [ ρ , H S + ∑ i , j J i j σ ieg σ jge ] + ∑ i , j γ i j / ( σ ige ρσ jeg − σ jeg σ ige ρ − ρσ jeg σ ige ) . The collective in-teractions J i , j , γ i j are: γ i j + iJ i j = N ∑ k | G ( k ) | ω e − ω ( k ) + i + e i k · ( n i − n j ) , (10)whose integrand is directly connected with the asymptoticemission pattern described in Eq. 5. This tells us, for example,that using the couplings G D ( k ) or G chi ( k ) we will be able tosimulate standard or chiral [73] waveguide QED couplings in two-dimensional baths, as well as other QE interactions withno counterpart in other setups, i.e., V-type collective decays. Conclusions.
Summing up, we propose a method to en-gineer effective non-local light-matter couplings using ultra-cold atoms in dynamical state-dependent optical lattices. Con-trolling the confinement and relative position of two opticalpotentials, one can simulate giant atoms coupled to structuredphotonic baths in one, two and three dimensions. Irrespec-tive of the implementation, we also numerically illustrate thepotential of giant emitters to yield unconventional quantumoptical behaviour when coupled to a two-dimensional struc-tured bath. In particular, we exploit the interplay between thestructured energy dispersion and non-local couplings to ob-tain exotic emission patterns and collective dissipative inter-actions. These recipes can be immediately adapted to otherplatforms where such non-local couplings can be engineered,or to higher dimensions [70, 74].Beyond the fundamental interest of the phenomena ex-plored along the manuscript, there are many possible follow-up applications. From the quantum simulation perspective,giant atoms provide a very flexible playground to probe equi-libirum [46, 47] and non-equilibrium many-body physics [75,76] with no analogue in other setups. Besides one can in-crease their tunability exploiting the interplay with the polar-ization degree of freedom [77–79], or through additional bathengineering [80]. Other possibilities, if one is able to engi-neer it with optical photons, is to exploit the multi-directionalchiral emission to transfer simultaneously quantum informa-tion into several nodes, or for generating high-dimensionalphotonic entangled states [81], which can be used for fault-tolerant measurement based quantum computation [82].
ACKNOWLEDGEMENTS
JIC acknowledges the ERC Advanced Grant QENOCOBAunder the EU Horizon 2020 program (grant agreement742102). C.S.M. is supported by the Marie Sklodowska-CurieFellowship QUSON (Project No. 752180). AGT acknowl-edges very useful discussions with J. Kn¨orzer, and the criticalreading of the manuscript of T. Ramos. ∗ [email protected][1] E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. , 37(1946).[2] S. Haroche and D. Kleppner, Physics Today , 24 (1989).[3] R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D.Boozer, and H. J. Kimble, J. phys. B.: At. Mol. Phys. , S551(2005).[4] H. J. Kimble, Nature , 1023 (2008).[5] E. Vetsch, D. Reitz, G. Sagu´e, R. Schmidt, S. T. Dawkins, andA. Rauschenbeutel, Phys. Rev. Lett. , 203603 (2010).[6] J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V.Akimov, M. Gullans, A. S. Zibrov, V. Vuletic, and M. D. Lukin,Science , 1202 (2013). [7] A. Goban, C.-L. Hung, S.-P. Yu, J. Hood, J. Muniz, J. Lee,M. Martin, A. McClung, K. Choi, D. Chang, O. 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In this Supplementary Material, we provide more detailson: i) the derivation of the time-averaged Hamiltonian in thehigh-frequency limit; ii) how to calculate the radiation pat-terns from spontaneous emission; iii) experimental setup andfeasibility analysis; iv) how to extend some of the phenomenapredicted in the manuscript to three-dimensional systems.
DERIVING THE FLOQUET HAMILTONIAN
The starting point of the derivation is the time-dependentinteraction Hamiltonian of Eq. 4 of the main text: H int , mov ( t ) = N p ∑ α = (cid:0) Ω n , α ( t ) a † n α σ ge + H . c . (cid:1) . (SM1)To do the Floquet derivation, we assume that we move thelattices such that the QE excitation probes N p positions, eachof them during a time T / N p , such that the global period of themovement is ω = π / T . Thus, Ω n , α ( t ) = g n , α f α ( t ) , where: f α ( t ) = , ( α − ) T / N p < t < α T / N p , (SM2)or 0 otherwise. We are aware that in a practical situation thetransition from one lattice site to the other will be smooth and,in fact, must satisfy the adiabaticity condition at any time [59–61]. However, the step function f α ( t ) will allow us to captureanalytically the most relevant features of the effective dynam-ics without worrying about the particular details of Ω n , α ( t ) .The key point is that step functions can be easily expanded in their Fourier components as follows [69]: f α ( t ) = N p + ∑ j C j , α e i j ω t , (SM3) C j , α = π i j e − i πα j / N p (cid:16) e i π j / N p − (cid:17) (SM4)Using this expansion, the Hamiltonian H int , mov ( t ) can beseparated into a time-independent part, V ( ) , which contains athe time-averaged interaction of H int , mov ( t ) , V ( ) = N p ∑ α (cid:0) g n α a † n α σ ge + H . c . (cid:1) , (SM5)that is the part we want to obtain, plus all the periodic modu-lation introduced by the harmonics: H int , mov ( t ) − V ( ) = ∑ j (cid:54) = V ( j ) e i j ω t , (SM6) V ( j ) = ∑ α (cid:0) C j , α g n α a † n α σ ge + H . c . (cid:1) , (SM7)In the high-frequency limit, an effective time-independentHamiltonian can be derived [69], which to first order in 1 / ω reads: H int , eff ≈ V ( ) + ω ∑ j > [ V ( j ) , V ( − j ) ] j (SM8)With it, we can calculate explicitly the first order correctionto the time averaged Hamiltonian V ( ) : H ( ) eff = ω ∑ j > [ V ( j ) , V ( − j ) ] j = N p ∑ α , β ∞ ∑ j = (cid:34) ig n α g ∗ n β π j ω sin (cid:18) j π N p (cid:19) sin (cid:18) π ( β − α ) jN p (cid:19) a † n α a n β σ z (cid:35) with σ z = ( σ ee − σ gg ) /
2. Since we typically restrict to situ-ations where the number of excitations, ˆ N = ∑ n a † n a n + σ ee ,is conserved, and in the single-excitation regime, the norm ofthis operator can be upper-bounded by: || H ( ) eff || < g N p π ω ∞ ∑ j = j = g N p π ω ζ [ ] (SM9)where g = max {| g n |} . It must be noted that when consid-ering the full dynamics with H S + H B , the density of stateswill also enter into play in the discussion, i.e., suppressing (orenhancing) the contributions of the different sidebands at fre-quencies j ω . In the examples considered along the text, sincethe density of states is peaked around ω a , the sideband contri-butions are suppressed compared to the time-averaged com-ponent. The opposite behaviour (enhancement of sidebands)can also be used an extra degree of freedom to design moreexotic quantum optical phenomena beyond the time averagedterms of H int , mov ( t ) . CALCULATING THE EMISSION PATTERNS
The global Hamiltonian of the system: H = H S + H B + H int conserves the number of excitations: ˆ N = ∑ j σ jee + ∑ k a † k a k ,no matter whether H int is time-dependent or not. Thus, ifwe consider a single QE initially excited as the initial state: | Ψ ( ) (cid:105) = | e (cid:105) ⊗ | vac (cid:105) , the global state at any time can be writ-ten as: | Ψ ( t ) (cid:105) = (cid:20) C e ( t ) σ eg + ∑ n C n ( t ) a † n (cid:21) | g (cid:105) ⊗ | vac (cid:105) , (SM10)where the coefficients can be always obtained by numericallysolving i | Ψ ( t ) (cid:105) dt = H ( t ) | Ψ ( t ) (cid:105) . This is how we obtain the C n ( t ) plotted in the Figs. 2-4 of the main text. Moreover, by Fouriertransforming C n ( t ) we can obtain the wavefunction in mo-mentum space: C k ( t ) = N ∑ n C n ( t ) e − i k · n . (SM11)With C k ( t ) it is easy to define the fraction of light emitted in each of the four directions of Fig. 2(a) at any time: F ( t ) = ∑ k x > , k y > | C k ( t ) | C k ( t ) , (SM12) F ( t ) = ∑ k x < , k y > | C k ( t ) | C k ( t ) , (SM13) F ( t ) = ∑ k x < , k y < | C k ( t ) | C k ( t ) , (SM14) F ( t ) = ∑ k x > , k y < | C k ( t ) | C k ( t ) . (SM15)This is what we use to characterize the fraction of lightemitted in one or two-directions in Fig. 4(c-d). For the chiralemission we plot 1 − F ( t ) , and for the V -shape emission1 − F ( t ) − F ( t ) at time tJ = N / EXPERIMENTAL CONSIDERATIONS
In this Section we give a more detailed explanation on theexperimental setup that could be used to observe the phenom-ena predicted in the manuscript, and analyze the feasibility ofour proposal using realistic experimental parameters.
Atomic level configuration
One possibility consists in using Rubidium atoms as in therecent experiment by Krinner et al [22], where quantum op-tical phenomena was simulated for the first time using Ru-bidium matter-waves in state-dependent optical potentials. Asschematically explained in Fig. SM1, in that experiment twostates in the ground state manifold of Rb atoms are used tosimulate the quantum emitter and bath. Let us review some ofthe parameters of that experiment: • They use the | F = , m F = − (cid:105) = | b (cid:105) and | F = , m F = (cid:105) = | a (cid:105) , as the emitter/bath state,respectively, which are separated in energies by6 . • They transfer the excitations directly from a to b usinga microwave field with strength of the order of Ω / π ∼ • They are interested in observing one-dimensional band-edge physics, such that they enforce the two atomicstates to live within one-dimensional tubes through acommon radial confinement. The state-dependent opti-cal potential along the other direction is generated witha σ − -polarized laser beam with λ =
790 nm. Theychoose that combination of polarization/wavelengthsuch that the a atom does not feel any potential alongthat direction, while the emitter-like state is stronglyconfined with a trap frequency ω t / ( π ) ∼
40 KHz.However, as they mention in their Sup. Material byeither rotating the polarization and/or changing wave-length, they can also induce different trapping condi-tions for a . • The advantage of using Rb hyperfine states is that theyhave very long coherence times. Possible sources ofdecoherence such as thermal fluctuations or the scatter-ing rates introduced by the trapping potentials are verywell understood and under control in these setups. Forexample, the main source of these spin-dependent lat-tices will be the scattering rates introduced by the op-tical potential, which for that particular experiment, weestimate to be Γ ∗ / ( π ) ∼ −
100 Hz (even though itwas not explicitly mentioned in the paper). By usingdifferent atomic states, and/or wavelengths one couldoptimize these decoherence rates for the particular ex-periments we are considering.A variation of this setup can be used to implement ourideas. One would require: i) extra laser fields to create theoptical confinement in other directions depending on whetherwe want to simulate two or three dimensional baths; ii) a wayof dynamically change the phase of the laser to displace theemitter-state optical potential. Besides, if want the bath stateto create a fully independent tuneable optical potential for thebath state, one should add independent laser fields with otherpolarization/frequencies.Another interesting possibility consists of using Alkali-Earth atoms [64], such as Ytterbium [66] or Strontium [65],to create such state-dependent optical lattices. These atomsare characterized by having optically excited mestable states, P / with very narrow linewidths which can be as small as Γ e / ( π ) ∼ .
01 Hz. This allows one to use these excited statesto store excitations with very long coherence times. The ad-vantage is that since the ground and excited states are sep-arated by optical frequencies, one can engineer completelyindependent potentials for both states. For a particular real-ization of such independent state-dependent optical latticeswith Strontium one can check Ref. [64], where it was alsoexplained how to dynamically move the relative position be-tween the two potentials, and how to transfer the excitationsbetween the states.
Feasibility conditions
As discussed in the main text, the approximate set ofinequalities to obtain the desired phenomena is lower and
Ground state level structure
Figure SM1. Ground state level structure of Rb with two hyperfineground state manifolds ( F = F =
2) with three and five stateslabeled by its m F quantum number, respectively. The two groundstates manifolds are separated in energies by 6 . | F = , m F = − (cid:105) and | F = , m F = (cid:105) ,as the quantum emitter/bath states, respectively, and a microwavefield, Ω , to couple them. upper bounded by the decoherence rate and trap frequencies,respectively. Thus, the difference between these two mag-nitudes determine how much room we have to implementour proposal, for example, limiting how many bath positionsthe emitter can probe without breaking the adiabaticity con-dition. In the previous Section, we have shown that typicalexperimental parameters for, e.g., Alkali-Earth atoms, canbe ω t / π ∼ Hz, and Γ ∗ / ( π ) ∼ .
01 Hz, such that onein principle has many orders of magnitude available to playwith. In any case, we want to note that when implementingour ideas in a particular setup, a full analysis of all possibleerror sources and limitations should be performed to fullyunderstand the limits of the experiment.
GIANT ATOMS IN THREE-DIMENSIONAL BATHS
As we argue in the concluding paragraph of the main text,many of the phenomena and recipes that we illustrate fortwo-dimensional photonic baths can be exported to three-dimensional ones with an adequate choice of the bath andemitter-bath couplings. For example, as shown in Ref. [74], ifthe three-dimensional bath has a body-centered-cubic geome-try, the energy dispersion is given by: ω ( k ) = − J [ cos ( k x ) + cos ( k y ) + cos ( k z ) + cos ( k x + k y + k z )] . (SM16)This energy dispersion leads to a Van-Hove singularity inthe middle of the band, ω ( k ) =
0, which occurs for the planes k a ± k b = ± π , where a , b is any combination of x , y , z . Thisleads to a highly directional emission pattern in eight lines, ascompared to four lines in 2D, which we can exploit in combi-nation with giant atoms in a similar fashion. For example, • Coupling an emitter to two positions, e.g., ( , , ) , ( , , ) , the k -dependent coupling will read G ( k ) ∝ + e i ( k + k ) that will vanish when k + k = ± π , cancel-ing the emission into the direction defined these planes.Using these tricks, or directly applying the reverse engi-neering that we explain in the main text, one can obtainthe connectivity that one desires along these eight emit-ting lines defined by energy dispersion of the bath. ••