Supplementary Information for Observation of Parity-Time Symmetry in. Optically Induced Atomic Lattices

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1 Supplementary Informaton for Observaton of Party-Tme Symmetry n Optcally Induced Atomc attces Zhaoyang Zhang 1,, Yq Zhang, Jteng Sheng 3, u Yang 1, 4, Mohammad-Al Mr 5, Demetros N. Chrstodouldes 5, Bng He 1, Yanpeng Zhang 1, 6 and Mn Xao 1 Department of Physcs, Unversty of Arkansas, Fayettevlle, Arkansas 7701, USA Key aboratory for Physcal Electroncs and Devces of the Mnstry of Educaton & Shaanx Key ab of Informaton Photonc Technque, X an Jaotong Unversty, X an , Chna 3 Department of Physcs and Astronomy, The Unversty of Oklahoma, Norman, OK 73019, USA 4 College of Physcs, Jln Unversty, Changchun 13001, Chna 5 CREO, College of Optcs and Photoncs, Unversty of Central Florda, Orlando, Florda 3816, USA 6 Natonal aboratory of Sold State Mcrostructures and School of Physcs, Nanjng Unversty, Nanjng 10093, Chna Contrbuted equally to ths work; Correspondng author: mxao@uark.edu 1. Theoretcal dervatons of the refractve ndex profle wth PT-symmetrc confguraton n the four-level N-type atomc system The densty-matrx equatons for the four-level N-type atomc system (see Fg. 1(b) n the manuscrpt) under the rotatng-wave approxmaton are gven by & ρ =Γ 4ρ44 +Γ3ρ33 Γ 1ρ + ( ρ3 ρ3) Ωc, & ρ33 =Γ43ρ44 Γ3ρ33 Γ 31ρ33 + [( ρ3 ρ3) Ω c + ( ρ13 ρ31) Ωs], & ρ44 = Γ ( 43 +Γ 4 +Γ 41) ρ44 + ( ρ14 ρ41) Ωp, & ρ1 = % 1ρ1 + ( ρ31ωc ρ4ωp ρ3ωs), & ρ31 = % 31ρ31 + [ ρ1ωc ρ34ω p + ( ρ11 ρ33) Ωs ], & ρ41 = % 41ρ41 + [ ρ43ω s + ( ρ11 ρ44) Ωp], & ρ3 = % 3ρ3 + [ ρ1ω s + ( ρ ρ33) Ωc], & ρ4 = % 4ρ4 + ( ρ1ωp ρ43ωc), & ρ43 = % 43ρ43 + ( ρ13ωp ρ4ωc ρ41ωs), ρ + ρ + ρ + ρ = Here, Ω s =μ 13 E s /ħ, Ω c =μ 3 E c /ħ and Ω p =μ 14 E p /ħ are the Rab frequences correspondng to the sgnal, (S1) 1

2 couplng and pump felds, respectvely, μ j s the dpole momentum between levels and j. Γ j s the decayng rate between and j, j =(Γ +Γ j )/ s the decoherence rate. % 1 = 1 ( Δ Δ ), % 31 = 31 Δs, 41 = 41 Δp %, % 3 = 3 Δc, % 3 = 3 Δc, % 3 = 3 Δc, s c % 4 = 4 ( Δ c +Δp Δs), % 43 = 43 ( Δp Δs). Δ s =ω s ω 31, Δ c =ω c ω 3 and Δ p =ω p ω 41 are defned as the frequency detunngs of the sgnal, couplng and pump felds, respectvely. Accordng to the relaton Nμ 13 ρ 31 =ε 0 χe s, the correspondng susceptblty can be obtaned by numercally solvng ρ 31 n Eq. (S1) under steady-state approxmaton. The ntally calculated susceptblty s shown n Fg. S1. By comparng Fgs. S1(b) and 1(d), we can see that the presence of pump feld can gve rse to smultaneous gan and loss n the system, and the zero pont of the magnary part keeps constant at dfferent Ω c. Fgure S1. The theoretcally calculated susceptblty χ versus Δ s. (a) Real part and (b) magnary part of the susceptblty versus Δ s wth Ω p =0. (c) The real and (d) magnary parts of the susceptblty versus Δ s wth Ω p =π 6 MHz. The presence of the pump feld can result n postve and negatve magnary susceptblty at dfferent Δ s values. Other parameters are Ω s =π 0.1 MHz and Δ p =Δ c =0. By properly settng the parameters as Δ s π 15 MHz, Δ c = 100 MHz, Δ p 40 MHz, Ω s =π 0.[1+cos(πx/)] MHz and Ω c =π 0. MHz, the real and magnary parts of the susceptblty can meet the condton for PT-symmetrc potental,.e. χ has symmetrc profle whle the correspondng χ becomes antsymmetrc along the x drecton. The theoretcal real and magnary parts of the susceptblty correspondng to current expermental setup are shown as Fg. n the manuscrpt.

3 . Band structures of the perodcal coupled gan-loss wavegude system The above calculated complex spatal ndex refracton (susceptblty) represents the potental V(x) n the paraxal wave equaton,.e. the Schrödnger-lke equaton [1-4] E E + + V( x) E = 0. (S) z x Consderng the potental s unform along the propagaton drecton, the PT-symmetrc potental descrbes a perodc coupled-wavegude structure. In the potental, the electrc feld can be wrtten as [1] E( x, z, t) = exp( β z)[ A( x) E ( z) + A ( x) E ( z) + A ( x) E ( z) + A ( x) E ( z) A ( x) E ( z) + A ( x) E ( z) + A ( x) E ( z) + A ( x) E ( z) A ( x) E ( z) + A ( x) E ( z), where A m (x) s the egenmode of each wavegude and E m (z) s the ampltude of the mode, m=1,,,10 s the number of the ten coupled wavegudes. As a consequence, the couplng equatons wth 10 wavegudes nvolved are gven as [5, 6] de1 E1+ κ E = 0, dz de + E + κ( E1+ E3) = 0, dz de3 E3 + κ( E + E4) = 0, dz de4 + E4 + κ( E3 + E5) = 0, dz (S4) de5 E5 + κ( E4 + E6) = 0, dz de6 + E6 + κ( E5 + E7) = 0, dz de7 E7 + κ( E6 + E8) = 0, dz de8 + E8 + κ( E7 + E9) = 0, dz de9 E9 + κ( E8 + E10) = 0, d z de10 + E10 + κ E9 = 0, dz where and are the gan and loss experenced by two adjacent wavegudes (for example, the ffth (A 5 (x)) and sxth (A 6 (x)) wavegudes) and κ s the couplng coeffcent. The three coeffcents n Eq.(S4) explctly are gven as: (S3) 3

4 = = κ = V6( x) A5( x) A6( x), A ( x) A ( x) 5 6 V5( x) A6( x) A5( x), A ( x) A ( x) 6 5 V5( x) A6( x) A6( x). A ( x) A ( x) 5 6 Also, we would lke to note that the egenmodes have the followng relatons: and A ( x) = A ( x), A ( x) = A ( x), A ( x) = A ( x), A ( x) = A ( x), A ( x) = A ( x), 5 6 A ( x 4 x ) = A ( x x ) = A ( x) = A ( x+ x ) = A ( x+ 4 x ), A ( x 4 x ) = A ( x x ) = A ( x) = A ( x+ x ) = A ( x+ 4 x ) Here x 0 s the space between two adjacent wavegudes. (S5) (S6) (S7) Accordng to the couplng equatons n Eq. (S4), we can obtan the correspondng band structures (under balanced gan/loss case = = ) shown n Fgs. S(e1) and S(e) n the manuscrpt, whch clearly ndcate that the excepton pont s at about /κ 0.84 when N wavegude =10 wavegudes are coupled. Also, accordng to Fg. S, we can see that the excepton pont value decreases wth ncreasng N wavegude. Fgure S. Band structures of the real and magnary parts wth N wavegude (=, 4, 6, 8, 10) gan-loss wavegudes coupled n an array. The PT-symmetry breakng threshold decreases as the number of wavegudes ncreases. 3. Phase dfference between the adjacent gan/loss wavegudes 4

5 The phase dfference (both below and above the PT-symmetry breakng threshold) between two neghborng wavegudes s calculated accordng to the wavegudes couplng equatons n Eq. (S4). Fgure S3 schematcally shows how the phase dfference s measured n experment. Fgure S3. Schematc dagram for measurng the relatve phase dfference between the gan/loss channels. (a) The nterference between the sgnal feld E S and a reference beam (both of whch are from the same laser) n the y drecton. The phase dfference between the two sold lnes s defned as π. The phase dfference between the black dotted lne (located at the center of the two sold lnes) and one of the two sold lnes s π. (b) The nterference pattern between the ntensty modulated E S feld (after dffracton) and the reference beam, so that the square-lke lattce s obtaned and the phase dfference can then be measured. 4. Three sets of nterference n the experment. There exst three nterference patterns n the current experment. Frst, the couplng beams E c and E c (wth vertcal polarzaton) from the same contnuous-wave dode laser (ECD) are coupled by two polarzaton beam spltters (PBSs) and ntersect at the center of the vapor cell to establsh the frst nterference pattern n the x drecton, namely, the standng-wave couplng feld. The half-wave plates placed n front of the correspondng PBSs can adjust the powers of E c and E c. Second, the two pump beams E p and E p (wth horzontal polarzaton) from the same ECD3 are coupled nto the vapor cell by two reflectve mrrors and buld the standng-wave pump feld wth the powers of E p and E p adjusted by rotatng ther correspondng half-wave plates. Thrd, we establsh an nterference pattern for 5

6 reference outsde the cell n the y drecton by usng the horzontally-polarzed sgnal beam and the reference beam both from ECD1. References 1. El-anany, R., Makrs, K.., Chrstodouldes, D. N. & Musslman, Z. H. Theory of coupled optcal PT-symmetrc structures. Opt. ett., 3, (007).. Makrs, K.., El-anany, R., Chrstodouldes, D. N. & Musslman, Z. H. Beam dynamcs n PT symmetrc optcal lattces. Phys. Rev. ett., 100, (008). 3. Ruter, C. E. et al. Observaton of party-tme symmetry n optcs. Nat. Phys., 6, (010). 4. Regensburger, A. et al. Party tme synthetc photonc lattces. Nature, 488, (01). 5. uo, A. et al. Observaton of PT-symmetry breakng n complex optcal potentals. Phys. Rev. ett., 103, (009). 6. Feng,. et al. Expermental demonstraton of a undrectonal reflectonless party-tme metamateral at optcal frequences. Nat. Mater., 1, (013). 6