Bound and Subradiant Multi-Atom Excitations in an Atomic Array with Nonreciprocal Couplings
aa r X i v : . [ qu a n t - ph ] F e b Bound and Subradiant Multi-Atom Excitations in an Atomic Array with Nonreciprocal Couplings
H. H. Jen ∗ Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan (Dated: February 9, 2021)Collective decays of multiply-excited atoms become subradiant and bound in space when they are stronglycoupled to the guided modes in an atom-waveguide interface. In this interface, we analyze their average density-density and modified third-order correlations via Kubo cumulant expansions, which can arise and sustain forlong time. The shape-preserving dimers and trimers of atomic excitations emerge in the most subradiant cou-pling regime of light-induced dipole-dipole interactions. This leads to a potential application of quantum infor-mation processing and quantum storage in the encoded nonreciprocal spin diffusion, where its diffusion speeddepends on the initial coherence between the excited atoms and is robust to their relative phase fluctuations. Thestate-dependent photon routing can be viable as well in this interface.
Introduction. –Quantum correlation features the essence ofquantum mechanical systems and distinguishes them from theclassical world [1]. This nontrivial correlation provides the es-sential resource with which quantum information processing[2] and quantum computation [3] gain supremacy over theirclassical counterparts. Atom-waveguide interface [4] presentssuch potential of superiority, which has been proposed to cre-ate mesoscopic entanglement [5] and quantum spin dimers[6, 7]. It has also been predicted to manifest strong photon-photon correlations [8], and only recently single photon stor-age and retrieval [9] is realized in an atomic array coupled toa nanofiber. Owing to the guided modes of the waveguide, thelong-range dipole-dipole interactions emerge in light-mattercouplings [10], which leads to superradiant emissions evenfrom two distant atom clouds [11]. This collective light-matter coupling is also responsible for the subradiant emis-sions [12, 13] with a lifetime longer than the intrinsic one, andan enhanced performance of photon storage can be achievedvia the tailored collective states [14].The light-matter coupling can be engineered under externalmagnetic fields to be nonreciprocal with controlled fractionsof left- to right-propagating decay rates [15], which breaks thetime-reversal symmetry [16, 17] that is preserved in most free-space quantum optical settings. Equipped with nonreciprocalor unidirectional light couplings [18–20], chiral quantum op-tics [21] offers many opportunities in generating path-encodedphotons [22], simulating Mach-Zehnder interferometer to re-alize single-photon diodes and circulators [23], creating local-ized edge or delocalized hole excitations [24], and renewingthe study of disorder-assisted single excitation localization un-der dissipations [25].It is interesting and challenging to investigate and generatefew-photon quantum correlations. Recently, a long-lived pho-ton pair [26] has been theoretically proposed in an atomic ar-ray, two-photon and three-photon bound states are observed ina quantum nonlinear medium with the atomic Rydberg states[27], and the few-photon scattering property can be recon-structed from the statistics of light in a single quantum emit-ter coupled to a nanophotonic waveguide [28]. The photonicbound states can also be observed in superconducting qubit ar-rays [29], making the atom-waveguide interface promising tosimulate quantum many-body states of light with long-range quantum correlations.In this Letter, we present the subradiant and bound density-density and third-order correlations, which arise in the dif-fusion of atomic excitations in an atom-waveguide interfacewith nonreciprocal couplings. We construct a Hilbert spaceof multiply-excited states, under which the quantum correla-tions can be obtained via Kubo cumulant expansions [30, 31].We further propose to use the encoded excitations diffusion toconvey and manipulate quantum information in this quantuminterface, which is found to be robust to the phase fluctuationsof the initialized states.
Model of a chirally-coupled atomic array. –The atom-waveguide system can be schematically shown in Fig. 1,where we consider an array of two-level quantum emitters( | g i and | e i for the ground and excited states as an effectivespin / system [32]) which strongly couple with the guided L (cid:74) ge R (cid:74) L (cid:74) ge R (cid:74) L (cid:74) ge R (cid:74) ˆ z ˆ- z ˆ x R L (cid:74) (cid:74)(cid:33)
FIG. 1. Schematic plot of an atomic array with propagating boundatomic excitations. The atomic array of two-level quantum emitters(an effective spin / systems of up and down representing for theexcited | e i and ground states | g i , respectively) with nonreciprocalcouplings ( γ R = γ L ) transport a correlated bound dimer (in the leftstaircase) or trimer of atomic excitations (in the right staircase) tothe right side of the array when γ R > γ L . A strong coupling regimecan be reached when the atoms are positioned close to the waveguideand interact with the guided modes, leading to the subradiant boundmultimers. modes in the waveguide. An effective model of this interface,which allows the nonreciprocal couplings [7], can be writtenin Lindblad forms ( ~ = 1 ), and the density matrix ρ evolvesas dρdt = − i [ H L + H R , ρ ] + L L [ ρ ] + L R [ ρ ] , (1)where, respectively, the coherent and dissipative parts are H L ( R ) = − i γ L ( R ) N X µ< ( > ) ν (cid:16) e ik s | r µ − r ν | σ † µ σ ν − H.c. (cid:17) (2)and L L ( R ) [ ρ ] = − γ L ( R ) N X µ,ν e ∓ ik s ( r µ − r ν ) (cid:0) σ † µ σ ν ρ + ρσ † µ σ ν − σ ν ρσ † µ (cid:1) . (3) σ † µ ≡ | e i µ h g | and σ µ = ( σ † µ ) † are dipole operators, k s de-notes the wave vector in the guided mode, and γ L ( R ) quan-tifies the left(right)-coupling rate. Equation (1) is obtainedwith Born-Markov approximation [33] in an interaction pic-ture (energy difference ω eg between the levels | e i and | g i isabsorbed) under the reservoirs in the allowed dimension [5].An intuitive and normalized directionality factor can be de-fined as [15] D = γ R − γ L γ R + γ L , (4)where γ R + γ L = Γ ≡ | dq ( ω ) /dω | ω = ω eg g k s L [5] is the totaldecay rate. The inverse of group velocity is | dq ( ω ) /dω | witha resonant wave vector q ( ω ) , the coupling strength is g k s , andthe quantization length is L . A fractional D ∈ [ − , quan-tifies the tendency and the amount of light exchange in theatomic array. For a periodic array of atoms with equal spac-ings, we use ξ ≡ k s | r µ +1 − r µ | to characterize the strength ofthe resonant one-dimensional dipole-dipole interactions. Wenote that in Eq. (2), the atomic location has been ordered as r < r < ... < r N − < r N for N atoms. Bound and subradiant spin diffusion. –When multiple atomsare excited initially, the system time dynamics can be solveddirectly from Eq. (1). Under the multi-atom excitations space, | φ p i = σ † j σ † k>j ...σ † l σ † m>l | i as the labeled p th bare state basisfor M excitations in general, the corresponding probabilityamplitudes a p ( t ) can be directly solved from ˙ a p ( t ) = C NM X q =1 V pq a q ( t ) , (5)where V pq denotes the matrix elements of the interaction ker-nel V under a total of C NM bare states with C denoting thebinomial coefficient. We further group these bare states into ( N − M + 1) sectors [34, 35], where each sector denotes oneincrement of the index of the first atomic excitation. There-fore, for example, there will be ( N − sectors for doubleexcitation, where the first and the last bare states in the first FIG. 2. Time evolutions of the bound dimers and trimers of atomicexcitations in an atomic array. (a) Atomic excited state populations P m ( t ) evolve from two atomic excitations side-by-side (left panel)and separated by two lattice sites (right panel) at D = 0 . and . respectively for N = 40 . (b) The time evolutions of the excitationpopulations P m ( t ) from three atomic excitations side-by-side at D = 0 . for N = 21 . In all plots, ξ = π . sector should be | e e g ...g N i and | e g g ...g N − e N i , re-spectively, while the last sector involves only one bare state | g g ...e N − e N i . This is particularly useful for few atomicexcitations space, where the excitation populations can be cal-culated in a systematic way and can be extended to a larger M under a hierarchy relation [34]. For an even larger M . N/ ,the cost of computation time increases exponentially as ex-pected. From a p ( t ) , we then obtain the excitation population P m ( t ) = h ψ ( t ) | σ † m σ m | ψ ( t ) i with | ψ ( t ) i = P C NM p =1 a p ( t ) | φ p i [34], and P Nm =1 P m ( t ) = M is the conserved quantity of to-tal spin excitations.In Fig. 2, we plot the time dynamics of the doubly- andtriply-excited state populations from the initialized productstates σ † j σ † k | i with j − k = 1 and respectively in Fig. 2(a)and σ † j σ † j +1 σ † j +2 | i in Fig. 2(b), with j chosen near the cen-ter of the array. They propagate toward the end of the chainat long time and preserve their respective shapes. The long-time dynamics is particularly significant when ξ is chosenclose to π , where alternate minus and plus signs of hoppingrates manifest in the excited atoms with a mutual separationof (2 n + 1) λ/ and nλ for an integer n , respectively. Thissubradiant dynamics has been investigated in a singly-excitedatomic array [13], and superradiance, by contrast, can showup when ξ is close to or π .The spin propagation is ballistic since its speed of spin dif-fusion is ∝ D within each initialized configuration of the ex-cited product states. For doubly-excited states, the bound spinexcitations diffusion is faster when | j − k | is smaller, whichreflects the blockade of spin exchange between the nearest-neighbor atoms, effectively leading to a pushing force for thebound pair and the expedition of it at a shorter distance. Thediffusion speed saturates as | j − k | increases and approachesthe single excitation limit as if there are no correlations in theindependent atoms. For more spin excitations side-by-side inFig. 2(b), an enhanced diffusion shows up but suffers froma larger intrinsic decay of − M Γ / for M excitations. Theinterference patterns in Fig. 2 is typical and common in theexcitation propagation in the atom-waveguide interface [25],which results from multiple light reflections and transmissionsthroughout the atomic array. Long-range and long-time correlations. –To reveal the long-range and long-time characteristics of the multi-atom excita-tion transport, we employ the correlation functions via Kubocumulant expansions [30]. Essentially the cumulant expan-sion reduces the mean values of the high order quantum cor-relations to the lower order cumulants of correlations. In par-ticular, we take the average density-density and third-ordercorrelation functions as the bulk properties of the atomic ar-ray [36]. They are, respectively, h G (2) ( r ) i ≡ X j h n j n j + r i − h n j ih n j + r i N − r , (6) h G (3) i ≡ X j h n j n j +1 n j +2 i − h n j ih n j +1 ih n j +2 i N − , (7)where the site r denotes the correlation length for doubly-excited spin diffusion.For the modified h G (3) i we consider here as the third ordercorrelation function side-by-side, it actually involves the thirdorder cumulant and three other second order cumulants, whichare [30] ( h n j n j +1 i − h n j ih n j +1 i ) h n j +2 i , ( h n j n j +2 i − h n j ih n j +2 i ) h n j +1 i , ( h n j +1 n j +2 i − h n j +1 ih n j +2 i ) h n j i . Therefore, h G (3) i represents the correlation function morethan just the third order cumulant and can be regarded as the FIG. 3. Average density-density and third-order correlations. Thescaled and time-dependent density-density correlations h ˜ G (2) ( r ) i = h G (2) ( r ) i × arise at particular lattice separation r , when weinitialize double atomic excitations at r = 1 (solid line in red) in (a), (dotted line in blue) in (b), and (dash-dotted line in black) in (c), at D = 0 . , ξ = π , and N = 40 . The correlation functions at longer r =4 (dashed line in yellow) and (solid line in green close to and rightbelow the dashed line in yellow) are shown for comparison. In (d),the scaled third-order correlation h ˜ G (3) i = h G (3) i × is shownfor the case in Fig. 2(b) (solid line in red). As a comparison, thecases of ξ = 7 . π/ (dash-dotted line in green) and . π/ (dottedline in blue) are shown. upper bound of it if the sum of these second order cumulantsare positive.In Fig. 3, we numerically calculate the average and time-dependent density-density and modified third order correla-tion functions for various initialized product states of doubleexcitations σ † j σ † j + r | i . Significant h G (2) ( r ) i arises and sus-tains for long time at specific r , which highly correlates tothe initialized excitation configuration. At a longer r > , h G (2) ( r ′ < r ) i is suppressed initially but revives at a latertime. Interestingly, for most of the time dynamics before allcorrelations vanish, longer-range density-density correlationsdevelop, and h G (2) ( r ′′ ) i > h G (2) ( r ′′′ ) i when r ′′′ > r ′′ > r .This suggests a sequential spread of the correlations whenthe spin excitations traverse through the atomic array, where h G (2) ( r + 1) i grows preferentially over the others.The higher order correlation of the initialized product statesof triple excitations σ † j σ † j +1 σ † j +2 | i is shown in Fig. 3(d).Similar to the double excitations, a subradiant trimer of atomicexcitations manifests in the coupling regime close to ξ = π .The h G (3) i decays faster only slightly away from the subra-diant coupling regime, indicating the fragility in maintainingthe correlations in multiply-excited atomic excitations. Encoded spin diffusion. –Next we focus on the spin diffu-sion of the double excitations and propose an encoded spindynamics depending on its initial coherences. Other than theinitialized doubly-excited product states σ † j σ † j +1 | i with a nullentanglement entropy S , we can initialize the system with a fi-nite S . In general, the initial doubly-excited states that sharethe excitations separately can be expressed as (cid:16) cos φσ † j + sin φσ † j +1 (cid:17) ⊗ (cid:16) cos θσ † j +2 + sin θσ † j +3 (cid:17) | i , (8)which features the initial S = ln 2 and for ( φ, θ ) =( π/ , and ( π/ , π/ , respectively, at a cut right next tothe sites ( j + 2) and ( j + 3) . The most entangled doubly-excited state in the form of Eq. (8), on the other hand, shouldbe equally distributed within the subspace, √ (cid:16) σ † j σ † j +1 + σ † j σ † j +2 + σ † j σ † j +3 + σ † j +1 σ † j +2 + σ † j +1 σ † j +3 + σ † j +2 σ † j +3 (cid:17) | i , with S = ln 6 .In Fig. 4(a), we plot P N ( t ) for four different initializeddoubly-excited states as a probe of the spin diffusion at theedge of the array. They can be classified by the maximal P N ( t c ) at which a small (large) t c corresponds to a null orlower (higher) S . This distinction of spin diffusion dynamicsoffers an opportunity for the state-dependent photon routing.In the insets of Fig. 4(a), the space-time dynamics of dou-ble spin diffusion presents different propagation speeds andinterference patterns, where the respective maximal excitationpopulation appears at the forefront and the tail in P N ( t ) , re-spectively. The particular long period of the oscillations in P N ( t ) for S = 2 ln 2 and ln 6 before reaching their maximums FIG. 4. Encoded spin diffusion. (a) At D = 0 . , the end atomic ex-citation population probes the speed of spin diffusion for various ini-tialized entanglement entropy S = 0 (solid line in blue), ln 2 (dashedline in red), (dash-dotted line in green), and ln 6 (dotted line inblack). The insets present a faster (upper) and slower (lower) pairedspin diffusion for the cases of S = ln 2 and , respectively. (b)The θ dependence of the characteristic time t c for D = 0 . ( ◦ ), . ( (cid:3) ), and ( ⋄ ). ξ = π and N = 40 are used in both plots. can be seen as a precursor, in contrast to the case of a lower S with afterglow fringes. These phenomena are also presentin the case for a larger D [34]. This suggests that more en-tangled initial excitations propagate more subradiantly closeto the unidirectional coupling regime.For a smaller D . . , the interferences from the bi-directional light couplings are more involved in the spin dif-fusion dynamics, where the initially suppressed h G (2) ( r ) i canrevive and decline again, or even longer-range h G (2) ( r ) i canemerge, making the encoded spin diffusion indistinguishable.For the purpose to demonstrate encoded spin diffusion, we fo-cus on D & . .From Eq. (8), we plot the the dependence of t c on θ at φ = π/ in Fig. 4(b). As expected, the time to reach the endof the atomic array is ∝ D − , and the maximal t c appearsat θ ≈ π/ , corresponding to the most entangled state in oursetting. The various θ dependence plateaus also indicate therobustness to the relative phase fluctuations in θ . Therefore,a scheme using atom-waveguide interface to manipulate thestate-dependent photon routing becomes feasible. Discussion and conclusion. –The atom-waveguide interfaceprovides rich opportunities in processing quantum informa-tion [9] and quantum many-body simulations [29]. The ca-pability of this interface to reach the strong coupling regime[37–39] makes viable our prediction of the bound and sub-radiant dimers and trimers of atomic excitations. Althoughthe subradiant spin diffusion is fragile to the position fluctua-tions of the periodic array, they can be mitigated by applyingan optical lattice near the waveguide [9]. Moreover, it wouldbe intriguing to further look into the dynamics of the shape-preserving multimers, which can benefit from long-time spindiffusion aided by the guided modes in the waveguide and thusmay host quantum nonlinear interactions in photons [4, 40]with controlled strengths of dipole-dipole interactions and di-rectionality of light couplings. In conclusion, we theoretically investigate the time-evolveddensity-density and third-order correlations in the atom-waveguide system from the initialized product or entangledstates of multiple atomic excitations. Significant correlationscan arise and sustain for long time, along with the shape-preserving characteristics in dimers and trimers of the exci-tations. We demonstrate that the encoded nonreciprocal spindiffusion is robust to their relative phase fluctuations, whichcan be useful and advantageous in quantum information pro-cessing, quantum storage and transport, and state-dependentphoton routing.We acknowledge support from the Ministry of Science andTechnology (MOST), Taiwan, under the Grant No. MOST-109-2112-M-001-035-MY3. We are also appreciated forinsightful discussions with Jhih-Shih You and Ying-ChengChen. ∗ [email protected][1] W. H. Zurek, Physics Today , 36 (1991).[2] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev Mod Phys. , 1041 (2010).[3] D. P. DiVincenzo, Fortschritte der Physik: Progress of Physics , 771 (2000).[4] D. E. Chang, J. S. Douglas, A. Gonz´alez-Tudela, C.-L. Hung,H. J. Kimble, Rev. Mod. Phys. , 031002 (2018).[5] A. Gonz´alez-Tudela and D. Porras, Phys. Rev. Lett. ,080502 (2013).[6] T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, Phys. Rev. Lett. , 237203 (2014).[7] H. Pichler, T. Ramos, A. J. Daley, and P. 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The construction of multi-atom excitation space follows ourprevious work on multi-photon subradiant states [35]. Theidea is simply that we first order N atomic positions as r 2) = Sort ( | e e g g i , | e g e g i ) gives (2 , , showing that thedipole-dipole interaction lowers the third atomic excited statewhile raising the second atom to the excited one. If ( s , s ) gives (0 , , it leaves a null V n,m = n when more than one dis- tinct atomic indices appear in the p th or q th bare states. Usingthe same example, V , ( V , ) = 0 , since there is no dipole-dipole interaction coupling between the states | e e g g i and | g g e e i ( | e g e g i and | g e g e i ) [35]. Derivations of atomic excitation populations To calculate the atomic excitation population P m ( t ) , wegroup the bare states introduced above into ( N − M + 1) sec-tors, where each sector denotes one increment in the label µ .Therefore, there will be ( N − sectors for double excitations.The first bare state in the first sector is | e e g ...g N i , while thelast sector involves only one bare state | g g ...e N − e N i . Thisis particularly useful for few atomic excitations space, wherethe excitation populations can be calculated in a systematicway and can be extended to a larger M under a hierarchy re-lation.Within the n th sector, we use the labels s ( M ) n to denote ~µ n in the above, where s ( M ) n ( l ) denotes the l th bare state basis inthis sector, and dim ( s ( M ) n ) denotes the total number of the el-ements in the sector. The number of elements in the first sector s for M excitations is denoted as N ( M ) s, = C N − M − . Within each n th sector, we use the label N ( M ) s,n = N ( M ) s,n − + dim ( s ( M ) n ) asthe total number of elements up to the n th sector. We alsodefine N (1) s,α = 1 , N ( α ) s, = 0 for arbitrary α < N .The final results for general atomic excitation populations P ( M ) m ( t ) are P ( M ) m ( t ) = s ( M ) m [ dim ( s ( M ) m )] X n = s ( M ) m (1) | a n ( t ) | + δ M> M − X k =1 m − k> X n =1 X l = l I | a l | + δ M =2 δ k =1 + δ M> N − X k = M − ! m − k> X n =1 | a l ′′ | (14)where l I = 1 + N ( M ) s,n − + P m − n − l ′ = k ( N − m + 1 + l ′ ) ( M − s, and l ′′ = k − ( M − 2) + N ( M ) s,n − + P m − n − l ′ = k ( N − m + 1 + l ′ ) ( M − s, . The Kronecker delta function is δ . The hierarchyrelation is embedded in the sums of index k , which becomescumbersome as M increases owing to the large Hilbert spaceof a total C NM of them. Encoded spin diffusion for a larger D Here we demonstrate the encoded spin diffusion for a dif-ferent and a larger D in Fig. 5. The case for a larger D indi- cates a faster spin propagation in general. Similar observationis shown here, where the initialized double excitation with anull or lower entanglement entropy S propagates faster thanthe one with a higher S . This indicates a distinction of spindiffusion from different initialized coherence properties of thestates and suggests a potential application to state-dependentphoton routing. FIG. 5. Encoded spin diffusion for a larger D = 0 . . The atomicexcitation population P N ( t ) probes the speed of spin diffusion forvarious initialized entanglement entropy S = 0 (solid line in blue), ln 2 (solid line in red), (solid line in green), and ln 6 (solid linein black). The insets present a faster (upper) and slower (lower) spindiffusion for the cases of S = ln 2 and , respectively. ξ = π and N = 40= 40