Dynamics of Many-Body Photon Bound States in Chiral Waveguide QED
Mahmoodian, Sahand; Calajó, Giuseppe; Chang, Darrick E.; Hammerer, Klemens; Sørensen, Anders Søndberg
Published in: Physical Review X
DOI: 10.1103/PhysRevX.10.031011
Publication date: 2020
Document version Publisher's PDF, also known as Version of record
Citation for published version (APA): Mahmoodian, S., Calajó, G., Chang, D. E., Hammerer, K., & Sørensen, A. S. (2020). Dynamics of Many-Body Photon Bound States in Chiral Waveguide QED. Physical Review X, 10, [031011]. https://doi.org/10.1103/PhysRevX.10.031011
Download date: 27. sep.. 2021 PHYSICAL REVIEW X 10, 031011 (2020) Featured in Physics
Dynamics of Many-Body Photon Bound States in Chiral Waveguide QED
Sahand Mahmoodian ,1 Giuseppe Calajó ,2 Darrick E. Chang,2,3 Klemens Hammerer,1 and Anders S. Sørensen4 1Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Appelstraße 2, 30167 Hannover, Germany 2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 3ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain 4Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
(Received 29 November 2019; revised 7 May 2020; accepted 11 June 2020; published 14 July 2020)
We theoretically study the few- and many-body dynamics of photons in chiral waveguides. In particular, we examine pulse propagation through an ensemble of N two-level systems chirally coupled to a waveguide. We show that the system supports correlated multiphoton bound states, which have a well- defined photon number n and propagate through the system with a group delay scaling as 1=n2. This has the interesting consequence that, during propagation, an incident coherent-state pulse breaks up into different bound-state components that can become spatially separated at the output in a sufficiently long system. For sufficiently many photons and sufficiently short systems, we show that linear combinations of n-body bound states recover the well-known phenomenon of mean-field solitons in self-induced transparency. Our work thus covers the entire spectrum from few-photon quantum propagation, to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. Finally, we demonstrate that the bound states can undergo elastic scattering with additional photons. Together, our results demonstrate that photon bound states are truly distinct physical objects emerging from the most elementary light-matter interaction between photons and two-level emitters. Our work opens the door to studying quantum many-body physics and soliton physics with photons in chiral waveguide QED.
DOI: 10.1103/PhysRevX.10.031011 Subject Areas: Quantum Physics
I. INTRODUCTION frequencies [7–9]. Such an interface creates a highly nonlinear medium as photons propagating in a waveguide Generating quantum many-body states of light remains interact deterministically with atoms. Systems of this kind one of the outstanding challenges of modern quantum optics [1]. Such many-body states of light are of funda- have thus far been used to propose or demonstrate the generation of states of photons with strong two- or three- mental physical interest as they arise from nonequilibrium – systems with strong interactions between light and matter. body correlations [10 20]. On the other hand, they also promise to form novel Studies of photon correlations in these systems typically resources for quantum technologies, for example, in consider steady-state driving, and photon correlations are quantum-enhanced metrology [2]. The main obstacle in subsequently measured in relative coordinates. On the other the pursuit of generating such many-body states has been hand, here we show that pulse propagation through two- the difficulty in developing one-dimensional systems with a level systems (TLSs) strongly coupled to a waveguide leads sufficiently strong nonlinear response at the few-photon to very distinct temporal features which reveal the under- scale [3]. Recently however, significant progress has been lying dynamics and has the potential to generate temporally made in creating an ideal light-matter interface between ordered many-body states of light. In particular, we atoms or artificial emitters coupled to a one-dimensional theoretically consider the propagation of pulses of coherent continuum of photons at optical [4–6] and microwave and Fock states of light through a waveguide to which N TLSs are chirally coupled. We show that photons undergo a Wigner delay [21,22]—a delay of the pulse center due the optical excitation transferring to the atom and back to the Published by the American Physical Society under the terms of waveguide—which is dependent on the number of photons. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Therefore, incident pulses break up into a state that is the author(s) and the published article’s title, journal citation, temporally ordered by its photon correlations. For example, and DOI. as shown in Fig. 1(a), an incident coherent field can
2160-3308=20=10(3)=031011(14) 031011-1 Published by the American Physical Society SAHAND MAHMOODIAN et al. PHYS. REV. X 10, 031011 (2020)
(a) with atoms coupled to a cavity [23,24]. A key difference is that the delay in VIT mainly depends on the total photon number inside the entire system, and not on the details of the pulse shape. The nonlinear effect we consider is spatially localized to the individual atoms and occurs for the simplest possible configuration of TLSs. This leads to a different spatiotemporal behavior which we evaluate in a full multimode theory of the dynamics. Our theory features (b) a full quantum many-body treatment of the system, where the photon time delay is examined by considering the many-body photonic scattering eigenstates. We also point out that in mean-field theories, solitons are FIG. 1. (a) N two-level atoms (blue circles) are chirally known to be highly stable objects which are unaffected by Γ coupled to a waveguide with decay rate anddrivenbyan external perturbations. Here we show that similar properties input Gaussian pulse, which can be a coherent or Fock state. The exist for few-photon bound states. We outline how one can light pulse propagates with a correlation-number-dependent groupvelocityleadingtoanoutputstatewhereone-,two-, conduct scattering experiments between photon bound and three-photon bound states are spatially separated. (b) Sche- states and individual photons. In the considered scattering matic of the bound-states propagation. When an n-photon experiments, the bound state is deflected by the interaction bound state is scattered by an atom, it reemits the absorbed but is otherwise unperturbed by it. Together, the results we photon with a stimulated emission rate Γn, coinciding with the obtain here demonstrate that the bound states should be inverse of its width. Since only a single photon out of n is considered as truly distinct physical entities emerging from delayed by an amount 4=ðΓnÞ, the pulse preserves its shape but the underlying light-matter interaction between photons τ 4 Γ 2 is delayed by n ¼ =ð n Þ. and two-level emitters. This manuscript is arranged as follows: In Sec. II,we produce a pulse with three-photon correlations followed introduce the model for chiral waveguide QED (WQED) by two-photon correlations states followed by uncorrelated and outline various theoretical approaches for computing photons. The underlying physics that causes this time the dynamics through Sec. II A mean-field theory, Sec. II B delay is examined by considering the many-body photonic the photon-scattering eigenstates, and Sec. II C the matrix- scattering eigenstates of TLSs chirally coupled to a product states (MPSs). In Sec. III, we compute how the waveguide. input pulse propagates through the medium and compute Of central importance are the class of bound eigenstates, the representation in terms of photon bound states. This is where two or more photons propagate together. We show followed by Sec. IV, which shows that the many-body that the photon bound states propagate past each atom with photon bound states can be used to construct the mean-field 2 soliton solutions obtained in self-induced transparency. a photon-number-dependent delay of τn ¼ 4=ðn ΓÞ, where Γ is the decay rate into the waveguide, as can be understood In Sec. V, we show that photon bound states can undergo in terms of absorption and stimulated emission of a photon elastic scattering with individual photons modifying the as shown in Fig. 1(b). Since the interaction is chiral, the delay of the bound state but otherwise leaving it unaltered, delay is also proportional to the number of emitters N in the much like classical solitons. In Sec. VI, we show that waveguide. We also show that the pulse distortion encoun- few-photon bound-state propagation can potentially be tered by an n-photon bound state scales as n−6. Pulses of observed in state-of-the-art experiments with a few emit- higher-number bound states therefore propagate with ters, and we discuss potential future applications. Finally, negligible distortion, i.e., much like a soliton. Moving we conclude in Sec. VII. beyond the few-photon–few-atom limit, we obtain a simple description of photon propagation with a mesoscopic II. MODEL number of photons n but a large number of atoms N ≫ 1. Here, the number-dependent group delay breaks We consider a system of N TLSs chirally coupled to a the input pulse apart and produces a many-body ordered linearly dispersive one-dimensional bath of photons. Here, state of light. Finally, we show that the system approaches chiral coupling means that the TLSs couple only to right- the classical limit for n ≫ 1 and n ≫ N, where our full propagating photons. The Hamiltonian for this system ℏ 1 quantum description captures the quantum-to-classical ( ¼ )is transition and reproduces the mean-field results of soliton Z ffiffiffi XN propagation in self-induced transparency. † p − † Hˆ ¼ −i dxaˆ ðxÞ∂ aˆðxÞþ Γ ½σˆ aˆ ðx Þþσˆ þaˆðx Þ ; The effect we discuss here has features similar to x i i i i i¼1 vacuum-induced transparency (VIT) where photons undergo a photon-number-dependent delay after interacting ð1Þ
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ℜ σˆ where all integrals are over , i are Pauli operators for the medium. In particular, the spin dynamics can be efficiently ith TLS, aˆðxÞ [aˆ †ðxÞ] is a photon annihilation (creation) solved by making use of a MPS ansatz [31,32], as recently operator at position x, the group velocity is set to unity described in Ref. [27]. As we show in the following, this vg ¼ 1, and the energy is renormalized to that of the TLS. approach allows us to fully explore the limit of many In the limit of ideal chiral coupling, the system dynamics photons and large atomic arrays, a scenario that is chal- lenging to simulate with standard numerical techniques. are not influenced by the positions of the TLSs xi. In this Hamiltonian, the first term captures the free propagation of photons in the waveguide with linear dispersion, while the A. Mean-field theory and self-induced transparency next terms take into account the interaction between the Before considering the full many-body dynamics of photons and the emitters. In the absence of the interaction the Hamiltonian (1), we consider the system within the terms, the photonic eigenstates of the Hamiltonian are mean-field limit. We present this mean-field limit to ω plane waves with wave vector k and frequency ¼ k. contrast its predictions with the full many-body theory Equation (1) constitutes the typical scenario for chiral that we present below. WQED [25]. In this manuscript, we are interested in The first treatment of Eq. (1) within mean-field theory describing the propagation of a multiphoton input field dates back to the work on self-induced transparency (SIT) through the system. This goal can be achieved by exactly [33–35]. In these early experiments, gasses of two-level computing the scattering eigenstates of Hamiltonian (1),as atoms were excited by short intense laser pulses. Although we illustrate in Sec. II B. the atoms are not ideally coupled to a single-waveguide In addition to computing the eigenstates of Eq. (1),we mode in such systems, the laser pulses are sufficiently short also introduce an equivalent formulation that is well suited so that decay channels to modes other than the laser mode for using the MPS technique that we introduce later in can be neglected. Furthermore, both the weak coupling of Sec. II C. This approach is based on the observation that the atoms to this mode and the high intensity of the laser ’ the transmitted field depends directly on the emitters means that one can consider the atoms as a spin continuum evolution. This can be seen by formally integrating the under the mean-field approximation where quantum cor- ˆ Heisenberg equation for the field operator aðx; tÞ, that relations between the atoms and the light field can be provides, within the Born-Markov approximation, the neglected. Under these approximations, the equations of following generalized input-output relation for the trans- motion give the SIT equations mitted field [26,27] ffiffiffi X pffiffiffi ∂ ∂ p ˆ E Γ σˆ − þ aðx; tÞ¼−i Γσ−ðx; tÞ; aoutðtÞ¼ inðtÞþi j ðtÞ; ð2Þ ∂t ∂x j ∂ pffiffiffi σ−ðx; tÞ¼i Γσ ðx; tÞaðx; tÞ; ˆ ˆ þ ∂ z where we define aoutðtÞ¼aðxN;tÞ as the output field t measured right after the last atom. Note that, within this ∂ pffiffiffi σ 4 Γ σ E zðx; tÞ¼ Im½aðx; tÞ −ðx; tÞ : ð5Þ approach, we assume the input field inðtÞ to be a classical ∂t coherent field on resonance with the atomic transitions. P ˆ σˆ σˆ −=z δ − With these assumptions, the emitter dynamics driven by the Here, aðx; tÞ¼haðx; tÞi, −=zðx; tÞ¼ i i ðtÞ ðx xiÞ, input field is known to be described by a purely dissipative and σ−ðx; tÞ¼hσˆ −ðx; tÞi, σzðx; tÞ¼hσˆ zðx; tÞi are the chiral master equation (ME) of the form [28,29] (see the expectation values of the spin operators in the continuum Supplemental Material [30] for more information): limit. These nonlinear equations of motion have SIT soliton X solutions. Following the treatment in Ref. [35], for a ρ_ − ρ − ρ † Γ σˆ −ρσˆ þ ¼ iðHeff HeffÞþ i j : ð3Þ resonant pulse the field can be taken to be real, and one ij can map the equations of motion onto a nonlinear pendu- lum equation that can be solved exactly. This treatment Here, leads to the fundamental soliton solution for the field Γ X X pffiffiffi H ¼ −i σˆ þσˆ − þ H − iΓ σˆ þσˆ − ð4Þ Γn¯ Γn¯ x eff 2 j j drive l j aðx; tÞ¼ sech − t ; ð6Þ j l>j 2 2 V0 is the effective Hamiltonian, which provides the non- where n¯ is the number of photons in the field, (or more Hermitian collective evolution of the emitters, while the P pffiffiffi precisely, the total energy in the original SIT work). The Γ E σˆ þ term Hdrive ¼ j ½ inðtÞ j þ H:c: gives the coupling pulse velocity within the medium in the laboratory frame is of the emitters to the input field. V0 ¼ n¯ 2Γ=ðn¯ 2Γ þ 4νÞ, where ν is the gas density (see the The combination of Eqs. (2) and (3) provides a full Supplemental Material [30] for details). Transforming to a description of the photon propagation through the chiral frame comoving with the pulse in the absence of emitters,
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1 i.e., at velocity vg ¼ , gives the relative velocity in the off all N emitters, theQ eigenstates are multiplied by 2 N n N 4ν ¯ Γ 4ν the eigenvalue tk ¼ 1 t , where t ¼ðk − iΓ=2Þ= backwards direction V ¼ =ðn þ Þ. This value for j¼ kj k the relative velocity corresponds to each emitter imparting a ðk þ iΓ=2Þ. Since, by assumption, the system does not 2 delay of τn¯ ¼ 4=ðn¯ ΓÞ on the pulse. contain any dissipation and is chiral, all the transmission An important feature of the solitonic solution (6) is that 1 pffiffiffi R coefficients have jtkj¼ . This means that the transmission the integrated Rabi frequency Ω ¼ 2 Γ dtaðx; tÞ, which coefficients simply multiply the eigenstate by a phase. For is proportional to the area under the pulse, is fixed by the example, a single photon on resonance undergoes a π-phase relationship between the pulse amplitude and pulse width, shift with t0 ¼ −1. Importantly, the phase that is imparted and always evaluates to be 2π. This corresponds to a full on the eigenstate varies with k; i.e., the TLSs introduce Rabi cycle of complete excitation and subsequent deexci- dispersion to the system. As we soon show, different types tation. The SIT soliton therefore can be physically inter- of eigenstates also accumulate different phases. We note preted as a rapid excitation and deexcitation of the atoms, that the output states are described using the position which suppresses spontaneous emission of the excited state coordinates x. This is equivalent to using the time variable and thus makes the medium transparent. −t in Eq. (2) as we set the group velocity to unity. Inspired by the apparent dependence of the velocity on In addition to different eigenstates for different values of photon number, it is interesting to ask whether this property n, there are also different possible types of eigenstates extends to the few-photon limit, thus enabling, e.g., photon- within different excitation-number manifolds. In the single- number separation at the output. A full quantum treatment excitation subspace, there is only one type of eigenstate is necessary to answer this question, which we turn to in the and it is characterized by a real wave number k and the following sections. eigenstate is fully extended in space. For n ¼ 2, the wave numbers k1 and k2 can either both be real which gives rise B. Many-body scattering eigenstates to a fully extended solution of the form (7). They can also assume complex values k1 ¼ E=2 þ iΓ=2 and k2 ¼ E=2 − In contrast to the mean-field treatment, we now consider iΓ=2 which are called strings [37]. These values give the full many-body eigenstates. Since the Hamiltonian (1) additional valid solutions. Since the wave vectors can form preserves the combined number of atomic and photonic complex conjugate pairs, the eigenstates become localized excitations, eigenstates with different numbers of excita- in a relative coordinate while being fully extended within tions decouple. By computing the eigenstates in the one- the center-of-mass coordinate. This localization is associ- and two-excitation subspaces, one can generalize the result ated with the formation of photon bound states that to an arbitrary number of excitations. This technique is manifest in the bunching of two photons which travel often referred to as Bethe’s ansatz [36] and is used to together during their propagation [39,40]. diagonalize a class of one-dimensional many-body For n>2, the m-body string (m ≤ n) has the wave Hamiltonians [37]. In particular, it has previously been vector k ¼ K=m − i½m þ 1 − 2j Γ=2 with j ¼ 1; 2; …;m used to diagonalize the Hamiltonian in Eq. (1) [38]. Since j where K is the total energy of the m string. In total, the we are interested in the state that emerges after interaction n-photon manifold has p n string combinations where with the TLSs, we are interested in the scattering eigen- ð Þ p n is the number of partitions of n. For example, for three states. These are photon eigenstates that interact with the ð Þ photons, there is a completely extended scattering eigen- TLSs and emerge unchanged apart from an overall trans- state, a completely bound eigenstate, and a hybrid state mission coefficient. The n-body scattering eigenstates have with two bound photons and one extended photon. The the form different string combinations give the different types of Z 1 scattering eigenstates S by substituting the complex wave ffiffiffiffiffi nxaˆ † x 0 jSkin ¼ Ck;n;S p d ð Þj i vectors into Eq. (7). The transmission coefficient is also n! obtained by substituting the complex wave vectors into the Yn Yn expression for tk. × ½k − k þ iΓsgnðx − x Þ eikjxj þ ↔; l m l m For n photons, the n-string state is a fully localized l 031011-4 DYNAMICS OF MANY-BODY PHOTON BOUND STATES IN … PHYS. REV. X 10, 031011 (2020) 2 E − iΓn =2 D1 ¼ 1 and DN ¼ 1. The bond dimension reflects the tE;n ¼ 2 : ð9Þ 1 E þ iΓn =2 entanglement entropy. For instance, if Dj ¼ for all j, the matrices Aij are scalars and the state (11) reduces to a Importantly, the phase of the transmission coefficient varies product state with no entanglement. For arbitrary states, the with n; i.e., the system has a photon-number-dependent bond dimension grows exponentially with the number of dispersion. particles. The advantage of the MPS ansatz is that, in many With all the eigenstates at hand, the scattering matrix for physical scenarios as the one considered here, the entan- interacting with all N emitters in the n-photon manifold can glement grows slowly with the system size allowing an be formally written as efficient description of the state in terms of a smaller X X bond dimension [32]. An important figure of merit of ˆ N N Sn ¼ tk jSkinnhSkj; ð10Þ the efficiency of the MPS ansatz is given by Dmax, the S k maximum bond dimension that is needed to faithfully represent the system during the entire time evolution (see where the sum over S is a sum over the different string the Supplemental Material [30] for more information). We combinations of the n-photon manifold. We note that the make use of this quantity in Sec. IV to quantify the amount eigenstates are orthogonal; thus, the scattering matrix for N of many-body correlations present in the system. emitters simply requires taking the eigenvalue to the Nth power. Since the number of string combinations increases III. MANY-BODY PULSE PROPAGATION as pðnÞ, the number of terms in the sum increases exponentially for large n [38,41]. In this manuscript, we We are interested in studying multiphoton propagation compute the full output states for up to n ¼ 3 using through the chirally coupled array. Here, we consider this formalism. We note that a formalism exists where coherentR and Fock input states with mode creation operator one does not have to sum over string combinations [42]. ˆ † E ˆ † ain ¼ dx ðxÞa ðxÞ, and we evaluate the transmitted field Nevertheless, the form used here is particularly insightful with the two methods described in the previous section. as it gives direct access to the number-dependent trans- In particular, for the exact solution we compute the mission coefficient which, as we show, plays a central role transmitted photon state using the eigenstates for up to in understanding the many-body pulse propagation. three photons, while for higher excitations we make use of the MPS ansatz. In Fig. 2, we consider the propagation C. MPS ansatz of a Gaussian photonic mode with amplitude EðxÞ¼ 2 2 pffiffiffi In order to study the dynamics for stronger input pulses eik0x−x =ð2σ Þ=ð σπ1=4Þ where, throughout the manuscript, 3 (n> ) than the one computed with the S-matrix formal- resonant pulses are considered k0 ¼ 0. ism, we solve Eqs. (2) and (3) using a MPS ansatz. Figures 2(a)–2(c) show the power PðxÞ¼haˆ †ðxÞaˆðxÞi Specifically, the system evolution can be solved either for one-, two-, and three-photon Fock states after propa- by directly solving the ME (3) [43] [method used for gating through N ¼ 20 TLSs. Here, x ¼ 0 is chosen to be Figs. 2(d) and 3] or by using a quantum trajectories the reference frame of the pulse propagatingffiffiffi in the absence algorithm where the state of the system evolves under p of emitters. A pulse width of Γσ ¼ 3 2 is chosen to have the effective Hamiltonian (4) and stochastically experiences appreciable overlap with all the different types of scattering quantum jumps [27] [method used for Fig. 2(f)]. In both eigenstates while remaining sufficiently narrow to observe cases, a MPS representation is applied either to the the photon-number-dependent velocities. The magnitude of quantum state or to a vectorized form of the density the overlap of an input Gaussian pulse with two- and three- matrix. Here for simplicity, we limit the discussion to photon bound states versus Gaussian pulse width σ is the former, while the latter is discussed in the Supplemental shown in the Supplemental Material [30]. In Fig. 2(b),we Material [30]. see that the two-photon bound state comes out earlier than The MPS ansatz consists of reshaping the generic P the extended state. The two-photon bound state thus clearly quantum state jϕi¼ … ψ … ji1;i2; …;i i (with i1; ;iN i1;i2; ;iN N propagates with a faster velocity than the extended state. ∈ ij fg; eg) into a matrix-product state of the form: The bound state also undergoes significantly less broad- X ening and distortion. For the three-photon transport in ϕ … … j i¼ Ai1 Ai2 ; ;AiN ji1;i2; ;iNi; ð11Þ Fig. 2(c), again the extended state is distorted and delayed, … i1; ;iN while the three-photon bound state has significantly less distortion and delay. In the three-photon manifold, there is where, for each specific set of physical indices also a string combination that forms a hybrid state where fi1;i2; …;i g, the product of the A matrices gives N ij one photon is completely extended while the other two are ψ … A back the state coefficient i1;i2; ;iN . Each matrix ij has bound. The evolution of this state is determined by the dimension Dj−1 × Dj known as the bond dimension, and individual components of the state: The two bound photons finite-edge boundary conditions are assumed by imposing propagate in a similar manner to the two-photon bound 031011-5 SAHAND MAHMOODIAN et al. PHYS. REV. X 10, 031011 (2020) (a) One-photon Fock state Two-photon Fock state Output Output Input field Bound state Extended state (c)Three-photon Fock state (d) Coherent state Output Bound state Hybrid state Extended state (e) (f) FIG. 2. Transmitted power P x (solid curves) for (a) one- (b) two-, and (c) three-photon transport through N 20 TLSs for a pffiffiffi ð Þ ¼ resonant pulse with Γσ ¼ 3 2 computed using the photon-scattering theory. In (b) and (c), dashed lines show the contributions from the different eigenstates. The input mode in all panels is the Gaussian curve shown in (a). (d) Transmitted power and correlation functions for a coherent state with n¯ ¼ 0.5. Solid lines show photon-scattering results truncating at three photons, while marker points show MPS 150 calculations performed by solving the ME with a Runge-Kutta algorithm and using maximum bond dimension Dmax ¼ . (e) Second- ð2Þ order correlation function G ðx1;x2Þ for the transmitted coherent state computed using photon-scattering theory. The two- and three- photon bound-state contributions appear as peaksffiffiffi that are localized in the relative coordinate. (f) MPS calculation for input coherent state p 2 with n¯ ¼ 8, N ¼ 60 emitters, and Γσ ¼ 2 2. Dashed vertical lines show N times the Wigner delay T n ¼ Nτn ¼ 4N=ðΓn Þ. With PnðxÞ, we indicate the contribution to the power coming from the emission of n photons within the space bin Δx ¼ 1. To perform this calculation, we use a quantum trajectories MPS algorithm fixing the maximum bond dimension to Dmax ¼ 40. state in Fig. 2(b), while the extended photon propagates like the center-of-mass coordinate. This can be done by using a single photon. This separation of the propagation of the the form of the n-photon bound state and its transmission bound and extended parts of this state can be shown coefficient given in Eqs. (8) and (9). Within the n-photon explicitly in the long pulse limit σΓ ≫ 1 (see the manifold, the projection of an input Gaussian state on the 2 2 Supplemental Material [30]). This is significant because −ðE−nk0Þ σ =ð2nÞ bound state is nhBEjini¼cne , where cn is a it implies that in order to understand the pulse evolution, constant in E. Here,P it is convenientP to use Jacobi one does not have to understand the behavior of all the n J i − coordinates xc ¼ j xj=n, xi ¼ 1 xj=i xiþ1, where S j¼ different string combinations . Rather, one can focus on i ∈ f1; 2; …;n− 1g. The resulting bound-state contribu- simply understanding the behavior of the bound states. On tion is then the other hand, for short pulses this separation is not completely true because one needs to include the effect of interactions between the components; see Sec. V. Z c J jouti ¼ pffiffiffiffiffin dnxaˆ †ðxÞj0ie−ðΓ=2Þgðx Þ n;bound ! A. Evolution of bound states Zn − 2σ2 2 To better quantify the difference in propagation N iExc E =ð nÞ × dEtE;ne ; ð12Þ observed above, let us compute the pulse evolution in 031011-6 DYNAMICS OF MANY-BODY PHOTON BOUND STATES IN … PHYS. REV. X 10, 031011 (2020) P xJ − where gð Þ is the exponent i This gives the Wigner delay imparted on an n-photon B. Influence of imperfections bound state by a single emitter in WQED: The photons We show that the hallmark of many-body photon propagate with a number-dependent velocity. propagation through an ensemble of quantum emitters in By taking higher-order derivatives, we also compute chiral WQED is the number-dependent velocity of the 2 2 2 2 the pulse broadening bðk0Þ¼−32k0Γ=½nð4k0 þ n Γ Þ , photon bound states. Here we analyze the influence of which is zero on resonance. The third-order pulse distortion imperfections such as losses, imperfect chirality, and 6 3 term on resonance is d ¼ −32=ðn Γ Þ. The pulse distortion inhomogeneous broadening on this propagation. We first is thus drastically reduced for higher-order bound states. consider the influence of losses, where each emitter couples This indicates that many-photon bound states suffer neg- to an additional decay channel out of the waveguide with a ligible pulse distortion while propagating through the array rate Γ0. It is possible to obtain an analytic criterion for when of nonlinear and dispersive atoms. these losses can be neglected. The form of the transmission In order to verify that indeed the physics of the bound coefficient of the bound state can be obtained in the states dominates the wave-packet evolution, we also presence of loss by mapping the total energy to a complex compute the evolution of a coherent input pulse as shown → Γ 2 pffiffiffi energy using the replacement E E þ in 0= . The reduc- in Fig. 2(d). Here, the pulse width Γσ ¼ 3 2 is the same as tion in probability of the output state then implies that the in Figs. 2(a)–2(c), while the average photon number in the state can undergo one or more quantum jumps. If the pulse is n¯ ¼ 0.5. We compute the output both by truncating probability remains close to unity, the output state is only the coherent state to three photons, and solving exactly weakly affected. Defining the efficiency or β factor as β Γ Γ Γ Γ Γ using Eq. (10), or by solving Eqs. (2) and (3) with the MPS ¼ = tot, where tot ¼ þ 0, the transmission coeffi- algorithm. The evolution of the bound states is seen in the cient (9) in the presence of loss is position of the peaks of the power distribution haˆ †ðxÞaˆðxÞi as well as in the difference between the power and the mth- E þ inΓ ½1 − βð1 þ nÞ =2 t ¼ tot : ð14Þ ðmÞ † m m E;n Γ 1 − β 1 − 2 order correlation functions G ðxÞ¼h½aˆ ðxÞ ½aˆðxÞ i. E þ in tot½ ð nÞ = The localized nature of the bound states in the relative coordinates is shown in Fig. 2(e) where we show the After scattering off all N emitters the magnitude of ð2Þ N two-point second-order correlation function G ðx1;x2Þ¼ the resulting state is jtE;nj . For small imperfections † † N haˆ ðx2Þaˆ ðx1Þaˆðx1Þaˆðx2Þi. Here the photons tend to local- 1 − β ≪ 1, this gives jtE;nj ¼ 1–2Nð1 − βÞ=n þ ize around the diagonal at small values of the relative Oðð1 − βÞ2Þ, where the notation OðMÞ indicates a term coordinate x1 − x2, while they are delocalized about the of order M and higher. This means that a sufficient center-of-mass coordinate ðx1 þ x2Þ=2. This reflects that condition for neglecting losses is Nð1 − βÞ=n ≪ 1. This the photons are tightly bound together in the bound state, implies that losses have a reduced influence on higher- but the bound state itself is free to propagate. While exact order bound states. If this condition is not met, there is a analytical calculations become infeasible for n¯ ≫ 1, the sizable probability that one or more of the photons in a validity of our arguments and the importance of the bound photon bound state is lost. states can still be seen in MPS simulations. For example, in If a photon is lost at one point along the ensemble, the Fig. 2(f), we calculate the transmitted power for an input remaining photons propagate through the rest of the atoms coherent state n¯ ¼ 8 and N ¼ 60. Here, for this particular with a different effective group velocity and dispersion. In system length, the photon-number-dependent Wigner addition to a reduced amplitude, the fact that the remaining delay clearly manifests itself as separate peaks for up to transmitted photons have effectively propagated with a mix six-photon bound states. of velocities and dispersions causes the peaks associated 031011-7 SAHAND MAHMOODIAN et al. PHYS. REV. X 10, 031011 (2020) with the different bound states in Fig. 2 to broaden and propagation. In this section, we aim to bridge the gap eventually overlap (see the Supplemental Material [30]). between these two regimes: First we show that indeed In addition to photon loss, we note that a nonzero value the many-body theory reduces to the mean-field result in of Γ0 ≠ 0 also affects the delay τn, which is also com- the limit of large photon number. Second, we derive the 3 2 puted analytically, τn ¼ 4β =fΓ½βð2þβðn −1ÞÞ−1 g ¼ quantum corrections to the mean-field results which 2 2 2 4=ðΓn Þ−4ð1−βÞ=ðΓn ÞþO½ð1−βÞ . Values where τn become relevant when both the number of photons n diverges occur when jtE;nj approaches zero; i.e., no light is and the number of emitters N are large. Finally, we push transmitted. the numerical simulations to the many-photon limit to In the limit of large losses, the photon bound states suffer verify the analytical predictions. exponential damping. The transmission of a steady-state Let us consider a wave packet composed of a linear input field was considered for up to two photons in Ref. [19], combination of many-body bound states. This state is where it was shown that the output shows strong photon expressed with the ansatz bunching. This bunching, however, arises from the extended X Z states and does not reflect the bound-state dynamics. Even ψ j i¼ dEcnðEÞjBEin: ð16Þ with finite-pulse durations, it is likely that the bound-state n dynamics will be hard to discern in this limit. The introduction of imperfect chirality also influences the We later show that this is the expected form of the state for a nature of pulse propagation through the ensemble. Imperfect high-power coherent input with a large spectral width. chirality occurs when, in addition to coupling to the forward- Unlike the previous sections where we select an input pulse propagating mode, each atom also couples to the backward- and propagate it through the medium, here we simply propagating mode with a rate ΓL. The influence of imperfect consider a linear combination of bound eigenstates and chirality cannot be considered analytically in a straightfor- compute the observables for this state. A localized function ward manner. We perform numerical MPS calculations (see cnðEÞ ensures that the bound-state ansatz is localized in the the Supplemental Material [30])forN ¼ 20 emitters. When center-of-mass coordinate. ΓL ¼ 0.05 Γ, the shape of the output state remains quali- Such a state can be probed by measuring either the ψ ˆ ψ tatively unchanged. However, when ΓL ¼ 0.2 Γ the output field h jaðxÞj i or the mth normally ordered observable, state is completely distorted. These computations indicate which we consider in the center-of-mass coordinate that the effect observed in the ideal chiral case is robust to hψj½aˆ †ðxÞ m½aˆðxÞ mjψi. We compute these observables in imperfect chirality, provided that the imperfections are not the limit where the average photon number is large n¯ ≫ 1, too large. the pulses are spectrally broad σ ≪ 1=Γ, and the order We also consider the influence of inhomogeneous of the correlation function is much less than the photon broadening in the two-level systems. The two-level atoms number m ≪ n. Within these limits (see the Supplemental are considered to have a normally distributed resonance Material [30] for full calculations), ς Γ frequency with dimensionless standard deviation = . This pffiffiffi inhomogeneous broadening affects the transmission coef- ¯ Γ ¯Γ ψ ˆ ψ n n x ficient of the bound states, which most notably affects the h jaðxÞj i¼ 2 sech 2 ; ð17Þ pulse delay. An expression for the mean pulse delay can be pffiffiffi computed analytically, and for small broadening, gives to n¯ Γ n¯Γx 2m ψ ˆ † m ˆ m ψ leading order h jðaðxÞ Þ ½aðxÞ j i¼ 2 sech 2 ; ð18Þ 4ς2 ς4 τ τ 1 − O n¯ h ni¼ n 2 2 þ 4 : ð15Þ where is the average number of photons in the pulse. n Γ Γ These observables reproduce the fundamental soliton solution of SIT [34,35,44] given in Eq. (6). Self-induced The reduction in the delay imparted by each emitter transparency is thus the classical limit of the photon bound ς Γ therefore scales quadratically in =ðn Þ. Inhomogeneous state when the photon number becomes large, or con- broadening therefore has a reduced impact on higher-order versely, photon bound states are simply the quantum ≲Γ bound states provided that the broadening is limited . limit of a soliton, a quantum soliton. Just like the SIT We note that the influence of inhomogeneous broadening solitons,ffiffiffi the integrated Rabi frequency of the pulse is can be compensated by introducing more emitters. p R Ω ¼ 2 Γ dxhaˆðxÞi ¼ 2π; i.e., the intense pulse of light rapidly excites and deexcites the emitters resulting in a 2π IV. CONNECTION TO THE SIT SOLITON Rabi oscillation. We note that, the expressions in Eqs. (17) In the previous sections, we show how the Hamiltonian and (18) scale with Γ, which is the coupling to the one- (1) leads to the SIT solitonic solutions in the mean-field dimensional continuum. These expressions are therefore limit, and that the full quantum-mechanical treatment of unchanged in the presence of coupling to an external this Hamiltonian predicts correlation-ordered photon reservoir. This implies that, provided the ensemble does 031011-8 DYNAMICS OF MANY-BODY PHOTON BOUND STATES IN … PHYS. REV. X 10, 031011 (2020) (a) Mean field (b) FIG. 3. (a) Output power for N ¼ 20 atoms for coherent input pulses with a solitonic shape (see text) and different mean photon ¯ 150 number n. The results are computed by directly solving the ME (3) (solid lines) with the MPS algorithm where we fix Dmax ¼ . In the second panel, we include the results given in the many-body limit (19) (dashed lines), and in the third the mean-field solitonic ansatz (dotted line). We note that the x axis is limited to focus on larger photon-number components. (b) Maximum bond dimensions Dmax of the MPS calculations required to obtain a tolerance less than 10−4 versus input power n¯ for different TLS number N. The blue shading highlights the regions for which panels (1)–(3) in (a) show the output power. not act like a Bragg mirror, SIT is expected to occur even in When n¯ ≫ 1 and N ≳ n3=2, one can consider a wave the presence of backscattering in accordance with mean- function composed of bound states with a number distri- field results. bution given by a coherent state. In the limit where mean- field theory breaks down, one must explicitly consider A. Beyond mean-field theory the Fock-state-dependent delay. In this limit, the expression As a lowest-order approximation, the variation in photon for the field and the power give (see the Supplemental number n making up the pulse (16) can be ignored, and the Material [30]), state will simply propagate at a reduced speed dictated by pffiffiffi X 2n the mean photon number. In Eqs. (17) and (18), this lowest- 2 α n Γ nΓ 4N aˆ x e−jαj x ; order approximation for the SIT soliton simply maps h ð Þi ¼ ! 2 sech 2 þ 2Γ 2 n n n x → x þ 4N=ðn¯ ΓÞ. However, for a coherent state, theffiffiffi p X 2n 2 2 Δ ∼ ¯ 2 α n Γ nΓ 4N uncertainty in the photon number scales as n n, aˆ † x aˆ x e−jαj x ; h ð Þ ð Þi ¼ ! 4 sech 2 þ 2Γ which leads to a gradual broadening of the pulse due to n n n the different photon-number components accumulating different time delays. This difference in time delay can ð19Þ become significant for a sufficiently large number of with equivalent expressions for higher-order correlation emitters N. This broadening is not captured by the pffiffiffi mean-field theory. The breakdown of the mean-field theory functions. Here, α ¼ n¯ is the coherent field amplitude, therefore occurs when the difference in the delay of the n¯ which is assumed real. We note that similar expressions and the n¯ þ Δn photon bound states becomes on the order exist for the bound state of the nonlinear Schrödinger of the width of the n¯-photon bound state. The ratio of the equation with an attractive interaction [45,46]. difference in delay and the width of the n¯-photon bound Equations (19) provide a simple description of the T − T Δ ∼ 3=2 state is given by ð n¯þΔn nÞ= tn N=n . This means observables of a quantum many-body state of light. In that when N=n3=2 ≳ 1 the mean-field theory breaks down order to investigate the full transition from the multiphoton even for large input photon number n¯. This inequality bound-state propagation to the formation of the SIT provides the boundary between mean-field theory and solitons, we make use again of the master equation genuine quantum many-body dynamics. simulation. In Fig. 3(a), we plot the transmitted power 031011-9 SAHAND MAHMOODIAN et al. PHYS. REV. X 10, 031011 (2020) pffiffiffi E ¯ Γ 2 ¯Γ 2 (a) (b) (c) of input pulses with inðxÞ¼ðn = Þsechðn x= Þ for different amplitude strength n¯. This pulse shape is chosen such that its electric field matches the SIT criterion. In the first box, we see again the formation of the bound-state peaks which tend to reduce their time delay as the input power is increased. For intermediate input pulses (second box), the bound states with a large number of photons get more populated, and they accumulate toward a single peak as the difference in delay times for large photon numbers FIG. 4. Evolution of the state jini (see text) demonstrating the interaction between a two-photon bound state centered at becomes less distinguishable. In this regime, the trans- a2 ¼ −10=Γ and a singleffiffiffi photon centered at a1 ¼ 15=Γ both mitted power starts to be well described by Eq. (19).For p † with width σΓ ¼ 3 2. The output observables (a) haˆ ðxÞaˆðxÞi, even higher input power (third box), the individual bound (b) Gð2ÞðxÞ, and (c) Gð3ÞðxÞ are plotted versus the number of states are no longer recognizable and a solitonic pulse well emitters N. described by mean-field theory emerges. These results show how the SIT solitons emerge from a superposition of photon bound states that, in the limit of few photons, can the pulse delay for this state as it propagates through the be indeed interpreted as quantum solitons. On the other ensemble, i.e., for different N. As in previous figures, a hand, it is important to emphasize the difference in physical frame comoving with the pulse in the absence of inter- effects. While the SIT solitons can be fully described by a actions (i.e., N ¼ 0) is assumed. Here, since a2 031011-10 DYNAMICS OF MANY-BODY PHOTON BOUND STATES IN … PHYS. REV. X 10, 031011 (2020) (a) -3 (b) (c) 10 x10 0.1 0.08 0 5 0.05 0.04 -10 -20 0 0 0 -20 -10 0 10 -20 -10 0 10 -20 -10 001 (d) (e) (f) 0.08 0.08 0.08 0.04 0.04 0.04 0 0 0 -20 -10 001 -20 -10 001 -20 -10 001 ¯ 0 5 2 FIG. 5. Propagation of a coherent pulse withpffiffiffi average photon number n ¼ . through N ¼ emitters. (a) Two-time correlation ð2Þ function G ðx1;x2Þ for pulse width Γσ ¼ 3 2 for ideal chiral coupling computed using the three-photon theory. Dashed lines show N τ ð2Þ times the Wigner delay of the two-photonpffiffiffi bound state 2. (b) Power PðxÞ and second- and third-order correlation functions G ðxÞ and Gð3ÞðxÞ for pulse width Γσ ¼ 2 with ideal coupling. Power and correlations when also coupling to a backward mode with rate ΓL ¼ 0.1Γ where the emitters are separated by (c) kd ¼ π, (d) kd ¼ π=2, and (e) kd ¼ π=4. (f) Power and correlation functions for unidirectional coupling, but with each emitter also coupled to a loss mode with Γ0 ¼ 0.1Γ. Panels (b)–(f) are computed using a full numerical treatment. are the signature of the extended states. These tails are Here, the single-photon component suffers the largest loss, clearly not bunched in comparison to the bound state. while the bound states suffer a reduced loss. The intro- Figure 5(b) shows the equal-time correlation function for a pffiffiffi duction of the loss does not spoil the difference in the shape narrower pulse width Γσ ¼ 2. Here, a narrower pulse of the power and the correlation functions. width is chosen so it dominantly projects on the two-photon The results here demonstrate that the propagation of few- bound state. The hallmark of the photon-photon inter- photon bound states can be observed with as few as two actions is observed in the difference between the power quantum emitters chirally coupled to a reservoir and is PðxÞ and the correlation functions Gð2ÞðxÞ and Gð3ÞðxÞ. robust to imperfections. This means that the phenomena we Clearly, the leftmost peak corresponding to the single- show here can be realized by several platforms currently photon component undergoes a larger time delay than the under investigation. Optical quantum dots have demon- bound states. The difference in time delay between two- strated chiral coupling between a single emitter and a and three-photon bound states is also visible in the slight waveguide [47,48], while coupling between two diamond difference between the peak centers of Gð2ÞðxÞ and Gð3ÞðxÞ. impurity centers and a photonic nanostructure has been We note that it is also possible to observe a difference achieved [49]. At microwave frequencies, circuit QED between PðxÞ and Gð2ÞðxÞ for a single quantum emitter. platforms can also achieve strong coupling between a We also investigate the robustness of this effect when superconducting qubit and a transmission line [8] and a imperfections are introduced. Figures 5(c)–5(e) consider scheme for unidirectional coupling has been proposed [50]. additional coupling to the left-propagating mode at a rate Multiple qubits have also been coupled to a single mode or propagating modes [9,51–54]. Finally, gasses of Rydberg ΓL ¼ 0.1 Γ for different emitter spacings kd, where k is the propagation wave number, and d is the distance between atoms under the conditions of electromagnetically induced each of the emitters. Note that unlike the fully chiral transparency also exhibit strong nonlinearities [14] and possess bound eigenstates [55]. Under certain limits regime, when ΓL ≠ 0 the distance between emitters influences the dynamics of the system. Although the output they can also be mapped onto a nonlinear Schrödinger field is slightly reduced in all three cases, the difference in equation with an attractive interaction [18]. Such a the shape of the power and the correlation functions is Hamiltonian possesses bound eigenstates [37,46], and preserved as is the difference in the peak positions. Finally, therefore, sufficiently long samples of Rydberg gasses Fig. 5(f) considers ideal chiral coupling, but with each can also potentially exhibit the bound-state propagation emitter coupling to a loss reservoir at a rate Γ0 ¼ 0.1 Γ. shown here. Alternatively, ensembles can be engineered 031011-11 SAHAND MAHMOODIAN et al. PHYS. REV. X 10, 031011 (2020) using Rydberg blockade to mimic chirally coupled emitters ACKNOWLEDGMENTS [20]. The effects we predict here are thus widely applicable The authors thank Mantas Čepulkovskis, Peter Lodahl, and can be observed in many different physical systems Sebastian Hofferberth, Hanna le Jeannic, Johannes Bjerlin, spanning vastly different energy scales. L. Henriet, and J. Douglas for fruitful discussions. S. M. and Throughout this manuscript, we focus on understanding K. H. acknowledge funding from DFG through Grant the fundamental physics of photonic bound-state propaga- No. CRC 1227 DQ-mat, Projects No. A05 and A06, and tion. In addition to its fundamental interest, the dynamics of “Niedersächsisches Vorab” through the “Quantum-and this system is highly interesting from an applied perspec- Nano-Metrology.” G. C. and D. E. C. acknowledge support tive. As a particular example, we note that, for the output from ERC Starting Grant FOQAL, MINECO Severo Ochoa state shown in Figs. 2(a)–2(d), if one selects a temporal Γ − Grant No. SEV-2015-0522, CERCA Programme/Generalitat window centered at x0 ¼ N, a state most likely con- de Catalunya, Fundació Privada Cellex, Fundació Mir-Puig, taining either zero or two photons is produced. Launching Fundación Ramón Areces Project CODEC, European the output on a beam splitter and conditioning on the Quantum Flagship Project QIA, QuantumCAT (funded detection of a photon will thus produce a single-photon within the framework of the ERDF Operational Program Fock state. For a small-amplitude initial coherent state, the of Catalonia), and Plan Nacional Grant ALIQS, funded by main contribution to this process will arise from the two- Ministerio de Ciencia, Innovación, y Universidades (MCIU), photon component of the incoming field. Since the output Agencia Estatal de Investigación (AEI), and Fondo Europeo is in a pure state, the outgoing single photon will also be in de Desarrollo Regional (FEDER). A. S. S. acknowledges a pure state, although the temporal mode function will support from the Danish National Research Foundation depend on the detection time. A detailed investigation of (Center of Excellence Hy-Q). using photon bound-state propagation as a photon source is beyond the scope of this work. We can however estimate that for the parameters in Figs. 2(a)–2(d), which are in no way optimized for the application, choosing a window with [1] D. E. Chang, V. Vuletić, and M. D. 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