Dissipation in Few-Photon Waveguide Transport [Invited]

110 Photon. Res. / Vol. 1, No. 3 / October 2013 E. Rephaeli and S. Fan

Dissipation in few-photon waveguide transport [Invited]

Eden Rephaeli1,* and Shanhui Fan2 1Department of Applied Physics, Stanford University, Stanford, California 94305, USA 2Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA *Corresponding author: [email protected]

Received April 15, 2013; revised June 9, 2013; accepted June 9, 2013; posted June 10, 2013 (Doc. ID 188761); published August 30, 2013 We develop a formulation of few-photon Fock-space waveguide transport that includes dissipation in the form of reservoir coupling. We develop the formalism for the case of a two-level atom and then show that our formalism leads to a simple rule that allows one to obtain the dissipative description of a system from the nondissipative case. © 2013 Chinese Laser Press OCIS codes: (270.2500) Fluctuations, relaxations, and noise; (270.4180) Multiphoton processes; (270.5580) Quantum electrodynamics; (290.5850) Scattering, particles; (270.5585) Quantum information and processing; (270.0270) Quantum optics. http://dx.doi.org/10.1364/PRJ.1.000110

1. INTRODUCTION photons to interfere coherently, which can give rise to signifi- – The topic of dissipation in quantum optics is both important cant capabilities in controlling transport [6,7,11,16,19 23], and subtle. Dissipation is important because many quantum switching [17,24], and amplification [9,15,25] processes for optical systems are open; therefore tracking all of the out few-photon states. – flowing energy is often not possible, but accounting for it is A formulation of input output formalism has been recently often quite crucial. Dissipation is subtle because energy in developed that enables analytical calculation of few-photon quantum systems cannot simply disappear, and naively de- scattering matrices in these systems of waveguide coupled scribing dissipation with phenomenological damping factors to quantum multilevel atoms. In this paper we extend this in the system’s equations of motion is formally incorrect, lead- formalism to include dissipation. We model loss by introduc- ing to a non-Hermitian Hamiltonian, violation of commutation ing an auxiliary reservoir that transports energy away from relations [1], and time-reversal invariance [2]. the atom. – A correct treatment of dissipation entails coupling the sys- In applying the input output formalism to treat two-photon tem under study, which usually contains discrete energy lev- transport [8,18], a key step is to insert a complete set of basis els, to a reservoir that is modeled as having a continuum of states for the single-excitation Hilbert space. Here we show energy states [1]. When this is done, the presence of dissipa- that, to obtain the correct result for two-photon transport tion in the system is accompanied by a source of fluctuation in the presence of dissipation, such a basis set needs to from the reservoir, enabling two-way energy exchange. include the states of the reservoir. In quantum optics, the system–reservoir model is treated The paper is organized as follows. In Section 2 we present with many different approaches. In the Schrödinger picture, the Hamiltonian describing a two-level atom with reservoir this model has been treated in the master equation approach, coupling. In Section 3 we obtain the one and two-photon – where multiple-time averages are calculated via the quantum S-matrices of the system using the input output formalism, regression theorem [2]. Alternatively in the Schrödinger pic- showing that excluding the reservoir from the system ture one can use the stochastic approach of the Monte Carlo description leads to violation of outgoing particle exchange wave function [3], which involves evolving the wave function symmetry. Finally, in Section 4 we conclude with a general of the system with a non-Hermitian Hamiltonian term. In the recipe for the inclusion of dissipation in a general class of Heisenberg picture, the input–output formalism [4] is widely important and relevant systems. used. In typical use of the input–output formalism, one assumes that the input is a weak coherent state. The nonlinear 2. SYSTEM HAMILTONIAN response is then often treated approximately in the weak We consider a two-level atom side-coupled to a waveguide excitation limit [5]. The input–output formalism is attractive with right and left moving photons, as described by the because the external degrees of freedom, such as those of Hamiltonian the reservoir, are integrated out, allowing one to focus on the Z dynamics of the system alone. † d † d Hwvgatom dx −ivgc x cRxivgc x cLx Recently, there has been great interest in describing the R dx L dx h i properties of photons propagating in a waveguide coupled p † † κδx c xσ− σ c xc xσ− σ c x to quantum multilevel atoms either directly [6–16] or through R R L L the use of a microcavity [17,18]. The one-dimensional nature Ω σ : (1) of waveguide propagation allows incident and scattered 2 z

† † Here cRxcRx and cLxcLx create (annihilate) a right or a x − a −x κ a x R p L ; (3b) left moving photon with group velocity vg, respectively. is o 2 the coupling rate between the atom and waveguide. The form of the interaction here assumes the rotating-wave approxima- with the above operators, the even subspace Hamiltonian He tion. σσ− raises (lowers) the state of the two-level atom and the odd subspace Hamiltonian Ho follow whose transition frequency is Ω. We note that the waveguide Z model in Hwvgatom can describe any single-mode dielectric † d † d or plasmonic waveguide in the optical regime [13] or a trans- H dx −iv a x ax − iv b x bx e g dx r dx mission line in the microwave regime [26]. Since the atomic linewidth is typically quite narrow, a linear approximation δ † σ σ δ † σ σ V xa x − ax V r xb x − bx of the waveguide dispersion relation, which we adopt here, is generally sufficient to describe the waveguide. We further 1 Ωσz; note that the two-level atom in Hwvgatom can describe either a 2 real atom or an artificial atom, such as a quantum dot [13]or Z superconducting qubit [15]. † d Ho −ivg dxaox aox. To model dissipation we introduce reservoir–atom cou- dx pling as described by Hr and consider the composite Here H describes a one-way propagating chiral waveguide system H Hwvgatm Hr as shown schematically in Fig. 1, e where mode that interacts with the atom. Ho describes a one-way propagating waveguide mode without any interaction with Z the atom. Once the solutions for He are known, the solution † d p † Hr dx −ivrb x bx γδxb xσ− σ bx : to H H can be straightforwardly obtained by using dx wvgatm r Eqs. (3a) and (3b). In what follows we consider only the even (2) Hamiltonian, since it contains all of the nontrivial physics of the system. Here bx and b†x are the reservoir photon operators The Hamiltonian He can be recast in k-spacep by and γ is the atom–reservoir coupling rate. We assume the res- ≡ 1∕ 2π definingR momentum-spacep photonR operators ak ervoir has a continuum of modes centered around the atomic dxaxe−ikx and b ≡ 1∕ 2π dxbxe−ikx, transition frequency Ω. We further assume that within the k r spectral range of interest the dispersion of these reservoir Z † † κ † H dk kv a a kv b b a σ− σ a modes is well described by a group velocity vr. The reservoir e g k k r k k 2π k k will initially be set to the vacuum state. As a result of interac- r tion with the atom, photons originating in the waveguide may γ † 1 b σ− σbk Ωσz: leak back into the waveguide, described in Eq. (1), or they 2π k 2 may leak into the reservoir as described in Eq. (2), resulting in loss. WeR note thatR the total number of excitations Nexc † † σ σ To proceed, we exploit the spatial inversion symmetry of H dkakak dkbkbk − commutes with He, Ho, allowing for the solution to proceed independently in each excitation and decompose it to even and odd subspaces H He Ho, number manifold. where He;Ho0. We transform from right- and left-moving waveguide photon operators to even and odd photon operators. We omit from here on the subscripts of the even mode operators: 3. INPUT–OUTPUT FORMALISM In the absence of dissipation, the one- and two-photon scat- a xa −x tering matrices of this system were solved in [10], to which ax R p L ; (3a) 2 we refer the reader for a detailed derivation of the formalism we use. Our focus here is on the addition of dissipation to the formalism. By defining input and output field operators for the waveguide, Z 1 −ikt−t0 a tp dka t0e ; in 2π k Z 1 −ikt−t1 a tp dka t1e ; out 2π k

and reservoir, Z 1 −ikt−t0 Fig. 1. Schematic of sample system considered. A waveguide side b tp dkb t0e ; κ in 2π k coupled with rate to a two-level atom with transition frequency Z Ω. The atom is additionally coupled with rate γ to a reservoir. The 1 −ikt−t1 initial state in the waveguide is either a one- or a two-photon Fock bouttp dkbkt1e ; state, whereas the auxiliary waveguide is initially in the vacuum state. 2π 112 Photon. Res. / Vol. 1, No. 3 / October 2013 E. Rephaeli and S. Fan we obtain (in the limits t0 → −∞ and t1 → ∞) the input–output incoming photons with momenta k1 and k2 to scatter into two equations, outgoing photons with momenta p1 and p2 and can be expressed as p aouttaint − i κσ−t; (4)

− − † † hp ;p jk ;k ih0ja p2a p1a k1a k2j0i: p 1 2 1 2 out out in in bouttbint − i γσ−t; (5) To proceed, we insert the resolution of the identity in the dσ−t κ γ p single-excitation subspace −i Ω − i − i σ−ti κσ ta t dt 2 2 z in Z Z pγσ i ztbint: (6) ˆ I1 dkjk ihk j dkjk irrhk j; We see that the presence of dissipation in Eq. (6) in the form of damping rate γ is accompanied by the input field operator p which comprises of a sum of both waveguide and reservoir i γσ tb t from the reservoir. Below we use Eqs. (4)–(6) z in interacting eigenstates outer products to obtain one and two-photon S-matrices.

A. One-Photon S-Matrix − − hp1 ;p2 jk1 ;k2 i The one-photon S-matrix relates an incoming free-photon Z − † † state jki with momentum k to an outgoing free-photon state dkhp1 jk ihk ja p2a k1a k2j0i − out in in jpi with momentum p and is given by hpjSjkihp jki, where Z † S is the scattering operator and jk i ≡ a kj0i is an incoming − † † in dkhp1 jk i hk ja p2a k1a k2j0i one-excitation interacting eigenstate that evolved from a rr out in in − † ≡ 0 r free-photon state jki in the distant past. jp i aoutpj i is an tp1 hp1 jaoutp2jk1 ;k2 itp1 rhp1 jaoutp2jk1 ;k2 i; outgoing one-excitation interacting eigenstate that will evolve into a free photon state jpi in the distant future. The one- photon S-matrix can be reexpressed as follows: where we have used the definitions of the one-photon scattering amplitudes hp−jkit δk − p and hp−jki p 1 k r 1 − 0 † 0 δ − − κ 0 σ hp−jki trδk − p. Inserting the Fourier transform hp jk ih jaoutpainkj i k p i h j −pjki; 1 r k 0 of Eq. (4) and noting that rhp1 jainp2jk1 ;k2 i ,we where we have used the input–output relation in Eq. (4) have and h0jainpjk iδk − p [10]. To obtain the matrix element h0jσ−pjk i, we solve for its time-domain counterpart δ − δ − δ − δ − 0 σ 0 σ 0 tp1 k1 p1 k2 p2 k1 p2 k2 p1 h j −tjk i using Eq. (6), noting that h j ztbintjk i p † −ikt p p r and h0jσztaintjk i−1∕ 2πe , where jk ir b kj0i − κ σ − κ σ in i tp1 hp1 j −p2jk1 ;k2 i i tp1 rhp1 j −p2jk1 ;k2 i: is an incoming reservoir eigenstate. The resulting one-photon (7) transport matrix elements are p κδk − p 0 σ ≡ δ − We now need to solve for the matrix elements h j −pjk i κ γ sk k p; k − Ω i i hp1 jσ−p2jk1 ;k2 i and hp1 jσ−p2jk1 ;k2 i.Todosowe 2 2 r pγδ − sandwich Eq. (6) between the appropriate states, getting k p r h0jσ−pjk i γ δk − p ≡ s δk − p; the equations r k − Ω i κ i k 2 2 γ k − Ω − i κ − hp−jki 2 2 δk − pt δk − p; κ γ κ γ k d k − Ω i 2 2 hp1jσ−tjk ;k i−i Ω − i − i hp jσ−tjk ;k i dt 1 2 2 2 1 1 2 − pκγδ − − i k p p p k rh j i κ γ i κhp1 jσztaintjk1 ;k2 i; (8) k − Ω i 2 2 ≡ rδ − tk k p: ≠ d κ γ We note that, as a result of reservoir damping, sk tksk . hp1jσ−tjk1 ;k2 i−i Ω − i − i hp1 jσ−tjk1 ;k2 i This seemingly innocuous difference between the dissipative dt r 2 2 r p case and the dissipation-free case stems from the fact that i κrhp1 jσztaintjk1 ;k2 i; (9) the set jki is not complete in the presence of dissipation. We will see the crucial importance of this fact in the two- photon calculation. where we note that hp1 jσztbintjk1 ;k2 i0 and σ 0 rhp1 j ztbintjk1 ;k2 i . To proceed, we solve for the B. Two-Photon S-Matrix source terms in Eqs. (8) and (9) by using σ 2σ σ− − 1 ˆ z The transport of two waveguide photons is given by the two- and inserting a resolution of the identity I0 j0ih0j in the photon S-matrix, which describes the probability amplitude of zero-excitation Hilbert space, E. Rephaeli and S. Fan Vol. 1, No. 3 / October 2013 / Photon. Res. 113 e−ik2t 1 This form of the S-matrix violates the particle exchange ip1−k2t hp jσ ta tjk ;k ip s s e − δp1 − k2 1 z in 1 2 2π π p1 k2 symmetries mentioned above. If we examine the form of the S-matrix in Eq. (13), it e−ik1t 1 ip1−k1t p s s e − δp1 − k1 ; might seem odd that, even though the result is expressible 2π π p1 k1 by single-photon scattering amplitudes, it violates exchange −ik2t e 1 r − symmetry of outgoing particle momenta while the one- hpjσ ta tjk;kip s s eip1 k2t r 1 z in 1 2 2π π p1 k2 photon S-matrix does not. This is explained as follows.

−ik1t The physical origin of the symmetry violation is in the scat- e 1 r p s s eip1−k1t: σ 0 2π π p1 k1 tering amplitude hp1 j j i. This amplitude is proportional ’ to the Fourier coefficient sp1 of the atom s spontaneous emission amplitude into the waveguide. However, in the Having calculated the source terms, we may now solve presence of dissipation, the atom may also spontaneously Eqs. (8) and (9) and Fourier transform their solution to emit a photon into the reservoir, corresponding to the am- obtain σ 0 plitude rhp1j j i. This amplitude is provided by the term in 1 Eq. (10) and originates in our inclusion of the reservoir σ − r δ − − when inserting the two-excitation resolution of the identity rhp1 j −p2jk1 ;k2 i sp2 sp1 sk1 sk2 p1 p2 k1 k2; R R π ˆ I2 dkjk ihk j dkjk i hk j. (10) rr

4. A GENERAL RULE FOR DISSIPATION hp1 jσ−p2jk1 ;k2 i When examining the correct form of the two-photon S- δ − δ − δ − δ − sp2 f p2 k1 p1 k2 p2 k2 p1 k1 matrix in Eq. (12), we see that the presence of dissipation 1 γ∕2 is seen as the addition of a i term in the one-excitation − s s s δp1 p2 − k1 − k2g: (11) π p1 k1 k2 transmission and atomic excitation amplitudes. It follows that the one- and two-photon S-matrices of a two-level Finally, we insert Eqs. (10) and (11) into Eq. (7) to get the atom with reservoir coupling rate γ, as described by two-photon S-matrix, Hwvgatm Hr, can be obtained from the dissipation-free S-matrices by making the replacement Ω0 Ω − iγ∕2. − − hp1 ;p2 jk1 ;k2 i However, we stress that this substitution should be made in the S-matrix and not in the Hamiltonian, since, as de- δ − δ − δ − δ − tp1 tp2 p2 k1 p1 k2 p2 k2 p1 k1 p scribed above, doing so would leads to an incorrect result κ in the two-photon case. i s s s s δp1 p2 − k1 − k2: (12) π p1 p2 k1 k2 This result is even more general. In the presence of N reservoir dissipation channels with coupling rates We note that the two-photon S-matrix is invariant under ex- γ1; γ2; …; γN , the resulting S-matrix is obtained from the ↔ 0 change of incoming and outgoing momenta k1;k2 p1;p2 as dissipation-freeP S-matrix by making the substitution Ω Ω − N γ ∕2 required by time-reversal symmetry, and also with respect i n1 n . Moreover, this result holds for a general to exchange of incoming or outgoing momenta k1↔k2 and quantum multilevel system as long as the Hamiltonian light- p1↔p2, as required by the Bose statistics of photons. matter interaction term can be described in the rotating- Instead of the correct approach outlined above, one might wave approximation. Furthermore, the quantum multilevel instead consider an ad hoc approach where an imaginary system can be either in a cavity or directly coupled to a part is added to the frequency of the atom, resulting in the waveguide. following Hamiltonian: Finally, the eigenstate of the ad hoc Hamiltonian turns out to be the same as the true scattering eigenstate—other than Z r κ the reservoir degree of freedom, which it is missing. This fact H k kv a†a a†σ σ a ad-hoc d g k k 2π k − k allows a correct real-space interacting eigenstate solution [6,7,27,28] for the nonreservoir dynamics that account for 1 γ Ω − σ the loss of energy that accompanies reservoir coupling, in 2 i 2 z: complete agreement with the solutions presented here. In particular, the existence of two-photon bound states, which From this ad hoc Hamiltonian, one then follows the steps in is one of the important predictions made by using the [10]. With this approach, the Hilbert space consists of only wave-function method, is still valid in the presence of dissipa- waveguide and atom states—with no reservoir. For one pho- tion. In conclusion, the method described in this paper ton, this approach yields the correct S-matrix. For two pho- provides a straightforward and correct construction of the tons, the term of Eq. (10) would have been missing, and S-matrix in the presence of dissipation while working with the resulting incorrect two-photon S-matrix would have been the full Hermitian Hamiltonian of the system in the context of the input–output formalism. − − hp1 ;p2 jk1 ;k2 i δ − δ − δ − δ − tp1 tp2 p2 k1 p1 k2 p2 k2 p1 k1 ACKNOWLEDGMENTS 1 p The work is supported by an AFOSR-MURI program on i κt s s s s δp1 p2 − k1 − k2: (13) π p1 p1 p2 k1 k2 quantum metamaterial (grant FA9550-12-1-0488). 114 Photon. Res. / Vol. 1, No. 3 / October 2013 E. Rephaeli and S. Fan

REFERENCES 15. O. V. Astafiev, A. A. Abdumalikov, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Ultimate on-chip quantum ampli- 1. I. R. Senitzky, “Dissipation in quantum mechanics. the harmonic fier,” Phys. Rev. Lett. 104, 183603 (2010). oscillator,” Phys. Rev. 119, 670–679 (1960). 16. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon trans- 2. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations port in low-dimensional systems: interaction-induced radiation (Springer, trapping,” Phys. Rev. Lett. 104, 023602 (2010). 2003). 17. T. Shi, S. Fan, and C. P. Sun, “Two-photon transport in a wave- 3. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach guide coupled to a cavity in a two-level system,” Phys. Rev. A 84, to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, – 063803 (2011). 580 583 (1992). 18. E. Rephaeli and S. Fan, “Few-photon single-atom cavity QED 4. C. W. Gardiner and M. J. Collett, “Input and output in damped with input-output formalism in Fock space,” IEEE J. Sel. Top. quantum systems: quantum stochastic differential equations Quantum Electron. 18, 1754–1762 (2012). and the master equation,” Phys. Rev. A 31, 3761–3774 19. J.-T. Shen, M. L. Povinelli, S. Sandhu, and S. Fan, “Stopping (1985). single photons in one-dimensional circuit quantum electrody- 5. E. Waks and J. Vuckovic, “Dipole induced transparency in drop- namics systems,” Phys. Rev. B 75, 035320 (2007). filter cavity-waveguide systems,” Phys. Rev. Lett. 96, 153601 20. L. Zhou, Z. R. Gong, Y.-x. Liu, C. P. Sun, and F. Nori, “Control- (2006). “ lable scattering of a single photon inside a one-dimensional 6. J.-T. Shen and S. Fan, Coherent single photon transport in a resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008). one-dimensional waveguide coupled with superconducting 21. Z. Wang, Y. Li, D. Zhou, C. Sun, and P. Zhang, “Single-photon quantum bits,” Phys. Rev. Lett. 95, 213001 (2005). scattering on a strongly dressed atom,” Phys. Rev. A 86, 7. J.-T. Shen and S. Fan, “Strongly correlated multiparticle trans- ” 023824 (2012). port in one dimension through a quantum impurity, Phys. Rev. 22. J.-F. Huang, T. Shi, C. Sun, and F. Nori, “Controlling single- A 76, 062709 (2007). photon transport in waveguides with finite cross-section,” 8. E. Rephaeli, Ş. E. Kocabaş, and S. Fan, “Few-photon transport in ” arXiv:1303.1981 (2013). a waveguide coupled to a pair of colocated two-level atoms, “ 84 23. H. Zheng and H. U. Baranger, Persistent quantum beats and Phys. Rev. A , 063832 (2011). long-distance entanglement from waveguide-mediated inter- 9. E. Rephaeli and S. Fan, “Stimulated emission from a single actions,” Phys. Rev. Lett. 110, 113601 (2013). excited atom in a waveguide,” Phys. Rev. Lett. 108, 143602 24. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and (2012). “ Ş ş “ H. J. Kimble, A photon turnstile dynamically regulated by one 10. S. Fan, . E. Kocaba , and J.-T. Shen, Input-output formalism atom,” Science 319, 1062–1065 (2008). for few-photon transport in one-dimensional nanophotonic 25. D. Valente, Y. Li, J. P. Poizat, J. M. Gérard, L. C. Kwek, waveguides coupled to a qubit,” Phys. Rev. A 82, 063821 M. F. Santos, and A. Auffèves, “Optimal irreversible stimulated (2010). ” 14 “ ” 81 emission, New J. Phys. , 083029 (2012). 11. D. Roy, Few-photon optical diode, Phys. Rev. B , 155117 26. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. (2010). Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong cou- 12. D. Roy, “Two-photon scattering by a driven three-level emitter in pling of a single photon to a superconducting qubit using circuit a one-dimensional waveguide and electromagnetically induced ” 431 – ” 106 quantum electrodynamics, Nature , 162 167 (2004). transparency, Phys. Rev. Lett. , 053601 (2011). 27. J.-T. Shen and S. Fan, “Theory of single-photon transport in a 13. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, single-mode waveguide. I. Coupling to a cavity containing a P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single two-level atom,” Phys. Rev. A 79, 023837 (2009). optical plasmons in metallic nanowires coupled to quantum “ ” 450 – 28. H. Zheng, D. J. Gauthier, and H. U. Baranger, Waveguide QED: dots, Nature , 402 406 (2007). many-body bound-state effects in coherent and Fock-state 14. D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A scattering from a two-level system,” Phys. Rev. A 82, 063816 single-photon transistor using nanoscale surface plasmons,” (2010). Nat. Phys. 3, 807–812 (2007).