Input-Output Formalism For Few-Photon Transport in One-Dimensional Nanophotonic Waveguides Coupled to a Qubit
aa r X i v : . [ qu a n t - ph ] N ov Input-Output Formalism For Few-Photon Transport inOne-Dimensional Nanophotonic Waveguides Coupled to a Qubit
Shanhui Fan, ∗ S¸ ¨ukr¨u Ekin Kocabas¸, and Jung-Tsung Shen Ginzton Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305 Department of Electrical & Systems Engineering, Washington University, St. Louis, MO 63130 (Dated: November 2010)We extend the input-output formalism of quantum optics to analyze few-photon transport in waveguideswith an embedded qubit. We provide explicit analytical derivations for one and two-photon scattering matrixelements based on operator equations in the Heisenberg picture.
PACS numbers: 03.65.Nk, 32.50. + d, 42.50.Ct, 42.50.Pq I. INTRODUCTION
In the context of quantum information technology, includ-ing quantum computing devices, understanding the interac-tion between a few-photon state and a two-level atom playsan important role [1–3]. The photons are a possible candi-date for the ‘flying qubit’ that carries the information, and thetwo-level atom constitutes the ‘stationary qubit’ where the fly-ing qubits are generated on demand and correlated with eachother.Recently, there has been an increased activity in analyzingthe properties of photons propagating in a waveguide coupledto a qubit—a two-level quantum mechanical system. Exper-imental demonstration of the control of single photons wasmade in a waveguide coupled to an optical cavity with anatom in its near field [4]. Similar e ff ects were observed in themicrowave domain, when low frequency photons in a trans-mission line were coupled to a superconducting qubit [5, 6],which later was shown to act as a photon amplifier [7].To theoretically model such systems one needs to considera continuous set of waveguide modes that are free to propa-gate in one dimension, either directly coupled to a multi-levelsystem (referred to as an ‘atom’ in the paper), or indirectlycoupled through an optical cavity with a discrete set of modes.Photon transport properties are non-trivial in these structures[8–11] which can be tailored to perform logic operations [12]or form a diode [13]. Exact solutions of one and two-photonscattering have first been reported in [9, 11].The most widely used theoretical approach is to treat theset of equations in the Schr¨odinger picture, and apply theLippmann-Schwinger formalism to calculate the reflectionand transmission properties of the single and multi-photonstates [10, 14–17]. An alternative technique is to use the re-duction formulas from field theory to calculate the scatteringmatrix of the system [18, 19]. Time-domain simulations thattake the waveguide dispersion into account are also possible,and an interesting radiation trapping mechanism was recentlypredicted [20].In this paper, we extend the input-output formalism [21, 22]of quantum optics—an Heisenberg picture approach origi-nally introduced to analyze the interaction between an atom ∗ [email protected] in a cavity and a continuous set of electromagnetic states out-side of the atom-cavity system—to analyze the transport offew-photon states in a waveguide with an embedded qubit. Inthe input-output formalism one obtains a nonlinear set of op-erator equations based on the Hamiltonian of the system. Fora coherent or a squeezed state input, this formalism has beenextensively used to calculate various coherence properties ofthe output state of light. Here, we show that one can adopt thisformalism to obtain exact results regarding one or two photonproperties. To do so, we establish a relationship between theinput-output formalism and the scattering matrix elements ofthe system. Our approach complements the existing theoreti-cal literature and bridges di ff erent analytical techniques.The paper is organized as follows: In Section II we intro-duce the Hamiltonian of the system. In Section III we buildthe link between the scattering theory and the input-output for-malism and continue in Section IV with the derivation of theone-photon transport properties. In Section V we show howto extend the calculations to the two-photon case. In SectionVI we make observations on correlation function calculationsbased on coherent state inputs and end with our conclusionsin Section VII. II. HAMILTONIAN
We start by discussing the model Hamiltonian that we willuse in this paper. As an illustration of the formalism, we con-sider a two-level atom coupled to a single polarization, single-mode waveguide [9], and treat the transport properties of few-photon states in such a system (Fig 1). The Hamiltonian, ˜ H ,is defined as ( ~ =
1) ˜ H = ˜ H + ˜ H . Here ˜ H describes a chiral, i.e. one-way, waveguide wherephotons propagate in only one direction˜ H = Z ∞ d β ˜ ω ( β ) ˜ a † β ˜ a β and ˜ a β , ˜ a † β are the annihilation and creation operators for thephotons with a wavevector β respectively. In Appendix A wecalculate the reflection and transmission probabilities for pho-tons in a waveguide where the fields propagate in both direc-tions and show that the results are straightforward extensions | e i| g i ˜ Ω τ − e i kt e i pt FIG. 1. (Color online) Schematic representation of two photons in awaveguide, at frequencies k and p , moving to the right, towards a twolevel atom with energy levels | g i and | e i . ˜ Ω is the separation betweenthe energy levels. Coupling of the two level atom to the modes in thewaveguide is proportional to τ − . Long horizontal lines denote thewaveguide geometry. of the chiral case. The operators obey the commutation rela-tion [˜ a β , ˜ a † β ′ ] = δ ( β − β ′ ). ˜ H describes the atom as well as theatom-waveguide interaction˜ H =
12 ˜ Ω σ z + V Z ∞ d β (cid:16) σ + ˜ a β + ˜ a † β σ − (cid:17) . Here, ˜ Ω is the atomic transition frequency, σ ± are the rais-ing and lowering operators for the two level atom and σ z = σ + σ − − V denotes the coupling strength between theatomic states and the waveguide modes. The derivation ofthe Hamiltonian is based on the dipole and the rotating waveapproximations [23] as well as taking the continuum limit forfield operators. The details of taking the continuum limit arediscussed in Appendix B.It will be useful to have ˜ H in terms of the frequency of thephotons instead of their wavevector, therefore, we linearizethe waveguide dispersion around ( β , ω ) as ˜ ω ( β ) = ω + v g ( β − β ) (see Fig 2). Notice that the total excitation operator N E = Z ∞ d β ˜ a † β ˜ a β + σ z commutes with ˜ H , i.e [ ˜ H , N E ] =
0. We could thus equiva-lently solve a system as described by H = ˜ H − ω N E = H + H (1)where H = Z ∞−∞ d β v g ( β − β )˜ a † β ˜ a β H = Ω σ z + V Z ∞−∞ d β (cid:16) σ + ˜ a β + ˜ a † β σ − (cid:17) . Here
Ω = ˜ Ω − ω , and we also extended the lower limit ofintegration to −∞ so that we can define the Fourier transformof operators in the next section. Since we will be dealing withstates with wavevectors around β , the extension of the inte-gration limit is well justified [24, 25]. Finally, we completeour transition from wavevectors to frequencies by defining ω ≡ v g β , and the operator a ω ≡ ˜ a β + β / √ v g , which satisfiesthe commutation relation [ a ω , a † ω ′ ] = δ ( ω − ω ′ ). As a result of all these changes, we have H = Z ∞−∞ d ω ω a † ω a ω (2) H = Ω σ z + V √ v g Z ∞−∞ d ω (cid:16) σ + a ω + a † ω σ − (cid:17) . (3)Throughout the paper, the labels for photon degrees of free-dom, for example k , p , refer to photon frequency. III. CONNECTION BETWEEN THE SCATTERINGTHEORY AND THE INPUT-OUTPUT FORMALISM
In a typical scattering experiment, various input states areprepared and sent towards a scattering region. After the scat-tering takes place, the outgoing states of the experiment areobserved, and information about the interaction is deduced.The mathematical formulation of such a process is commonlymade using the scattering matrix with elements of the form S p p , k k = h p p | S | k k i where | k k i denotes the input states—here given as a two par-ticle state with energies (frequencies) k and k —and | p p i the outgoing states. These input and output states are assumedto be free states in the interaction picture that exist long before, t → −∞ , and long after, t → ∞ , the interaction takes place.The S operator, then, is equal to the evolution operator in theinteraction picture, U I , from time −∞ to + ∞ , S = lim t →−∞ t →∞ U I ( t , t ) = lim t →−∞ t →∞ e i H t e − i H ( t − t ) e − i H t where H is the non-interacting part of the Hamiltonian, and H = H + H is the total Hamiltonian. In order to have a morecompact notation, we will drop the limits and imply t → −∞ and t → ∞ .An equivalent way to describe the scattering is in terms ofthe scattering eigenstates | k k ± i that evolve in the interactionpicture from a free state either in the distant past or the distantfuture | k k + i ≡ U I (0 , t ) | k k i = e i Ht e − i H t | k k i ≡ Ω + | k k i| k k − i ≡ U I (0 , t ) | k k i = e i Ht e − i H t | k k i ≡ Ω − | k k i . The interaction picture time evolution operators that relatescattering and free states are called the
Møller wave opera-tors , Ω ± . The scattering operator can equivalently be writtenas S = Ω †− Ω + . It is also possible to write the scattering matrixelements as h p p | S | k k i = h p p − | k k + i . See [26] for more information about stationary scattering theory. [27] pro-vides a historical account of the developments related to the S -matrix. There is also an alternative definition of the scattering operator S ′ = Ω + Ω †− which relates the incoming and outgoing scattering eigenstates, | k + i = S ′ | k − i , such that h p | S | k i = h p − | k + i = h p − | S ′ | k − i = h p + | S ′ | k + i . See[28, 29] for details. We should note that scattering eigenstates and the free stateswith the same quantum numbers have the same energies, thatis H | k k i = E k k | k k i and H | k k ± i = E k k | k k ± i [26].It is possible to denote the scattering matrix elements by anappropriate definition of input and output operators such that h p p − | k k + i = h | a out ( p ) a out ( p ) a † in ( k ) a † in ( k ) | i (4)where a in ( k ) ≡ Ω + a k Ω † + = e i Ht e − i H t a k e i H t e − i Ht (5) a out ( k ) ≡ Ω − a k Ω †− = e i Ht e − i H t a k e i H t e − i Ht (6)have the property of creating input and output scattering eigen-states a † in ( k ) | i = | k + i a † out ( p ) | i = | p − i and the commutation relations[ a in ( k ) , a † in ( p )] = [ a out ( k ) , a † out ( p )] = δ ( k − p ) . We now relate the scattering theory, as briefly sketchedabove, to the input-output formalism [21, 22] of quantum op-tics. To do so, we start by recalling the definition of the inputfield operator [21] a in ( t ) = √ π Z d k a k ( t ) e − i k ( t − t ) where a k ( t ) ≡ e i Ht a k e − i Ht is an operator in the Heisenbergpicture. The relationship between a in ( t )—which is definedin the input-output formalism—and a in ( k )—which is definedabove in (5) as a result of the scattering theory—can be deter-mined by noting that a in ( t ) = √ π Z d k e i Ht a k e − i Ht e − i k ( t − t ) = √ π Z d k e i Ht e − i H t a k e i H t e − i Ht e − i kt = √ π Z d k a in ( k )e − i kt (7)where in the second line we used the fact that [ H , a k ] = − ka k to convert the a k e i kt term into e − i H t a k e i H t . As a result, a in ( k )provides the spectral representation of a in ( t ) in the limit t →−∞ . Similarly, the output field operator in the input-outputformalism a out ( t ) = √ π Z d k a k ( t )e − i k ( t − t ) is related to a out ( k ) in the scattering theory through a out ( t ) = √ π Z d k a out ( k )e − i kt (8)in the limit t → ∞ . We have thus established a direct con-nection between the input-output formalism, and the scatter-ing theory. We should note that a di ff erent set of input and β ˜ ω ( β ) β ω FIG. 2. (Color online) Linearization of a surface plasmon-like waveg-uide dispersion relation ˜ ω ( β ) around a wavevector β is shown. Theslope of the line is equal to the group velocity v g . The photon statesin the text are assumed to have frequencies in the vicinity of ω sothat the linearization is justified. output operators were defined in [30] with an aim to make aconnection to correlation functions. In [31], a similar set ofinput-output operators were defined in order to relate two dif-ferent quantization schemes in dielectric media. To the bestof our knowledge, the explicit link we provide above betweenthe input-output formalism and the scattering theory has notbeen previously published in the literature. IV. SINGLE-PHOTON TRANSPORT
Now that we know the relationship between the input-output formalism and the scattering theory, let us now calcu-late the S -matrix elements h p | S | k i between two single pho-ton states | k i and | p i . Following the standard procedure, (seeAppendix C), the input-output equations appropriate for theHamiltonian in (1) ared N d t = − i r τ ( σ + a in − a † in σ − ) − τ N (9)d σ − d t = i r τ σ z a in − τ σ − − i Ω σ − (10) a out = a in − i r τ σ − (11)where all operators are in the Heisenberg picture and hencethey are all time-dependent. τ − = π V / v g is proportional tothe spontaneous emission rate. N = σ + σ − = ( σ z + / S -matrix, which is related to the input and out-put operator by h p | S | k i = h | a out ( p ) a † in ( k ) | i = √ π Z d t h | a out ( t ) | k + i e i pt where we used (8) to write a out ( p ) in terms of a out ( t ). It istherefore su ffi cient to first calculate h | a out ( t ) | k + i and thenperform an inverse Fourier transformation to determine thesingle-photon S -matrix. In the calculations to follow in thisand the next section, we will go back and forth betweenFourier transforms of the operators, and we will explicitly use t , t ′ to imply time dependent operators and k , , p , to denotethe time independent Fourier transformed pairs.The quantity h | a out ( t ) | k + i can be obtained by sandwiching(10) and (11) between the two states h | and | k + i . We havedd t h | σ − | k + i = i r τ h | σ z a in | k + i − τ h | σ − | k + i− i Ω h | σ − | k + i (12) h | a out | k + i = h | a in | k + i − i r τ h | σ − | k + i . (13)Note that h | a in ( t ) | k + i = h | a in ( t ) a † in ( k ) | i = √ π e − i kt (14)by the use of (7) and h | σ z a in ( t ) | k + i = −h | a in ( t ) | k + i (15)since | i has an atomic part that is in the ground state. Using(14)–(15) in (12)–(13) results in a first order ordinary di ff eren-tial equation. By solving it, we get h | σ − | k + i = √ π e − i kt √ /τ ( k − Ω ) + i /τ (16) h | a out | k + i = √ π e − i kt ( k − Ω ) − i /τ ( k − Ω ) + i /τ . (17)After Fourier transforming (17), we obtain the single-photon S -matrix h p | S | k i = t k δ ( k − p ) (18)where t k ≡ ( k − Ω ) − i /τ ( k − Ω ) + i /τ is the single-photon transmission coe ffi cient. For subsequentcalculations, we also define s k ≡ √ /τ ( k − Ω ) + i /τ that measures the excitation of the atom by the single-photonwave when normalized against the incident wave amplitude. t k and s k are related by t k = − i r τ s k . These results for single-photon transport agree with [9, 11],where the scattering wavefunction was directly calculatedthrough a real space formalism.The crucial step in the derivation above is (15) which takesadvantage of the single-excitation nature of the input state.Formally, the same result can also be obtained by approxi-mately setting σ z = − weak ex-citation limit , where the atom is assumed to be mostly in theground state. Physically, in the case of single-photon trans-port, the weak excitation limit is valid, when a single-photonpulse has a duration that is much longer than the spontaneouslifetime of the atom. However, we emphasize that the weak-excitation limit is not always valid in general even for a single-photon pulse. It has been shown that for the Hamiltonian in(1), a single photon pulse with a duration comparable to thespontaneous emission lifetime can in fact completely invert anatom [35].The formalism here removes the need for the assumption ofweak-excitation limit when calculating single-photon proper-ties. In fact, we can directly calculate the excitation probabil-ity h k + | N | k + i for the scattering eigenstate | k + i . N = σ + σ − andusing (16) we have h k + | N | k + i = h k + | σ + σ − | k + i = h k + | σ + | ih | σ − | k + i = π | s k | = π /τ ( k − Ω ) + (1 /τ ) . Here, we again have taken advantage of the fact that | k + i is asingle-excitation state whereas σ + acting on any state except | i would result in a multi-excitation state leading to a zerooverlap with h k + | . More generally, we have h k + | σ z ( t ) | p + i = h k + | (2 σ + σ − − | p + i = h k + | σ + | ih | σ − | p + i − δ ( k − p ) = π e − i( p − k ) t s ∗ k s p − δ ( k − p ) (19)which will be useful when deriving the two-photon S -matrix. V. TWO-PHOTON TRANSPORT
Our aim in this section is to calculate the two-photon S -matrix based on the results we obtained for the single photoncase. We first introduced the two photon S -matrix element in(4). We will begin by inserting an identity operator in between a out ( p ) and a out ( p ) h | a out ( p ) a out ( p ) a † in ( k ) a † in ( k ) | i = h | a out ( p ) Z d k | k + ih k + | ! a out ( p ) a † in ( k ) a † in ( k ) | i and use the Fourier transform of (17) to simplify the expres-sion as = t p h p + | a out ( p ) a † in ( k ) a † in ( k ) | i . Using the Fourier transform of (11) we get = t p h p + | a in ( p ) − i r τ σ − ( p ) a † in ( k ) a † in ( k ) | i = t p δ ( p − k ) δ ( p − k ) + t p δ ( p − k ) δ ( p − k ) − i r τ t p h p + | σ − ( p ) | k k + i where we used the orthogonality of the scattering eigenstates.Thus, to determine the two-photon S -matrix, we will need tocalculate h p + | σ − ( t ) | k k + i and take its Fourier transform.Using (10), we obtain the di ff erential equation that de-scribes h p + | σ − ( t ) | k k + i dd t h p + | σ − ( t ) | k k + i = i r τ h p + | σ z ( t ) a in ( t ) | k k + i − τ + i Ω ! h p + | σ − ( t ) | k k + i . (20)If we can simplify the part that depends on σ z a in , we can thensolve the di ff erential equation. Since a in is an annihilationoperator for scattering states, by using (7) we can write h p + | σ z ( t ) a in ( t ) | k k + i = √ π h h p + | σ z ( t ) | k + i e − i k t + h p + | σ z ( t ) | k + i e − i k t i and then using (19) results in = √ π π e − i( k + k − p ) t s ∗ p ( s k + s k ) − √ π δ ( k − p )e − i k t − √ π δ ( k − p )e − i k t which is what we were after. We can now solve the first orderordinary di ff erential equation (20) in a way very similar to thederivation that led to (18). After some algebra and rearrange-ment we get h p + | σ − ( t ) | k k + i = − √ π π s k + k − p s ∗ p ( s k + s k )e − i( k + k − p ) t + √ π δ ( k − p ) s k e − i k t + √ π δ ( k − p ) s k e − i k t . Taking the Fourier transform of the expression above gives us h p + | σ − ( p ) | k k + i = − π δ ( k + k − p − p ) s p s ∗ p ( s k + s k ) + s k δ ( k − p ) δ ( k − p ) + s k δ ( k − p ) δ ( k − p ) . Lastly, using the relation t p s ∗ p = s p , we obtain h | a out ( p ) a out ( p ) a † in ( k ) a † in ( k ) | i = t k t k [ δ ( k − p ) δ ( k − p ) + δ ( k − p ) δ ( k − p )] + i 1 π r τ δ ( k + k − p − p ) s p s p ( s k + s k ) . (21)This final result agrees with previous calculations using ad-vanced techniques such as the Bethe ansatz in real space [10], Equations (118)-(119) in [10] and equation (21) in this paper are the samewith the following notational correspondence:
Γ = /τ , ∆ = ( k − k ) / ∆ = ( p − p ) / E = k + k , E = p + p . the algebraic Bethe ansatz [14], and the LSZ formalism inquantum field theory [18, 19]. The derivation here, however,is perhaps more elementary, and thus may serve to make suchresults more accessible. In addition, the results relate the pres-ence of the background fluorescence, to the excitation of theatoms. VI. COHERENT STATE COMPUTATION
A traditional use of the input-output formalism is to calcu-late the correlation function when the input is in a coherentstate. Here we briefly outline such a calculation for our sys-tem in order to contrast it with the single and two-photon cal-culations of the previous two sections. For this purpose, weconsider a coherent input state | α k i , such that a in ( t ) | α + k i = α e − i kt | α + k i and calculate, as an example, the G (1) correlation function G (1) ( t ′ , t ) = h α + k | a † out ( t ′ ) a out ( t ) | α + k ih α + k | α + k i . Using (11), we have G (1) ( t , t ′ ) = | α | e − i k ( t − t ′ ) + i α e − i kt r τ h σ + ( t ′ ) i− i α ∗ e i kt ′ r τ h σ − ( t ) i + τ h σ + ( t ′ ) σ − ( t ) i (22)where for any operator O , h O i ≡ h α + k | O | α + k i .Each of the expectation values in (22) can be calculated us-ing the input-output formalism. Taking the expectation valuesin (9) and (10) results indd t h σ z ( t ) i = − i2 r τ (cid:16) α e − i kt h σ + ( t ) i − α ∗ e i kt h σ − ( t ) i (cid:17) − τ h σ z ( t ) + i dd t h σ − ( t ) i = − i Ω − τ ! h σ − ( t ) i + i α e − i kt r τ h σ z ( t ) i dd t h σ + ( t ) i = i Ω − τ ! h σ + ( t ) i − i α ∗ e i kt r τ h σ z ( t ) i . Directly solving the equations above provides the values of h σ + ( t ′ ) i and h σ − ( t ) i in (22), while the h σ + ( t ′ ) σ − ( t ) i term canbe computed using the quantum regression theorem. Thesecalculations can be found in standard textbooks [22, 23], insections related to the properties of resonance fluorescence,and we will not repeat them here. Instead, based on the out-line above, we make a few observations about the coherentstate computations, as commonly done, and the one and two-photon computations as carried out in this paper.1. The input-output formalism provides a set of nonlinear operator equations. Therefore, all computations, by neces-sity, involve the conversion of such operator equations intoordinary di ff erential equations for various operator matrix el-ements. While the coherent state computations typically in-volve taking expectation values in terms of the input states,the one and two-photon computations involve matrix elementsthat have di ff erent photon numbers.2. It is certainly reasonable to expect that the one or two-photon S matrices can be obtained by analyzing various cor-relation functions for a weak coherent state input. Indeed, theconnection between the two-photon out wavefunction, and the g (2) correlation function, has been pointed out in [10] and itis likely that stronger connections exist. This will be carriedout in future work. However, if the aim is to determine the S -matrix in the few-photon Fock state Hilbert space, the com-putation as discussed here should be far more direct.3. We emphasize that the few-photon computations yieldthe S -matrix in the few-photon Hilbert space, and thus pro-vide a complete description of all physical processes in thefew-photon Fock state Hilbert space. In contrast, comput-ing G (1) or G (2) correlation functions alone do not completelyspecify the out state for a given incident coherent state in gen-eral. Certainly, in the majority of quantum optics experimentsat present, one probes a quantum system with a coherent inputstate, and obtains information about the system by measuringdi ff erent correlation functions. The coherent state computa-tions, as briefly reproduced above, are adequate to describethese experiments. However, these quantum systems are be-ginning to be considered as prospective devices which willeventually process quantum states [36, 37]. In such an engi-neering context, one ultimately has to be able to completelyspecify the output quantum states. It is in this respect that wehope the few-photon transport computations will prove to bevaluable for future engineering applications. VII. CONCLUSION
In this paper, we extend the input-output formalism ofquantum optics to analyze one and two photon scattering inwaveguides with a two-level atom inside. We develop therelationship between the input-output operators and the scat-tering theory which in turn enables us to analytically cal-culate the photon scattering matrix elements with minimumamount of algebra. We also contrast our calculations for few-photon Fock state transport with the conventional applicationof input-output formalism for coherent-state transport. Thiswork helps us go beyond the correlation function analysis ininput-output formalism, and leads to exact solutions for thescattering matrix elements.
ACKNOWLEDGMENTS
This work is supported by the David & Lucile PackardFoundation.
Appendix A: Two Mode Model
In this section we will write the Hamiltonian for the casewhen photons are allowed to propagate in both directionswithin the waveguide. We will refer to this case as the twomode model. After introducing the Hamiltonian, we will usethe results of Sections IV and V to calculate one and twophoton reflection and transmission coe ffi cients for right to leftmoving fields.When photons propagate to the r ight and to the ℓ eft, we willneed to add extra terms to the Hamiltonian. We begin as wedid in Section II and write˜ H = Z ∞ d β ω r ( β ) r † β r β + Z −∞ d β ω ℓ ( β ) ℓ † β ℓ β for the waveguide part of the Hamiltonian. The dispersionrelation for the left moving modes ω ℓ ( β ) is the mirror imageof the one for the right moving modes. We linearize the leftand right branches of the dispersion relationship at β = ± β toget ω r ≈ ω + v g ( β − β ) and ω ℓ ≈ ω − v g ( β + β ). Follow-ing linearization, we extend the limits of integration to ±∞ ,make a change of variables β β ∓ β for the right and leftwaveguides respectively and define ω = v g β , r ω ≡ r β + β / √ v g , ℓ ω ≡ ℓ β − β / √ v g to get H = Z ∞−∞ d ω ω (cid:16) r † ω r ω − ℓ † ω ℓ ω (cid:17) . (A1)The interaction part of the Hamiltonian is given by H = Ω σ z + V √ v g Z ∞−∞ d ω h σ + ( r ω + ℓ ω ) + ( r † ω + ℓ † ω ) σ − i . (A2)Since the total excitation operator N E = Z ∞ d β r † β r β + Z −∞ d β ℓ † β ℓ β + σ z commutes with the Hamiltonian, we subtracted the term ω N E from the Hamiltonian and set Ω = ˜ Ω − ω in the derivation,mimicking the steps in Section II.Now that we have the Hamiltonian, we can write down theHeisenberg equations of motion and define the input-outputoperators for the fields as illustrated in detail for a chiral modelin Appendix C. The equations for the annihilation operatorsare d r ω ( t )d t = − i[ r ω , H ] = − i ω r ω − i ˜ V σ − d ℓ ω ( t )d t = − i[ ℓ ω , H ] = + i ωℓ ω − i ˜ V σ − where ˜ V = V / √ v g . The definitions for the input and outputoperators for right going fields are the same as in Appendix Cand we get r out ( t ) = r in ( t ) − i r τ σ − ( t )where τ is defined in (C6). Left going modes have a groupvelocity which is negative that of the right going modes andthat leads to a negative sign in (A1). As a result, starting fromthe the definition of the input and output operators in (5)–(6),the input and output operators for left going modes have theform ℓ out ( t ) = √ π Z d ω ℓ ω ( t )e i ω ( t − t ) ℓ in ( t ) = √ π Z d ω ℓ ω ( t )e i ω ( t − t ) ℓ out ( t ) = ℓ in ( t ) − i r τ σ − ( t )where we note the change of sign in the frequency variable.Using these results we can show thatd σ − d t = i r τ σ z r in + i r τ σ z ℓ in − τ σ − − i Ω σ − which is in a form similar to those that we get in temporalcoupled mode theory [38].We now have all the tools to solve for the scattering thattakes place in the two mode model. Let us define even andodd combinations of the operators for the right and left propa-gating modes as a ω = r ω + ℓ − ω √ a ω = r ω − ℓ − ω √ H = Z d ω a † ω a ω + ˚ a † ω ˚ a ω ≡ H e , + H o , H = Ω σ z + √ V √ v g Z d ω (cid:16) σ + a ω + a † ω σ − (cid:17) ≡ H e , where we see that the interaction part of the Hamiltonian de-pends only on the even combination of modes. In Sections IVand V we solved for H = H e , + H e , for a rescaled value of V .The odd part H o , is interaction free and hence is also solved.From (A3) we get r in / out ( ω ) = a in / out ( ω ) + ˚a in / out ( ω ) √ ℓ out ( ω ) = a out ( − ω ) − ˚a out ( − ω ) √ h | ℓ out ( p ) r † in ( k ) | i = h | [ a out ( − p ) − ˚a out ( − p )][ a † in ( k ) + ˚a † in ( k )] | i = h | a out ( − p ) a † in ( k ) | i − h | ˚a out ( − p ) ˚a † in ( k ) | i =
12 ( t k − δ ( p + k ) ≡ ¯ r k δ ( p + k ) . Here we used (A4) and (18) to get the one photon reflectioncoe ffi cient ¯ r k . Similarly, the one photon transmission coe ffi -cient ¯ t k is given by h | r out ( p ) r † in ( k ) | i =
12 ( t k + δ ( p − k ) ≡ ¯ t k δ ( p − k ) . Two photon calculations require adding another input-output pair. For instance, the scattering matrix element as-sociated with one photon scattering to the right, another to theleft when two photons initially propagate to the right is givenby h | r out ( p ) ℓ out ( p ) r † in ( k ) r † in ( k ) | i = h h | a out ( p ) a out ( − p ) a † in ( k ) a † in ( k ) | i− h | a out ( p ) ˚a out ( − p ) a † in ( k ) ˚a † in ( k ) | i− h | a out ( p ) ˚a out ( − p ) ˚a † in ( k ) a † in ( k ) | i + h | ˚a out ( p ) a out ( − p ) a † in ( k ) ˚a † in ( k ) | i + h | ˚a out ( p ) a out ( − p ) ˚a † in ( k ) a † in ( k ) | i− h | ˚a out ( p ) ˚a out ( − p ) ˚a † in ( k ) ˚a † in ( k ) | i i = ¯ t k ¯ r k δ ( k − p ) δ ( k + p ) + ¯ r k ¯ t k δ ( k + p ) δ ( k − p ) + B δ ( k + k − p + p )where from (21) B = i 1 π r τ ′ s p s − p ( s k + s k ) . We note that τ ′ = τ/ √ V inthe definition of H . These results agree with equations (52)and (130) in [10]. Appendix B: Hamiltonian in the continuum limit
This section will summarize the steps taken to obtain thecontinuum form of the Hamiltonian from its discrete version.We will follow the approach in [24, 25].The discrete variables are assumed to be for those in a onedimensional cavity of length L . The mode spacing in the cav-ity is given by ∆ β = π/ L . In this 1D cavity, the free spaceelectromagnetic Hamiltonian, H , is given by H = X β ω β ˆ a † β ˆ a β with the commutator relationship [ˆ a β , ˆ a † β ′ ] = δ β,β ′ . Now,we will convert the sum into an integral by the equivalence (cid:16) ∆ β P β (cid:17) → (cid:16)R d β (cid:17) to get H = L π Z d β ω β ˆ a † β ˆ a β . The continuous mode operator ˜ a β is related to the discretemode ˆ a β by˜ a β = r L π ˆ a β which results in H = Z d β ω β ˜ a † β ˜ a β . The commutator relationship [˜ a β , ˜ a † β ′ ] = L π δ β,β ′ in the limit L → ∞ becomes [˜ a β , ˜ a † β ′ ] = δ ( β − β ′ ) . To see this result, define f ( β ) = L π δ β, . Integrating f ( β ) willgive Z d β f ( β ) → π L X β f ( β ) = π L L π = . As a result, the correct Hamiltonian in the continuum limit is H = Z d β ω ( β ) ˜ a † β ˜ a β with [˜ a β , ˜ a † β ′ ] = δ ( β − β ′ ). It is then easy to show that = Z d β | β ih β | where | β i = ˜ a † β | i , since h γ | Z d β | β ih β | ζ i = Z d β δ ( γ − β ) δ ( β − ζ ) = δ ( γ − ζ ) . In the discreet case H =
12 ˜ Ω σ z + V ′ √ L X β ( σ + ˆ a β + ˆ a † β σ − )where V ′ is the physical coupling constant. The factor L − / arises because the photon as created by ˆ a † β has a normalizationconstant L − / . In the continuum case we get H =
12 ˜ Ω σ z + V ′ √ L L π r π L Z d β ( σ + ˜ a β + ˜ a † β σ − ) =
12 ˜ Ω σ z + V ′ √ π Z d β ( σ + ˜ a β + ˜ a † β σ − ) . Thus, the coupling constants in the discrete ( V ′ ) and the con-tinuum ( V ) cases di ff er by a factor of (2 π ) − / . Appendix C: Derivation of the input-output formalism
Here we provide a derivation of the input-output equations(9)–(11). This derivation closely follows [21, 22]. Based onthe Hamiltonian (2)–(3), and the definition ˜ V ≡ V / √ v g , the Heisenberg equations of motion for the operators arei d a k d t = ka k + ˜ V σ − (C1)i d σ − d t = Ω σ − − ˜ V Z d k σ z a k (C2)i d σ z d t = V Z d k ( − a † k σ − + σ + a k ) . (C3)After multiplying (C1) by the integration factor exp(i kt ), weintegrate it from an initial time t < t to get a k ( t ) = a k ( t )e − i k ( t − t ) − i ˜ V Z tt dt ′ σ − ( t ′ )e − i k ( t − t ′ ) . (C4)We define the input operator as a in ( t ) = √ π Z d k a k ( t )e − i k ( t − t ) which satisfies the commutation relation[ a in ( t ) , a † in ( t ′ )] = δ ( t − t ′ ) . We further introduce a field operator Φ ( t ) = √ π Z d k a k ( t )and integrate (C4) with respect to k to get Φ ( t ) = a in ( t ) − i ˜ V √ πσ − ( t ) = a in ( t ) − i r τ σ − ( t ) . (C5)Here, notice that we integrate over half the delta-function [21]which results in a factor of 1 / τ is defined as1 τ ≡ π ˜ V . (C6)Furthermore, plugging (C5) into (C2) and (C3), results ind σ − d t = i r τ σ z a in − τ σ − − i Ω σ − d N d t = − i r τ ( σ + a in − a † in σ − ) − τ N . Here N = ( σ z + /
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