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controllable single - photon transport is of central importance in quantum information processing . there have been many recent experimental  @xcite and theoretical  @xcite works probing the photon transport properties of a wavelength - scale cavity that incorporates a two- or multi - level system . these works encompass both the weak and the strong coupling regime . traditionally , such a system is typically studied with quantum trajectory method  @xcite which , as a monte carlo approach , is inherently stochastic  @xcite . other approaches include employing a master equation  @xcite or input - output formalism  @xcite that assume a weak input coherent state , and usually involves uncontrolled approximations  @xcite . however , the recent experiments allow us to determine the response to the input of a single injected photon . thus , a theoretical framework that allows one to directly calculate the response of such a system to a single injected photon is valuable .    in this article , we provide a full quantum - mechanical and deterministic approach to solve the response of this system to a single photon , yielding a wealth of information on the transport properties , as well as the effects of dissipations to the relative phase between the excitations of the cavity mode and the two - level system . our formalism treats the problems in real space , which is particularly convenient for discussing photon transport from one space - time point to another one . moreover , the treatment is exact , and makes no assumptions on temporal behaviors of the constituents of the system . rather , our formalism allows direct computation of the temporal evolution of the system . we first describe a few configurations of the systems that is relevant to this paper . [ fi : schematics](a ) and ( b ) show schematically a single - mode waveguide side - coupled to a cavity interacting with a two - level atom , where the cavity can be a single - mode microcavity ( fig . [ fi : schematics](a ) ) , or a ring resonator that supports two degenerate counter - propagating modes ( fig . [ fi : schematics](b ) ) . several notable solid - state systems are of such geometry : a superconducting quantum bit - coplanar waveguide system  @xcite , an atom - microtoroidal resonator - waveguide geometry  @xcite , and a solid - state quantum dot - microdisk - waveguide system  @xcite . the second configuration has a single - mode cavity direct - coupled to the waveguide , as shown in fig . [ fi : schematics](c ) . many atomic cavity quantum electrodynamics ( cavity qed ) experiments use this configuration  @xcite . in this paper , we will focus on systems as described in fig . [ fi : schematics](a ) and ( c ) . the formalism can be generalized to treat the system in fig . [ fi : schematics](b ) which will be published in a separate article . the article is organized as follows : in sec . [ sec : hamiltonian ] the effective real - space hamiltonian of the systems is introduced , and the exact one - photon solution is given . [ sec : nodissipation ] discusses the non - dissipative case , give the transmission spectrum , the cavity and the atomic excitation amplitude , and point out the possibility of using the system as a single - photon switch by tuning the atomic transition frequency . in sec . [ sec : dissipation ] , we examine the effects of the dissipations in the system . our results indicate that the dissipation of the cavity and the atom , respectively , affects the transmission and the coherence between the cavity and the atom in qualitatively different manner . finally , in sec . [ sec : direct ] we show that the single - photon transport of the direct - coupled case of fig . [ fi : schematics](c ) can be mapped into those of the side - coupled case , thus the single - photon transport properties of the two systems are related . we discuss the transmission spectrum of this case , and compare our results with experimental data of a recent circuit qed experiment . in this section , we will derive an effective real - space hamiltonian to describe the systems of interest . as will be shown in later sections , such a real - space representation is particularly convenient for discussing the transport properties of the photons . the interaction between propagating photons and a cavity with a single mode can be described by the hamiltonian  @xcite : @xmath0where @xmath1 is the frequency of the mode of the propagating photon field corresponding to wave vector @xmath2 , _ i.e. _ , the dispersion relation . @xmath3 ( @xmath4 ) is the bosonic creation ( annihilation ) operator of the propagating photon mode , @xmath5 is the resonance frequency of the cavity mode , @xmath6 ( @xmath7 ) is the bosonic creation ( annihilation ) operator of the cavity mode , and @xmath8 is the coupling between the cavity and the waveguide , which leads to the decay of cavity mode into the waveguide . we now specialize eq . ( [ e : dicke ] ) to the case of a single - mode waveguide . by linearizing the dispersion , the effective real - space hamiltonian can be derived . the typical dispersion @xmath1 of a single - mode waveguide is shown in fig . [ fi : dispersion ] . for an arbitrary frequency @xmath9 that is away from the cut - off of the dispersion , with the corresponding wave vector @xmath10 , one approximates @xmath1 around @xmath11 and @xmath12 as    [ e : linearboth ] @xmath13    where the subscripts @xmath14 and @xmath15 are used to label the branch . since we will be interested in a narrow bandwidth in vicinity of @xmath9 , we can further extend the range of @xmath16 and @xmath17 to @xmath18 . thus , after linearizing the dispersion , one has @xmath19 each term on the right hand side of eq . ( [ e : afterlinearizing ] ) now can be easily represented in real space . by defining the fourier transform    @xmath20    where @xmath21 ( @xmath22 ) creates ( annihilates ) a right - moving photon at @xmath23 , the first term becomes @xmath24similarly , by defining    @xmath25    where @xmath26 ( @xmath27 ) creates ( annihilates ) a left - moving photon at @xmath23 , the second term becomes @xmath28    the interaction term transforms as @xmath29 in deriving eq . ( [ e : linearizinginteraction ] ) , @xmath8 , the coupling strength between the cavity and waveguide , is assumed to be independent of @xmath2 , and is denoted by @xmath30 . this assumption is equivalent to the markov approximation  @xcite . the linearization procedure is analogous to a commonly used procedure in electronic cases , where @xmath11 is chosen as the fermi wave vector @xmath31  @xcite . having described the cavity - waveguide coupling , we now include both the atom part , as well as the interactions of the atom and the cavity with the reservoirs . such interactions with reservoirs give rise to intrinsic dissipation  @xcite . the hamiltonian of the _ composite _ system @xmath32 is @xmath33 :    @xmath34\notag\\ & + \omega_c a^{\dagger } a + g\left(\sigma_+ a + a^{\dagger } \sigma_{-}\right)\notag\\ & + \omega_e a_e^{\dagger}a_e + \omega_g a_g^{\dagger}a_g,\\   h_r/\hbar & \equiv h_{r_1}/\hbar + h_{r_2}/\hbar \notag\\ & \equiv \sum_j \omega_{1j } r_{1j}^{\dagger}r_{1j}+\sum_j \omega_{2j } r_{2j}^{\dagger}r_{2j},\\ h_{sr}/\hbar & \equiv h_{sr_1}/\hbar + h_{sr_2}/\hbar\notag\\ & \equiv \sum_j\left(\kappa^{*}_j r_{1j}^{\dagger}a + \kappa_j a^{\dagger } r_{1j}\right)+   \sum_j\left(\eta^{*}_j r_{2j}^{\dagger}\sigma_{- } + \eta_j \sigma_+ r_{2j}\right).\end{aligned}\ ] ]    @xmath35 is the hamiltonian of the system @xmath36 of coupled waveguide - cavity - atom . this hamiltonian includes the atomic part , and the interaction between the atom and the cavity . @xmath37(@xmath38 ) is the creation operator of the ground ( excited ) state of the atom , @xmath39(@xmath40 ) is the atomic raising ( lowering ) ladder operator satisfying @xmath41 and @xmath42 , where @xmath43 describes the state of the system @xmath36 with @xmath44 propagating photons , @xmath45 photons in cavity mode , and the atom in the excited ( @xmath46 ) or ground ( @xmath47 ) state . @xmath48 is the atomic transition frequency . @xmath49 is the coupling strength between the cavity and the atom . @xmath50 describes the reservoir , which is composed of two subsystems : @xmath51 , where the cavity couples only to @xmath52 , and the atom couples only to @xmath53 . @xmath52 and @xmath53 are assumed to be independent . each @xmath54 and @xmath53 is modeled as a collection of harmonic oscillators with frequencies @xmath55 and @xmath56 , and with the corresponding creation ( annihilation ) operators @xmath57 ( @xmath58 ) , and @xmath59 ( @xmath60 ) , respectively . @xmath61 describes the interactions between the cavity and the atom with the reservoirs , respectively . the cavity @xmath6 couples to the @xmath62th reservoir oscillator @xmath58 in @xmath52 with a coupling constant @xmath63 , while the atom @xmath64 couples to the @xmath62th reservoir oscillator @xmath60 in @xmath53 with a coupling constant @xmath65 .    in appendix  [ a : reservoircoupling ] , we show that by incorporating the excitation amplitudes of the reservoir @xmath14 , the effective hamiltonian @xmath66 of @xmath36 can be obtained and is given by : @xmath67\notag\\ & + \left(\omega_c -i\frac{1}{\tau_c}\right ) a^\dagger a + \left(\omega_e - i\frac{1}{\tau_a}\right ) a_e^{\dagger } a_e + \omega_g a_g^{\dagger } a_g\notag\\ & +   \int dx\ , v\delta(x)\left(c_r^{\dagger}(x ) a + a^{\dagger}c_r(x ) + c_l^{\dagger}(x ) a + a^{\dagger}c_l(x)\right)\notag\\ & + g(a\sigma_+ + a^{\dagger}\sigma_{-}),\end{aligned}\]]where @xmath68 and @xmath69 are the dissipation rates of the cavity and the atom , respectively , due to coupling to the reservoir . we will call @xmath66 as @xmath70 in the following for brevity . note that @xmath71 , @xmath49 , @xmath72 , and @xmath73 all have the same unit as frequency . the temporal evolution of an arbitrary state @xmath74 describing the system @xmath36 is described by the schrdinger equation @xmath75 where @xmath70 is the effective hamiltonian of the system in eq .  , and @xmath74 can be expanded as @xmath76where @xmath77 is the vacuum , with zero photon in both the waveguide and the cavity , and with the atom in the ground state . @xmath78 is the single - photon wavefunction in the @xmath79 mode . @xmath80 is the time - dependent excitation amplitude of the cavity , and @xmath81 is the time - dependent excitation amplitude of the atom . the expansion of the state @xmath74 in eq . assumes that the atom was initially in the ground state and the cavity was empty when the incoming photon was at @xmath82 . the schrdinger equation ( eq .  ( [ e : schrodinger ] ) ) thus gives the following set of equations of motion :    [ e : teom ] @xmath83    for any given initial state @xmath84 , the dynamics of the system can be obtained straightforwardly by numerically integrating the set of equations , eqs . ( [ e : teom ] ) . in particular , one could study the time - dependent transport of an arbitrary single - photon wave packet .    in the following , we concentrate on the single - photon transport of constant frequency . when @xmath74 is an eigenstate of frequency @xmath85 , _ i.e. _ , @xmath86 , eq .  ( [ e : schrodinger ] ) yields the time - independent eigen equation @xmath87 and the interacting steady - state solution @xmath88 can be solved for . here @xmath89 is the total energy of the coupled system @xmath36 , with @xmath90 , and @xmath91 .    for an input state of one - photon fock state , the most general interacting eigenstate for the hamiltonian of @xmath70 takes the following form : @xmath92 , etc . the connection between the interacting eigenstate and a typical scattering experiment is described by the lippmann - schwinger formalism  @xcite . the time - independent schrdinger equation of eq . ( [ e : timeindependent ] ) for the state @xmath88 of eq . ( [ e : interactingeigenstate ] ) yields the following equations of motion :    [ e : eom ] @xmath93    with @xmath90 , and @xmath91 . our aim is to solve for the transmission and reflection amplitudes for an incident photon . for this purpose , we take @xmath94 , and @xmath95 , where @xmath96 is the transmission amplitude , and @xmath97 is the reflection amplitude  @xcite . solving eqs .   - for @xmath98 , @xmath96 , @xmath97 , @xmath99 , and @xmath100 gives :    @xmath101    which are valid in both strong and weak coupling regimes .    before we proceed , here we briefly outline some of the main features of the transmission amplitude @xmath96 of eq . ( [ e : t ] ) . in a waveguide - side - coupled cavity - atom system as shown in fig . [ fi : schematics](a ) , when the cavity mode and the atom are in - tune ( @xmath102 ) , the transmission spectrum is always symmetric with respect to the atomic transition frequency ( @xmath103 ) . moreover , the transmission spectrum has two dips with spectral separation proportional to the atom - cavity coupling , and an on - resonance photon attains local maximum . this is in contrast to the rabi splitting peaks in a direct - coupled cavity wherein an on - resonance attains local minimum . the transmission spectrum remains symmetric even when the cavity and the atom dissipations are present , and only becomes asymmetric when the cavity and the atom are detuned . also , a finite cavity dissipation does not destroy the phase relation between the atom and the cavity , but the atomic dissipation does . here as a side note we make a comment on a recent approach using the input - output formalism to obtain the transmission and reflection amplitudes . in ref . [ ] , in order to linearize the nonlinear term @xmath104 in the equations of motion of the heisenberg operators ( see eq . ( 2 ) in ref .  [ ] ) , the two - level atom is approximated to be in the ground state all the time , such that @xmath105 is subsequently substituted with @xmath106 . in fact , this approximation is non - physical , as can be seen from the exact solutions here , since the atom is strongly excited even with a single photon , in the absence of strong dissipation , which is the case of interest here . such excitation is crucially important for using the atom to control the photon transport . instead , one can show that the equations of motion in the heisenberg picture for the operators can give rise to a set of exact equations of motion of the amplitudes that is closed , provided we sandwich the operators with @xmath88 and vacuum state @xmath107 . this set of equations has solutions identical to eq . ( [ e : t])-([e : ea ] ) . in particular , the nonlinear term @xmath104 gives the matrix element @xmath108 , which , after dividing the common factor @xmath109 on both sides of eq . ( 2 ) in ref . [ ] , gives the @xmath110 term as in eq . ( [ e : eom4 ] ) . in doing so , one can then derive all the amplitudes , @xmath99 etc . , and hence determines @xmath111 the single - photon transport is exactly solvable in both schrdinger picture and heisenberg picture . below we investigate the effects of dissipations on the transmission spectrum . we start by discussing the non - dissipative case , followed by examining the dissipations of the cavity and of the atom . for each case , we will focus on the changes due to dissipations on the local maxima and minima determined from @xmath112 . ( 1)_atom - cavity in tune _ ( @xmath114 ) : the transmission spectrum for this case is shown in fig .  [ fi : nodissipations](a ) . the spectrum exhibits a maximum at @xmath115 . at @xmath115 , the on - resonance photon completely transmits ( @xmath116 ) , and the cavity is not excited ( @xmath117 ) , but only the atom is excited ( @xmath118 ) . we note that , in contrast , when only the atom or the cavity is present , an on - resonance photon is completely reflected  @xcite . the transmission spectrum shows two minima at @xmath119 , which correspond to the rabi - splitted frequencies . at @xmath120 , the photon is completely reflected ( @xmath121 ) . also , @xmath122 , thus the cavity excitation and the atom excitation are of equal amplitude and either in - phase or completely out of phase with each other . the full - width at half - minimum ( @xmath123 ) of each dip is exactly @xmath71 , independent of the atom - photon coupling constant @xmath49 . ( in comparison , for a side - coupled cavity with no atom embedded , the cavity decay rate is also @xmath71 ) . the spectral distance between the two local minima is @xmath124 . when @xmath49 is small , the transmission peak thus becomes very narrow , exhibiting a spectrum that is analogous to electromagnetically induced transparency phenomena  @xcite , as shown in fig . [ fi : nodissipations](b ) . ( 2)_atom - cavity detuned _ ( @xmath125 ) : when the photon is on - resonant with the atom(@xmath115 ) , the transmission amplitude is always 1 , regardless of the detuning between the atom and the cavity , as shown in fig . [ fi : nodissipations](c ) . on the other hand , when the atom is far detuned from the cavity resonance frequency ( @xmath126 ) , the single photon transmission spectrum has dips down to zero at the cavity frequency @xmath127 . this feature could be exploited to achieve a fast single - photon switch : for an incoming photon with frequency @xmath128 , the transmission is 1 when the atom is in - tuned with the cavity ( @xmath129 ) ; while the transmission is essentially 0 when the atom is far detuned . thus , by tuning the transition frequency of the atom , the single - photon transport can be regulated and the setup acts as a single - photon switch , as shown by fig . [ fi : nodissipations](d ) . this effect was pointed out in ref . [ ] , here we give an exact derivation of this effect . the unavoidable intrinsic dissipative processes in the system always result in the leakage of photons into non - waveguided degrees of freedom . in general , the dissipations affect the transmission properties , and change the phase relation between the excitation amplitudes of the cavity mode and the atom . however , as we show below , the dissipation of the cavity and of the atom affect these properties rather differently . ( 1)_atom - cavity in - tune , dissipative cavity _ ( @xmath114 ; @xmath130 and @xmath131 ) : the transmission spectrum is shown in fig . [ fi : allcases](a ) . the overall transmission spectrum remains symmetric with respect to @xmath115 . the locations of the local maximum @xmath115 , and the local minima @xmath119 , are independent of cavity dissipation @xmath72 and are the same as in the dissipationless case . at @xmath115 , one still has @xmath116 , @xmath132 , and @xmath118 . remarkably , a photon at this frequency still transmits perfectly and the cavity is not excited , even with the presence of the cavity dissipation @xmath72 . one can understand this result from the non - dissipative case : since the cavity is not excited for a photon at @xmath115 , the cavity dissipation makes no effect on the transmission . at @xmath119 , one has @xmath133 for both frequencies . @xmath134 approximates @xmath135 when the decay rate is small such that @xmath136 , _ i.e. _ , when the coupling rate between the waveguide and the cavity is larger compared with the cavity dissipation rate . moreover , at @xmath119 , @xmath137 . thus , while the excitation amplitudes are reduced from the previous case , both cavity and atomic excitations are still either in - phase or out - of - phase with each other with the presence of cavity dissipation and has the same amplitude . thus , the cavity on one hand largely insulates the atom from decohering interactions with external environment  @xcite , on the other hand its own dissipation does not affect either the perfect transmission of an on - resonace photon , or the phase relation between the cavity and the atom at the rabi - splitted frequencies ( @xmath119 ) . when the dissipation is small , the full - width at half - maxium ( @xmath138 ) of each dip is exactly @xmath139 , independent of the atom - photon coupling constant @xmath49 . ( 2)_atom - cavity in - tune , dissipative atom case _ ( @xmath114 ; @xmath140 and @xmath141 ) : the transmission spectrum is plotted in fig . [ fi : allcases](b ) . the overall transmission spectrum remains symmetric with respect to @xmath115 . the local maximum is still located at @xmath115 , independent of atom dissipation @xmath73 and is the same as in the dissipationless case . the local minima @xmath142 , accurate up to the third order of @xmath143 , are @xmath144 .    at @xmath115 , one has @xmath145 comparing with the dissipative cavity case above , we thus see that the atom dissipation has a stronger effect on the transmission of an on - resonance photon than the cavity dissipation does . the relative phase @xmath146 between the cavity excitation and the atom excitation is always @xmath147 for any value of non - zero atomic dissipation . accurate up to second order of @xmath143 , @xmath148 , and the transmission amplitude is @xmath149 thus @xmath150 when the dissipation is small . @xmath151 at the same limit . moreover , the cavity excitation and the atom excitation are no longer in - phase ( @xmath152 ) , even at the order of @xmath73 . the relative phase @xmath146 between the cavity excitation and the atom excitation is given by @xmath153 . similarly , accurate up to second order of @xmath143 , @xmath154 , and the transmission amplitude is @xmath155which is complex conjugate of @xmath96 at @xmath156 in eq . ( [ e : tkbig ] ) . the cavity excitation and the atom excitation are no longer completely out - phase ( @xmath157 ) even at the order of @xmath73 . the relative phase @xmath146 between the cavity excitation and the atom excitation is @xmath158 . nonetheless , at both rabi - splitted frequencies , @xmath159 . the full - width at half - maxium ( @xmath160 ) of each dip is @xmath161 , independent of the atom - photon coupling constant @xmath49 . ( 3)_atom - cavity in - tune , dissipative cavity and atom case _ ( @xmath114 ; @xmath72 , @xmath141 ) : the local maximum is still located at @xmath115 exactly , while the rabi - splitted frequencies are shifted to @xmath162 . the transmission spectrum remains symmetric with respect to @xmath115 . at @xmath115 , @xmath163 thus @xmath134 approximates @xmath164 , when @xmath165 , and @xmath72 , and is independent of cavity decay rate @xmath72 . @xmath166 at this limit . the cavity is slightly excited with @xmath167 . the relative phase @xmath146 between the cavity excitation and the atom excitation is always @xmath147 for any value of non - zero atomic dissipation . at @xmath168 , @xmath169 . the cavity excitation and the atom excitation are not in - phase at the order of @xmath73 . similarly , at @xmath170 , @xmath169 , and the cavity excitation and the atom excitation are not completely out - phase even at the order of @xmath73 . also , the transmission amplitudes at two local minima are complex conjugate to each other , @xmath171    ( 4)_detuned , dissipative cavity and atom case _ ( @xmath172 ; @xmath72 , @xmath141 ) : when the cavity and the atom are in - tune , the single - photon transmission spectrum is always symmetric with respect to @xmath115 , regardless of the dissipations . when the atomic transition frequency @xmath103 is detuned slightly away from the cavity frequency @xmath5 , the transmission spectrum becomes asymmetric . the frequency of the local maximum is now located between @xmath103 and @xmath5 . remarkably , there exists an optimal frequency for the incident single photons such that at which frequency the magnitude of the transmission is _ insensitive _ to the frequency detuning @xmath173 . this `` cavity protected '' optimal frequency can be obtained by expanding @xmath174 with respect to @xmath175 and numerically solving the resultant polynomial equation . when the cavity and atom dissipations are small , retain only the first order term is enough , and the optimal frequency is essential at @xmath115 , as demonstrated in fig . [ fi : allcases](c ) . such optimal frequency could be useful to stabilize the single - photon transmission when the cavity frequency experiences drifting . also , in the far - detuned case , the atom is essentially decoupled from the cavity , thus the transmission properties at @xmath127 are determined by the cavity only . another configuration often employed in photon transport experiments is to directly place the cavity in the waveguide , as shown in fig . [ fi : schematics](c ) . in this case , the transmitted photons in the forward direction can only do so by tunneling in and out of the cavity . in comparison , the reflected photons in the side - coupled case also has only contribution from the cavity . thus , one anticipates that the _ transmission _ amplitude of the direct - coupled case is related to the _ reflection _ amplitude of the side - coupled case . we show in the following that it is indeed so .    to describe the direct - coupled case in fig . [ fi : schematics](c ) , we first construct the hamiltonian for the system , which involves the fields in the left ( @xmath176 ) and right ( @xmath97 ) branches , and their interaction via the cavity , as shown in fig . [ fi : folding ] . for the left branch , we define a field @xmath177 such that @xmath178 describes a photon that is moving to the right at @xmath179 in the left branch , and @xmath180 describes a photon that is moving to the left at @xmath179 in the left branch . thus , @xmath178 describes an incoming photon and @xmath180 describes an outgoing photon in the left branch . in order to take into account the phase shift that occurs during the reflection at the end of the waveguide , we write the following hamiltonian @xmath181 here @xmath182 is the phase shift due to reflection , as shown below . @xmath183 is a switch - on function with the general property that @xmath184 and @xmath185 in a short spatial extent , otherwise the specific form is unimportant . for computational purpose , one can take , for example , @xmath186 , where the spatial turning range is controlled by @xmath7 .    the hamiltonian in eq . ( [ e : leftbranch ] ) is in fact equivalent to a free hamiltonian via the the following canonical ( gauge ) transformation : @xmath187__i.e . _ _ , the @xmath176-branch mode of the direct - coupled case is folded from the @xmath14-mode of the side - coupled case . ( [ e : leftbranch ] ) becomes @xmath188 the corresponding eigen wavefunctions of @xmath189 ( eq .  ) and of @xmath50 ( eq .  ) transform to each other according to @xmath190thus , in the left branch , for an incoming wave @xmath191 at @xmath192 , the outgoing wave ( _ i.e. _ , reflected wave ) acquires a phase @xmath182 to become @xmath193 at @xmath194 . similarly , for the right ( @xmath97 ) branch in fig . [ fi : folding ] , we define a field @xmath195 such that @xmath196 describes a photon that is moving to the right at @xmath197 in the right branch , and @xmath198 describes a photon that is moving to the left at @xmath197 in the right branch . thus , @xmath198 describes an incoming photon and @xmath196 describes an outgoing photon in the right branch . the field in the right branch is described by the hamiltonian @xmath199via the canonical transformation @xmath200 _ i.e. _ , the @xmath97-branch mode of the direct - coupled case is folded from the @xmath15-mode of the side - coupled case , the hamiltonian @xmath201 of eq . can be transformed into a free hamiltonian @xmath202with the wavefunction transformed as @xmath203    when the cavity is present to allow the tunneling between the @xmath176 and @xmath97 branches , the hamiltonian is @xmath204which , using eq . ( [ e : ltor ] ) and ( [ e : rtol ] ) , can be written as : @xmath205 similar to the hamiltonian of eq . ( [ e : hamiltonian ] ) but with the additional phase terms in the cavity - waveguide coupling terms .    it can be shown straightforwardly that the single - photon transmission and reflection amplitudes for systems described by the hamiltonian of eq . ( [ e : hamiltonian ] ) and of eq . ( [ e : hamiltonianwithphase ] ) respectively are identical and are both independent of @xmath182 and @xmath206 . only the cavity and atomic excitation amplitudes depend upon @xmath182 and @xmath206 . the relations between each eigenfunctions are given by @xmath207 , and @xmath208 , _ i.e. _ , in the direct - coupled case the _ reflection _ amplitude is @xmath209 and the _ transmission _ amplitude is @xmath210 , where @xmath96 and @xmath97 are the transmission and reflection amplitudes of the side - coupled case discussed above ( eq .  ( [ e : t ] ) and ( [ e : r ] ) ) . this relation establishes the intuitive physical picture at the beginning of this section .      to demonstrate the validity of our approach , we apply our formalism to the transmission spectrum of single photons in a solid - state circuit qed system  @xcite , where a superconducting josephson junction qubit is embedded in a cavity capacitively direct - coupled to a transmission line waveguide ( fig . [ fi : schematics](c ) ) , thus the transmission amplitude is described by @xmath210 ( eq .  ( [ e : r ] ) ) . [ fi : fitting ] shows the results , where it is clearly seen that the experimental data can be fitted extremely well by @xmath211  @xcite . importantly , our results indicate that the asymmetry in the two transmission peaks at the rabi frequencies is likely due to very slight frequency detuning between the cavity and the qubit . dissipation and decoherence processes limit the performances of quantum information processing devices . a thorough understanding on such processes therefore are of vital importance in the realistic implementations of any such quantum devices . we have shown the impacts of the dissipations on the entanglement in a waveguide - cavity - atom system . here we make some final remarks on our approach . our approach could be generalized to take into account of other atomic degrees of freedom , such as multi - level atomic system , as well as the motion and the polarizability of the atom . this allows the alternative treatment of a number of interesting one - dimensional problems , such as the resonant radiation pressure on neutral particles in a waveguide  @xcite , strong optical interactions between particles  @xcite , resonance cooling of polarizable particles  @xcite , and low - light - level optical interactions with atomic vapor in fiber  @xcite . shen acknowledges the informative discussions with j. shin at stanford and k. srinivasan at nist . s. fan acknowledges financial support by the david and lucile packard foundation . in the effective hamiltonian of eq .  , the dissipation rates of the cavity and the atom are characterized by @xmath72 and @xmath73 , respectively . here we show explicitly how these intrinsic dissipation rates emerge as the system @xmath36 couples to a reservoir @xmath14 . to illustrate the physics , we will use as an example a system that consists of an atom in a waveguide , in which the atom also couples to a reservoir , to derive the effective hamiltonian and to show that coupling to the reservoir yields a damped atom . the same procedures carry through for the cavity - atom case as well . the hamiltonian of the composite system @xmath32 is @xmath33 : @xmath212\notag\\ & + \omega_e a_e^{\dagger}a_e + \omega_g a_g^{\dagger}a_g,\\ h_r/\hbar & \equiv \sum_j \omega_j r_j^{\dagger}r_j,\\ h_{sr}/\hbar & \equiv \sum_j\left(\eta^{*}_j r_j^{\dagger}\sigma_{- } + \eta_j \sigma_+ r_j\right).\end{aligned}\ ] ]    @xmath35 is the hamiltonian of the system @xmath36 of coupled waveguide - atom . the reservoir @xmath14 is modeled as a collection of harmonic oscillators with frequencies @xmath213 , and with the corresponding creation ( annihilation ) operators @xmath214 ( @xmath215 ) . @xmath61 describes the interactions between the atom and the reservoir . an arbitrary time - independent one - excitation state in @xmath32 is given by latexmath:[\]]where @xmath238 , and we have absorbed @xmath239 into @xmath240 . the state is entirely in @xmath36 and is given by @xmath241|\emptyset\rangle + e_a \sigma_{+}|\emptyset\rangle.\ ] ] this approach thus yields a damped atom .
the single - photon transport in a single - mode waveguide , coupled to a cavity embedded with a two - leval atom is analyzed . 
 the single - photon transmission and reflection amplitudes , as well as the cavity and the atom excitation amplitudes , are solved exactly via a real - space approach . 
 it is shown that the dissipation of the cavity and of the atom respectively affects distinctively on the transport properties of the photons , and on the relative phase between the excitation amplitudes of the cavity mode and the atom .