Photon Scattering from a System of Multi-Level Quantum Emitters. II. Application to Emitters Coupled to a 1D Waveguide
In a preceding paper we introduced a formalism to study the scattering of low intensity fields from a system of multi-level emitters embedded in a D dielectric medium. Here we show how this photon-scattering relation can be used to analyze the scattering of single photons and weak coherent states from any generic multi-level quantum emitter coupled to a D waveguide. The reduction of the photon-scattering relation to D waveguides provides for the first time a direct solution of the scattering problem involving low intensity fields in the waveguide QED regime. To show how our formalism works, we consider examples of multi-level emitters and evaluate the transmitted and reflected field amplitude. Furthermore, we extend our study to include the dynamical response of the emitters for scattering of a weak coherent photon pulse. As our photon-scattering relation is based on the Heisenberg picture, it is quite useful for problems involving photo-detection in the waveguide architecture. We show this by considering a specific problem of state generation by photo-detection in a multi-level emitter, where our formalism exhibits its full potential. Since the considered emitters are generic, the D results apply to a plethora of physical systems like atoms, ions, quantum dots, superconducting qubits, and nitrogen-vacancy centers coupled to a D waveguide or transmission line.
Efficient light-matter interfaces at the few to single-photon level are crucial for quantum information processing and future quantum technologies Kim08 (); Tey08 (); Brien09 (); Chang07 (); Hwang09 (). Traditionally, such interfaces have been pursued with atoms coupled to a single mode of an optical cavity with a high Q factors, in the regime of cavity quantum electrodynamics (QEDs) Haroche_rmp (). The strong confinement of light in optical cavities, however, also poses a limitation to their integration into quantum networks, which relies on the efficient out-coupling of light Rempe_rmp (). As such, currently a wide variety of physical systems are being studied where one achieves good light-matter interface, which can be integrated in future with opto-electronics Kurt20 (); Brouri20 (); Yuan05 (); Shields07 (); Shen_prl07 (); Houck07 (); Fu08 (); Rebic09 (); Shi09 (); Babi10 (); Liew10 (); Shi11 (); Bamba11 (); Maju12 (); Pey12 (); Loo13 (); Shi13 (); Baur14 (); Tiecke14 (); Giesz16 (). Among these, waveguides coupled to quantum emitters have turned out to be a viable alternative Peter_rmp ().
The study of photon scattering in waveguides traditionally considers an emitter either coupled to a continuous set of freely propagating waveguide modes or coupled to a discrete set of modes via an optical cavity. A key question in such system is then, how to efficiently evaluate the photon reflection and transmission amplitudes, which are due to the medium’s response corresponding to different pathways of scattering. In the past decades several approaches have been introduced to solve this problem. For example, one of the early approaches uses the Lippmann-Schwinger formalism in a Schrödinger picture to evaluate the reflected and transmitted field amplitudes Shen07 (); Yud08 (); With10 (); Zheng10 (); Liao10 (). This formulation, even though exact, cannot be applied for propagating photons interacting with separated multi-level emitters. Alternatively, some studies have used the transfer matrix method which is particularly useful in the weak excitation regime, where the emitters can be considered to be linear scatterers Deutsch95 (); Chang12 ().
To solve the problem of photon scattering from nonlinear emitters, an input-output formalism was developed although only for a two-level emitter coupled to a D waveguide Fan10 (). An analogous approach was later introduced for superconducting qubits coupled to a D transmission line Lalu13 (). There are several other frameworks to solve the scattering problem for nonlinear emitters coupled to D waveguides Shi09 (); Shi11 (); Zheng11 (); Roy11 (); Laakso14 (). Recently the formalism of Ref. Fan10 () was generalized to multi-level emitters coupled to a D waveguide Can15 (). Furthermore, in a related work a path integral formalism-based scattering matrix was developed to study few-photon scattering dynamics in the non-Markovian regime Shi15 (). Typically, all these approaches reduces to setting up the problem by either linearization, or by restricting the system to two-level emitters and a D waveguide and then numerically solving it. Even then, the solution of the full photon-scattering problem from multi-level emitters in the paradigm of waveguide QED, remains quite tedious even for a single photon.
In a preceding paper we developed a general photon-scattering relation from a system of multi-level quantum emitters embedded in a -dimensional dielectric medium Das17 (). The theoretical framework for this problem involved a set of excited and ground-state subspaces and respectively. Each of these subspaces are spanned by the manifold of the excited () and ground () states of the emitters. The theory is applicable to incident fields with a sufficiently low intensity, e.g., single-photon or weak coherent states, so that saturation effects can be ignored. In this limit, the coupling between the two subspaces can be treated perturbatively. We showed that our theory provides a solution for the amplitudes of the scattered fields, in terms of the input-photon amplitude and the dynamical response of the emitters. As a continuation of Ref. Das17 (), in this paper we apply the formalism to the particular case of -dimensional waveguides and show how it can be used to solve a variety of scattering problems. Following Ref. Das17 (), we derive a photon-scattering relation for a system of multi-level emitters coupled to a D waveguide in the form
Here and are the input and the output field-mode operators in the waveguide, is an operator in the Heisenberg picture giving the dynamics within the ground-state manifold of the emitters, while the superscripts signify the directionality (forward, backward propagation) of photons in the waveguide. The kernel is the scattering amplitude which can be evaluated once the coupling of the emitters has been determined.
In the following section we give a detailed derivation of Eq. (1) and discuss how to evaluate the ground-state dynamics in terms of the operator . Furthermore, it will also be apparent that Eq. (1) has the following salient features it provides a direct solution of the scattering problem assuming Markovian dynamics for weak input fields, it can include any kind of dipole emitters coupled to the D mode of a waveguide and it uses effective operators (EOs) to give a full solution of the emitter dynamics keeping track of all the phases and scattering component. The introduction of the EOs basically amounts to adiabatic elimination of the excited states and describing the system dynamics solely in terms of the ground-states evolution Reiter12 (). Thus, by using EOs, the complications arising from multiple emitters in the scattering problem, can be reduced to solving the dynamics for the ground-state coherences and populations.
The article is organized as follows: In Sec. II we give the detailed derivation of Eq. (1) starting from the photons scattering relation developed for a general dielectric medium in Ref. Das17 (). In Sec. III we then elaborate on the physical processes that contribute to the non-Hermitian Hamiltonian, which is the key quantity for determining the scattering relation, and explain what the different terms in this Hamiltonian correspond to. Readers primarily interested in the application of the photon scattering formalism are encouraged to visit Sec. IV directly to avoid the technical details laid out in Secs. II and III. In Sec. IV we elaborate on our results by solving different examples of photon scattering from a single emitter coupled to a one-dimensional waveguide. We start with a simple example of a two-level emitter in Sec. IV.A and continue with a more complicated example of an emitter in a V-level configuration in Sec. IV.B. In Sec. IV.C we then consider several different cases of photon scattering from a system of multiple emitters coupled to a one-dimensional waveguide. In Sec. V we then give an example that demonstrates the versality of our formalism. We consider scattering from an emitter with multiple ground-states and study several aspects including the formation of ground-state superpositions conditioned on photon scattering. Finally, in Sec. VI we summarize our results and give an outlook. Several details of our calculations are relegated to the appendices. In Appendix A we provide the derivation of the photon-scattering relation for the D waveguide. In Appendix B we present the derivation of the decay rate into the D mode of the waveguide. In Appendix C we give details of the effective detunings and decays for the two-emitter systems.
Ii photon-scattering relation for emitters coupled to a one-dimensional waveguide
In this section we derive the photon-scattering relation for a system of multi-level emitters coupled to a double-sided D waveguide. To achieve this we first invoke the general photon-scattering relation in a dielectric medium
that was derived in Ref. Das17 (). Here is the point of observation, while corresponds to the spatial positions of emitter and , respectively. The dipole moments and correspond to the transition and for the emitters and . The Green’s function, gives the response of the field at the characteristic frequency of the dielectric medium containing the emitters. Here is the central frequency of the input field and is the difference in frequency between states in the ground-state subspace. The input field in the above equation is defined as , where is the mode function while is the mode operator for the mode of the field. The second term in Eq. (2) represents the whole scattering event. It gives the scattered field including the dynamical response of the emitters. It is formulated in terms of the operator and the non-Hermitian Hamiltonian , which describes the dynamics in the excited-state subspace . The non-Hermitian Hamiltonian is well known in the theory of Montecarlo wave-functions . In Sec. III. we will describe in detail the meaning of this for our model. The states and belong to the ground-state manifold of the emitters as shown in Fig. (1). Note that our definition of the operator can be considered unconventional since the order is reversed. As we will see later, this definition gives us a simple relation to the density matrix and simplifies the notation below.
To proceed we first rewrite Eq. (1) in a more convenient form. We expand in terms of the Green’s function
in Eq. (2) and writing the frequency-dependent as the Fourier transform of the time-dependent Green’s function we get,
Here is the space-dependent electric permittivity of the waveguide. The first term on the right hand side of Eq. (4) represents the freely propagating field with the Green’s function being simply a propagator.
We want to derive the photon-scattering relation for a double-sided D waveguide. As such, we assume that the waveguide modes allow for the scattered photons to travel both in the forward and backward directions with wave-numbers and , respectively. Furthermore, to account for the scattering into the waveguide and to the outside we divide into a waveguide and a radiative part. To treat this formally, we decompose the electric field in the form with , such that
represent the field in the forward and backward propagating modes of the waveguide. Here are the modes representing the field in the waveguide, z is the co-ordinate along the waveguide, while are the radiative modes representing the scattered light to the outside.
we arrive finally (see Appendix A for details) at the photon-scattering relation in the D waveguide
Here is group velocity of the photon in the waveguide, while is a noise operator that corresponds to the and and is associated with the loss of photons out of the waveguide. The mode operators and correspond to the output and input light field, respectively. Note that the sign stands for photons travelling in the forward (backward) direction. The scattering amplitude is defined as
where we have defined the directional coupling of the emitters to the waveguide mode as
with . The wave vectors in the forward and backward direction follow the relation for and , respectively. The photon-scattering relation in Eq. (7) is the key result of this work and has the generic form stated in Eq. (1). Note that, the coupling defined in Eq. (9) has a directional dependence and in principle its strength can be different for the field-mode propagating along two different directions (forward or backward) in the waveguide. This leads to an interesting and emerging question of chiral light-matter interaction Peter17 (). Even though we do not explicitly address this, our general formalism is already equipped with such possibilities. As such the photon-scattering relation in Eq. (7) is applicable even to the study of chiral interactions in waveguides.
It is worth emphasizing that in the derived photon-scattering relation all the system properties are included through the non-Hermitian Hamiltonian while the evolution of the emitters, response is through the operator defined in the ground-state manifold . To get the complete photon-scattering dynamics using the photon-scattering relation introduced above we need to find . This can be quite cumbersome for complex systems involving multiple levels. However, by exploiting the formulation of EOs Reiter12 (), which again involves the inverse of the non-Hermitian Hamiltonian , we can solve for using the master equation derived explicitly in the preceding paper Das17 ()
Here all the operators are defined in the Heisenberg picture and the subscript “eff” symbolizes EO’s. The symbol in Eq. (10) stands for normal ordering, the significance of which will be discussed in details in section VI.C. Note that, Eq. (10) is a Heisenberg-picture generalization of the result of Ref. Reiter12 () to quantum fields. Solving the above master equation for a given system is a straightforward algebraic/numerical exercise whose complexity simply depends on the size of the Hilbert space of the emitters. Later in section IV.C we consider an example where the emitters have multiple ground-states and show how one can use the master equation in Eq. (10) to solve for the dynamics of the emitter’s ground-state.
It is important to point out that for the examples we discuss in Sec. IV, the noise term in Eq. (7) is typically neglected. This is justified by the fact that in those examples we are only interested in the click probability where the vacuum noise does not contribute to any photodetector clicks. However, we would like to remind the readers that in general particular care should be taken for Heisenberg equations as the noise can play a crucial role in the system dynamics. We account for this in our formalism through the effective Lindblad operators in the master equation, which includes the noise contribution. Hence for problems where the scattering is influenced by the coherence dynamics of the ground-states, the crucial effect of noise is taken care of in the master equation. We show this in detail in the example in Sec. VI.C. Thus we discuss explicitly how to deal with the noise and treat it via the effective-operator master equation.
Iii The non-Hermitian Hamiltonian
Note that this non-Hermitian Hamiltonian includes all possible interactions that the emitters can have within the excited-state manifold. In the following we discuss each of the terms in Eq. (III). The first term is the Hamiltonian of the system defined in the single excitation manifold as shown in Fig. 1. Note that this term is completely general and can in principle also include effects like the long-range Rydberg interactions among emitters. The second and third term and arise from the dynamics induced by the quantized field and are related to the decay from the manifold to , and shifts of the states in the manifold due to light induced coupling between the emitters. They are defined as
where the excited , and ground states belong to the excited and ground subspaces and , respectively. Note that to write Eq. (12) and Eq. (III) we have used the general form of these expression derived in Ref. Das17 ().
We rewrite in Eq. (14) in the form . Here corresponds to the first term on the right-hand side of Eq. (14) and represents decay-induced coupling between the emitters mediated by the D waveguide mode. represents the second term and arises due to collective decay to the non-waveguide modes (decay to the outside of the waveguide). For , corresponds to spontaneous decay of the emitter into the D waveguide mode while gives spontaneous decay of the emitter to the outside of the waveguide. Similarly, Eq. (III) for can be defined as , where represent the first term on the right-hand side of Eq. (III) and stands for waveguide-mediated coupling of the emitters while represents the second term and corresponds to coupling via other processes like dipole-dipole interactions. For , the coupling gives a contribution to the Lamb shift of the excited state of a single emitter. Note that in Ref. Das17 () these terms were derived within the rotating wave approximation, which does not produce the correct form of the dipole-dipole interaction for emitters separated by less than a wavelength. Care should therefore be taken to use the correct shifts beyond the rotating wave approximation for nearby emitters.
In the following we derive an exact expression for the waveguide-mediated coupling between the emitters, by solving for the first terms on the right-hand side of Eq. (14) and Eq. (III). For this purpose we invoke the relation Novobook06 ()
and do an inverse Fourier transform of it to get
Furthermore, substituting Eq. (18) into the first term on the right-hand side of Eq. (III) and then performing the principal value integral over an anticlockwise contour and invoking Cauchy’s residue theorem (see Appendix B for details) gives us
If we refer to the expression for the non-Hermitian Hamiltonian in Eq. (III) and consider the contribution to the second and the third term due to the waveguide-mediated interactions, we find, using Eq. (III) and Eq. (III), that Chang12 (); Can15 (); Hak05 ()
Note that for the case of a single two-level emitter, and . Eq. (21) becomes
where is the total decay of energy level into the D mode of the waveguide for the emitter transition .
We can now rewrite the non-Hermitian Hamiltonian of Eq. (III) as a combination of two parts, one comprising of all the interactions mediated by the waveguide (w) while the other one concerning all other processes not mediated by the waveguide (nw). The non-Hermitian Hamiltonian then takes the form , where
Here , with being the energy of the ground-state involved in the excitation process while is the frequency of the incoming photon. The waveguide-mediated off-diagonal term in Eq. (23) can also be re-written in terms of as,
where we have used and the definition of directional decay into the waveguide in terms of the coupling constants from Eq. (22).
On using the general form of and Eq. (25) we find that the non-Hermitian Hamiltonian has a simple diagonal part spanned by the excited states of the emitters as
where and is the natural line width of an excited state in the single-excitation manifold . Here is the total decay rate to the outside of the waveguide and is a redefined excited-state Hamiltonian formed by absorbing the Lamb-shift contribution in . Note that Eq. (26) can also be written in the standard form of a non-Hermitian Hamiltonian
where the Lindblad operators model decay of an excited emitter both into and outside of the waveguide.
We next discuss the contribution to the non-Hermitian Hamiltonian from the non-waveguide part in Eq. (III). These terms can have contributions both for inter- and intra-emitter couplings. In the Dicke superradiant limit, where the separation between the emitters is less than a wavelength, the gives rise to collective decay and dipole-dipole couplings. For most of this article we will ignore the part of the non-Hermitian Hamiltonian. However, we do use this in two particular examples to illustrate the wide range of applicability of our formalism.
Iv Application of the formalism to emitters with a single ground-state
In the previous sections we have introduced a formalism for photon scattering from quantum emitters in a D waveguide, and elaborated on the non-Hermitian Hamiltonian that is central to the response of the emitters interacting with the incoming field. In the following sub-sections IV.A - IV.C we focus on, a number of paradigmatic physical situations that demonstrates the effectiveness of our formalism for solving photon scattering problems in waveguides. In this section we restrict ourselves to examples where the emitters have a single ground-state. In the next section we consider in detail an example of emitters with multiple ground-states. It is worth emphasizing that even the simple and generic examples of scattering that we treat here are in some cases rather tedious to solve with the existing methods. However, using our formalism we can immediately provide the solution to these problems. Note that for notational convenience, in all further discussion we will label the photons incoming from the left and moving to the right with subscript (R) and the photons moving to the left as (L), such that now .
iv.1 A two-level emitter coupled to a one-dimensional waveguide
We first analyze the simplest possible system. We consider an emitter comprising two levels with a single optical transition between a ground level and an excited level as shown schematically in Fig. 2 (a). The emitter is located at a position along the axis of a D waveguide. The transition is coherently coupled to a waveguide. Such a system is generally described by a Hamiltonian , where
with the free-energy Hamiltonian , and the Hamiltonian of the field being given by , while the excitation (de-excitation) is represented by (. Here, and are the energies of levels and , respectively. Furthermore, as above we have used the definition of the atomic operator such that the density matrix is given by . The coupling strength of the emitter transition to the field is given by , with being the corresponding annihilation (creation) field-mode operator and . Here, signifies that is the coupling strength of the transition to modes outside the waveguide, while represents the directional coupling to the D waveguide mode with strength . For the rest of this example we drop the subscripts from the coupling constants as it involves only a single transition. We can then write the non-Hermitian Hamiltonian for this system in the form , where the Lindblad operators are given by
corresponding to decay out of and into the waveguide . Note that in writing Eq. (30) we have used the definition of from Eq. (22), and defined the rate of decay out of the waveguide as . The non-Hermitian Hamiltonian can then be written similar to that in Eq. (26) as
where is the total decay rate of the level into and is given by , while the detuning is . Here is the frequency of the incoming field. Combining the decay with the detuning we then define as the complex energy of the state . Inverting the is then straightforward and we find
For a single photon incident from left and propagating towards the right in the waveguide, Eq. (7) straightway gives the complete scattering dynamics of the photon from the two-level emitter. Let us write Eq. (7) in terms of the field-mode operators on the left and right of the emitter, after scattering of a photon as
where we have used that and is the point of observation to the right (left) of the emitter spatially situated at . Here is an additional phase that the reflected photon picks up as it propagates towards the left of the emitter. Note that in writing Eq. (34) and Eq. (35) we have neglected the noise term as we are mainly concerned with the photon click probability at a detector.
Substituting for and assuming that , we get
where we have used that for a emitter initially in the ground-state. We can do this because, once we eliminate the excited state the emitter can only be in the ground-state. For an emitter tuned into resonance we get the well-known results of photon scattering in waveguides, with transmission and reflection amplitudes of and , respectively Shen05 (), where . This is illustrated in Fig. 3 (a) where we plot the transmitted intensity which shows a Lorentzian dip at resonance. The corresponding FWHM is found to be . Thus, for a waveguide with strong coupling to the emitter such that , scattering leads to complete reflection of the photon with the atom behaving as a mirror Shen07 (); Shen05 (); Zhou08 ().
iv.2 A three-level emitter in V-configuration coupled to a one-dimensional waveguide
Above we considered the simplest possible situation which could also easily be solved by other means. We now consider a situation, where the result is less obvious. We choose an emitter in a V-configuration comprising a ground-state and two excited states and located at some point in the waveguide (see Fig. 2 (b) for the schematic level structure). It is worth emphasizing that single photon scattering from such three-level emitters have been studied extensively in the past With10 (). The purpose of addressing this problem here is to illustrate how the results of these previous works can be obtained directly with our method. To demonstrate the versatility of our approach, we assume that the exited states are coherently coupled by a (generally complex-valued) coupling . This then corresponds to a nonzero contribution to the non-Hermitian Hamiltonian . Furthermore, we assume that the transitions from to and to are coupled to the waveguide mode with strengths and and decay with a total decay rate of and respectively. The Hamiltonian of the system is then given by