Any quantum system inevitably interacts with its surroundings, which possess a huge amount of uncontrollable degrees of freedom. Such interaction causes the rapid destruction of quantum coherence, which is an essential requirement for quantum information processing to fully exploit the new possibilities opened by quantum mechanics. For example, the information stored in two-level systems (we refer to atoms hereafter) can be destroyed by their surrounding electromagnetic field. Spontaneous emission (SE) is one of the most prominent effects in the interaction of atoms with vacuum. It is an atomic radiation as follow: an atom is initially in an excited state relax to its ground state and emit a quanta of energy to its surrounding vacuum electromagnetic (EM) field, which carries away the difference in energy between the two levels.

SE is not only useless but also harmful to quantum information process. However, recently studies have shown that it is useful to build a device in a quantum network for controlling single photon by a local atom, e.g. the atomic radiation leads to the total reflection of the single-photon propagating in one quantum channel,^[1–2] the frequency converter for single photon,^[3] and the transfer of single photon from one quantum channel to the other.^[4] Nowadays, great interest has been paid on the use of atoms to act as a quantum node in extended communication networks and scalable computational devices.^[15–18] As the SE rate of a single atom can be modified by a succession of short and strong pulses or measure to the quantum system,^[19–22] a dynamical Quantum-Zeno-Effect (QZE) switch for single photon is proposed.^[5] The quantum interference between multiple transition pathways of atomic internal states has been exploited to modify the transport property of the single photon in a quantum channel.^[9] With the well-known result that the atomic SE depends on the electromagnetic vacuum environment that the atom is subjected to, a boundary has been used to increase the efficiency of the quantum router.^[12]

Actually, the SE rate of a single atom is one of the basic topic of quantum electrodynamics, numerous studies of the SE rates^[23] have been carried out for atoms in free space (e.g. Ref. [24]), in a cavity (e.g. Ref. [25]), near a metallic mirror (e.g. Ref. [26]) or a dielectric interfacer (e.g. Ref. [27]), between two mirrors (e.g. Ref. [28]) or two dielectric interfaces (e.g. Ref. [29]). However, photons used to transmit information or distribute the entanglement along the network, are confined in a one-dimensional (1D) waveguide. Similar to cavities, 1D waveguide has a well-defined mode spectrum and a relatively loss-free environment. However, unlike cavities, modes are available in waveguide for photons to propagate. The geometry constraint not only confines the propagating direction of photon, but also gives rise to an increasing of the interference effects. Since photons do not interact with each other, atoms implanted in waveguide are necessary to mediate the photon-photon interaction or redirect the possible propagating directions. The coupling strength of atoms to the waveguide is enhanced by decreasing the mode volume. Consequently, the atomic radiation in 1D waveguide plays an important role in controlling photons in quantum networks. And the studies on the atoms in 1D waveguide are now referred to by the term “waveguide quantum electrodynamics (QED).”

Since the radiative properties of an atom in a confined space differ fundamentally from that in free space, considerable interest has been paid on atoms in a semi-infinite or infinite 1D waveguide. However most works focus on 1D waveguide without a cross section.^[30–34] In this paper, we study the radiative properties of an atom in a semi-infinite waveguide of rectangular cross section, which is a typical 1D QED system. The termination of the waveguide is regarded as a perfect mirror, which reflects emitted photons back to the atom. We analyze the interaction of an initially excited two-level atom with the waveguide in vacuum. The Markovian approximation is first used to analyze the dependence of the SE rate on the density of states and the spatial profile of the waveguide. To find the influence of the time that one-photon wave packet requires to bounce back and forth between the atom and the mirror, we perform the linear expansion of the dispersion relation for the atomic transition frequency far away from the cutoff frequencies, and obtain a delay-differential equation. Then the atomic SE dynamics is studied by varying the cross section of the waveguide as well as the atomic location.

This paper is organized as follows. In Sec. 2, we introduce the system we have studied. In Sec. 3, we derive the relevant equations describing the dynamics of the system in single-excitation subspace. In Sec. 4, we do the Markovian approximation to study the effect of the mode profile on the spontaneous rate. In Sec. 5, the atomic dynamics is studied by linearly expanding the dispersion relation around the transition frequency, which is valid far from the branch threshold, where the delay time is introduced. Finally, we conclude this work in Sec. 6.

The system we studied is shown in Fig. 1. A waveguide made of ideal perfect conducting walls is formed from surfaces at x = 0, x = a, y = 0, y = b, z = 0, and is placed along the z axis. The waveguide is assumed to be infinite along the z direction. The boundary condition restricts photons to travel without loss of power in two independent guiding modes:^{[35, 36]} TE modes whose electric field has no longitudinal component, and TM modes whose magnetic field has no longitudinal component. Let be the wave vector. The relations and with positive integers n, m can be imposed by the condition that the tangential components of the electric field vanish at all the conducting wall, however, there is no constraint on k. Therefore, the waveguide allows a continuous range of frequencies described by the dispersion relation

Fig. 1

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	Fig. 1 (Color online) Schematic illustration for (a) the two-level atom embedded in a semi-infinite waveguide of rectangular cross section, and (b) the dispersion relation of the guiding modes, which interact with the atom at with .

At is an atom with transition frequency between upper level and lower level , which is excited initially. The atom sits inside the hollow waveguide. z₀ is the distance between the atom to the wall (or the mirror) at z = 0. The free Hamiltonian for the atom is described by

In this section we study the dynamics of the system when the atom is initially in the excited state and the field is in the vacuum state . Since the number of quanta is conserved in this system, we can write the wavefunction of the system as:

To see how the mode distribution of the quantum vacuum fluctuation modifies the atomic spontaneous rate, we set on the right-hand side of Eq. (10), and the amplitude of level reads

Fig. 2

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Fig. 2 (Color online) Atomic excitation probability as function of with . (a) The transition frequency . The atom is located at three different positions (black dot-dashed line), (blue solid line), (red dashed line). (b) The transition frequency . The atom is located at . The red dashed line presents the contribution of the TM11 mode. The blue solid line gives the contribution of both TM11 and TM31 modes.

One can also obtain the spontaneous rate in Eq. (18) by replacing with in Eq. (10) and making the up limit of integral tend to infinite. Hence, the Markovian approximation yields the same phenomenon in the context below Eq. (18), which means that retardation effect is neglected.

An excited atom relaxes to its ground state accompanied by a release of a photon to the EM vacuum. In this hollow waveguide, the emitted photon propagates along the positive and negative z directions. Since the termination of the waveguide imposes a hard-wall boundary condition on the field, which behaves as a perfect mirror, the photon traveling along the negative z axis is retroreflected to the atom, and re-excites the atom, which leads to a non-Markovian type dynamics of the system.^[40] In this section, we study the spontaneous emission dynamics involving the retardation time for atom located at with . For the convenience of discussion, we denote the transversally confined propagating modes, which couple to the atom as TM_j with according to the ascending order of the cutoff frequencies. By assuming that the transition frequency is far away from the cutoff frequencies Ω_j, we can expand frequency ω_jk around the transition frequency up to the linear term

5.1 SE Dynamics in Single Mode

In the frequency band between Ω₁₁ and Ω₃₁, the waveguide is said to be single-moded. The atom with the transition frequency only interacts with the TM₁₁ () guiding mode, the delay-differential equation reduces to

where given in Eq. (20). For the case that the retarded argument , the memory effects inherent in the system disappear. The amplitude of state becomes

The SE rate and the frequency shift are presented by the real part and imagine part , respectively. In the limit , the second term of Eq. (25) vanishes. Since the waveguide becomes infinite, the atomic population decays exponentially and the SE rate Γ is independent of the coordinate z₀.

It can be seen from Eq. (25) that the time axis is divided into intervals of length τ₁. We can formally integrate Eq. (25) and change the dummy integration variable, which is then substituted into the integrand. Proceeding indefinitely with iteration, the time behavior of the atomic state populations reads

A step character is presented in Eq. (26). For , the atomic amplitude decays exponentially, which coincides with the behavior of an excited atom in an infinite waveguide. The underlying physics is that the atom requires at least the time τ₁ to recognize the mirror. For , due to the emitted radiation reflected back to the atom, term has been included, which gives rise to the interference for finding the atom in the excited state. In Fig. 3, we have plotted the norm of the atomic amplitude versus with delay (a) and (b). The exponential decay of the atom inside an infinite waveguide is plotted with the black solid line. When , an exponential decay law is found, however, the SE can be either increased or descreased by the phase. The SE is completely suppressed when phase . When , the atom first decays exponentially, after the atom recognizes the mirror, it displays a behavior deviating from the exponential decay law, and a partial revival of the atomic population can be found. It is the interference between the emitted wave and the radiation wave reflected back to the atom, which makes the atom-mirror separation z₀ has significant influence on the atomic dynamic via the phase. When the distance between the atom and the termination are further large (i.e., ), it is found from Fig. 5(a) that there is also a partial revival of the atomic population, however, the atom-mirror separation z₀ does not make any sense. In this case, the atom has already decayed to the ground state at the time that the photon bounces back to the atom, so there is no emitted wave to be interference with the wave reflected back to the atom, which means that the atomic revival is due to the atom being partially re-excited by the radiation. Since the light emitted in the positive direction has departed from the atom, the probability that the atom is re-excited becomes lower and lower.

Fig. 3

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	Fig. 3 (Color online) Amplitude versus with delay (a) and (b) in the following cases: no termination (black solid line), phase (blue dotted line), (red dashed line), (green dash-dotted line). We have set the following parameters: , . Since both φ1 and τ1 are related to the distance z0, different phases are achieved by adjusting the ratio .

5.2 SE Dynamics in Multiple Modes

An excited atom radiates waves into the continua of all modes resonant with the atom. When the cross area becomes larger, more modes are included in the resonance, then the atomic dynamics is not only affected by the time τ₁ that light needs to bounce back and forth between the atom and the termination in the TM₁₁ mode, but also by other time τ_j required for a photon emitted by the atom to propagate in the TM_mn mode and reabsorbed by the atom. The definition of delay time τ_j told us that .

In this section, we assume that the atomic transition frequency is in the regime , which means that only two TM modes (i.e. TM₁₁, TM₃₁) in resonance with the atom, the delay-differential equation reduces to

The space dependence enters via both the phases φ₁, φ₂ and the delay time of the different modes. If both arguments , the amplitude of state becomes

Two terms are consisted of in the above equation, the SE rate and frequency shift . Comparing to the single mode case, the SE rate is enhanced, however the frequency shift can be either increased or decreased due to the space dependence. In the limit , the second and third terms of Eq. (27) vanish. The amplitude shows that the atomic population decays exponentially, is the SE rate that the atom interacts with the continuum of the TM₁₁ and TM₃₁ modes of an infinite waveguide, which is also independent of the coordinate z₀. In the case that , the delay-differential equation reads

Using Laplace transformation and geometric series expansion, the solution reads

If the atom is located at z₀ satisfying , the SE that the TM₁₁ mode contributes to is completely suppressed, then one obtains the SE dynamics due to the emitted photon propagating only via the continuum of the TM₃₁ mode. In the case with finite τ₁ and , the upper state amplitude becomes

However, it is impossible for an atom inside a realistic waveguide to appear the dynamics described by Eq. (31).

In Fig. 4, we have numerically plotted the amplitude as a function of with (a) and 1 (b). It can be seen that in the interval , the upper state population of the atom decays exponential with a rate . After time τ₁, photons emitted by the atom are reflected back to the atom by the mirror so that the atom-mirror distance has great influence on the SE dynamics via phase φ_j and τ_j. Phase φ₁ first gives arise to deviation from the decay with a rate in the interval . As soon as , the wave propagating in the TM₃₁ mode is reflected back to the atom by the mirror, phase φ₂ deviates the atomic dynamics from that of finite τ₁ and . In the weak coupling case (see, Fig. 4(a), the excited state probability decreases as the time increases. However, in the strong coupling case, several peaks can be observed in Fig. 4(b), which present partial revivals of the atom probability when . Comparing to the time evolution in Fig. 3, the SE is enhanced for a given phase φ₁. Phase φ₂ and retardation time τ₂ shift the position of the peak and the dip.

Fig. 4

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	Fig. 4 (Color online) Amplitude versus with delay (a) and (b) in the following cases: no termination (black solid line), phase (blue dotted line), (red dashed line), (green dash-dotted line). We have set the following parameters: , .

Fig. 5

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	Fig. 5 (Color on line) Probability versus with delay with phase (blue dotted line), (red dashed line), (green dash-dotted line) in the following cases: (a) single mode , (b) two modes . Here, .

In Fig. 5, we have numerically plotted the probability as a function of with delay for (a) single-moded and (b) double-moded cases. When the photon returns back to the atom, the atom has already decayed to its ground state, it is impossible for interference to occur so that the phases φ_j have no effect on the atomic dynamics. The peaks at time owe to the reflected light reabsorbed by the atom. By comparing two figures in Fig. 5, we found that the probability that the atom is reexcited by the radiation wave is lower in the multiple-mode case than that in the single-mode case, and more peaks appear in the interval for the multiple-mode case. Such observations are easy to understand because there are more transverse modes to interact with the atom.

We have studed the dynamics of an atom inside a hollow waveguide of rectangular cross section , made of ideal perfect conducting walls. Such 1D waveguide generally consists of both TE and TM waves, the atom with dipole along the z-direction interacts only with the TM_mn transverse modes, their coupling strength depends on the atomic location. A two-level atom with location fixed at is considered, which decouples to fields of the TM modes with even integer m, n. We have first discussed the dependence of the SE rate on the atom-mirror separation and the density of states by Markovian approximation (i) Since the density of state vanishes below Ω₁₁, the SE is completely suppressed when ; (ii) Since the density of state tends to infinite, the excited state population decays very rapidly when , (iii) Away from the cutoff frequencies, the SE rate is increased when more TM modes in resonance with the atom, and the upper state population could be unchanged all the time for single mode case with (i.e., ). However, the distance gives rise to the time delay that radiation emitted by the atom return to the emitter. To study this backaction of the ideal perfect conducting at on the atom, we perform a linear approximation, which is valid for far from the cutoff frequencies, and phase φ_j and retardation time τ_j are introduced into the dynamics of the atom via and z₀. The SE dynamics is studied for both single-moded and double-moded cases. We find that (i) the upper state population is less than its initial as long as . The discussion in Markov approximation corresponds to the case with the retardation time . (ii) The probability for finding the atom in its excited state is lowered when more transverse modes are resonant with the atom. (iii) In the interval , the atomic behavior is the same to that of an excited atom in an infinite waveguide, and the SE rate is independent of z₀. (iv) After , two situations should be distinguished. For short retardation time, the interference between the radiation wave and the emitted wave makes the dynamics is strongly dependent on φ_j. For long retardation time, the atom has already decay to its ground state when the photon returns back to the atom, the partial revivals and collapses are due to the photon reabsorbed and re-emitted by the atom.