Control of the radiative level shift and linewidth of a superconducting artificial atom through a variable boundary condition

We investigate the dynamics of a superconducting qubit strongly coupled to a semi-infinite one-dimensional microwave field having a variable boundary condition. The radiative level shift and linewidth of the qubit are controllable through the boundary condition of the field, and the spectral properties of the output microwave are modified accordingly. The current scheme provides a compact method for controlling the radiative characteristics of quantum few-level systems that is useful in single-photon engineering.


Introduction
In the early days of quantum mechanics, the transition frequencies and the spontaneous emission rates of atoms were regarded as inherent properties of individual atoms determined by their valence electrons. However, these quantities are sensitive to the dielectric environment such as the spatial profile of the field modes and their density of states, since the optical transitions occur as a result of interaction between the electrons and the light field. When an atom is in a free space, its transition frequencies are shifted slightly through the vacuum fluctuation of the field, which is known as the radiative (Lamb) shift [1,2]. On the other hand, when an atom is placed in a cavity, its radiative decay can be enhanced or suppressed drastically, which is known as the Purcell effect [3][4][5][6][7]. The changes in the transition frequency and the radiative decay rate are inseparable in general, since they are determined by the real and imaginary parts of the complex frequency of the atom renormalized by interaction with the light field.
In recent years, atoms (including artificial ones) coupled to one-dimensional (1D) photon fields have been realized in a variety of physical systems: trapped atoms [8][9][10][11][12][13] and molecules [14], quantum dots [15,16], color centers [17,18] and superconducting qubits [19][20][21][22][23]. They are classified into two types by their spatial configuration: the directcoupling type, in which the atom interacts directly with the 1D field [8-11, 14, 15, 17-19], and the cavity quantum electrodynamics (QED) type, in which the atom is coupled with a cavity 3 mode and then to the propagating field indirectly. The cavity QED systems are further classified by their internal dynamics into the weak-coupling regime [12,13,16] and the strong-coupling regime [20][21][22][23]. The former (latter) regime is characterized by a shorter (longer) lifetime of a cavity photon than the period of the vacuum Rabi oscillation. The weak-coupling cavity QED systems exhibit essentially the same dynamics as those of the direct-coupling systems, since the degrees of freedom of cavity photons may be eliminated adiabatically [12]. The charm of the strong-coupling cavity QED systems lies in the coherent quantum phenomena played by the atoms and cavity photons, which can be applied to various areas of quantum-state engineering. On the other hand, the charm of the direct-coupling systems lies in their peculiar optical response that originates from inevitable interference between the incident light and the radiation from the atom due to one dimensionality of the field. Recent studies revealed that (i) incident light can be reflected perfectly by a single atom [19,24], (ii) the nonlinear interaction between single photons can be enhanced drastically [25] and (iii) the Raman-type transitions may become nearly deterministic [26][27][28].
In this study, we investigate a circuit QED system in which a superconducting qubit is coupled directly to a semi-infinite microwave transmission line and analyze the microwave response of the qubit to stationary pumping. A similar situation was realized by a single ion trapped near a mirror, and the interference of radiation from the ion and its mirror image was confirmed [8,29]. Compared to this, the current situation has the following three merits in the light of quantum-state engineering. (i) The spontaneous emission into the transmission line overwhelms that into other spatial modes 6 . Therefore, the excitation in the qubit is transformed into a microwave photon nearly perfectly. In other words, the radiation mode of the qubit is matched with the incident wave. (ii) The qubit is motionless and its position can be controlled precisely (typically, 10 −4 of the wavelength of relevant photons). (iii) The radiative decay rate overwhelms the pure-dephasing rate. Therefore, the radiative decay of the qubit occurs cleanly. In this study, we treat three kinds of dissipation of the qubit (radiative decay, pure dephasing, non-radiative decay) in a non-perturbative manner and observe their interplay.
The physical setup considered here is schematically illustrated in figure 1. The microwave transmission line is terminated by a superconducting quantum interference device (SQUID). This enables us to change the boundary condition at the terminal through the magnetic flux threading the SQUID ring [30][31][32] and therefore to vary the effective position of the qubit continuously. We theoretically show that the radiative level shift and linewidth of the qubit can be controlled through the boundary condition, and that they are reflected in the properties of the output radiation. This scheme will offer an experimentally compact method for controlling the radiative properties of the qubit and will be useful in future single-photon engineering.
The rest of this paper is organized as follows. The theoretical model is presented in section 2, and the Heisenberg equations are derived in section 3. The microwave response of the qubit is discussed in section 4. The properties of the output microwave radiation, such as the coherent amplitude and the power spectrum, are respectively discussed in sections 5 and 6. The boundary condition of the microwave transmission line is derived for the self-containedness of the paper in appendix A, and the details of the calculation are presented in appendices B and C.  Shape of the microwave eigenmode at the qubit frequency: mod(ω b l + θ b , π )/π = 0 and ±1/2.

The Hamiltonian
The experimental setup considered is illustrated schematically in figure 1. The system is composed of a semi-infinite microwave transmission line terminated by a SQUID ring and a superconducting two-level system (qubit). The transmission line extends in the r > 0 region and is terminated at r = 0 by the ring, and the qubit is coupled to the line at r = l. The eigenmodes of the transmission line in the absence of the qubit are given by where k is the wave number and θ b is the phase shift determined by the magnetic flux threading the ring (see appendix A). We may set −π/2 < θ b < π/2 without loss of generality, since θ b + nπ with integer n yields the same mode function as equation (1). Although θ b is dependent on k rigorously, we can neglect this dependence in the narrow frequency range near the resonance of the qubit. Furthermore, to account for pure dephasing and non-radiative decay induced by environmental degrees of freedom, the qubit is coupled to two independent bosonic reservoirs, both of which are assumed to have the same dispersion relation as the microwave photon for simplicity. Settingh = v = 1, where v is the microwave velocity, the Hamiltonian of the overall system is given by where σ is the lowering operator of the qubit, b k is the annihilation operator of the microwave photon with wave number k and c k and d k are the annihilation operators of the reservoir excitations: c k accounts for pure dephasing and d k for non-radiative decay. The qubit-photon coupling g k and the qubit-reservoir couplings h k and i k are given by The meanings of the parameters are as follows. ω b is the transition frequency of the qubit, and r , p and n , respectively, denote the rates for radiative decay, pure dephasing and non-radiative decay.
For later convenience, we introduce the field operator in the real-space representation. Dividing the eigenmode function of equation (1) into incoming and outgoing components, the input and output field operators are given by b in,r = (2π) −1/2 ∞ 0 dk e −i(kr +θ b ) b k and b out,r = (2π) −1/2 ∞ 0 dk e i(kr +θ b ) b k , respectively. Since only high-energy excitations (k ∼ ω b ) are relevant, we can formally introduce the negative-energy excitations and extend the lower limit of k to −∞. Then we have b out,r = e iθ b b r , where b r is the Fourier transform of b k , as given by Note that the index r in b r runs over −∞ < r < ∞, and the r < 0 (r > 0) region represents the incoming (outgoing) field. We can formally define c r and d r in the same manner. Several remarks should be made regarding this model. (i) Regarding the b and d fields, high-energy excitations (k ∼ ω b ) are relevant in the dynamics. Although negative-energy excitations are introduced in the Hamiltonian, they are highly out of resonance with the qubit and do not affect its microwave response [33]. (ii) In contrast, regarding the c field, lowenergy excitations (k ∼ 0) are relevant, and the generation of negative-energy excitations implies absorption of energy from the environment. h k would depend not only on the bare interaction strength and the density of states but also on the Boltzmann factor. One may approximately treat h k as a constant when the energy of relevant excitations is much smaller than the background temperature. This condition is satisfied in circuit QED experiments: since the energy exchange with the c field occurs as a result of inelastic scattering, the relevant energy is determined by the 6 detuning between the pump photon and the qubit, which is usually of the order of r ∼ 0.5 mK. In contrast, the background temperature is typically at 50 mK. (iii) When one realizes the setup of figure 1 by using real atoms, the radiative decay into the lateral direction becomes significant [8]. This damping plays the same role as the non-radiative decay employed here. Therefore, the present model can cover the optical cases by interpreting the non-radiative decay as the radiative decay into the lateral direction.

Initial state vector
In this study, we investigate the microwave response of the qubit to stationary pumping. At the initial moment (t = 0), we assume that the qubit is in the ground state, no excitation exists in both reservoirs and the input microwave has not arrived at the qubit. The initial state vector is then written as where |0 is the overall vacuum state and N = exp(− dr |E in (r )| 2 /2) is a normalization constant. E in (r ) represents the input classical microwave at the initial moment, as given by where E and ω p are the amplitude and frequency of the pump, respectively. Note that |ψ i is in a coherent state and is therefore an eigenstate of the initial field operators, satisfying b r (0)|ψ i = e iθ b E in (−r )|ψ i and c r (0)|ψ i = d r (0)|ψ i = 0. Throughout this paper, we denote ψ i |A(t)|ψ i by A(t) , where A may be any operator.

Heisenberg equations
From the Hamiltonian of equation (2), we can rigorously derive the following equations (see appendix B for derivation): The microwave photons formally propagate only in the positive direction and interact with the qubit at both r = −l and +l. This makes the equation of motion of the qubit, equations (14) and (15), non-local in time [29]. However, the position l of the qubit is of the order of the microwave wavelength in the present situation and therefore the dynamics of the qubit is negligible during the short time interval of the order of l/v, except for the natural phase factor. (Typically, l ∼ 1 cm, v ∼ c/3, and therefore l/v ∼ 10 −10 s. In contrast, the time scale of the qubit dynamics is determined by −1 r ∼ 10 −8 s [19].) We may then safely replace σ (t + t) with σ (t) e −iω b t . Furthermore, since we consider the narrow frequency range near resonance, we can also replace b ±l−t (0) with e ±iω b l b −t (0). Then, equations (11), (14), (15) and (17) are rewritten as follows: where α = sgn[cos(ω b l + θ b )], ω b and r are the transition frequency and the radiative linewidth of the qubit, which are renormalized by coupling to the semi-infinite field as and 1 and 2 are the overall rates for energy relaxation and dephasing, as given by The commutation relations between the noise and qubit operators are summarized in appendix B. Here, the radiative level shift originates from the coupling to continuous photonic modes as found by Lamb and Retherford [1], in contrast with the shifts originating from the coupling to a discrete cavity mode [23]. Equations (24) and (25) indicate that both the radiative shift and linewidth of the qubit are controllable through the boundary condition. We will observe in the following how these renormalized values are reflected in the microwave response of the qubit.

Basic equations
In this section, we observe the response of the qubit to a continuous pump field of equation (10). The equations of motion for s 1 (t) ≡ σ (t) e i(ω p t−θ b ) and s 2 (t) ≡ σ † (t)σ (t) are given, from equations (9), (21) and (22), by d dt where δω = ω p − ω b is the detuning between the pump and the renormalized qubit frequency and = 2 1/2 r E is the Rabi frequency. The response of the qubit can be pursued by solving equation (28) with the initial condition of s 1 (0) = s 2 (0) = 0. The transient behavior before reaching the stationary state is determined by the 3×3 matrix appearing in equation (28). When the pump field is resonant (δω = 0), the three eigenvalues of the matrix are given by Therefore, if the pump field is strong enough to satisfy 1,2 , the upper-state population exhibits oscillatory behavior known as the Rabi oscillation. The qubit reaches its stationary state for t  (28) as

Numerical results
Transient behavior of s 2 under resonant pumping (ω p = ω b ) is shown in figure 2 for several different boundary conditions. The Rabi frequency is given by = 2 1/2 r E, whereas the damping rate is given by 1,2 ∼ r , both of which are sensitive to the boundary condition through the renormalization of r . The Rabi oscillations can be observed clearly when the Rabi frequency is much larger than the damping rate, namely, E 2 / r cos 2 (ω b l + θ b ). This tendency can be confirmed in figure 2.
In order to observe the effects of dissipation, the stationary populations 2 of the excited state is plotted as a function of θ b in figure 3, for several different dissipation rates, p and n . It is observed that both dissipative mechanisms decrease the excited state population as expected. Their difference appears in the limit of mod(ω b l + θ b ) → ±0.5π , where the qubit lies at the node of the corresponding eigenmode and therefore cannot be seen by the microwave.s 2 is nonvanishing in this limit in the ideal cases of n = 0. However, in reality,s 2 vanishes in this limit due to finite n (see the dashed line in figure 3). . The pump is tuned to the renormalized resonance (ω p = ω b ) and its intensity is fixed (E 2 = r ). Althoughs 2 is non-vanishing at mod(ω b l + θ b , π )/π = 0.5 in the ideal cases of n = 0,s 2 vanishes in reality due to finite n (( p , n )/ r = (0.2, 0.01), dashed line).

Amplitude of the output field
In the following sections, we investigate the properties of the output microwave. In this section, we discuss the coherent amplitude of the output wave, E out (r, t) = b out,r (t) , which can be measured by homodyne-type measurements. From equations (20) and (30), E out is given, in the  → 0) and in the dissipation-free limit ( p = n = 0). The origin of frequency is set at the bare resonance frequency, ω b . The boundary condition is set at mod(ω b l + θ b , π) = 0 (solid), ±0.25π (dotted) and ±0.5π (dashed). Note that the natural phase factor e 2iθ b is removed in (a) and that all lines are overlapping in (b). stationary state, by where E in (−r, t) = E e iω p (r −t) . This expression for E out holds for any input intensity.

Linear response
The effects of the boundary condition and dissipations are most clearly reflected in the linear response of the qubit. By taking the weak-field limit ( → 0) in equation (32), the linear response is given by We first discuss the dissipation-free limit of p = n = 0. The phase shift φ = arg(E out /E in ) and the amplitude |E out | of the output wave are plotted in figure 4 for several different boundary conditions. (In order to observe the boundary effects clearly, the natural phase factor e 2iθ b is removed.) We can confirm that |E out | = |E in | from equation (33). This implies that the input wave is reflected coherently without attenuation. The input wave is subject to a linear phase shift of φ(ω p ), which changes steeply around ω b within a narrow frequency width of r . Therefore, φ(ω p ) is sensitive to the boundary condition as observed in figure 4(a). When mod(ω b l + θ b , π ) = ±0.5π and therefore r = 0, the qubit lies at the node of the corresponding eigenmode. Therefore, the input wave cannot see the qubit and is reflected as it is, E out = E in .
Next, we discuss the effects of dissipation. Equation (33) implies that, regarding the linear response, the roles of the two dissipative mechanisms, p and n , are equivalent. The dissipative version of figure 4 is shown in figure 5. A clear difference from the dissipation-free case is the  Figure 5. The same plots as figure 4 in the dissipative case ( p + n /2 = 0.15 r ). mod(ω b l + θ b , π ) = 0 (solid), ±0.373π (dotted) and ±0.5π (dashed). r = 2 p + n is satisfied for dotted lines.
appearance of a dip in |E out (ω p )|, which is located at ω b and has a width of r . In particular, when r = 2 p + n is satisfied (dotted lines in figure 5), the coherent amplitude vanishes in the output port under a resonant pump. In this case, the fate of input photons depends strongly on the nature of dissipation (see section 6.2 for detailed discussion): when pure dephasing is dominant ( p n ), all input photons are reflected into the output port after inelastic scattering. In contrast, when non-radiative decay is dominant ( n p ), all input photons are dissipated into environments. This is known as the critical coupling [34].

Coherent and incoherent components
The power spectrum S(ω) of the output radiation is defined by where r (> l) represents the detector position. Since the correlation function b † r (t 1 ) b r (t 2 ) depends only on t 1 − t 2 when stationary, S(ω) is recast into the following form: where t should satisfy t r . S(ω) can be calculated analytically using the Heisenberg equations (see appendix C for details). It is composed of the coherent and incoherent parts as S i (ω) = ( r /π )ReI 3 (ω).

Fluxes of three components
The input photons have the following three possibilities after interacting with the qubit, as shown in figure 1: (i) coherent reflection into the output port, (ii) incoherent reflection into the output port and (iii) dissipation into the environment. The photon fluxes for (i) and (ii) are evaluated by F c = dωS c (ω) and F i = dωS i (ω). From equations (36) and (C.10), we have The flux for (iii) is defined by F d = dωS d (ω), where S d (ω) is obtained by replacing b r with d r in equation (34). It is given by From equations (38)-(40), we have This represents the flux conservation law, since the input flux is E 2 .
In the linear-response regime (E 2 → 0), F i and F d are reduced to F i = 2 p 2 r 1 | 2 +iδω| 2 E 2 and F d = 2 n 2 r 2 1 | 2 +iδω| 2 E 2 . Figure 6 shows the dependences of F c , F i and F d on the boundary condition, assuming that the pump is always tuned to the renormalized resonance (ω p = ω b ). If dissipations are small enough to satisfy n + 2 p < 2 r , one may set the boundary condition to satisfy r = n + 2 p . Then, the coherent component vanishes in the output, whereas F i = p p + n E 2 and F d = n p + n E 2 . Therefore, the fate of input photons becomes sensitive to the type of dissipation. When pure dephasing is negligible ( n p ) as in optical experiments, all input photons are dissipated into the environment. This is known as critical coupling [34]. In contrast, when pure dephasing is dominant ( p n ) as in circuit QED experiments, all input photons are reflected back to the output port after inelastic scattering. When mod(ω b l + θ b , π) = ±0.5π and therefore r = 0, the qubit is located at the node of the corresponding eigenmode. Therefore, the pump field cannot see the qubit and is reflected as it is.  Since equations (38)-(40) are derived non-perturbatively and are valid for any input intensity, we can readily observe the strong field effects. Figure 7 shows the dependences of F c , F i and F d on the input power E 2 , fixing the boundary condition. It is worth noting that, if n + 2 p < r is satisfied, the coherent component may vanish at a certain input intensity, E 2 = 1 ( r − 2 )/4 r . In the limit of strong pumping (E 2 → ∞), the qubit becomes saturated (s 2 → 1/2) and does not interact with the field. Therefore, the pump field is reflected as it is in this limit.

Spectral shapes of S i (ω)
In the previous subsection, we clarified how the ratios of coherent and incoherent components are determined. Here we investigate the spectral shape of the incoherent component. In the dissipation-free limit ( p = n = 0), the spectral shape is identical to that of a two-level atom in free space, having the renormalized parameters ω b and r . Figure 8 shows how the pump intensity E 2 affects the spectral shape. Under a weak pump ( r ), S i (ω) has a single peak at the pump frequency ω p . In contrast, under a strong pump ( r ), two sidebands appear symmetrically at ω p ± . This is known as the Mollow triplet [19,35,36]. As the intensity is increased, the two sidebands shift keeping their linewidths. Figure 9 shows how the boundary condition affects the spectral shape, fixing the pump intensity and frequency. The boundary condition affects the spectrum through renormalization of r . The linewidths of the three peaks are proportional to r , while the level splitting is proportional to 1/2 r . Therefore, the triplet structure becomes clearer for smaller r [mod(ω b l + θ b , π) → ±0.5π ]. This tendency can be confirmed in figure 9.
Next we discuss the effects of dissipation. Here we fix the boundary condition at mod(ω b l + θ b , π ) = 0, where ω b = ω b and r = 2 r , but qualitatively similar results are obtained for general boundary conditions. Figure 10 shows the effects of dissipation on S i (ω) when the pump is tuned to the resonance (ω p = ω b ). We can observe that dissipations broaden the peaks, preserving the symmetry of the triplet structure about the pump frequency. The roles of p and n are qualitatively the same under resonant pumping. However, the symmetry of the triplet structure is not necessarily conserved for off-resonant pumping. Figure 11 shows the results when the pump frequency is below the resonance (ω p = ω b − 5 r ). If pure dephasing is absent ( p = 0), the symmetry of the spectrum about the pump frequency is preserved. This point is in agreement with the optical results reported in [29] in the Markovian limit. In contrast, if pure dephasing is present ( p > 0), the symmetry of the spectrum is broken. The sideband closer to the resonance ω b becomes larger than the other one. (Although not presented in figure 11, the lower sideband becomes larger than the higher one when ω p > ω b .) Since conservation of photon number is guaranteed by equation (41), asymmetry in the spectrum implies that conservation of photon energy is broken between the input and the output. These facts can be understood as follows. The excitation in the atom is necessarily converted into a single photon in the absence of pure dephasing (σ † |0 → b † k |0 ), whereas the atom can emit a photon accompanying low-energy excitations in the environment in the presence of pure dephasing (σ † . Namely, energy conservation is relaxed by the pure-dephasing-type interaction with the environment [37]. The atom can then emit photons dominantly near its resonance to break the symmetry.

Discussion
We analyzed a superconducting qubit coupled to a semi-infinite transmission line and observed that its transition frequency and linewidth become sensitive to the boundary condition of the line. Since the boundary condition can be varied continuously in circuit QED systems (see appendix A), the current scheme thus provides an experimentally compact method for controlling the radiative property of the qubit. We make two remarks here from the viewpoint of single-photon engineering. (i) The physical origin of the present phenomenon is interference among the following three waves: reflected wave (the first term in equation (11)), radiation emitted into the positive r -direction (the second term in equation (11)) and radiation emitted into the negative r -direction and reflected afterwards (the third term in equation (11)). Therefore, the conditions for obtaining the present effects by a single photon with pulse length d are as follows: the radiative decay rate is slow enough to satisfy r v/l, where v is the microwave velocity in the transmission line, and the pulse length is long enough to satisfy l d. Both conditions can be readily satisfied in the circuit QED systems. The giant optical nonlinearity sensitive to individual photons, which is peculiar to such a 1D system, is expected to appear in the present system [25]. (ii) The position dependence of r is determined by its bare transition frequency ω b (see equation (25)). Therefore, in multi-level quantum systems which have several decay paths with different transition frequencies ( -type three-level system, for example [27]), the present scheme will serve as a method for controlling the ratio of radiative decay rates. This would enable us to control the Raman-transition efficiency and would open up a new type of single-photon engineering, which will be discussed elsewhere.

Summary
We studied the resonance fluorescence from a superconducting qubit that interacts with a semiinfinite 1D microwave field and is subject to two types of dissipation, pure dephasing and nonradiative damping ( figure 1). When coupled to a semi-infinite field, the transition frequency and the radiative linewidth of the qubit are renormalized as equations (24) and (25), and become sensitive to the boundary condition of the field. We analyzed the properties of the output microwave radiation, such as the coherent amplitude and the power spectrum, and observed how the boundary condition and the two types of dissipation modify the properties of output radiation. The present scheme would offer an experimentally compact method for controlling the radiative properties of quantum few-level systems, which would be useful for future singlephoton engineering. where (h/2e)φ ex is the external magnetic flux threading the loop. The overall Lagrangian is given by We switch to new phase variables ϕ 0 = (φ a + φ b )/2 + πθ [−cos(φ ex /2)] and ϕ j = φ j + φ ex /2 + πθ[−cos(φ ex /2)] for j 1, where θ is the Heaviside step function. The Lagrangian is then recast in the following form: where C s = 2C s and E s (φ ex ) = 2E s | cos(φ ex /2)|. Therefore, the SQUID works as a single Josephson junction having a tunable Josephson energy E s (φ ex ) through the external magnetic flux φ ex . The equations of motion for ϕ 0 and ϕ j ( j 1) are, respectively, given by We hereafter treat the phase variable as a continuous one ϕ(r, t), which is related to ϕ j (t) by ϕ j (t) = ϕ( j x, t), where x is the distance between the neighboring inductances. We also introduce the inductance and capacitance per unit length by L 0 = L/ x and C 0 = C/ x. Equation (A.9) is then reduced to the wave equation where v = (L 0 C 0 ) −1/2 is the velocity of the microwave. On the other hand, equation (A.8) is recast in the following form after linearization: which determines the boundary condition at r = 0. The eigenmode function at the resonance frequency is given by ϕ( where f k (r ) is given by equation (1). Substituting this into equation (A.11), we obtain the equation that determines θ b as a function of the external magnetic flux φ ex : where Z 0 = √ L 0 /C 0 is the characteristic impedance of the transmission line. Typically, Z 0 ∼ 50 , ω b ∼ 10 GHz, C s ∼ 100 fF and (2e/h)E s ∼ 5 µA. Therefore, we may vary θ b largely through the external magnetic flux φ ex .