Nonclassicality of open circuit QED systems in the deep-strong coupling regime

We investigate theoretically how the ground state of a qubit-resonator system in the deep-strong coupling (DSC) regime is affected by the coupling to an environment. We employ as a variational ansatz for the ground state of the qubit-resonator-environment system a superposition of coherent states displaced in qubit-state-dependent directions. We show that the reduced density matrix of the qubit-resonator system strongly depends on how the system is coupled to the environment, i.e., capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator-waveguide coupling. In contrast, when the two couplings are of the same type (for instance, both are inductive), by increasing the resonator-waveguide coupling strength, the average number of virtual photons increases and the quantum superposition realized in the qubit-resonator entangled ground state is partially degraded. Since the superposition becomes more fragile with increasing the qubit-resonator coupling, there exists an optimal coupling strength to maximize the nonclassicality of the qubit-resonator system.


Introduction
The interaction between a two-level system (qubit) and a harmonic oscillator (resonator) has been widely studied, originally as one of the simplest systems to study light-matter interaction [1,2], and later as a platform for quantum optics [3][4][5] and quantum information processing [6][7][8][9]. The deep-strong coupling (DSC) regime of the qubit-resonator (Q-R) interaction, where the coupling strength g is comparable to or even larger than the transition energies of the qubit (Δ) and the resonator (ω r ), has recently been achieved using artificial atoms in superconducting circuit QED systems [10][11][12][13] and THz metamaterials coupled to the cyclotron resonance of a 2D electron gas [14], as reviewed in references [15,16].
In the DSC regime, the ground state of the Q-R system is quite different from that for weaker coupling [17,18]. First, it is an entangled state between qubit and resonator. Second, such a Schrödinger's cat-like state has a nonzero expectation value of the photon number, n = |g/ω r | 2 . These photons are referred to as virtual photons, since the system is in the ground state and therefore the photons cannot be spontaneously emitted. Such nonclassical properties of the ground state are proposed to be useful for quantum metrology [19] and the preparation of nonclassical states of photons [20][21][22].
Any quantum system realized in an actual experimental setup, however, is coupled to external degrees of freedom, intentionally or unintentionally. Particularly, in a superconducting circuit QED system,

Model
In this section, we describe the model considered in this paper. The relationship among the system, the environment, and the total system is summarized in figure 1.

Qubit and resonator
We consider the quantum Rabi model, which is described by the Hamiltonian (1) Figure 1. Energy diagrams of the qubit-resonator waveguide (Q-R-W) system. (a) The Q-R system (described by H S ) for g > ω r /2 and the environmental waveguide system (described by H E ) are coupled with H SE . (b) We denote the enlarged ground state of H tot = H S + H E + H SE by |EGS = |ψ . Its reduced matrix to the Q-R space is referred to as the zero-temperature state and is denoted by ρ ZTS = tr E |ψ ψ| .
Here, a(a † ) is the annihilation (creation) operator of a resonator photon with energy ω r , σ j (j = x, y, z) is the Pauli operator of the qubit with transition energy Δ(>0), and g is the coupling strength between them. The Planck constant is set to unity throughout this paper. Here, we assume that X I = a + a † is proportional to the flux operator, so that the Q-R coupling is inductive. When the coupling is capacitive, X I is replaced with X C = (a − a † )/i as after choosing an appropriate basis for qubit states. Both H S and H S are unitary equivalent through a gauge transformation a ↔ ia, where the unitary operator is explicitly given by U = exp ±iπa † a/4 . We note that there is a subtlety originating from the choice of gauge in the theoretical treatment of the system [28][29][30][31]. Although the microscopic model of the circuit should possess gauge invariance, the approximation in which only two levels are retained for the qubit part of the circuit often breaks gauge invariance. Specifically, the truncated Hamiltonian depends on the choice of gauge. We do not discuss this aspect in detail here. We simply assume that the qubit and resonator are described by the quantum Rabi model. In other words, our analysis and results apply to physical systems for which the quantum Rabi Hamiltonian provides a good approximation for the low-energy physics.
When ω r Δ or g ω r , Δ, the low-lying eigenstates of the quantum Rabi model are approximately described by [17,18] which is also denoted as |φ (±) n . Here, |↑ (|↓ ) is the qubit eigenstate corresponding to σ z = +1 (−1), α = g/ω r is the amplitude of displacement in the resonator, and D(α) = e αa † −α * a is the displacement operator. These parameter regions are referred to as the adiabatic oscillator limit [17,[32][33][34] or the perturbative DSC regime [16,18], in the sense that the perturbation in terms of Δ is valid. The eigenenergy is then, up to first order in Δ/ω r , given by where L n (x) is the Laguerre polynomial of the nth order. We note that the ground state, is the superposition of two coherent states displaced in opposite directions depending on the qubit state.

Coupling to the environment
The environment can be modeled by an ensemble of harmonic oscillators [35] as Figure 2. Circuit diagrams of a Q-R system coupled to a waveguide. The blue shaded area is an arbitrary circuit representing the qubit. The R-W coupling is mediated by (a) a capacitance C c or (b) an inductance L c .
where b k (b † k ) is the annihilation (creation) operator of the kth mode with energy ω k . Each mode of the environment interacts with a system operator X with strength ξ k as The total system is then described by the Hamiltonian H tot = H S + H E + H SE . As a concrete model of an open DSC system, we investigate a DSC system coupled to a waveguide through the resonator (figure 2). The blue shaded area can be an arbitrary circuit constituting a qubit, such as a Cooper pair box or a flux qubit. The waveguide is coupled to the Q-R system through (a) a capacitance C c or (b) an inductance L c . When the interaction is mediated by an inductance (a capacitance), the operator X through which the system is coupled to the waveguide is the quadrature operator of the resonator, given as X I = a + a † (X C = (a − a † )/i). We note that in the case of the capacitive coupling, the interaction Hamiltonian is given by . This is equivalent to equation (7) under the unitary transformation b k → −ib k . While the phase of the quadrature operator for b k does not affect the physical result because of the gauge invariance of H E , the phase of the quadrature operator for a has the physical influence since it is used to couple the resonator to not only the waveguide but also the qubit. In the following, we always assume that the Q-R coupling is inductive, and we analyze the cases where the R-W coupling is inductive or capacitive, except for section 5, where we discuss the relativity of the coupling.
Assuming a finite length L of the waveguide, the wavenumber k of the waveguide modes is discretized as k = nπ/L (n ∈ N), and the energy is given by ω k = vk, where v is the speed of microwave fields in the waveguide. The coupling constant ξ k is given by in both the capacitive [36] and inductive coupling case. We do not have to put the cutoff factor by hand, since it is naturally included in the Hamiltonian derived from the circuit, and the cutoff energy ω cutoff is determined from the circuit parameters (equations (A.26) and (A.29)). In the small frequency region (ω k ω cutoff ), the squared coupling strength |ξ k | 2 is proportional to ω k , which corresponds to the Ohmic case in the spin-boson model (see appendix B for details).
The loss rate κ of a bare resonator photon into the waveguide is determined by the Fermi Golden rule as

Coherent variational state
In this section, we introduce the CVS and analyze the ground state of H tot . In analogy to the approximate ground state of the quantum Rabi model (equation (5)), we define the CVS of the total system as where |α; {β k } is the product of coherent states of the resonator and waveguide modes, satisfying a|α; The variational parameters for the CVS are α and β k 's. We note that a more general form c 0 |↑ ⊗ |α; {β j } + c 1 |↓ ⊗ |α ; {β j } leads to the same results as the simpler form in equation (10) due to the parity symmetry of the quantum Rabi Hamiltonian, as discussed in appendix C. We also note that the renormalization of the qubit energy and the Rabi oscillation were analyzed using a similar ansatz by performing a polaron transformation [37]. The total energy for the CVS is given by where the plus (minus) sign represents the inductive (capacitive) coupling. The approximate ground state of the total system is the CVS |ψ C (ᾱ, {β k }) , whereᾱ andβ k 's are the variational parameters that minimize the total energy E CVS . Although there are a large number of degrees of freedom due to the numerous waveguide modes, the problem can be simplified into a stationary state problem with only two unknown parameters, α and S = k |β k | 2 , as discussed in appendix D. Here, S is a collective variable for the waveguide modes, representing the total number of virtual photons in the waveguide modes. Onceᾱ andS = k |β k | 2 are obtained, the reduced density operator for the system, , which we call the zero-temperature state, is completely characterized by two parametersᾱ and C = exp −2S . It is explicitly expressed as This equation implies that the R-W coupling has two effects. First, the displacementᾱ is modified from α = g/ω r , which means that the average number of virtual photons |ᾱ| 2 is changed. In fact, as we will see in section 4, the number of virtual photons increases as the R-W coupling increases. Second, ρ ZTS includes the first excited state of H S with a fraction P e = 1−C 2 . A similar behavior can be found in the case of a single two-level system coupled to an environment [35].
The quantity C serves as a measure of coherence, which is a real quantity within the range of 0 C 1. Ifβ k = 0 and hence C = 1, the system is in a pure state. On the other hand, if C = 0, the quantum superposition is completely destroyed and the reduced density matrix ρ S is maximally mixed. When we represent ρ S in the basis of |↑ ⊗ | −ᾱ and |↓ ⊗ |ᾱ , the quantity C appears in the off-diagonal element: In this sense, R-W coupling reduces the coherence realized in this basis. The purity of ρ ZTS can be calculated as which means that the purity is a simple function of the total number of virtual photons in the waveguide, and vice versa. In this way, the analysis based on the CVS clarifies the relation between the decoherence in the Q-R system and the virtual photons in the waveguide. As for the validity of the CVS, we note that it gives the exact ground state of the total Hamiltonian if the qubit energy Δ and the counter-rotating terms ξ k (ab k + a † b † k ) are neglected. In reference [11], it is argued that the Q-R state |φ (−) 0 gives a rather accurate description of the ground state of the quantum Rabi Hamiltonian even when the qubit energy Δ is finite. Furthermore, we compare the result of CVS in the inductive coupling case with that of the numerical diagonalization for a few waveguide mode case in appendix E, and show that the CVS describes not only the virtual photons but also the nonclassical properties well in the presence of the R-W coupling. We also compare the results from the CVS and perturbation theory with an infinite number of waveguide modes, and confirm that they are in good agreement in appendix E.

Numerical calculations
In this section, we numerically investigate the properties of ρ ZTS based on the CVS. We adopt the bare resonator photon loss rate κ (equation (9)) as a measure of the R-W coupling strength. The Q-R coupling is set to g/2π = 3 GHz or 6 GHz, which are achievable in circuit QED systems [12,13]. The other parameters are set to ω r /2π = 6 GHz and Δ/2π = 1.2 GHz. See appendix A for the details of parameter settings. We again note that the Q-R coupling is assumed to be inductive in this section, and we refer to the coupling between the resonator and the waveguide when we mention the inductive or capacitive coupling.

Inductive R-W coupling
We first calculate the average number of virtual photons in the resonator, tr[ρ ZTS a † a] = |ᾱ| 2 in the inductive coupling case. Figure 3(a) shows that, in the inductive coupling case, the number of virtual photons increase as the R-W coupling κ increases. Indeed, the interaction term acts as a shifter on the resonator phase space in the real direction. Furthermore, we can show that the energy gradient ∂E/∂α takes a negative value at α =α, whereα is the stationary solution in the absence of the R-W coupling, which implies that the average number of virtual photons always increases by the R-W coupling, independently of the details of the interaction spectrum ξ k . In contrast, in reference [26], where the matter is modeled by an ensemble of bosons rather than a qubit, the number of virtual photons decreases by the coupling to the environment. Therefore, whether it increases or decreases depends on the details of the model, e.g. how the matter part is modeled.
Next, we calculate the purity γ of ρ ZTS . Figure 3(b) shows that, in the inductive coupling case, the purity decreases as the loss rate κ increases. By comparing the results for g/2π = 3 and 6 GHz, we see that the purity also decreases as the Q-R coupling g increases. In other words, in the DSC regime, the quantum coherence of the ground state becomes fragile when the Q-R interaction g is extremely large. It implies that, even though the exact ground state of the quantum Rabi model becomes increasingly useful with increasing g, for instance, for quantum metrological tasks [19], the maximum performance is achieved at a moderate strength of the interaction g when the coupling to the environment is taken into account. Indeed, the nonclassicality, measured by the metrological power [27], has the maximum at a certain value of g, as we discuss below.
Let us evaluate the 'quantumness' of the ground state of the system. There are many ways of defining and quantifying the quantumness [27,[38][39][40][41]. Here, we project the state onto the eigenstates of some qubit operator, and calculate the quantumness from the reduced density matrix of the resonator, by exploiting the resource theory of the nonclassicality in continuous variable systems. One of the measures of the nonclassicality is the metrological power [27], which quantifies the maximum achievable quantum enhancement in displacement metrology based on the quantum Fisher information [42,43]. For a resonator state with the spectral decomposition ρ R = i λ i |i i|, the elements of quantum Fisher information matrix for quadrature operators are defined as where 2i are the quadrature operators. Then the metrological power of the resonator state ρ R is given by where λ max (F) is the maximum eigenvalue of the quantum Fisher information matrix. We consider a projective measurement of a qubit operator The measurement outcome σ θ,φ = ±1 is obtained with probability and the post-measurement state for the resonator is where P σ θ,φ =±1 is the projection operator corresponding to the eigenvalue σ θ,φ = ±1. Physically, the post-measurement resonator state is a partially decohered cat state possessing interference fringes around the origin in the phase space representation. These fringes become clearer as α and |C| increase, which enables us to measure the displacement more precisely than coherent states (figure 4). We can define the average metrological power as Finally we define the metrological power of the Q-R system state by optimizing the measurement axis of the qubit as In figure 5, the metrological power of ρ ZTS (equation (21)) is plotted against (a) the loss rate κ and (b) the Q-R coupling g. In our setting, the average metrological power is found to be maximized at θ = φ = π/2, corresponding to the measurement of σ y . Figure 5(a) shows that for each value of g, the metrological power rapidly decreases to zero when κ becomes larger than a certain value. This critical value of κ becomes small as the Q-R coupling g increases. Figure 5(b) shows that the average metrological power has a maximum at some finite value of g. This maximum is achieved when the loss rate κ is comparable to the energy gap Δe −2g 2 /ω 2 r , or g opt ∼ ω r log(Δ/κ)/2, so that the optimal coupling strength g opt increases only logarithmically by decreasing κ and increasing Δ. In practice, the loss rate κ cannot be too small because the measurement and control need time duration T ∼ 1/κ, during which decoherence occurs. Therefore, this result implies that it is important to design a circuit to have a proper strength of the Q-R coupling g and the loss rate κ to obtain an optimal metrological advantage.

Capacitive R-W coupling
The capacitive coupling affects the system much less than the inductive coupling does, as we see below. Indeed, in the capacitive coupling case, the CVS cannot capture the effect of R-W coupling, since it gives the exactly same result as the noninteracting case regardless of the coupling strength, as proved in appendix D. Therefore, we compare the result from the CVS and the numerical diagonalization in the capacitive coupling case, and the CVS in the inductive coupling case. To perform the numerical calculation, the total Hamiltonian is truncated as follows. We take up to 14 photons and 3 photons into account for the resonator mode and each waveguide mode, respectively. As the waveguide modes, we consider four modes with energies ω k /2π = 5, 10, 15, 20 GHz. Although this truncation is not sufficient to quantitatively discuss   the effect of the coupling to the environment, the results below clearly show the qualitative difference between the inductive coupling and the capacitive coupling.
In figure 6, the average number of virtual photons (a) and the purity (b) are plotted against the loss rate κ. In the capacitive coupling case, the average number of virtual photons is much less sensitive to the R-W coupling compared to the inductive coupling case. This fact can be qualitatively understood as follows. The coupling operator X C = (a − a † )/i acts as a shifter of the displacement α in the imaginary direction. However, since α = g/ω r is real without the environment, the amplitude |α| 2 is much less sensitive to the imaginary shift than the real shift. We also see that in figure 6(b), the purity is less affected by the capacitive coupling to the waveguide compared to the inductive coupling case. We will discuss the origin of this stability against the capacitive coupling to the waveguide in section 5.
In figure 7, the average metrological power is plotted against (a) the bare loss rate κ and (b) the Q-R coupling g. Since the effect of the W-R coupling is much underevaluated due to the truncation and the small number of environmental modes, we choose a rather huge value of κ/2π = 1000 MHz, which is formally obtained from equation (9). We see that in the capacitive coupling case, the metrological power calculated from the CVS and the numerical diagonalization agrees very well, and also that the nonclassicality is hardly affected by the capacitive coupling to the waveguide, compared to the inductive coupling case.

Stability in the R-W capacitive coupling case
In this section, we discuss the origin of the stability of the ground state when the R-W coupling is capacitive. To obtain some physical insight, let us first consider the case where a bare resonator is coupled to a waveguide. The fraction of the first excited state |1 contained in the total ground state is proportional to | 1|X|0 | 2 at the lowest order, where X is a quadrature operator. This transition amplitude is completely insensitive to the type of coupling, i.e. inductive X I = a + a † or capacitive X C = (a − a † )/i, as This is due to the fact that the resonator Hamiltonian H = ω r a † a, and the energy eigenstates |n are invariant under the rotation around the origin in the phase space, represented by the unitary transformation U = exp iθa † a . On the other hand, since the eigenstates of a DSC system are not invariant under this rotation, the transition amplitude strongly depends on the type of coupling as Therefore, when the Q-R coupling is mediated by the inductance, i.e. α is real, the system is stable against capacitive coupling to the waveguide. A similar argument is applied in references [44,45] to protect a qubit from noise based on the fact that the transition amplitude depends on the qubit operator (σ x , σ y , σ z ) and can be exponentially small in g. In contrast, in our case, the transition amplitude is exactly zero when the R-W coupling is capacitive. To see this coupling-type-dependence more directly, we perform a numerical diagonalization of the truncated total Hamiltonian of the Q-R-W system. The truncation is the same as in the previous section. Figure 8 shows the fraction of the excited states of the quantum Rabi model contained in the ground state of H tot . In the inductive coupling case, the most dominant excitation is the first excited state |φ (+) 0 . On the other hand, the fraction of |φ (+) 0 is not dominant in the capacitive coupling case, as is expected from equation (24). Instead, the most dominant excited state is |φ (−) 1 , which is the only excited state with a nonzero transition amplitude as | φ (−) 1 (α)|X C |φ (−) 0 (α) | 2 = 1. The reason why the system is not changed in the capacitive coupling case in the CVS analysis in section 4 is that the CVS considers only the two lowest eigenstates |φ (−) 0 and |φ (+) 0 . From the analysis performed in section 4, we cannot determine whether the metrological power monotonically increases as a function of g or peaks at a certain value of g in the capacitive coupling case. However, the peak, if it exists, is expected to occur at a much larger value of g than in the inductive coupling case, so that a higher metrological power is achievable in the capacitive coupling case.
We need to be careful to conclude from our results that the capacitive R-W coupling is superior to the inductive coupling, since there is a tradeoff between the gate speed and the relaxation time [46]. In the capacitive coupling case, this tradeoff relation indicates that the long relaxation time implies a slow control between the ground state and the first excited state.
Finally, we stress that only the relative phase between the resonator quadrature operators coupled to the qubit and the waveguide is relevant to this stability. When the Q-R coupling is assumed to be capacitive, the system is sensitive to the capacitive R-W coupling and insensitive to the inductive R-W coupling. These results are summarized in table 1.

Conclusion
In this paper, by analyzing the ground state of a Q-R-W system, we have investigated the effect of an environment on the ground state of the quantum Rabi model in the DSC regime. We have introduced the qubit-state-dependent CVS (equation (10)). This variational ansatz is easy to analyze and is consistent with the results from numerical diagonalization and perturbation theory. We have shown that the zero-temperature state ρ ZTS strongly depends on the type of the R-W coupling because of the broken rotational symmetry in the eigenstates of the DSC system.
When the resonator couples to the qubit and the waveguide in the same way (for instance, both are inductive), the number of virtual photons increases due to the R-W coupling, which might be advantageous to detect virtual photons experimentally [47,48]. We have also shown that, even at zero temperature, the Q-R system is in a mixed state and contains the excited states of the quantum Rabi Hamiltonian, which implies the fragility of the quantum superposition realized in the ground state. As a result, the nonclassicality of the resonator system, measured by the metrological power, is maximized at a certain coupling strength g, when the environment is taken into account. We note that the analysis based on the multi-polaron expansion [49] suggests that the CVS underestimate the coherence in the system. To obtain a more accurate result in the inductive coupling case, we may modify the CVS to include more than one polaron.
On the other hand, when the resonator couples to the qubit and the waveguide in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected, so that a higher metrological power than the same coupling case is achievable in the presence of environment. It is worth considering a better variational ansatz that can quantitatively describe the ground state in such case.
Our results offer guiding principles to obtain a better metrological advantage when we design superconducting circuit QED systems. Since it is necessary to perform projective measurements on the qubit to exploit this metrological advantage, our results also demonstrate the advantages of achieving dynamically controllable coupling between qubit and resonator.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.

Appendix A. Derivation of the Hamiltonian of the circuit coupled to two waveguides
In this section, we derive the Hamiltonians of the circuits in figures 2(a) and (b). For that purpose, we consider a Q-R circuit coupled to two waveguides, one inductively and one capacitively, as shown in figure A1. The circuits in figures 2(a) and (b) can be obtained by taking the limits L c → ∞ and C c → 0, respectively. We note that a similar circuit is discussed in reference [50].
We assign a flux variable to each vertex. Here, variables ψ, φ, Φ j and Ψ j represent degrees of freedom of the qubit, the resonator, the waveguide coupled inductively (left), and the waveguide coupled capacitively (right), respectively. The shaded area in figure A1 can be an arbitrary circuit constituting a qubit, such as a Cooper pair box or a flux qubit. We do not specify the details of the qubit circuit, but assume that it involves only ψ andψ. Then the total Lagrangian is given by Canonical conjugate variables are defined as By performing the Legendre transformation, we obtain the Hamiltonian of the total circuit as Figure A1. Circuit diagram of the Q-R system coupled to two waveguides, one inductively (left) and one capacitively (right).
where L R = L c L R L c +L R and C c = C c C R C c +C R . The first line represents the Hamiltonian for the qubit, the resonator, and the interaction between them. The second (third) line represents the Hamiltonian for the waveguide inductively (capacitively) coupled to the resonator. The final line represents the interaction between the resonator and the waveguide modes. We note that the qubit variables {ψ,q} do not appear except for the first term line. To diagonalize the waveguide modes, let us consider the equations of motion for the waveguide variables:Q From these equations, by taking the continuous limit, Δx → 0, Φ j → Φ(x) = Φ(jΔx), and Ψ j → Ψ(x) = Ψ(jΔx), we obtain the wave equations for x > 0 as with boundary conditions Let us consider the inductively-coupled waveguide modes. The eigenfunction with frequency ω is given by where L is the length of the waveguide, v = 1/ √ L T C T is the speed of microwaves in the waveguide, and ζ(ω) = L T v/L C ω is a dimensionless quantity defining the phase of the eigenfunction. Then the positive frequency part of the flux variable Φ(x) is quantized as where Z T = L T /C T is the characteristic impedance, b k is the annihilation operator of the eigenmode f ω k (x), and ω k = πkv/L. Therefore, the flux variables Φ 0 and φ are expressed in terms of the creation and annihilation operators as where the limit L → ∞ is taken in the second line. Here, a(a † ) is the annihilation (creation) operator of microwave photons in the resonator. The characteristic impedance for the resonator Z R is defined as In a similar manner, we obtain the capacitive coupling term as and d(ω) is the annihilation operator of the capacitively-coupled waveguide mode with frequency ω. We note that the interaction spectrums ξ I k and ξ C k satisfy the inequality 30) which ensures that there is no bosonic mode with negative energy. In this sense, the circuit used in our analysis shows no superradiance-like instability.
In the numerical calculation, the parameters are set to be ω r /2π = 6 GHz, Δ/2π = 1.2 GHz, Z W = 50 Ω, and Z R = 30 Ω. In figure A2, the values of κ and ω cutoff are plotted as a function of the coupling inductance or capacitance in each case. Since the left waveguide is directly connected to the Q-R system through the coupling inductance L c , κ I takes a large value unless L c is sufficiently large, and is a decreasing function of L c . This property is special to the circuit in figure A1, and different from the circuits realized in references [12,13], where the waveguide is coupled to the resonator through mutual inductance.

Appendix B. Relation to the spin-boson model
In this section, we show that the Hamiltonian describing the circuit in figure 2(b) can be effectively mapped to the spin-boson model [35]. We note that the mapping can be exactly performed if we first diagonalize the bosonic (R-W) part and consider it as the environment [37,51]. The difficulty of this approach seems that it is difficult to obtain the information on the reduced density matrix of the Q-R system. Instead, we chose a different approach where we first (approximately) diagonalize the Q-R system and then consider the interaction with the waveguide, by which we can easily discuss the state of the Q-R system.
The total Hamiltonian is given by To map the total Hamiltonian to the spin-boson model, we project the state space of the Q-R system onto the space spanned by the two low-lying energy states, {|φ (−) 0 (α) , |φ (+) 0 (α) }. Letσ j (j = x, y, z) be the Pauli matrix in the basis of {|↑ | − α , |↓ |α }. Then the truncated Hamiltonian becomes the spin-boson model, Here, the 'localized state' corresponds to |↑ | − α and |↓ |α . The spectral function of this truncated Hamiltonian can be calculate as which corresponds to the Ohmic case.

Appendix C. Symmetry of the total Hamiltonian and the CVS
The total Hamiltonian is invariant under a parity transformation a ↔ −a, b k ↔ −b k , and |↑ ↔ |↓ . The ground state is expected to have the same symmetry, so that Then we have From the last two equations, we have e iθ = ±1 and |c 0 | = |c 1 | = 1/ √ 2 from the normalization condition. The choice of sign only affects the qubit energy term Δσ x , and e iθ = −1 is chosen so that the qubit energy term reduces the total energy. Figure E1. The average number of virtual photons in the resonator (a), the purity (b), and the average metrological power, plotted against the bare loss rate κ. (d) The average metrological power plotted against the Q-R coupling strength g. All the quantities are calculated from the CVS (black solid line) and the numerical diagonalization (red dashed line). The loss rate is set to κ/2π = 1000 MHz, and the Q-R coupling is set to g/2π = 6 GHz.
In the capacitive coupling case, we have

Appendix E. Validity of CVS in inductive coupling
In this appendix, we confirm the validity of the CVS in the inductive coupling case. First, we compare the result from the CVS and the numerical diagonalization for a few waveguide mode case. We see that in figures E1(a) and (b), the number of virtual photons and the purity calculated from the CVS and the numerical diagonalization agree quantitatively. We also see that from figures E1(c) and (d), the nonclassical property of the system, measured by the metrological power can also be described by the CVS. Next, we compare the result from the CVS and perturbation theory. At the leading order in ξ k , ρ ZTS is given by ρ ZTS = (1 − 4α 2 f 2 (Δe −2α 2 ) − f 2 (2α 2 Δe −2α 2 + ω r ))|ψ (−) 0 (α) ψ (−) 0 (α)| + 4α 2 f 2 (Δe −2α 2 )|ψ (+) 0 (α) ψ (+) 0 (α)| + f 2 (2α 2 Δe −2α 2 + ω r )|ψ (−) 1 (α) ψ (−) 1 (α)| + o(ξ 2 k ), (E.1) Figure E2. The impurity (=1 − γ) plotted against the bare loss rate κ, calculated from the CVS (black) and perturbation theory (blue). The Q-R coupling is set to g/2π = 3 GHz (dashed line) and 6 GHz (solid line). All the quantities are calculated from the CVS (black solid line) and the numerical diagonalization (red dashed line). Figure F1. Increase of virtual photons from the noninteracting (ξ k = 0) case, plotted against the bare loss rate κ. The black (red) line shows the result from the numerical diagonalization with (without) the RWA in the R-W coupling. The Q-R coupling is set to g/2π = 6 GHz.
where α = g/ω r . Figure E2 shows that the impurity (1 − γ) calculated from the CVS and perturbation theory have the same scaling to κ and are in good agreement. The small deviation is attributed to the fact that the qubit energy Δ is finite, and equation (3) is no longer exact. Note that equation (3) remains a good approximation for the eigenstates of the quantum Rabi model when ω r Δ or g ω r , Δ.

Appendix F. Effect of counter-rotating terms in R-W coupling
In this section, we discuss the effect of counter-rotating terms and the RWA in the R-W coupling. In quantum mechanical analysis, the co-rotating and counter-rotating terms describe transitions to virtual intermediate states. If the virtual state is far off resonance, its effect is suppressed. Since the counter-rotating terms typically involve the virtual states far off resonant, they give smaller contributions than the co-rotating terms, which justify the RWA. In contrast, since the CVS is a semiclassical approach, where the annihilation operators a and b k are replaced with the c-numbers, α and β k , respectively, there is no similar distinction between the co-rotating and counter-rotating terms. Indeed, in the inductive coupling case, noting that α and β k are real quantities, the co-rotating terms (ab † k + a † b k ) and the counter-rotating terms (ab k + a † b † k ) give the same contribution to the CVS energy functional in equation (11). Then, performing the RWA is equivalent to changing the coupling constant ξ k → ξ k /2 in the CVS analysis. Therefore, the number of virtual photons increases even if we perform the RWA. Furthermore, the increase in virtual photons with RWA is ∼4 times smaller than without RWA, since the increase of virtual photons is O(ξ 2 k ) at the lowest-order perturbation. This simple analysis based on the CVS is in good agreement with the result from the numerical diagonalization as shown in figure F1, which justifies the symmetric treatment of the co-rotating terms and the counter-rotating terms in the DSC regime.