Scattering of coherent states on a single artificial atom

Consider a semi-infinite TL with characteristic inductance L₀ and capacitance C₀ per unit length. We discretize the TL [35] in units of the small length Δx, which we take to zero at the end of the calculation. The TL nodes are numbered with negative integers, while the SCB island node has index J and its Josephson junction has a capacitance C_J to ground and a Josephson energy E_J. The SCB is coupled to the TL at the zeroth node, through the capacitance C_c, as depicted in figure 1.

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To describe the circuit dynamics, we use the node fluxes $\Phi _\alpha (t)=\int ^t\,\mathrm {d}t' V_\alpha (t')$ as coordinates [36]. They are the time integrals of the node voltages and although less intuitive than the voltages, this choice greatly simplifies the description of the Josephson junction. Starting from a circuit Lagrangian, we can derive the discrete circuit Hamiltonian [37]

$$\begin{equation} H_{\mathrm{d}}=\frac{(p_0+p_{\mathrm{J}})^2}{2C_{\mathrm{J}}}+ \frac{p_0^2}{2C_{\mathrm{c}}} - E_{\mathrm{J}}\cos{\left(\frac{2e}{\hbar}\Phi_{\mathrm{J}}\right)}+\frac{1}{\Delta x} \sum_{n< 0} \frac{p_{n}^2}{2C_0} + \frac{(\Phi_{n+1} -\Phi_{n} )^2}{2L_0}, \end{equation} \tag{ 1 }$$

where the charges p_α are the conjugate momenta to the node fluxes Φ_α, fulfilling the canonical commutation relations

$$\begin{eqnarray*} &&[\Phi_\alpha,p_\beta]=\mathrm{i}\hbar \delta_{\alpha,\beta}, \ \ [\Phi_\alpha,\Phi_\beta]= [p_\alpha,p_\beta]=0, \end{eqnarray*}$$

where δ_α,β denotes Kronecker's delta. From the Hamiltonian, we obtain Heisenberg's equations of motion for the TL operators (n < 0)

$$\begin{equation} \partial_t \Phi_n = \frac{p_n}{\Delta x C_0},\quad \partial_t p_n = \frac{\Phi_{n-1}-2\Phi_n+\Phi_{n+1}}{\Delta x L_0} \end{equation} \tag{ 2 }$$

and for the SCB operators

$$\begin{equation} \partial_t p_0=\frac{\Phi_{-1}-\Phi_0}{\Delta x L_0}, \end{equation} \tag{ 3 }$$

$$\begin{equation} \partial_t\Phi_0=\frac{p_0+p_{\mathrm{J}}}{C_{\mathrm{J}}}+\frac{p_0}{C_{\mathrm{c}}}=\frac{C_\Sigma}{C_{\mathrm{c}}C_{\mathrm{J}}}p_0+\frac{p_{\mathrm{J}}}{C_{\mathrm{J}}}, \end{equation} \tag{ 4 }$$

$$\begin{equation} \partial_t p_{\mathrm{J}}=-E_{\mathrm{J}} \frac{2e}{\hbar}\sin{\left(\frac{2e}{\hbar}\Phi_{\mathrm{J}}\right)}, \end{equation} \tag{ 5 }$$

$$\begin{equation} \partial_t \Phi_{\mathrm{J}} = \frac{p_0+p_{\mathrm{J}}}{C_{\mathrm{J}}}, \end{equation} \tag{ 6 }$$

where C_Σ = C_c + C_J.

In the continuum limit Δx → 0, the charge of each TL node will go to zero together with the node capacitance. Thus, we define a charge density field p(x_n,t) = p_n(t)/Δx and a flux field Φ(x_n,t) = Φ_n(t), where we define the spatial coordinate x_n = nΔx for n < 0, along the TL. The continuum equations of motion for the TL (x < 0) are

$$\begin{equation} \partial_t p(x) = \frac{ \partial^2_{x} \Phi(x)}{L_0} ,\quad \partial_t\Phi(x)=\frac{p(x)}{C_0}. \end{equation} \tag{ 7 }$$

These are the equations of motion for the massless Klein–Gordon field, having freely propagating left- and right- moving solutions with velocity $v=1/\sqrt {L_0C_0}$. Therefore, we can write the general solution for x < 0 as a linear combination of right- and left- moving second-quantized fields

$$\begin{equation} \begin{array}{l} \Phi_{\rightleftarrows }(x,t)= \displaystyle\sqrt{\displaystyle\frac{\hbar Z_0}{4\pi}}\int_{0}^\infty \displaystyle\frac{\mathrm{d}\omega}{\displaystyle\sqrt{\omega}}\left( a^{\rightleftarrows}_\omega \mathrm{e}^{-\mathrm{i}(\omega t \mp k_\omega x)} + \mathrm{h.c.} \right),\\ p_{\rightleftarrows }(x,t) = -\mathrm{i} \displaystyle\sqrt{\displaystyle\frac{\hbar Z_0}{4\pi}}\int_{0}^\infty \mathrm{d}\omega \displaystyle\sqrt{\omega}\left( a^{\rightleftarrows}_\omega \mathrm{e}^{-\mathrm{i}(\omega t \mp k_\omega x)} - \mathrm{h.c.} \right), \end{array} \end{equation} \tag{ 8 }$$

where k_ω = ω/v and $Z_0=\sqrt {L_0/C_0}$ is the characteristic impedance of the TL. The operators $a_{\omega }^{\leftrightarrows }$ annihilate a left/right-moving photon with frequency ω, and obey the bosonic canonical commutation relations, [a^←_ω,(a^←_ω')^†] = [a^→_ω,(a^→_ω')^†] = δ(ω − ω') and $[a^\leftarrow _\omega ,(a_{\omega '}^\rightarrow )^{ \dagger }] = [a^\rightleftarrows _\omega ,a_{\omega '}^\rightleftarrows ]=0$. Finally, we note that in the continuum limit, (3) changes into

$$\begin{equation} \partial_t p_0=-\frac{\partial_x \Phi(0^-)}{L_0}. \end{equation}\noindent \tag{ 9 }$$

To describe the system dynamics, we first need to specify the incoming, right-moving field

$$\begin{eqnarray*} &&\Phi^{\mathrm{in}}(t)=\Phi_{\rightarrow}(0^-,t). \end{eqnarray*}\noindent$$

Given this initial condition, we can then calculate the SCB dynamics, as well as the outgoing field

$$\begin{eqnarray*} &&\Phi^{\mathrm{out}}(t)=\Phi_{\leftarrow}(0^-,t), \end{eqnarray*}\noindent$$

propagating to the left in the line. The flux at x = 0 is simply the sum of the incoming and outgoing flux fields

$$\begin{equation} \Phi_0(t)=\Phi(0^-,t)=\Phi^{\mathrm{in}}(t)+\Phi^{\mathrm{out}}(t)+V_{\mathrm{dc}}t, \end{equation}\noindent \tag{ 10 }$$

where for simplicity we also explicitly extracted the dc voltage bias V_dc, implying that Φⁱⁿ and Φ^out have no dc components. Now, solving for p₀ from (4) gives

$$\begin{equation} p_0=\frac{C_{\mathrm{c}} C_{\mathrm{J}}}{C_\Sigma} \left[ V_{\mathrm{dc}}+\partial_t (\Phi^{\mathrm{in}}+\Phi^{\mathrm{out}})\right] -\frac{C_{\mathrm{c}}}{C_\Sigma}p_{\mathrm{J}} \end{equation}\noindent \tag{ 11 }$$

and inserting this expression into (6), we arrive at

$$\begin{equation} \partial_t \Phi_{\mathrm{J}} = \frac{p_{\mathrm{J}}+C_{\mathrm{c}}\left[V_{\mathrm{dc}}+\partial_t (\Phi^{\mathrm{in}}+\Phi^{\mathrm{out}}) \right]}{C_\Sigma}. \end{equation}\noindent \tag{ 12 }$$

We then insert Φ₀ from (10) in (13) and arrive at

$$\begin{equation} \partial_t p_0 =-\frac{\partial_x \Phi(0^-)}{L_0}=\frac{\partial_t (\Phi^{\mathrm{in}}-\Phi^{\mathrm{out}})}{Z_0}, \end{equation}\noindent \tag{ 13 }$$

where we used the relation $\partial _x\Phi _\rightleftarrows (0^-)=\mp v^{-1}\partial _t\Phi _\rightleftarrows (0^-)$ to change the spatial derivative into a time derivative. Inserting the expression for p₀ from (11) into the left-hand side of this equation and integrating once with respect to time leads to

$$\begin{equation} \Phi^{\mathrm{out}}=\Phi^{\mathrm{in}}+Z_0 \frac{C_{\mathrm{c}}}{C_\Sigma}p_{\mathrm{J}}-\tau_{\mathrm{RC}} \partial_t (\Phi^{\mathrm{in}}+\Phi^{\mathrm{out}}), \end{equation}\noindent \tag{ 14 }$$

where the time τ_RC = C_cC_JZ₀/C_Σ is the characteristic RC time for discharging the SCB through the TL. Equations (5), (12) and (14), in principle, give the full time evolution of the SCB operators Φ_J and p_J as well as the out-field, in terms of the in-field. However, to solve these nonlinear equations straightforwardly, we need to make some approximations.

In the following, we will neglect the last term in (14). Since the time derivative enters the product with τ_RC, this will be a good approximation as long as the relevant frequencies of the incoming field Φⁱⁿ and of the SCB dynamics (p_J) are much lower than the inverse RC time. Under this approximation, the final equations of motion are

$$\begin{equation} \partial_t \Phi_{\mathrm{J}} = \frac{p_{\mathrm{J}}+ C_{\mathrm{c}} \left(V_{\mathrm{dc}}+2 \partial_t \Phi^{\mathrm{in}} +\frac{\tau_{\mathrm{RC}}}{C_{\mathrm{J}}} \partial_t p_{\mathrm{J}} \right)}{C_\Sigma}, \end{equation} \tag{ 15 }$$

$$\begin{equation} \partial_t p_{\mathrm{J}}=-E_{\mathrm{J}} \frac{2e}{\hbar}\sin{\left(\frac{2e}{\hbar}\Phi_{\mathrm{J}}\right)}, \end{equation} \tag{ 16 }$$

$$\begin{equation} \Phi^{\mathrm{out}} = \Phi^{\mathrm{in}} + \frac{\tau_{\mathrm{RC}}}{C_{\mathrm{J}}} p_{\mathrm{J}}. \end{equation} \tag{ 17 }$$

Here we also note that this approximation is valid in recent experiments [22, 23], where Z₀ = 50 Ω, C_c ∼ 10 fF and C_J ∼ 25 fF, giving an inverse RC timescale of 1/(2πτ_RC) ∼ 400 GHz, which is around 50 times higher than the relevant frequency of Φⁱⁿ and p_J, set by the qubit frequency ∼7.5 GHz.

The above set of equations (15)–(17) corresponds to the Hamiltonian

$$\begin{equation} H = H_{\mathrm{sys}}+H_{\mathrm{int}}+H_{\mathrm{bath}}, \end{equation} \tag{ 18 }$$

$$\begin{equation} H_{\mathrm{sys}} = \frac{\left[p_{\mathrm{J}}+C_{\mathrm{c}} V_{\mathrm{dc}}\right]^2}{2C_\Sigma}-E_{\mathrm{J}} \cos{\left(\frac{2e}{\hbar}\Phi_{\mathrm{J}}\right)}, \end{equation} \tag{ 19 }$$

$$\begin{equation} H_{\mathrm{int}}= \frac{C_{\mathrm{c}}}{C_\Sigma} (p_{\mathrm{J}}+C_{\mathrm{c}}V_{\mathrm{dc}}) \partial_t \Phi(0^-,t), \end{equation} \tag{ 20 }$$

$$\begin{equation} H_{\mathrm{bath}}= \frac{\left[C_{\mathrm{c}} \partial_t \Phi(0^-,t)\right]^2}{2C_\Sigma}+ \int_{-\infty}^0 \frac{p(x,t)^2}{2C_0} + \frac{\left[\partial_x \Phi(x,t)\right]^2}{2L_0} \,\mathrm{d}x. \end{equation} \tag{ 21 }$$

Thus, we have arrived at the Hamiltonian of a voltage-biased SCB, weakly coupled to the TL voltage at x = 0, i.e. V₀(t) = ∂_tΦ₀(t). (Here, we note that for the uncoupled TL, without SCB, Φ₀(t) = 2Φⁱⁿ(t) due to the perfect reflection.) Truncating the Hilbert space of H_sys to two levels, (18) is just the spin-boson Hamiltonian. From this point, we can proceed with a Bloch–Redfield derivation of a master equation for the SCB only [38]. By comparing to section 3.2 in [34], we also note that the equations of motion (15)–(16) can be interpreted as quantum Langevin equations (QLE) of the form

$$\begin{equation} \dot{Y}=\frac{\mathrm{i}}{\hbar}[H_{\mathrm{sys}},Y]+\frac{\mathrm{i}}{2\hbar}[\gamma \dot{X}-2\sqrt{\gamma v}\dot{A}^{\mathrm{in}},[X,Y]]_+, \end{equation} \tag{ 22 }$$

whereas (17) stands for the input–output relation

$$\begin{equation} A^{\mathrm{out}}(t)=A^{\mathrm{in}}(t)-\sqrt{\frac{\gamma}{v}}X(t) \end{equation} \tag{ 23 }$$

using the identifications Y =Φ_J, X = −(p_J + C_cV_dc), $A^{\mathrm {in}}=\sqrt {C_0} \Phi ^{\mathrm {in}}$ and where

$$\begin{equation} \gamma=Z_0\left(\frac{C_{\mathrm{c}}}{C_\Sigma}\right)^2 \end{equation} \tag{ 24 }$$

is the damping constant that accounts for spontaneous emission.

From the QLE (22), we can derive a master equation for the reduced density matrix of the SCB in the transmon regime. As the input field, we first consider a thermal background at temperature T, giving rise to a photon occupation number of

$$\begin{equation} n_\omega=\frac{1}{\exp{\left(\hbar\omega/k_{\mathrm{B}} T\right)}-1}. \end{equation} \noindent \tag{ 25 }$$

We assume that the density matrix initially can be written as a direct product, as well as Markovian properties and short correlation times for the TL variables. In the case when the damping (24) is much smaller than the system eigenenergies, we arrive, after also employing the rotating wave approximation, at the following quantum optical master equation,

$$\begin{eqnarray} &&\fl \dot{\rho}(t)=-\frac{\mathrm{i}}{\hbar} [ H_{\mathrm{sys}},\rho ] +\frac{2\gamma}{\hbar}\sum_m \omega_m [(n_{\omega_m}+1) {\cal D}(X^-_m)\rho + n_{\omega_m} {\cal D}(X^+_m)\rho] \end{eqnarray} \noindent \tag{ 26 }$$

with the Lindblad operator defined by $\mathcal {D}(c)\rho =c\rho c^{\dagger }-\frac {1}{2}\left (c^{\dagger }c\rho +\rho c^{\dagger }c\right )$. Also, X has been decomposed into eigenoperators of H_sys,

$$\begin{equation} \left[ H_{\mathrm{sys}} , X_m^{\pm}\right]=\pm\hbar\omega_mX_m^{\pm},\quad\omega_m>0, \end{equation} \noindent \tag{ 27 }$$

which is always possible as long as the eigenstates of H_sys form a complete set.

Projecting the master equation onto the SCB eigenstates |i〉, H_sys|i〉 = ω_i|i〉 (i∈{0,1,2,...}), we arrive at the following equation for the diagonal elements,

$$\begin{equation} \dot{\rho}_{ii}=\sum_{j\neq i} \Gamma_{ji} \rho_{jj} - \Gamma_{ij} \rho_{ii}, \end{equation} \noindent \tag{ 28 }$$

where the relaxation (ω_ij = ω_i − ω_J > 0 ) rates are

$$\begin{equation} \Gamma_{ij}= \frac{2\gamma}{\hbar} \omega_{ij} \left(1+n_{\omega_{ij}}\right) |\left\langle i\right|X\left|j\right\rangle|^2 \end{equation}\noindent \tag{ 29 }$$

and the excitation (ω_ij < 0) rates are

$$\begin{equation} \Gamma_{ij}= \frac{2\gamma}{\hbar} |\omega_{ij}| n_{\omega_{ij}}|\left\langle i\right|X\left|j\right\rangle|^2. \end{equation}\noindent \tag{ 30 }$$

Noting that X = −(p_J + C_cV_dc) is the charge operator, the matrix elements can be calculated numerically from the SCB Hamiltonian in (19). Denoting the SCB charging energy by E_C = e²/2C_Σ, the transmon regime is found for E_J ≫ E_C [33]. Here, the SCB spectrum approaches a linear oscillator with the junction plasma frequency $\omega _{\mathrm {p}}=\sqrt {8E_{\mathrm {J}}E_{\mathrm {C}}}/\hbar$, and the charge operator asymptotically couples only neighboring eigenstates [33]. We find the non-zero relaxation rates

$$\begin{equation} \Gamma_{(j+1)j}= \pi (j+1) \kappa^2 \frac{E_{\mathrm{J}}}{\hbar} \frac{Z_0}{R_K} (1+n_{\omega_{\mathrm{p}}}) \end{equation} \tag{ 31 }$$

and excitation rates

$$\begin{equation} \Gamma_{j(j+1)}= \pi (j+1) \kappa^2 \frac{E_{\mathrm{J}}}{\hbar} \frac{Z_0}{R_K} n_{\omega_{\mathrm{p}}}, \end{equation} \tag{ 32 }$$

where R_K = h/e² ≈ 25 k Ω denotes the quantum of resistance. The off-diagonal (i ≠ j) elements are subject to a pure exponential decay,

$$\begin{equation} \dot{\rho}_{ij}=-\gamma_{ij} \rho_{ij} \end{equation} \tag{ 33 }$$

with dephasing rates

$$\begin{equation} \gamma_{ij}=\Gamma^{i}_\phi+\Gamma^{j}_\phi+\frac{1}{2}\left(\sum_{k\neq i} \Gamma_{ik} + \sum_{k\neq j} \Gamma_{jk}\right), \end{equation} \tag{ 34 }$$

equal to half the sum of all rates for transitions from states |i〉 and |j〉, as well as the pure dephasing rates

$$\begin{equation} \Gamma^{k}_\phi=\frac{2\gamma}{\hbar}\frac{k_{\mathrm{B}} T}{\hbar} |\langle k|X|k\rangle|^2. \end{equation} \tag{ 35 }$$

The pure dephasing rates depend on the dc voltage, through the SCB spectrum, according to

$$\begin{equation} |\langle k|X|k\rangle|=\frac{e}{4E_{\mathrm{C}}}\left|\frac{\partial\omega_k(n_{\mathrm{g}})}{\partial n_{\mathrm{g}}}\right|, \end{equation}\noindent \tag{ 36 }$$

where n_g = C_cV_dc/2e is the dimensionless gate charge of the SCB. In the transmon regime the spectrum is well approximated by

$$\begin{equation} \omega_k(n_{\mathrm{g}})=\omega_k(n_{\mathrm{g}}=1/4)-\frac{\epsilon_k}{2}\cos{(2\pi n_{\mathrm{g}})}, \end{equation} \tag{ 37 }$$

where

$$\begin{equation} \epsilon_k \simeq(-1)^kE_{\mathrm{C}}\frac{2^{4k+5}}{k!}\sqrt{\frac{2}{\pi}}\left(\frac{E_{\mathrm{J}}}{2E_{\mathrm{C}}}\right)^{\frac{k}{2}+\frac{3}{4}}\mathrm{e}^{-\sqrt{8E_{\mathrm{J}}/E_{\mathrm{C}}}}, \end{equation}\noindent \tag{ 38 }$$

giving a maximum thermal pure dephasing rate (for n_g = ± 1/4) of

$$\begin{equation} \max{\Gamma^{k}_\phi}=\kappa^2 \frac{Z_0}{R_K} \frac{k_{\mathrm{B}} T}{\hbar}\frac{\pi^3}{8} \left|\frac{\epsilon_k}{E_{\mathrm{C}}}\right|^2. \end{equation}\noindent \tag{ 39 }$$

Here, we also note that in addition to small-amplitude thermal charge noise there can also be a slow but large-amplitude charge drift. In some cases, the effect of this drift can be taken into account by averaging over the range of transition frequencies involved. In the transmon regime, for the transition from |k〉 to |k + 1〉 this is given by _k+1 − _k ≈ _k+1.

In the next section, we will examine the scattering of coherent signals on the transmon in the two-level and three-level approximations. To include a coherent drive in the description, we take the input field Φⁱⁿ(t) to consist of a classical part Φⁱⁿ_cl(t) on top of the thermal background. Deriving the master equation for this case, it turns out that (26) is modified by adding the following time-dependent term to the system Hamiltonian,

$$\begin{equation} H_{\mathrm{d}}(t)=-2\sqrt{\frac{\gamma}{Z_0}}\dot{\Phi}_{\mathrm{cl}}^{\mathrm{in}}(t)X. \end{equation} \tag{ 40 }$$

In this section, we generalize the above master equation by adding more semi-infinite TLs to the SCB. First, by adding one more semi-infinite line, we arrive at the important case of an SCB capacitively coupled to an infinite TL. The discretized circuit is shown in figure 2(a), and the corresponding Hamiltonian is obtained from (1) by adding the TL terms for x > 0

$$\begin{equation} H'_{\mathrm{d}}=H_{\mathrm{d}} + \frac{1}{\Delta x} \sum_{i > 0} \left(\frac{p_{i}^2}{2C_0} + \frac{(\Phi_{i-1} -\Phi_{i} )^2}{2L_0}\right). \end{equation} \tag{ 41 }$$

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From a similar analysis as that above, we arrive at exactly the same master equation for the transmon's reduced density matrix, with the replacements

$$\begin{equation} \Phi^{\mathrm{in}}=\frac{1}{2}\left(\Phi^{\mathrm{in}}_{\mathrm{L}}+\Phi^{\mathrm{in}}_{\mathrm{R}}\right), \quad \tau_{\mathrm{RC}}=\frac{Z_0}{2}\frac{C_{\mathrm{c}} C_{\mathrm{J}} }{C_\Sigma},\ \gamma=\frac{Z_0}{2}\left(\frac{C_{\mathrm{c}}}{C_\Sigma}\right)^2 \end{equation} \tag{ 42 }$$

and the output fields are obtained from

$$\begin{equation} \Phi^{\mathrm{out}}_{\mathrm{L/R}}= \Phi^{\mathrm{in}}_{\mathrm{R/L}} + (\tau_{\mathrm{RC}}/C_{\mathrm{J}}) p_{\mathrm{J}}. \end{equation}\noindent \tag{ 43 }$$

We note that the damping constant γ as well as the RC time τ_RC are both halved compared to the semi-infinite case, since the impedance to ground is halved to Z₀/2. The in-field is the sum of the fields incoming from the left and right, but compared to the semi-infinite case the coupling coefficient is halved, since there is (almost) no reflection at x = 0. Indeed, for a more general scenario with N symmetrically coupled incident fields, as illustrated in figure 2(b), the mapping would be

$$\begin{equation} \Phi^{\mathrm{in}}=\frac{1}{N}\sum_{n=1}^N\Phi^{\mathrm{in}}_n,\quad \tau_{\mathrm{RC}}=\frac{Z_0}{N}\frac{C_{\mathrm{c}} C_{\mathrm{J}}}{C_\Sigma},\quad \gamma=\frac{Z_0}{N}\left(\frac{C_{\mathrm{c}}}{C_\Sigma}\right)^2 \end{equation} \tag{ 44 }$$

and using the relation Φ₀ = Φⁱⁿ_n + Φ^out_n (∀n), the output fields are given as

$$\begin{equation} \Phi^{\mathrm{out}}_n=\Phi_0-\Phi^{\mathrm{in}}_n=\left(\frac{2}{N}-1\right)\Phi^{\mathrm{in}}_n+\frac{\tau_{\mathrm{RC}}}{C_{\mathrm{J}}}p_{\mathrm{J}}+\frac{2}{N} \sum_{m\neq n}^N \Phi^{\mathrm{in}}_m. \end{equation} \tag{ 45 }$$

In this section, we examine the scattering of coherent signals on the transmon in an open TL. The input field is a constant coherent signal with a single frequency ω_p, close to resonance with the first transition frequency ω₁₀ of the transmon. Thus, we can safely describe the transmon as a two-level system. The master equation is given by (26) with a coherent drive and generalized to the case of an infinite TL (see sections 2.5 and 2.6), in the special case of only one system eigenfrequency ω₁₀. Moreover, we include an additional term due to pure dephasing, so that the total dephasing rates are given by (34). We represent our operators by the following Pauli matrices (using the notation X_ij ≡ 〈i|X|j〉):

$$\begin{equation} H_{\mathrm{sys}}=-\hbar\frac{\omega_{10}}{2}\sigma_z, \quad X^{\pm}=\pm \mathrm{i}|X_{10}|\sigma^{\pm}. \end{equation} \tag{ 46 }$$

Below, we will determine reflection and transmission coefficients for coherent signals scattered on the transmon. In the previous section, the incoming and outgoing fields were described in terms of the flux, since that gives a simpler description of the transmon. However, the voltage is a more intuitive quantity than the flux and is also usually what is measured in experiments. Therefore, in this section, we will describe the inputs and outputs in terms of the voltage.

We consider an incoming coherent voltage field,

$$\begin{equation} V^{\mathrm{in}}_{\mathrm{L}}(t)=\Omega_p\sin{\omega_{\mathrm{p}}t}, \end{equation} \tag{ 47 }$$

impinging on the transmon from the left. For simplicity, we set the temperature to zero (n_ω10 = 0). The reflected voltage field is the output to the left of the transmon. Using (42) and (43), we have

$$\begin{equation} V^{\mathrm{out}}_{\mathrm{L}}(t)=-\sqrt{\frac{\gamma Z_0}{2}}\langle\dot{X}(t)\rangle, \end{equation} \tag{ 48 }$$

where the expectation value can be written as

$$\begin{equation} \langle \dot{X}(t)\rangle={\rm i}\omega_{10}\left(\langle X^+(t)\rangle-\langle X^-(t)\rangle\right)=-\omega_{10}|X_{10}|\langle\sigma^x\rangle. \end{equation} \tag{ 49 }$$

Inserting (49) into (48) yields

$$\begin{equation} V^{\mathrm{out}}_{\mathrm{L}}(t)=\frac{1}{2}\sqrt{\hbar\omega_{10}\Gamma_{10}Z_0}\langle\sigma^x\rangle=\sqrt{\hbar\omega_{10}\Gamma_{10}Z_0}\it{Re}\left[\rho_{01}\right], \end{equation} \tag{ 50 }$$

where ρ₀₁ is a density matrix element in the transmon eigenbasis.

To solve the master equation, we perform a unitary transformation to a frame rotating with the driving frequency ω_p. In this frame, the equation becomes time independent after employing the rotating-wave approximation. Solving the equation in the steady state ($\dot {\rho }=0$) and transforming back to the non-rotaing frame yields the following expression for the desired density matrix element,

$$\begin{equation} \rho_{01}=\frac{1}{2}\frac{\sqrt{\hbar\omega_{10}\Gamma_{10}Z_0}\left( \Delta+\mathrm{i}\gamma_{10}\right)\Omega_p}{\hbar\omega_{10}Z_0\gamma_{10}^2+\hbar\omega_{10}Z_0\Delta^2+\gamma_{10}\Omega_{\mathrm{p}}^2}\mathrm{e}^{\mathrm{i}\omega_{\mathrm{p}}t}, \end{equation} \tag{ 51 }$$

where Δ ≡ ω_p − ω₁₀ is the detuning. Now, plugging this expression into (50) results in

$$\begin{equation} V_{\mathrm{L}}^{\mathrm{out}}(t)=-\frac{\Omega_p}{2}\frac{\sin{\omega_{\mathrm{p}}t}-\frac{\Delta}{\gamma_{10}}\cos{\omega_{\mathrm{p}}t}}{\frac{\gamma_{10}}{\Gamma_{10}}+\frac{\Delta^2}{\Gamma_{10}\gamma_{10}}+2\frac{N_{\mathrm{in}},}{\Gamma_{10}}}, \end{equation} \tag{ 52 }$$

where N_in = Ω²_p/(2Z₀ℏω₁₀) is the average number of incoming photons per second. Thus, the reflection coefficient for the negative frequency part of the field is given by

$$\begin{equation} r=-r_0\frac{1-\mathrm{i}\frac{\Delta}{\gamma_{10}}}{1+\left(\frac{\Delta}{\gamma_{10}}\right)^2+2\frac{N_{\mathrm{in}}}{\gamma_{10}}}, \end{equation} \tag{ 53 }$$

with r₀ ≡ Γ₁₀/2γ₁₀. For the transmitted field, (43) yields

$$\begin{equation} V^{\mathrm{out}}_{\mathrm{R}}(t)=V^{\mathrm{in}}_{\mathrm{L}}(t)-\sqrt{\frac{\gamma Z_0}{2}}\langle\dot{X}(t)\rangle \end{equation} \tag{ 54 }$$

which directly gives us the following expression for the transmission coefficient:

$$\begin{equation} t=1+r=\frac{1-r_0+\left(\frac{\Delta}{\gamma_{10}}\right)^2+2\frac{N_{\mathrm{in}}}{\gamma_{10}}+\mathrm{i}r_0\frac{\Delta}{\gamma_{10}}}{1+\left(\frac{\Delta}{\gamma_{10}}\right)^2+2\frac{N_{\mathrm{in}}}{\gamma_{10}}}. \end{equation} \tag{ 55 }$$

In figure 3, we plot the reflectance R = |r|² and transmittance T = |t|² as a function of the detuning, in the case of a weak input signal and no pure dephasing. For a resonant drive (Δ = 0), we see that perfect reflection is approached, in agreement with [12, 13, 39].

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In section 3.1, we showed that a low-amplitude input signal is totally reflected when it resonantly scatters off a transmon in the two-level approximation. In this section, we instead study the scattering off of a transmon in the three-level approximation. By strongly driving the second transition, the transmon becomes transparent to frequencies in resonance with the first transition. This effect is due to the Autler–Townes splitting and has been observed in recent experiments [22].

We consider an incoming voltage field from the left, consisting of a probe field Ω_p sin ω_pt close to resonance with the first transition (with detuning Δ_p = ω_p − ω₁₀) and a control field Ω_c sin ω_ct close to resonance with the second transition (with detuning Δ_c = ω_c − ω₂₁). Figure 4 shows the energy levels of the transmon in this approximation. In the transmon eigenbasis, the relevant operators are (with the ground state energy ω₀ = 0)

$$\begin{equation} H_{\mathrm{sys}}=\hbar\sum_{i=1}^2\omega_i \left|i\right\rangle\left\langle i\right|, \end{equation} \tag{ 56 }$$

$$\begin{equation} X=\mathrm{i}\sum_{i=1}^{2}|X_{i(i-1)}|(\sigma_i^+-\sigma_i^-) \end{equation} \tag{ 57 }$$

with σ⁺_i = |i〉〈i − 1| and σ⁻_i = (σ⁺_i)^†. In the same way as in the two-level case ((49)–(50)), we obtain the following expression for the reflected signal:

$$\begin{equation} V^{\mathrm{out}}_{\mathrm{L}}(t)=-\sqrt{\frac{\gamma Z_0}{2}}\langle\dot{X}(t)\rangle=\frac{1}{2}\sum_{i=1}^2\sqrt{\hbar\omega_{i(i-1)}Z_0\Gamma_{i(i-1)}}\langle\sigma_i^x\rangle, \end{equation} \tag{ 58 }$$

with σ^x_i = σ⁺_i + σ⁻_i. Thus, the reflected field consists of one part with frequencies around the probe frequency ω_p and one part with frequencies around the control frequency ω_c. Since we are interested in the reflectance and transmittance properties of the probe, we concentrate on the corresponding part of the reflected field

$$\begin{equation} V_{\mathrm{p}}^{\mathrm{ref}}(t)=\frac{1}{2}\sqrt{\hbar\omega_{10}Z_0\Gamma_{10}}\langle\sigma_1^x\rangle=\sqrt{\hbar\omega_{10}Z_0\Gamma_{10}}\it{Re}(\rho_{10}). \end{equation} \tag{ 59 }$$

The master equation is given by (26) for the case of two system eigenfrequencies, again with a coherent drive and generalized to the case of an infinite TL (see sections 2.5 and 2.6). Also, terms accounting for pure dephasing are added. To transform the master equation into a time-independent picture, we use the following unitary transformation matrix,

$$\begin{equation} U(t)= \left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ 0 & \mathrm{e}^{-\mathrm{i}\omega_{\mathrm{p}} t} & 0 \\ 0 & 0 & \mathrm{e}^{-\mathrm{i}(\omega_{\mathrm{p}}+\omega_{\mathrm{c}})t} \end{array} \right) \end{equation} \tag{ 60 }$$

and employ the rotating-wave approximation. As before, we solve the master equation in the steady state to determine ρ₁₀, but we now consider two different cases.

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Firstly, by setting Ω_c = 0, we recover exactly the same expression for the reflected field as in the two-level case. Thus, with the control field turned off, we see almost full reflection for weak probe fields in resonance with the first transition frequency of the transmon.

Secondly, we consider the case of a strong control field (Ω_c ≫ Ω_p). Solving the master equation and expanding ρ₁₀ to first order in (Ω_p/Ω_c), we obtain the following expression:

$$\begin{equation} \rho_{10}^{(1)}=-\frac{2\mathrm{i}\hbar\omega_{21}Z_0\sqrt{\frac{\Gamma_{10}}{\hbar\omega_{10}Z_0}}\left(\gamma_{20}-\mathrm{i}\left(\Delta_{\mathrm{c}}+\Delta_{\mathrm{p}}\right)\right)\Omega_{\mathrm{p}}}{4\hbar\omega_{21}Z_0\left(\gamma_{10}-\mathrm{i}\Delta_p\right)\left(\gamma_{20}-\mathrm{i}\left(\Delta_{\mathrm{c}}+\Delta_{\mathrm{p}}\right)\right)+\Gamma_{21}\Omega_{\mathrm{c}}^2}\mathrm{e}^{-\mathrm{i}\omega_{\mathrm{p}}t}. \end{equation}\noindent \tag{ 61 }$$

Inserting (61) into (59), we can determine the reflection coefficient. For a resonant control field (Δ_c = 0), the result is

$$\begin{equation} r=-\frac{2\Gamma_{10}\left(\gamma_{20}^2+\Delta_{\mathrm{p}}^2\right)\left(\gamma_{10}-\mathrm{i}\Delta_{\mathrm{p}}\right)+\Gamma_{10}\Gamma_{21}\left(\gamma_{20}+\mathrm{i}\Delta_{\mathrm{p}}\right) N^{\mathrm{c}}_{\mathrm{in}}}{4\left(\gamma_{10}^2+\Delta_{\mathrm{p}}^2\right)\left(\gamma_{20}^2+\Delta_{\mathrm{p}}^2\right)+4\Gamma_{21}\left(\gamma_{10}\gamma_{20}-\Delta_{\mathrm{p}}^2\right) N_{\mathrm{in}}^{\mathrm{c}}+\Gamma_{21}^2N_{\mathrm{in}}^{\mathrm{c}2}}, \end{equation}\noindent \tag{ 62 }$$

where N^c_in = Ω²_c/(2Z₀ℏω₂₁) is the average number of incoming photons per second in the control field. The transmission coefficient is again given by t = 1 + r. Figure 5 shows the transmittance T = |t|² for different probe detunings and control field strengths. In these plots, we have neglected pure dephasing and used (29) to express Γ₂₁ in terms of Γ₁₀.

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We can clearly see that, for strong control fields, the transmittance of a resonant probe approaches unity. Thus, by turning on and off a strong resonant control field, we can switch between the cases of full transmission and full reflection for the resonant probe.

In a recent experiment [23], the second-order statistics of the field scattered off a transmon was measured. In this section, inspired by the experiment, we analyze the second-order correlation functions in our system. The normalized second-order correlation function is in the steady state given as [40]

$$\begin{equation} g^{(2)}(\tau)=\frac{\langle V^+(t)V^+(t+\tau)V^-(t+\tau)V^-(t)\rangle}{\langle V^+(t)V^-(t)\rangle^2} \end{equation} \tag{ 63 }$$

and is proportional to the conditional probability of detecting a photon at time t + τ, given that one was detected at time t. Here, V^±(t) are the positive and negative frequency parts of the voltage field.

We calculate g⁽²⁾(τ) for the transmitted and reflected fields from a transmon driven by a resonant coherent signal. We treat the transmon as a two-level system and use the same notation as in section 3.1. To be able to compare with the experiments in [23], we perform the calculations for finite temperatures and a finite detection bandwidth on the output signal. For zero temperature and infinite bandwidth, we recover the results of [41]: perfect antibunching in the reflected field and bunching in the transmitted field.

Including the effect of a finite detection bandwidth is straighforward by including a filter in the calculations. The approach we have taken is to model the filter by a single-mode TL resonator in resonance with the transmon, with the Hamiltonian

$$\begin{equation} H_{\mathrm{res}}=\hbar\omega_{10}a^{\dagger}a \end{equation} \tag{ 64 }$$

and cascade it with the transmon. We start from the QLE (22) for the transmon, generalized to the case of an infinite TL (see section 2.6), and a similar equation for the resonator. Our coherent input signal is the voltage field Vⁱⁿ_L(t) = Ω_d sin ω_dt, just like in section 3.1. We can then use the formalism of cascaded quantum systems in [34] to arrive at a master equation for the joint density matrix of the transmon and the resonator. In this formalism, the output from the transmon (reflected or transmitted) is taken as input to the resonator, without any signals going the opposite way (see figure 6). For the field reflected from the transmon, the resulting master equation is

$$\begin{eqnarray} &&\fl \dot{\rho}=\frac{\mathrm{i}}{\hbar}[\rho,H_{\mathrm{sys}}+H_{\mathrm{res}}]+\Gamma_{10}\mathcal{D}(\sigma^-)\rho+\Gamma_{01}\mathcal{D}(\sigma^+)\rho+\gamma_{\mathrm{BW}}\left[\left(\frac{n_{\omega_{10}}}{2}+1\right)\mathcal{D}(a)\rho+\frac{n_{\omega_{10}}}{2}\mathcal{D}(a^{\dagger})\rho\right]\nonumber\\ &&+\frac{1}{2} \mathrm{i}\sqrt{\Gamma_{10}(n_{\omega_{10}}+1)\gamma_{\mathrm{BW}}}\left( [a,\rho \sigma^+]+[a^{\dagger},\sigma^-\rho]\right)\nonumber\\ &&+\frac{1}{2} \mathrm{i}\sqrt{\Gamma_{01}n_{\omega_{10}}\gamma_{\mathrm{BW}}}\left( [\sigma^+\rho,a]+[\rho\sigma^-,a^{\dagger}]\right)+\mathrm{i}\sqrt{\frac{\Gamma_{10}N_{\mathrm{in}}}{2(n_{\omega_{10}}+1)}}[\rho,\sigma^x], \end{eqnarray} \tag{ 65 }$$

where we have denoted the filter bandwidth by γ_BW. For the field transmitted through the transmon, (65) is modified by simply adding the following term to the right-hand side:

$$\begin{equation} \sqrt{\frac{\gamma_{\mathrm{BW}}N_{\mathrm{in}}}{2}}[\rho,a^{\dagger}-a]. \end{equation} \tag{ 66 }$$

The output we are interested in is the voltage field leaking out at the right side of the resonator, whose positive and negative frequency parts are proportional to a(t) and a^†(t), respectively. Thus, g⁽²⁾(τ) can be calculated as

$$\begin{equation} g^{(2)}(\tau)=\frac{\langle a^{\dagger}(t)a^{\dagger}(t+\tau)a(t+\tau)a(t)\rangle}{\langle a^{\dagger}(t)a(t)\rangle^2} =\frac{{\rm Tr}\left[ a^{\dagger}aP(\tau)(a\rho_{\mathrm{s}} a^{\dagger})\right]}{{\rm Tr}\left[ a^{\dagger}a\rho_{\mathrm{s}}\right]^2}, \end{equation} \tag{ 67 }$$

where ρ_s is the steady-state density matrix and P(τ) the propagator super-operator, defined by ρ(t + τ) = P(τ)ρ(t). Both ρ_s and P(τ) are obtained by solving the master equations (65) and (66). For the case without filter, g⁽²⁾(τ) for the reflected field is given by (67) with a replaced by σ⁻. Since (σ⁻)² = 0, it directly follows that g⁽²⁾(0) = 0, i.e. perfect antibunching.

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In figure 7(a), we plot g⁽²⁾(τ) for the reflected field for different temperatures and detection bandwidths. Typical parameter values from recent experiments [23] are used. For zero temperature and large bandwidth we see perfect antibunching, as expected. For a decreased bandwidth the full time dynamics of the antibunching cannot be resolved, which results in a less pronounced antibunching dip. For finite temperatures, we see even less antibunching, due to a nonzero probability of detecting bunched thermal photons. In figure 7(b), we plot g⁽²⁾(τ) for the transmitted field for different temperatures and detection bandwidths. Here we see a decrease of the superbunching for higher temperatures and smaller bandwidths. These results explain the qualitative features of the experimental data in [23] well.

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