Correlated Dephasing Noise in Single-photon Scattering

We develop a theoretical framework to describe the scattering of photons against a two-level quantum emitter with arbitrary correlated dephasing noise. This is particularly relevant to waveguide-QED setups with solid-state emitters, such as superconducting qubits or quantum dots, which couple to complex dephasing environments in addition to the propagating photons along the waveguide. Combining input-output theory and stochastic methods, we predict the effect of correlated dephasing in single-photon transmission experiments with weak coherent inputs. We discuss homodyne detection and photon counting of the scattered photons and show that both measurements give the modulus and phase of the single-photon transmittance despite the presence of noise and dissipation. In addition, we demonstrate that these spectroscopic measurements contain the same information as standard time-resolved Ramsey interferometry, and thus they can be used to fully characterize the noise correlations without direct access to the emitter. The method is exemplified with paradigmatic correlated dephasing models such as colored Gaussian noise, white noise, telegraph noise, and 1/f-noise, as typically encountered in solid-state environments.


Introduction
The field of waveguide-QED describes a variety of experimental setups where a quantum emitter interacts preferentially with a family of guided photonic modes, so that the emission rates γ ± into the waveguide approaches or even surpasses decay γ loss into unwanted modes (see Figure 1). This regime is achieved in experiments with superconducting circuits [1][2][3][4], neutral atoms [5,6] and quantum dots in photonic crystals [7]. With a few exceptions, such as [8,9], most experiments work in the Rotating-Wave Approximation (RWA) regime, allowing for a simple description in terms of one-and few-photon wavefunctions [10,11], inputoutput theory [12][13][14] and path integral formalism [15,16]. Those descriptions usually do not account for other sources of error, such as dephasing, but we know that 1/f-noise severely affects all solid state devices [17], including quantum dots and superconducting circuits. There have been some experimental attempts at characterizing noise sources outside actual circuits, directly exploring the dynamics of the quantum scatterer using time-resolved methods [18][19][20][21][22][23][24][25][26][27][28][29] or Fourier transform spectroscopy [30][31][32][33][34][35]. Those detailed studies require time-resolved measurements and direct control of the quantum scatterer in many cases, something which may be unfeasible or undesirable in waveguide-QED setups. Figure 1. A noisy qubit with arbitrary correlated dephasing noise ∆(t) couples with rates γ ± to right-and left-propagating photons along a 1D waveguide. The qubit also decays with rate γ loss into unguided modes. We model dephasing as a stochastic process, and photon scattering via input-output theory with operators a ± in (t) and a ± out (t).
The purpose of this work is to develop a framework of waveguide-QED and scattering theory that accounts for general correlated dephasing, teaching us how to probe a qubit's noise and environment using few-photon scattering experiments. There are earlier works connecting noise with spectroscopy: Kubo's fluctuation-dissipation relations links dephasing to lineshapes in nuclear magnetic resonance (NMR) [36], as do later works in the field of quantum chemistry [37]. Our study complements those works, focusing on the quantum mechanical processes associated to single-and multiphoton scattering in waveguide-QED. We develop a stochastic version of the waveguide-QED input-output formalism that includes dephasing noise in the energy levels of the quantum emitter. We relate the correlations in that noise to the average scattering matrix of individual photons and coherent wavepackets, and develop strategies to extract those correlations from actual experiments, in conjunction with earlier approaches to scattering tomography [38].
The paper and our main results are organized as follows. In Sec. 2 we introduce the model for a noisy two-level emitter in a waveguide. We describe dephasing noise as a stationary stochastic process ∆(t) and derive stochastic input-output equations. In Sec. 3 we solve these equations in the case of a standard Ramsey sequence. We quantify the decay of the averaged coherence via the Ramsey envelope C φ (t) which provides time-resolved information about the noise correlations of the environment. Section 4 shows that the same information provided by Ramsey spectroscopy can be obtained from single-photon scattering experiments, where we only manipulate the qubit through the scattered photons. More precisely, we solve the stochastic input-output equations for a qubit that interacts with a single propagating photon, and show that the averaged single-photon scattering matrix can be related one-toone to the Ramsey envelope. We also discuss analytical predictions for scattering under realistic dephasing models such as colored Gaussian noise and 1/f noise. We show how the noise correlations modify the spectral lineshapes on each case, and recover simple limits such as the Lorentzian profiles that are fitted in most waveguide-QED experiments. Section 5 generalizes these ideas, showing how to measure the averaged scattering matrix using weak coherent state inputs together with homodyne or photon number measurements, and how to reconstruct the Ramsey envelope C φ (t) from such spectroscopic measurements. This opens the door to the reconstruction of more general correlated noise models that are non-Gaussian but common in many solid-state environments such as telegraph noise and 1/f noise due to ensembles of two-level fluctuators. This is treated separately in Appendix A due to the higher complexity of the stochastic methods needed for the analysis. We close this work in Sec. 6, discussing the conclusions and open questions.

Model for a noisy qubit in a photonic waveguide
Our study considers the setup depicted in Figure 1: a two-level quantum emitter or qubit relaxes into a 1D photonic environment, emitting photons with rates γ ± along opposite directions, while simultaneously interacting with an environment that induces both dephasing and unwanted dissipation. We model environment-induced dephasing as a stochastic fluctuation of the qubit frequency around a mean value ω 0 ω 0 (t) = ω 0 + ∆(t). (1) The stochastic process ∆(t) [39][40][41] has vanishing mean -i.e. the stochastic average . . . over noise realizations is zero ∆(t) = 0 -, and is stationary -i.e. all expectation values and noise correlations ∆(t 1 ) . . . ∆(t n ) are invariant under a global shift in time-. The simplest autocorrelation function ∆(0)∆(τ ) defines a characteristic correlation time τ c of the noise as, Those conditions and the machinery of stochastic methods [39][40][41] account for any realistic source of qubit dephasing, including arbitrary correlated Markovian and non-Markovian noise, or 1/f noise, among the examples considered below. We treat the usual regimes of qubit-waveguide interaction with the RWA and Markov approximations, under the input-output formalism [12,42]. In the Heisenberg picture, the operator a ± in (t) describe the input field of photons propagating to the right (+) and left (−) of the waveguide, which couple to the qubit with strengths γ ± , respectively (see Figure 1). After interacting with the qubit, the photons leave the waveguide through the right (+) and left (−) output ports, and are characterized by standard output operators a ± out (t) given by [12,42] The action of the qubit on the output photons is described by the lowering operator σ − = |g e|, defined for the qubit transition between ground and excited states, |g and |e . In addition, the qubit can also couple with rate γ loss to unguided photons outside the waveguide, which are characterized by independent input-output operators a loss in (t) and a loss out (t). The full dynamics of the noisy qubit coupled to the waveguide is governed by the quantum Langevin equations [12,42,43] The total decay of the qubit Γ = γ + γ loss , combines the emission of the qubit into guided γ = γ + + γ − and unguided modes γ loss . While typical qubit-waveguide couplings are symmetric γ ± = γ/2, our formalism with independent channels (µ = ±) naturally admits the possibility of a 'chiral' waveguide with different couplings to left-and right-moving photons γ − = γ + [44,45]. Due to the random classical field ∆(t), the equations of motion (4) and (5) are stochastic differential equations. In such equations, each particular realization of the noise provides different quantum expectation values σ − or σ z , and we need to average over all possible noise realizations to obtain more meaningful and measurable values -i.e. σ − or σ z , as well as higher order multi-time correlations if needed-. In the remainder of the paper we calculate this kind of stochastic averages to characterize the effect of correlated dephasing noise on both the time-resolved dynamics (Sec. 3) and the single-photon spectroscopy (Secs. 4-5) of the qubit.

Time-resolved characterization of correlated dephasing noise
Ramsey interferometry [18,19,24] is the most common way to characterize qubit decoherence. This and other time-resolved methods require full control and read-out of the qubit, while it is in contact with its environment [see Figure 2(a)]. These methods are difficult, but give detailed information about noise correlations, specially when combined with with dynamical decoupling [21,26] and other control techniques [22,25,27,28]. In the following we shortly summarize the procedure of a basic Ramsey sequence, and then discuss how it works for colored Gaussian noise [39][40][41] -a paradigmatic dephasing model that is analytically solvable for arbitrary noise correlation times τ c . Understanding these concepts will be essential to study the effect of correlated noise in the scattering dynamics of the next sections.

Ramsey interferometry
A standard Ramsey sequence consists of the five steps from figure 2(b): (i) Preparation of the qubit in its ground state |g , (ii) application of a Hadamard gate H(0) with a very fast π/2 pulse, (iii) evolution of the qubit for a time t, (iv) application of a second Hadamard gate H(t), and (v) measurement of the qubit population difference σ z . The purpose of steps (i)-(ii) is to produce the initial superposition state,  Figure 2. Time-resolved characterization of correlated dephasing noise. (a) A qubit coupled to a generic noisy environment causing pure dephasing ∆(t), and radiative relaxation with rate Γ. (b) Basic Ramsey sequence consisting of (i) ground state preparation, (ii) Hadamard gate H(0), (iii) free evolution, (iv) second Hadamard gate H(t), and (v) measurement of the qubit population difference σ z (t) . (c) Ramsey envelopes C φ (t) for a noisy qubit with colored Gaussian dephasing, characterized by the noise strength σ and the correlation time τ c = 1/κ (see Eq. (12)). For κ = 10σ the noise is in the white limit and C φ (t) is exponential (blue/solid line). For κ = 0 the noise is quasi-static and C φ (t) is Gaussian (red/dashed line). Finally, for κ = 2σ the decay interpolates between two previous behaviors.
for which the qubit coherence is maximal, namely σ − (0) = 1/2. Steps (iii)-(v) monitor the destruction of the qubit coherence σ − (t) , under the influence the noisy environment. Repeating this procedure for various waiting times t and averaging over many realizations, one obtains the average coherence σ − (t).
The dynamics of the qubit coherence under the influence of pure dephasing and radiative decay is obtained by taking expectation values on the quantum Langevin equation (4). For the initial condition (6), it reads which is a multiplicative stochastic differential equation with a random variable ∆(t) [39][40][41].
To solve for the average σ − (t), we integrate equation (7) formally and average the result over all stochastic realizations of the random trajectory ∆(t), obtaining In addition to the exponential decay due to the radiative coupling Γ, pure dephasing originates an extra decay factor C φ (t) known as Kubo's relaxation function [36] or "Ramsey envelope" [22]. For stationary noise, C φ (t) is the average of the random phase accummulated by the qubit after a time t, In general, this function depends on noise correlations of arbitrary order ∆(t 1 ) . . . ∆(t n ) that require sophisticated noise spectroscopy methods [27,46], but for Gaussian noise we will find that only first and second moments are required, as shown below.

Colored Gaussian noise, white noise, and quasi-static noise
For stationary Gaussian noise with vanishing mean, all cummulants and correlations can be expressed in terms of the autocorrelation ∆(0)∆(τ ) [36,40], and thus C φ (t) in Eq. (9) is reduced to If the noise is also Markovian, Doob's theorem [40] implies that the noise can be described as an Ornstein-Uhlenbeck process [39][40][41] with autocorrelation given explicitly by This rich model describes "colored Gaussian noise" with strength σ = ∆ 2 (0) 1/2 and a correlation time τ c = 1/κ, that covers both fast and slow noise limits. This includes white noise in the limit κ → ∞, with autocorrelation (11) ∆(0)∆(τ ) = 2γ φ δ(τ ), and a pure dephasing rate γ φ = σ 2 /κ. It also includes quasi-static noise in the opposite limit of κ → 0, in which the autocorrelation becomes constant ∆(0)∆(τ ) = σ 2 . Another advantage of the colored Gaussian noise (11) is that the Ramsey envelope (10) can be derived analytically This super-exponential envelope has been fitted to experimental data to quantify the strength and correlation of realistic environments [18,22]. Figure 2(c) shows the typical shape of this envelope in the the limits of fast and slow noise. In the white noise limit (κ σ, blue/solid line), the decay is exponential C φ (t) = exp(−γ φ t) with γ φ = σ 2 /κ; in the quasi-static limit limit (κ σ, red/dashed), the decay is Gaussian C φ (t) = exp(−σ 2 t 2 /2); and for intermediate values such as κ = 2σ, the curve clearly interpolates between both shapes (black/dotted).

Single-photon scattering from a qubit with correlated dephasing noise
In this section, we use the input-output formalism [12] to compute the average single-photon scattering matrix for a qubit with stationary dephasing noise ∆(t). We find expressions for the average single-photon transmittance and reflectance, relating them to the time-resolved Ramsey experiments and Kubo's relaxation functions from previous section. Finally, we evaluate the average transmittance for a qubit with colored Gaussian and 1/f noise and discuss the broadening of its spectral lineshape on each case.

Average single-photon scattering matrix
The single-photon scattering matrix S λµ νω describes the interaction between an isolated photon and a quantum emitter. It is defined as the probability amplitude for the emitter to transform an incoming photon with frequency ω in channel µ = ± into an outgoing photon with possibly different frequency ν and direction λ = ±: The monochromatic input-output photonic operators a µ in (ω) and a µ out (ω) are given by the Fourier transform F of the Heisenberg input-output field amplitudes defined above [12]: with for a test function f (t). Notice that these monochromatic operators (14) satisfy canonical bosonic commutation relations as well as . The scattering matrix of the noisy qubit is derived by combining equations (3), (13), and (14) to obtain Here, the scattering overlap, in † (ω)|0 satisfies an inhomogeneous stocahstic differential equation derived from Eqs. (4)- (5), with initial condition G ω (−∞) = 0. This is similar to the equation for the qubit's coherence (7), but now including a constant source term. As explained in Sec. 5, spectroscopic measurements are not related to S but to the average scattering matrix S λµ νω . Computing this quantity is a two-step process. First, equation (16) is integrated formally for a stationary noise ∆(t). Using the stationary noise property , we obtain the averaged solution Note how the noise correlations enter via the Kubo relaxation function C φ (t) in Eq.
The second step is to take a Fourier transform of G, which is trivially given by . The total averaged scattering matrix then reads and conserves the energy of the scattered photons on average. In practice, each experimental realization of the noise ∆(t) will cause a different energy shift and broadening of the scattered photons, and the total system must be described by a mixed state. However, this is not relevant when we focus on the averaged single-photon transmittance t µ ω and reflectance r µ ω , given by which measure the photons on the same (λ = µ) and opposite (λ = −µ) output channel with respect to the input beam µ. These relations have deep physical meaning as they allow us to predict the spectroscopic lineshape of a noisy qubit from the knowledge of the Ramsey envelope C φ (t) obtained via standard time-resolved noise experiments. In the following we illustrate this procedure with a qubit subject to dephasing due to colored Gaussian noise (Sec. 4.2), and 1/f noise (Sec. 4.3). We leave to Appendix A the study of more general non-Gaussian correlated noise models. Finally, we would like to remark that the present derivation may be extended to include multiple noise sources. In particular, Appendix B shows that adding a white noise background ∆ WB (t) to correlated noise, i.e. ∆(t) → ∆(t) + ∆ WB (t), amounts to a trivial replacement Γ/2 → Γ/2 + γ WB in the stochastic equation (16), where γ WB is the pure dephasing rate of the white noise background.

Average transmittance of qubit with colored Gaussian dephasing
In spectroscopy, correlated dephasing is typically referred to as spectral diffusion [31][32][33][34] because it broadens the lineshapes of emitters. In this subsection we analyze this broadening and the average single-photon transmittance t µ ω of a qubit with colored Gaussian dephasing noise (see Sec. 3.2 for details on the model), paying special attention to the limits of white and quasi-static noise where the transmittance exhibits qualitatively different behaviors.
Equation (19) provides an expression for the single-photon transmittance t µ ω in terms of the analytical Ramsey envelope in Eq. (12). For colored Gaussian noise with arbitrary correlation time τ c = 1/κ and noise strength σ we can either estimate numerically the Laplace transform, or expand the super-exponential function in a power series to obtain In the limit of white noise, κ → ∞, only the term with n = 0 survives in Eq. (21), and the average transmittance is a Lorentzian function ‡, with pure dephasing rate γ φ = σ 2 /κ. This is a well-known result, typically proven via the master equation formalism [47], which demonstrates that white noise dephasing maintains the natural Lorentzian lineshape of the qubit, while its width and depth get modified by γ φ [1,48,49]. This Lorentzian behavior is shown by the blue/solid transmittance in Figure  3(a), for typical waveguide QED parameters. If we now consider a finite but moderate correlation time σ κ < ∞, more and more terms in the series expansion (21) become important, resulting in a transmittance with a larger width and smaller depth, as shown by the black/dash-dotted curve in Figure 3(a). Finally, in the quasi-static limit of very long correlation times κ σ, all terms in Eq. (21) contribute and the series expansion fails to converge numerically. In this case, we make the approximation κ → 0 in which the relaxation function (12) is Gaussian, C φ (t) = exp(−σ 2 t 2 /2), and perform the required Laplace transform ‡ In the white noise limit, Eq. (12) becomes the exponential C φ (t) = exp(−γ φ t), so that the Lorentzian lineshape follows directly from the Laplace transform in Eq. (19). of a noisy qubit coupled to a 1D photonic waveguide with parameters γ ± = γ/2, γ = 0.9Γ, and γ loss = 0.1Γ. We consider a colored Gaussian dephasing model, characterized by the correlation time τ c = 1/κ and the noise strength σ = Γ. For κ = 10σ, the noise is in the white limit and the transmittance has a Lorentzian lineshape (blue/solid). For κ = 0 the noise is quasi-static and t µ ω is Gaussian-like as given in Eq. (23) (red/dashed). Finally, for κ = 2σ the transmittance interpolates between two previous behaviors (black/dash-dotted). (b) Measurement of t µ ω of a noisy qubit in a waveguide, using a coherent state input (Ω Γ), and homodyne or power measurements at the output.
analytically, obtaining § with the complementary error function erfc( (23) we conclude that in the slow noise limit κ σ, t µ ω is Gaussian-like and has a width proportional to the noise strength σ. This behavior is shown by the red/dashed transmittance from figure 3(a). Notice that the Lorentzian (blue/solid) and Gaussian-like (red/dashed) lineshape limits can be qualitatively distinguished in transmittance experiments by their width, curvature, and tails [50], suggesting that spectroscopy can be a simple approach to discover the noise correlation properties. More specifically, fitting arbitrary parameters κ and σ to experimental transmittance data t µ ω one may even quantify the correlation time τ c = 1/κ and the noise strength σ of a given environment as recently done in Ref. [3].
Colored Gaussian noise is a useful and powerful dephasing model, and thus it is tempting to assume that this is the real noise. Indeed, this is what is done in most common waveguide-QED experiments, where a Lorentzian profile is assumed and a single dephasing parameter γ φ is fitted [1]. In Sec. 5 we will show that there is a more general approach, using estimates of the transmittance t µ ω to extract the Ramsey profile and noise correlations, in a single-photon scattering protocol that generalizes current experiments [see Figure 3 (b)]. § This expression can also be obtained by a simple static average over noiseless Lorentzian transmittances with different qubit frequencies as t µ is a Gaussian probability distribution with standard deviation σ.

Average transmittance of a qubit with 1/f dephasing noise
In this subsection consider a noisy qubit with dephasing due to 1/f noise, a very slowly varying, highly correlated, and low-frequency noise that is ubiquitously encountered in electronics and solid-state devices such as superconducting qubits or quantum dots [17]. Nowadays there is still ongoing research on the microscopic origin and universal mechanisms behind this type of noise [51][52][53][54][55], but an unquestionable experimental evidence is that its noise power spectrum, , presents a power-law behavior S(ω) ∝ 1/ω η , with 0 < η < 2. In fact, it is exactly this low frequency divergence what makes 1/f noise so difficult to filter and to controllably observe in experiments [17].
There have been various proposals for phenomenologically modeling the effects of 1/f noise within a finite but broad frequency window κ min ω κ max [56][57][58][59][60]. The basic assumption is that it is produced by a sum of N uncorrelated noise sources, with noise components ∆ j (t) presenting correlations of the form ∆ j (0)∆ j (τ ) = σ 2 j e −κ j |τ | , and thus the total autocorrelation and noise spectrum read To achieve this situation, the noise components ∆ j (t) can be modeled as independent Ornstein-Uhlenbeck processes [60] (Sec. 3.2), but it is also typically assumed that ∆ j (t) are originated by an ensemble of two-level fluctuators [56][57][58], characterized by different noise strengths σ j and jumping rates κ j (see Appendix A.2). In either case, if the parameters κ j present an uniform distribution of log 10 (κ j /Γ) in a broad range from κ 1 = κ min to κ N = κ max , and if σ j = σ 1 (κ 1 /κ j ) (η−1)/2 , then in the limit N 1 the noise spectrum S(ω) in Eq. (25) approximates a power-law behavior [60], In Figure 4(a) we illustrate the effectiveness of this method with a numerical simulation of 1/f 0.99 noise with only N = 8 independent noise components. We see that that exact noise spectrum in Eq. (25) (blue/solid), approximates well the expected the power-law behavior (red/dashed) in the frequency range 10 −4 ω/Γ 10 4 . Now we solve for the average transmittance t µ ω and the Ramsey envelope C φ (t) for a noisy qubit subject to the above model of 1/f noise. We can simulate Gaussian or non-Gaussian 1/f noise depending if we choose the noise components ∆ j (t) in Eq. (25), and reads where κ j and σ j are chosen to simulate the 1/f model as explained above. for non-Gaussian 1/f noise requires more advanced stochastic methods for describing the dynamics of the two-level fluctuators. This is done in full detail in Appendix A, but here we discuss the results. In Figures 4(b) and (c) we display t µ ω and C φ (t) for a noisy qubit with dephasing due to the 1/f 0.99 noise simulated in Figure 4 The main difference between Gaussian and non-Gaussian 1/f noises are small bumps in t µ ω and C φ (t), which are signatures of the sparsity or granularity of the dephasing environment as treated in detail in Appendix A.3. Besides that, both predictions agree well and behave very similar to a single colored Gaussian noise in the quasi-static limit, except for the power-law spectrum S(ω).

Spectroscopic characterization of correlated dephasing noise
This section introduces a simple experimental protocol to measure the average single-photon transmittance and reflectance, and to recover the correlated dephasing noise from those quantities. This protocol only requires attenuated coherent states and either homodyne or power measurements at the output -the choice of which depends mainly on whether the experiment is performed with microwave [1,3] or optical photons [61][62][63]-.
The experimental procedure is sketched in Figure 3(b), where a monochromatic coherent state |α µ ω of amplitude α µ ω and frequency ω is injected on the input channel µ = ± of the waveguide. We study the evolution of the corresponding initial state which describes a coherently driven qubit from an arbitrary initial state |Ψ qb , and vacuum states in all photonic channels different than µ. We will work in the limit of weak driving, |Ω| Γ with Ω = −iα µ ω √ γ µ . We show below that in this limit we recover the single-photon transmittance from homodyne or power measurements, as follows with β µ = γ µ /Γ the directional β-factor. Note that, while homodyne measurements provide direct access to t µ ω , power measurements give us only its real part, but we can still reconstruct the full transmittance via the Kramers-Kronig relation, with P representing the Cauchy's principal value of the integral. Once we have the singlephoton transmittance, we can invert equation (19) to access to the time-resolved Ramsey envelope and characterize noise correlations of the environment. A convenient inverse formula can be derived when the dephasing fluctuation ∆(t) has a symmetric probability distribution around the average, which is very reasonable assumption in experiments. In this case, the Ramsey envelope defined in Eq. (9) is a real function of time, C φ (t) = [C φ (t)] * , and it can be directly related to Re t µ ω by This relation (32) has important physical consequences to single-photon scattering experiments in waveguide QED, as it demonstrates that applying a Fourier transformation on the usual transmittance data [1,3,48,49,[61][62][63], one can characterize noise correlations without requiring direct access and time-dependent control of the emitter. Moreover, Eq. (32) is particularly convenient in the case of power measurements (30) as it only requires the knowledge of Re t µ ω , and thus avoids the use of the Kramers-Kronig transformation (31). Let us briefly summarize how equations (29)-(30) are derived. We begin with the equations of motion for the noisy qubit, taking expectation values on (4)-(5) with the initial Inverting Eq. (19) in the general case leads to C φ (t) = (2π) −1/2 e (Γ/2)t F −1 [(1 − t µ ω )/γ µ ] (t), for t > 0, but this expression presents a slower numerical convergence compared to Eq. (32). Notice that the presence of a non-zero photon decay Γ > 0 allows us to mathematically replace the inverse Laplace transform L −1 by the more convenient inverse Fourier transform F −1 . condition (28). Using the property a λ in (t) |Ψ(0) = α µ ω δ λµ e −iωt |Ψ(0) with λ = ±, and going to a rotating frame with the driving frequency ω, we find Here, we have defined the slowly evolving coherence σ − (t) = e iωt σ − (t) and the strength of the coherent drive Ω = −iα µ ω √ γ µ . The qubit equations are stochastic Bloch equations that can combine correlated dephasing with saturation at strong drives |Ω| Γ [64]. The stochastic methods from section Appendix A provide a solution to this complex dynamics of the qubit, but noise spectroscopy only requires the steady state averaged coherence σ − ss = σ − t→∞ , which appears both in homodyne and power steady state measurements as ¶ ã λ out ss In the low driving limit |Ω| Γ, the qubit will remain close to the ground state σ z = −1 + O [|Ω|/Γ] 2 , and equations (33) and (16) From this relation we conclude that homodyne and power measurements give us full information about the average single-photon transmittance t µ ω and reflectance r µ ω , and allow a full spectroscopic characterization of the noise via Eqs. (29)- (32).
This finishes the presentation of our spectroscopic method to characterize correlated dephasing noise. In addition to this, Appendix C discusses measurements of single-photon reflectance and the conservation of the average photon flux in this model. Finally, in Appendix D we generalize the measurement relations (29)-(32) to case the noisy qubit sees Fano resonances [65,66], as for instance, in experiments with quantum dots in photonic crystals waveguides [49,61,67].

Conclusions and outlook
We have predicted the effect of correlated dephasing noise in single-photon scattering experiments with weak coherent inputs, introducing a stochastic version of input-output theory. Using this theory, we studied scattering subject to the typical noise models from solid-state and quantum optics -white noise, quasi-static noise, colored Gaussian noise (see Secs. 3.2 and 4.2), and 1/f noise (see Sec. 4.3), in addition to telegraph noise and non-Gaussian ¶ To derive the relation (36), we combined the input-output equations (3), the equation of motion (34), and used the exact relation σ z ss = −1 + Re {4Ω * σ − ss /Γ}, which results from integrating (34) and averaging in steady state. jump models in Appendix A-, illustrating how to calculate the single-photon transmittance t µ ω and reflectance r µ ω of each model. Complementary to these theoretical developments, we introduced a spectroscopic method that extracts the qubit noise correlations from standard homodyne or photon counting measurements. The method provides the same information as time-resolved Ramsey experiments, but does not require direct access or time-dependent control of the emitter. This method and the techniques developed in this work are suited not only for waveguide QED experiments -superconducting circuits [1][2][3]8], quantum dots in photonic crystals [49,61], SiV-centers in diamond waveguides [63,68], or nanoplasmonics [69]-, but also for generic experiments with two-level quantum emitters interacting with propagating photons, such as molecules in a 3D bath [62] or ions in a Paul trap [70].
There are still several open questions and extensions to consider in the interaction between few photons and noisy quantum emitters. For instance, our theory is valid for general stationary random fluctuations ∆(t), but we only analyzed classical noise models. Therefore, it would be interesting to study the effects of specific microscopic quantum models producing correlated dephasing on the quantum scatterers [71,72], and try to find the connections between its classical analogs. Moreover, we can combine the stochastic methods discussed here with our recent theory of scattering tomography [38] to characterize multi-photon processes or many-body scatterers [73][74][75] under realistic conditions of noise. due to correlated non-Gaussian Markovian noise models In the main text we explicitly calculated the average transmittance of a qubit with colored Gaussian noise and Gaussian 1/f noise, where C φ (t) is analytical and t µ ω can be directly obtained from (19). Although Gaussian noise models are very successfully applied in numerous experiments [71], the Gaussianity assumption breaks down in situations where the qubit is coupled to a sparse dephasing environment [27] such as a few frequency modes [46], or ensembles of few two-level fluctuators (TLFs) [51,58]. In the following we extend the analysis to arbitrary correlated non-Gaussian Markovian noise models which include telegraph noise caused by a single TLF (see Appendix A.2), tunable non-Gaussian noise caused by a sparse ensemble of TLFs (see Appendix A.3), and non-Gaussian 1/f noise (see Appendix A.4) typically found in solid-state devices [17]. To compute t µ ω and C φ (t) for a qubit under these types of dephasing, we require the stochastic methods introduced in the following subsection Appendix A.1.

Appendix A.1. Stochastic differential equations with arbitrary correlated Markovian noise
Here we state the equations to solve for the average transmittance t µ ω and the Ramsey envelope C φ (t) in the case of the most general correlated, stationary, and Markovian dephasing noise. In practice, we extend the method in page 418 of Ref.
[40] to inhomogeneous stochastic differential equations, and then apply it to the scattering equation (16).
Our first assumption is that the stochastic process ∆(t) is stationary and Markovian. The probability for the noise to be in realization ∆(t) = ∆ at time t, conditioned on being ∆(t 0 ) = ∆ 0 at time t 0 is denoted by P (∆, t) = P (∆, t|∆ 0 , t 0 ). The most general Markovian dynamics for the above conditional probability is governed by a differential Chapman-Kolmogorov equation [41], with initial condition P (∆, t 0 ) = δ(∆ − ∆ 0 ) and classical Liouvillian L given by 2) Here, D 1 (∆) is the drift function, D 2 (t) ≥ 0 the diffusion function, and W (∆, ∆ ) ≥ 0 for ∆ = ∆ are transition probabilities between different values of the noise. The conservation of total probability also requires d∆ LP (∆, t) = 0 and thus d∆ W (∆, ∆ ) = 0. We further assume L is time-independent to have an homogeneous stationary Markovian process with well-defined steady state LP ss (∆) = 0.
We want to study G ω (t) which is a stochastic process related to ∆(t) via Eq. (16). Since ∆(t) is Markovian, the joint process [∆(t), G ω (t)] is Markovian too [40] with joint probability denoted by P(G ω , ∆, t). For a multiplicative inhomogeneous stochastic differential equation of the form dG/dt = A(∆)G + B, the joint probability satisfies [40] with the initial condition P(G ω , ∆, 0) = δ(G ω −G ω (0))P (∆, 0). To compute the noise average G ω , the strategy is to convert the stochastic equation (16) into a set of ordinary differential equations for the marginal averages g ω (∆, t) = dG ω G ω P(G ω , ∆, t), and from its solution obtain the total average as G ω (t) = d∆g ω (∆, t). To do so, we use Eq. Eq. (A.3), multiply it by G ω and integrate it over G ω (t), obtaining For the scattering problem in Sec. 4.1, the differential equation (A.4) must be solved with the initial condition g ω (∆, −∞) = G ω (−∞)P (∆, −∞) = 0, which effectively corresponds to finding the steady state solution g ss ω (∆) = g ω (∆, t → ∞) or dg ω (∆, t)/dt = 0. Finally, when having the steady state marginal averages g ss ω (∆) for each frequency ω and each noise realization ∆, we obtain the average transmittance as We see that computing the average transmittance t µ ω for the most general non-Gaussian, correlated, stationary, and Markovian noise model amounts to solve for the steady state of the partial differential equation (A.4) and then to integrate it in Eq. (A.5) over all noise realizations. To simplify this solution, we now particularize the analysis to discrete jump noise models, where the stochastic process ∆(t) has a discrete number of realizations denoted by ∆ m . In this case, we can set D 1 = D 2 = 0 in Eq. (A.2), and the probability P (∆ m , t) for the noise to be in the realization ∆(t) = ∆ m at time t, conditioned on being ∆(t 0 ) = ∆ m 0 at t = t 0 is governed by the time-local rate equation [40], Here, the matrix coefficients W mn ≥ 0 for m = n describe transition rates of the noise to jump from realization ∆ n to ∆ m -which must satisfy m W mn = 0 to ensure the conservation of total probability m P (∆ m , t) = 1-. Importantly, the partial differential equation (A.4) reduces to a discrete set of ordinary differential equations for the discrete number of marginal averages g ω (∆ m , t) as, which now allows us for a much simpler steady state solution. In fact, setting dg ω (∆ m , t)/dt = 0 in Eq. (A.7) we can map the problem to a linear system of equations, n J mn g ss ω (∆ n ) = P ss (∆ m ), with (A.8) Here, the matrix J mn is of the same size as W mn , and P ss (∆ m ) denotes the steady state solution of the rate equations (A.6). Finally, solving this linear problem for different values of the input field ω, we obtain the average single-photon transmittance from the sum, On the other hand, to obtain the Ramsey envelope C φ (t) in Eq. (9), we can numerically extract it from t µ ω via the inversion formula (32). Alternatively, we can also obtain it by calculating the average solution C φ (t) = X(t) of the homogeneous stochastic differential equation, d dt X(t) = −i∆(t)X(t). which must be solved for the initial condition x(∆ m , 0) = P ss (∆ m ). Finally, we obtain the Ramsey envelope as C φ (t) = X(t) = m x(∆ m , t).
In the following three subsections, we evaluate t µ ω and C φ (t) for different forms and sizes of W mn corresponding to correlated telegraph noise, and more general non-Gaussian 1/f noise models.

Appendix A.2. Telegraph correlated noise
Charges or impurities in the materials of solid-state devices are modeled in many cases as localized double-well potentials or two-level fluctuators (TLFs) [17,51,76,77]. A strong resonant coupling between the qubit and an environmental TLS can lead to the observation of resonances [29,78,79], but a weak off-resonant coupling can induce fluctuating Stark shifts on the qubit and thus originate correlated dephasing as in Eq. (1). Although TLFs naturally appear in large ensembles of them [56][57][58], the telegraph noise produced by a single TLS is an instructive and exactly solvable model capturing many features of more complex correlated non-Gaussian noises.
Telegraph noise is the simplest jump model, where random variable ∆(t) can only take two possible values ∆ ± = ±σ [39,41], corresponding to an increase or decrease of the qubit resonance as ω 0 ±σ. The dynamics of this noise consists in random jumps with rate κ between the two possible realizations ∆ m with m = ±, as depicted in Figure A1(a). The probabilities P (∆ m , t) of being in ∆ m at time t, conditioned of being in ∆ m 0 at an initial time t 0 , are governed by the Markovian rate equations, which can be recast in the general form (A.6) with the transition matrix W mn = −mnκ/2 (m, n = ±). The above equations imply that in steady state the probabilities of being in either realization are equal P ss (∆ m ) = 1/2, the mean fluctuation vanishes ∆(t) = 0, and the autocorrelation has the same form ∆(0)∆(τ ) = σ 2 e −κ|τ | [39,41] as the colored Gaussian noise in Eq. (11). Notice that this is just a coincidence since higher order correlations highly differ due to the non-Gaussian character of the telegraph noise [41]. The simplicity of the telegraph noise allows us to analytically solve for the average transmittance t µ ω in Eq. (A.10), since the linear system (A.8) is of size 2-by-2 with the matrix J mn = [Γ/2 − i(ω − ω 0 ) + imσ]δ mn + mnκ/2 (m, n = ±), and P ss (∆ m ) = 1/2. For the steady state marginal averages we obtain , (A.14) and using Eq. (A.10) we find that t µ ω can be expressed in a form reminiscent to a Lorentzian, , but with a frequency-dependent pure dephasing rate γ φ (ω) given by The lineshape is thus not Lorentzian in general, except for the white noise limit κ → ∞ where the dephasing rate (A.15) becomes a constant γ φ = σ 2 /κ. This is illustrated by the blue/solid transmittance in Fig. A1(b) for standard waveguide QED parameters. For a finite but moderate correlation time σ κ < ∞, t µ ω gets broader than the Lorentzian (black/dash-dotted), and in the quasi-static limit of long correlation times κ σ, t µ ω develops two well separated dips centered at ω ≈ ω 0 ± σ whose widths are proportional to σ (red/dashed). To obtain the Ramsey envelope C φ (t) for the qubit under this telegraph noise, we can either use the inverse relation Eq. (32) on our known t µ ω or solve the differential Eq. (A.12), which gives 16) with v 0 = κ/ √ κ 2 − 4σ 2 and v ± = (−κ ± √ κ 2 − 4σ 2 )/2 [40]. As shown in Figure A1(c), C φ (t) is the exponential decay in the white noise limit κ → ∞, and for a finite correlation time κ < ∞, it shows damped oscillations with frequency ∼ σ and damping rate ∼ κ.

Appendix A.3. Correlated dephasing noise with tunable non-Guassianity
In this subsection we introduce a model of non-Gaussian correlated noise, whose non-Gaussianity can be tuned to describe situations such as the telegraph noise from previous subsection, all the way to the limit of colored Gaussian noise in Secs. 3.2 and 4.2.
We follow Ref. [80] and construct a discrete noise model from the sum of M independent and identical TLFs, Figure A2(a)). Here, each noise component ∆ l (t) corresponds to a telegraph noise as in the previous subsection, which flips between the values ∆ l (t) = ±σ at a rate κ and independently satisfies the Markovian rate equation , with a transition matrix W nm whose nonzero elements read [80],  19) becomes a continuous Fokker-Planck differential equation for the Ornstein-Uhlenbeck process [80], which is given by Eqs. (A.1)-(A.2) with D 1 (∆) = −κ∆, D 2 = 2κσ 2 , and W (∆, ∆ ) = 0. As a result of this connection, we conclude that by increasing the number M of independent telegraph noises, we can reduce the non-Gaussian character of the noise model until reaching the limit of standard colored Gaussian noise.
We exemplify this tuning of the non-Gaussianity by computing the average transmittance t µ ω for a qubit in dephasing environments with different values of M . To do so, we numerically solve the linear system (A.8)-(A.9) by using the W mn coefficients in Eq. (A. 19), and the steady state binomial distribution P ss (∆ m ) = 1 2 M M m . It is computationally simple to reach the Gaussian limit M 1 since the size of the matrix J mn grows linearly with M as (M + 1) × (M + 1). The results are shown in Figure A2(b) for M = [2, 3, 4, 5, 10], κ = 0.1σ, and typical waveguide QED parameters. The non-Gaussianity of the dephasing is manifested by the multiple dips in t µ ω which reduce with increasing M . Also notice that already for M = 10 (red/dashed) the Gaussian limit is well-established with a Gaussianlike transmittance as expected in the quasi-static limit κ σ. In addition, we compute the Ramsey envelopes C φ (t) for the parameters above by applying Eq. (32) on the numerical data for t µ ω . The results are shown in Figure A2(c), where the non-Gaussinity of the dephasing noise is manifested by the multiple oscillations in C φ (t) and whose amplitude reduce with M . In the Gaussian limit (red/dashed) there is only the Gaussian decay as expected in the quasi-static case κ = 0.1σ. Notice that we do not display the results in the white noise limit, κ σ, where the behavior is independent of M , the lineshapes are standard Lorentzians, and C φ (t) are exponential decays with pure dephasing rate γ φ = σ 2 /κ.

Appendix A.4. Simulation of non-Gaussian 1/f noise
The aim of this subsection is to construct a model for 1/f noise with tunable non-Gaussianity and show how to compute the non-Gaussian results for t µ ω in Figure 4(b)-(c). To similate non-Gaussian 1/f noise, we assume that each noise component ∆ j (t) for j = 1, . . . , N in Eq. (24) is represented by an independent ensemble of M identical TLFs as introduced in Appendix A.3. We therefore need to construct a more general jump model for the total noise, ∆(t) = N j=1 ∆ j (t)/ √ N , with permutation symmetry only within each ensemble ∆ j (t). As a result, there will be (M + 1) N distinguishable global realizations of the total noise ∆(t), which are given by Finally, we should evaluate W m n for the parameters κ j and σ j that simulate the desired 1/f noise model as stated in Sec. 4.3, replace this and Eq. (A.22) in the linear system (A.8)-(A.9), and numerically solve for the steady state marginal averages. With that result we can evaluate t µ ω via Eq. (A.10), and C φ (t) via Eq. (32). The size of the linear system scales exponentially with N as (M + 1) N , but as shown in Figure 4(a), already a moderate N = 8 is enough to properly simulate the 1/f noise spectrum.
Appendix B. Adding a white noise background to the dephasing model In this appendix, we use stochastic Ito calculus [39,41] to include dephasing due to a white noise background ∆ WB (t), in addition to the correlated noise ∆(t) in the scattering differential equation (16).
The stochastic differential equation for scattering that includes both noise sources reads, where the white noise background is specified by the autocorrelation function ∆ WB (0)∆ WB (τ ) = 2γ WB δ(τ ), with γ WB its pure dephasing rate. The multiplicative stochastic differential equation (B.1) must be physically interpreted in the Stratonovich form [39,41], with dW (t) = ∆ WB (t)dt/ √ 2γ WB the Wiener increment. To solve the average over the white noise background more easily, we use the Ito rules to convert Eq. (B.2) to the Ito form, obtaining where now dW (t) is uncorrelated with G ω (t) at equal times. We take the average over the white noise background . . . WB , which does not affect ∆(t) as we assume it is uncorrelated with ∆ WB (t), i.e. ∆(t)∆ WN (t) WN = 0 and ∆(t)G ω (t) WN = ∆(t) G ω (t) WN . Additionally using the Ito property G ω (t)dW (t) WN = G ω (t) WN dW (t) WN = 0, we obtain a stochastic differential equation that depends on the correlated noise ∆(t) only, d dt Therefore, we can solve this stochastic differential equation instead of (16) if we would like to include an extra uncorrelated white noise background with pure dephasing rate γ WB . In practice it just amounts to perform the replacement Γ/2 → Γ/2 + γ WB in Eq. (16), before starting to solve it.
Using the above replacements, we can generalize Eqs.