Path integral framework for characterizing and controlling decoherence induced by non-stationary environments on a quantum probe

Reliable processing of quantum information is a milestone to achieve for the deployment of quantum technologies. Uncontrolled, out-of-equilibrium sources of decoherence need to be characterized in detail for designing the control of quantum devices to mitigate the loss of quantum information. However, quantum sensing of such environments is still a challenge due to their non-stationary nature that in general can generate complex high-order correlations. We here introduce a path integral framework to characterize non-stationary environmental fluctuations by a quantum probe. We found the solution for the decoherence decay of non-stationary, generalized Gaussian processes that induce pure dephasing. This dephasing when expressed in a suitable basis, based on the non-stationary noise eigenmodes, is defined by the overlap of a generalized noise spectral density and a filter function that depends on the control fields. This result thus extends the validity to out-of-equilibrium environments, of the similar general expression for the dephasing of open quantum systems coupled to stationary noises. We show physical insights for a broad subclass of non-stationary noises that are local-in-time, in the sense that the noise correlation functions contain memory based on constraints of the derivatives of the fluctuating noise paths. Spectral and non-Markovian properties are discussed together with implementations of the framework to treat paradigmatic environments that are out-of-equilibrium, e.g. due to a quench and a pulsed noise. We show that our results provide tools for probing the spectral and time-correlation properties, and for mitigating decoherence effects of out-of-equilibrium -- non-stationary -- environments.

To deal with these outstanding problems, we here implement a path integral framework to describe the decoherence process on a controlled 1/2-spin, quantum-probe coupled to non-stationary noise sources.We set the pathintegral framework for the most general pure dephasing coupling with fluctuating fields that can be described by non-stationary, generalized Gaussian processes, and find the solution for the decoherence decay.Our results generalize the universal formula [19,20] for the dephasing decay that only depends on the overlap of a noise spectral density and a filter function that depends on the control fields.Specifically, we demonstrate its extension to the case of non-stationary Gaussian environments.
This generalization allows to implement two important applications for the deployment of quantum technolo-gies, dynamical decoupling noise spectroscopy for quantum sensing operations and mitigating by control methods the decoherence effects of out-of-equilibrium environments.Moreover, we provide simple interpretations of non-stationary noise spectrums and time-dependent correlations for a broad subclass of local-in-time noises and simple criteria to distinguish Markovian from non-Markovian noise dynamics.We also show how to implement our framework with two paradigmatic nonstationary noises, one derived from a quenched environment where excitations suddenly start spreading over a large number of degrees of freedom [13,50,56,64,74] and the other from a noise that acts near to a point in time [75][76][77][78].Overall we introduce a tool to characterize and control decoherence effects of out-of-equilibrium environments providing avenues of quantum information processing for the deployment of quantum technologies.
Our manuscript is organized as follows.In Sec.II we describe the quantum-probe interacting with fluctuating fields.In Sec.III we introduce the path integral framework to determine the qubit-probe dephasing induced by a general, non-stationary Gaussian noise in the time-and frequency-domain.In Sec.IV we define a generalized noise spectral density and control filter function in a suitable non-stationary, noise eigenmode basis, and show that in this basis the dephasing is given by the overlap between them.Based on this noise eigenmode basis, we show how to implement (i) quantum sensing of the generalized noise spectrum and (ii) optimal dynamical decoupling sequences to protect the quantum system against decoherence.In Sec.V we consider a broad subclass of non-stationary Gaussian noises that are local-in-time to provide more direct physical meanings of parameters that characterize these non-stationary noises within this path integral framework.In Sec.VI we apply the presented tools to two paradigmatic out-ofequilibrium environments: a noise acting near to a point of time by an analogy to a quantum harmonic oscillator and a quench on the environment that suddenly starts an Ornstein-Uhlenbeck diffusion process.Lastly, in Sec.VII are the concluding remarks.

II. QUBIT-PROBE INTERACTING WITH FLUCTUATING FIELDS
We consider a qubit system with spin S = 1/2 as a quantum probe experiencing pure dephasing due to the interaction with its environment (bath) [27,79].In the system rotating frame of reference, the Hamiltonian is given by where H B is the bath Hamiltonian and H SB is a general pure dephasing system-bath interaction Hamiltonian.The spin operator S z in the z axis is the qubit-probe operator, the bath-operators are represented by an ndimensional vector B = (B 1 , . . .B n ), and g = (g 1 , . . .g n ) are the system-bath coupling strengths.Notice that the index i of the components g i B i of the system-bath interaction, labels properties of the qubit-bath coupling network morphology.For example, it can label different components of an hyperfine interaction tensor [80,81], different spins on the environment [31,34,82] or spatial directions as in the case of anisotropic molecular diffusion [83][84][85].Figure 1(a) shows a schematic representation of this interaction.This type of interaction is encountered in a wide range of systems, as for example nuclear spin systems in liquid and solid state NMR [31,34,35,86,87], dephasing induced by spin bearing molecules in liquid and gas state NMR [84,[88][89][90][91][92], the hyperfine interaction of electron spins in diamonds [36,80,81,93,94], electron spins in quantum dots [95][96][97][98], donors in silicon [99], electron spins in nanoscale nuclear spin baths [54], superconducting qubits [40], trapped ultra-cold atoms [45,100], etc.
By using an interaction representation with respect to the evolution of the isolated environment, we eliminate the bath-Hamiltonian H B .The system-bath Hamiltonian becomes Since H B does not commute with H SB , the effective system-bath interaction SB is time-dependent and the qubit-probe experiences a fluctuating coupling to the bath.We use the semi-classical field approximation to replace the quantum environment operator e −i H B t Be i H B t with a classical stochastic function B(t) [Fig. 1 (a,b)].The corresponding self-correlation function is thus given by B i (t)B j (t ) = 1  2 Tr [ρ e {B i (t), B j (t )}], where • is the stochastic mean value of the semi-classical stochastic fields.The rhs of the equation is the quantum correlation function of the bath operators B i/j , determined by the trace Tr over the bath degrees of freedom with ρ e as the density matrix of the environment and {•} denoting the anticommutator.This approximation typically called the weak coupling approximation represents well a time-dependent quantum correlation of the SB interaction up to second order [19,20,37,67,79,101,102] that properly describe a wide-range of experimental setups [3,31,35,36,40,42,54,62,65,[67][68][69].
The evolution operator of a dynamically controlled qubit-probe combined with this fluctuating field interaction is e −iφ[B,f ]Sz , where φ[B, f ] is the accumulated phase by the probe during the evolution from an initial time t 0 to a final time t f , starting from an initial state polarized in the XY plane.Specifically, with dt ≡ +∞ −∞ dt and the vector f (t) = (f 1 (t), . . ., f n (t)) is a controlled modulation of each bath interaction term driven by dynamical control techniques during t ∈ [t 0 , t f ], being null outside this time interval [17-20, 27, 28].For example, for a free evolution t ∈ [t 0 , t f ], and , where Θ is the Heaviside step function, while for a dynamical decoupling sequence of π-pulses applied to the qubit, f i (t) switches between ±g i at the position of every pulse [27,28].Notice that we write explicitly the dependence on B and f of the qubit-probe phase, to highlight dependence of the phase on the stochastic fluctuating field that we then describe with the path integral approach and the control fields, respectively.
In this article we consider the general pure dephasing system-bath interaction Hamiltonian of Eq. (1).However, the path integral framework discussed below to describe non-stationary, general Gaussian noise that induces dephasing by this interaction, can be extended to a dissipation interaction based on replacing the spinprobe operator S z by S x or S y .For this dissipation interaction, the phase of Eq. (3) represents a population phase of a coherence exchange [19,20,102].This adaptation describes interactions found for example in chemical quantum solvation dynamics with a time-dependent structurally changing solvent [55,103,104] and nonequilibrium response of nano-and atomic-systems coupled to driven Caldeira-Leggett baths [51,105,106].

III. PATH INTEGRAL FRAMEWORK FOR THE QUBIT-PROBE DEPHASING
In this section, we introduce the general path integral framework for describing the decoherence effects on the qubit-probe, induced by a non-stationary, general Gaussian noise that describes an out-of-equilibrium environment.Under this framework, we define the generalized noise spectrum for non-stationary environments in terms of the inverse of the kernel operator that determines the probability of the noise field paths.We then show the implementation of this framework to stationary, general Gaussian noises, and recover the known expression for the induced dephasing.

A. Dephasing induced by non-stationary, general
Gaussian noise We use path integrals to calculate the decoherence effects on the qubit-probe, induced by the fluctuating field B(t).The notation φ[B, f ] of the accumulated phase in Eq. ( 3), shows the functional dependence of φ with the evolution path followed by the fluctuating field B(t), where the qubit probe is driven by the control fields f (t).Since the evolved phase involves real square-integrable functions that conform a Hilbert space, we use a bra-ket notation to simplify the calculations and interpretations.A function f (t) is associated to the ket |f ), and similarly the multidimensional function f (t) to the ket |f ).The inner product of this space is given by We also introduce the time-basis {|t) : t ∈ R} of the Hilbert space of scalar functions of time such that (t|f ) = f (t) is a scalar function over the real numbers R. Therefore (t|f ) = (f 1 (t), . . ., f n (t)) is a vector function in R n .This is analogous to the position basis {|x) : x ∈ R n } in quantum mechanics.Then the qubit-probe phase of Eq.
The qubit dephasing-decay is given by the ensemble average e iφ [B,f ] taken over all possible field path fluctuations of B. Figure 1(b) shows a schematic representation of the paths of the field fluctuations.Based on path integral theory, this ensemble average is where DB is a functional integral with the probability distribution of B given by exp {−A[B]}, and A[B] is the action, in analogy with quantum path integrals in imaginary time [107].
We consider the most general non-stationary Gaussian noise process, for which the action is where † denotes hermitian conjugation and D(t, t ) = (t|D|t ) is a n × n matrix-valued function of t and t that gives the kernel of the operator D in terms of times.The kernel operator D must be positive definite and real for the path integral to be well defined, and for the action to define a probability density exp {−A[B]}.We assume that D is hermitian without loss of generality, such that The dephasing-decay of the qubit-probe can be solved exactly using path integrals for this general nonstationary Gaussian noise process as [107] e iφ [B,f ] where the operator G defines the self-correlation functions of the stochastic process The operator G is defined by the inverse of the kernel operator or equivalently by the expression where δ(t) is the Dirac delta and I is an identity matrix of dimension n × n.
Using the bra-ket notation, one can also solve Eq. ( 8) in the frequency-basis of scalar functions {|ω) = dt √ 2π e −iωt |t) : ω ∈ R}, rather than performing the calculation using the time-basis {|t) : t ∈ R}.This frequency-basis, yields to the Fourier representation of the ensemble average of the qubit-probe signal Here S(ω 1 , ω 2 ) = (ω 1 |G|ω 2 ) defines a spectral density of two frequency dimensions equivalent to the double Fourier transform of the correlation functions G(t, t ) [Fig.1(c)], and F(ω) = (ω|f ) = (F 1 (ω), . . ., F n (ω)) is a filter function defined by the Fourier transform of Equations ( 8) and ( 12) are the qubit dephasing solutions for general -non-stationary-gaussian noises derived from the presented path integral framework.Using the frequency-basis, Eq. ( 10) becomes (14) where D(ω, ω ) = (ω|D|ω ).Therefore, this shows that the noise generated by the non-stationary environment is completely determined by the bispectrum S(ω 1 , ω 2 ) that is defined by the inverse of the Kernel operator D that describes the probability density of the field paths.Similar bispectra, functions of two frequencies, were shown to be a useful tool to characterize stationary, non-Gaussian noises [42,67].Here, we show that they are also useful tools to characterize non-stationary, Gaussian noises.

B. Dephasing induced by stationary, general Gaussian noise
For stationary noise, the kernel D(t, t ) and the correlation functions G(t, t ) are invariant under a time translation, therefore they only depend on the time difference t − t .We can consider then that D(t, t ) = D(t−t ) and G(t, t ) = G(t−t ), and the two dimensional Fourier transform of G(t, t ) is then block diagonal in the frequency-basis (ω|G|ω ) = S(ω)δ(ω − ω ).Similarly, the Fourier transform of D(t, t ) is (ω|D|ω ) = D(ω)δ(ω − ω ).Equation (12) then becomes According to Eq. ( 10), the inverse of the Kernel operator that describes the probability density of Eq. ( 7), defines the noise spectrum for stationary noise when expressed in the frequency-basis.Thus, the noise spectrum of stationary noises is further simplified with respect to Eq. ( 14) to the multiplicative inverse of the operator D(ω) in this basis, i.e. it is the inverse for each frequency mode.Equation ( 15) expresses the dephasing-decay induced by a stationary noise in terms of an integral over only one frequency, in contrast with Eq. ( 12) that integrates an overlap between the filter function and the spectral density of a non-stationary noise over two different frequencies.As a result, we here recovered the universal formula for the qubit dephasing under the weak coupling approximation mainly considered for the n = 1 case, i.e. when the dephasing is given by e 2 [19,20,29,30,32,34,108,109].This formula for the dephasing is typically used for designing optimal control [20,28,34] and for dynamical decoupling noise spectroscopy [35,36,40,41,45,65,110,111].

IV. NOISE SPECTRAL DENSITY ON THE NOISE EIGENMODE BASIS
In this section, we first introduce the noise eigenmode basis that defines the proper basis for non-stationary noises, to generalize the universal formula for the dephasing of open quantum systems that depends on the overlap between a noise spectral density and a qubit-control filter function.We then show how stationary noises are described in this noise eigenmode basis.Based on the eigenmode representation, we discuss two important applications of this dephasing generalization to non-stationary environments: we show how to implement dynamical decoupling noise spectroscopy for quantum sensing, and optimized control methods to mitigate decoherence effects.

A. Non-stationary noise modes
The kernel operator D is in general non-stationary, and therefore the frequency-basis does not diagonalize it, leading to the bispectrum S(ω 1 , ω 2 ) of Eq. (12).For this reason, in general the frequency-basis {|ω) : ω ∈ R} are not the eigenfunctions of D for non-stationary noises.As the kernel operator D is hermitian, we can introduce the basis {|Ω) : Ω ∈ I} that diagonalizes it where I is a set that indexes the elements of the eigenmode-basis and D(Ω) ∈ R >0 are the corresponding eigenvalues.This basis is normalized to satisfy the completeness relation I = dΩ |Ω)(Ω|, where dΩ = I dΩ represents an integral when the spectrum is continuous or a sum when it is discrete.Notice that |Ω) are vectors in the same space as |B), and they represent the non-stationary modes of the noise fluctuations (t|B) = B(t) = dΩ b Ω (t|Ω), where (t|Ω) is a vector-valued time dependent function and b Ω = (Ω|B) are scalars.
The correlation operator G is then diagonal in this basis G|Ω) = [D(Ω)] −1 |Ω), as I|Ω) = GD|Ω) = G|Ω)D(Ω) following Eq.(10) and considering that {|Ω) : Ω ∈ I} is a basis.As a result, we can now define a generalized noise spectrum for non-stationary Gaussian noises as the eigenvalues of the correlation operator where G|Ω) = S(Ω)|Ω).The attenuation argument of the dephasing-decay of Eq. ( 8) is then in this noise eigenmode basis.This Ω-basis provides then a natural way to define a generalized filter function for these non-stationary noises as F (Ω) = (Ω|f ), to obtain The generalized noise spectrum S(Ω) for a nonstationary Gaussian noise is thus the multiplicative inverse of the eigenvalues of the kernel operator D, indexed by the noise eigenmode parameter Ω.
Notice that as Ω is a parameter, in general its physical meaning depends specifically on the non-stationary process, and on the parametrization of the functional behaviour S(Ω), as it is always possible to reparametrize the eigenmodes by a different parameter.Conversely, the eigenvalues of the operator D and their corresponding noise eigenspaces are always independent of the parametrization.By identifying the basis that diagonalizes the kernel operator D, we have mapped the bidimensional spectral density that depends on two frequencies [Fig.1(c)] into a mono-parametrical spectral density based on the non-stationary noise eigenmodes [Fig.1(d)].This sets one of the main results of this article, where Eq. ( 20) is a generalization for the qubitprobe dephasing for non-stationary Gaussian noise that is only determined by the overlap between the generalized spectral density of the environment S(Ω) and the generalized filter function |F (Ω)| 2 determined by the control on the qubit-probe.This result thus extends the validity to non-stationary noises, of the universal formula for stationary pure dephasing noises of open quantum systems within the weak coupling approximation [19,20,29,30,32,34,35,108,109].

B. Stationary noise modes
In the stationary noise case, according to Eq. ( 15), the frequency dimension is separable from the morphological dimension of the qubit-bath coupling network represented by the vector indices i of B and F in Eq. ( 9), where we obtain D|ω) = D(ω)|ω).The noise eigenmodes in the stationary case are defined by the eigen- Therefore the eigenmodes of the Kernel operator are |ω, m) = b m (ω)|ω), and it acts on them as D|ω, m) = D m (ω)|ω, m).We consider the eigenmode basis index as Ω = (ω, m) and use the notation D(Ω) = D m (ω).The dephasing of Eq. ( 20) then becomes where S m (ω) = D −1 m (ω) are the eigenspectrums of the bath fluctuations on the qubit-probe.The index m for example defines the principal axes of anisotropic diffusion tensors in magnetic resonance imaging [83][84][85].
While the choice of the frequency ω as the relevant parameter to describe the noise eigenmodes for the stationary case seems natural, its selection is also arbitrary as the choice of Ω in general.For example one can keep the diagonal form of Eq. ( 21) but use, instead of the angular frequency ω, the frequency ν = ω/(2π).One can also use a different reparamerization like λ = ω 1 /3 , or replace the complex exponentials with trigonometric functions and describe the noise eigenmodes with two parameters (E, π), with E = ω 2 and π the parity of the trigonometric funtions (see Appendix B).

C. Dynamical decoupling noise spectroscopy of non-stationary environments
An important application of the presented framework, that leads to the generalized picture for the qubit-probe dephasing based on Eqs. ( 12) and (20), is that they can be used to probe the noise spectral properties for non-stationary, general Gaussian noises.Several methods have been designed to probe the noise spectrum of stationary noises based on Eq. ( 15), mainly for the n = 1 case, i.e. when the dephasing is given by 35,36,40,41,45,65,110,111].Control sequences generate filter functions that can allow only specific frequency components of the spectral density to produce dephasing on the qubitprobe system.The width of these "pass bands" filters can be made arbitrarily narrow.Therefore the spectral density can be reconstructed by performing series of measurement with different filter functions to scan the noise spectrum.This procedure termed dynamical decoupling noise spectroscopy [35] can be performed either by using continuous fields [45,110,111] or sequences of pulses [35,40,65].
The noise eigenmodes |Ω) that diagonalize the Kernel operator D are thus the natural basis to probe the non-stationary noise spectrum based on Eq. ( 20).In the simplest case of stationary noise, this basis becomes |ω, m), and therefore using a modulating function f m (t) = g m e −iω0t ∝ (t|ω 0 ) driven by control fields, one can probe the frequency modes of the noise spectrum for each morphological eigenmode m by scanning ω 0 .For such control modulations, the filter function |F m (ω)| 2 ∝ δ(ω − ω 0 ) senses single frequency modes according to Eq. ( 21), and leads to the so-called continuous-wave (CW) noise spectroscopy [45,[110][111][112][113][114].However, for non-stationary noises the frequency-basis in general cannot probe selectively the noise eigenmodes.In this case, the eigenvectors |Ω) define the proper basis to probe the single noise eigenmodes according to Eq. (17).Designing filter functions 1(d)] based on finding the control modulation functions that satisfy f Ω0 (t) = (t|Ω 0 ), one can probe the noise eigenspectrum by scanning it by changing Ω 0 based on Eq. ( 20) as (f |G|f ) ∝ S(Ω 0 ).Therefore, if the Ω-basis is known, dynamical decoupling noise spectroscopy approaches designed to scan the spectral density in the frequency domain for stationary noises [35,40,45], can now be adapted to probe non-stationary noises using the Ωdomain.
Alternatively, if information about the Ω-basis is not known, dynamical decoupling noise spectroscopy to scan multifrequency spectral densities can be implemented based on the general form derived in Eq. ( 12).Again, qubit noise spectroscopy control methods for estimating high-order noise spectra (so-called polyspectra), have been developed for stationary non-gaussian noises [42,67].The method is based on using dynamical control approaches based on frequency comb control modulations to design multidimensional filter functions to probe the polyspectra via repetition of suitable pulse sequences.Based on Eq. ( 12), this technique can now be straightforwardly adapted to estimate the non-stationary bispectrum S(ω 1 , ω 2 ) that induces the qubit-probe dephasing.Then, once the bispectrum is estimated, it can then be diagonalized to determine its eigenmode basis |Ω) and eigenspectrum S(Ω).

D. Mitigating decoherence effects of non-stationary noises
The reduction of decoherence effects on a qubit-probe coupled to non-stationary -out of equilibrium-noises is other important application of the presented framework.
The key result for attaining this goal, is the extension of the universal expression for the qubit dephasing induced by non-stationary, general Gaussian noises based on the overlap between a noise spectral density and a control filter function as demonstrated with Eq. ( 20).This extension allows implementing optimal control methods that were developed for mitigating decoherence induced by stationary noises on open quantum systems.These methods are based on finding an optimal control filter function F opt (ω) = (ω|f opt ) that minimizes the overlap between the noise spectral density and the control filter function in Eq. (15), and then implementing it experimentally in the time-basis as f opt (t) = (t|f opt ) [12,29,30,32].These control strategies can now be applied to nonstationary noises by designing an optimal filter F opt (Ω) that, analogously to the stationary case, minimizes the overlap with the non-stationary Gaussian noise spectrum S(Ω) in Eq. (20).This is only possible due to the introduction of the generalized non-stationary Gaussian noise spectrum based on determining the noise eigenmodes G|Ω) = S(Ω)|Ω), and is one of the main results of this paper.As described in Sec.IV C, S(Ω) can be inferred, so as to determine the proper modulation control |f opt ) that provides the optimal filter (Ω|f opt ) = F opt (Ω).This control, can then be expressed in the time-basis as f opt (t) = (t|f opt ) for its experimental implementation, as one does to decouple stationary environments.Therefore we have shown here, how the presented generalized path integral framework can be used for implementing dynamical control methods to mitigate decoherence effects induced by non-stationary, general Gaussian noises.

V. NON-STATIONARY NOISE PROCESSES LOCAL-IN-TIME
In this section we consider a broad subclass of nonstationary Gaussian noises that are local-in-time, as they allow more direct interpretations of the physical meaning that provides the path integral approach for the noise processes.We show that the dephasing for this type of non-stationary noises can be described by a differential operator based on constraints to the derivatives of the fluctuating field paths.We also show how these constraints are reflected on the functional behavior of the non-stationary noise spectrums.Therefore, the qubitprobe dephasing can be obtained by solving ordinary differential equations rather than integral equations as is the case for non local-in-time noises.This picture allows to obtain simpler expression for the noise spectrum of localin-time, stationary noises, as the inverse of the differential operator.In this case the differential operator is determined by a matrix polynomial of the frequency modes.We also show how this description for non-stationary, local-in-time noises, provides conditions for differentiating Markovian noises from non-Markovian ones.Moreover, we introduce a generalized Markovian process which includes all the derivatives of B in the stochastic process, that fully describe local-in-time non-stationary processes.The state of B is not only determined by the probability distribution of B, but by the joint probability distribution of B and its derivatives.These local-in-time noises while in general can be non-Markovian, in our generalized framework appear as a natural extension of Markovian noises, and maintain many of their properties while being able to model a greater variety of environments.

A. Local-in-time framework
We here consider a broad subclass of non-stationary Gaussian noises that are local-in-time.The kernel that defines the probability distribution for the possible paths that can take a local-in-time fluctuation B(t), satisfies D(t, t ) = 0 when t = t .The most general Kernel operator satisfying this condition can then be expressed as an expansion series where δ (k) denotes the k-th time derivative of the Dirac delta function and C k (t) are time dependent matrices.When the sum is finite, the most general action satisfying these conditions can be written as where We demonstrate in Appendix C that any local-in-time process can be described by an action of the form where D H k (t) are n × n real symmetric (Hermitian) matrices and D A k (t) are real antisymmetric (anti-Hermitian) matrices.These matrices define constraints on the values that the fluctuating field B and its derivatives can take.For example, the larger |D H k | is, the smaller the derivative B (k) must be or equivalently, the slower B (k−1) may change.The matrix norm |D H k | thus limits how fast B (k−1) can lose information of its previous state.In particular, for N > 1, information can now be stored in the derivatives of B, and it is this information storage that leads to these processes being non-Markovian.Based on Eq. ( 22), to achieve to Eq. ( 24) we have introduced the differential operator that gives the Kernel operator for local-in-time noise processes, where . Notice that if B is a one dimensional process, D A k = 0 for all k, since there are no non-zero real anti-Hermitian 1 × 1 matrices (scalars).
The inverse relation of Eq. ( 11) for local-in-time processes becomes under these assumptions.This relation shows that the time dependent operator D(t) can be diagonalized by simply solving a set of ordinary differential equations.This is in contrast to the case of nonlocal-in-time noise processes, where it is necessary to solve integral equations to obtain the noise eigenmodes.The autocorrelation functions G(t, t ) are the Green functions of the local-in-time differential operator D(t).Therefore a local-in-time noise process can be purely described by the constraints D H/A k (t) on the derivatives of the fluctuating field B(t) as seen in Eq. ( 24), providing more simple physical meanings for the form of the Kernel operator given in Eq. (25).A direct implication of this Kernel form is that it describes fluctuating field paths that are differentiable up to order N − 1.This means that B and its first N − 1 derivatives must be continuous functions [see Fig. 2 for an example] so as the action does not diverge and they contribute to the propagator (See Appendix D for a demonstration).Another implication of this Kernel form is that its characterization only requires to estimate the matrices D The Kernel operator of these local-in-time fluctuating field processes is described in the frequency basis as where (t)e it(ω1−ω2) are the Fourier transforms of the corresponding time dependent constraints D H/A k (t) on the k-th derivatives of the field fluctuations.Then, according to Eq. ( 14), its inverse gives the bispectrum S(ω 1 , ω 2 ).Based on Eq. ( 27), this spectrum is similar to a polinomial of ω 1 and ω 2 , except that in the non-stationary case, the "coefficients" ) depend on the frecuencies, and are determined by the temporal dependence of the constraints on the the k-th derivatives of the field fluctuations.The non-stationary evolution of the constraints is manifested for these local-in-time processes by functions that satisfy D(ω 1 , ω 2 ) = D(ω 2 , ω 1 ) † , and therefore have reflectionconjugation symmetry about the diagonal axis ω 1 = ω 2 in the frequency space.In stationary processes as we see below, D show the degree of non-stationarity of a process.
An alternative interpretation of the physical properties of these processes comes from employing a time discretization of the time evolution (see Appendix E).The argument of the integral of Eq. ( 24) depends only on the local time t, i.e. the Kernel only correlates the fluctuating field values and its derivatives instantaneously.However, the correlations imposed by the Kernel on the field derivatives effectively correlate the field at different times locally.Replacing the field derivatives in Eq. ( 24) with finite time diferences, the discrete version of the Kernel correlates the value of the fluctuating field at the nearest 2N + 1 times steps close to the time t, i.e.B(t ± k∆t) with k = −N...N and ∆t the time discretization step.Thus, this shows how a kernel operator that is local-in-time, correlates the fluctuating field values on a limited correlation time length, introducing therefore memory of the fluctuating field process B(t) on infinitesimally near times determined by the the contraints on the field derivatives.These processes are in general non-Markovian as discussed in Sec.V C, yet, the long-time correlations do not exits.

B. Stationary noise spectrum as the inverse of the differential operator
For stationary noises that are local-in-time, the differential operator D is time independent where we used The path integral framework allows then to express the noise spectrum S(ω) in terms of the operators D H,A k that define the probability density of field paths.Notice that D H,A k define now constant constraints on the derivatives of the fluctuating field B(t). Figure 2(a) shows typical field fluctuations with constraints up to the derivative of the field (N = 1) and up to the second derivative of the field (N = 2).Based on Eq. ( 16), these constraints then define the noise spectrum as where this matrix polynomial D(ω) is the expression in the frequency-basis of the differential operator D of a local-in-time stationary process.This equation is obtained from Eq. ( 27), where D for stationary processes.The polynomic functional behavior of the predicted noise spectrum is consistent with several experimental observations [35,93,98,115].As examples, a white noise correspond to N = 0, a Lorentzian spectrum corresponds to N = 1 representing Ornstein-Uhlenbeck one dimensional noise processes, and an spectrum given by the inverse of a quartic polynomial corresponds to N = 2 with a power law tail ∝ ω −4 [Fig.2(b)].This result thus provides a simple physical meaning for the power law exponents dependence of the noise spectrum based on the conditions D H k and D A k imposed to the derivatives of B(t) that the probability density of the possible paths can take.
An important application of Eq. ( 29) is that it facilitates the noise spectroscopy characterization, as the noise spectrum of local-in-time processes is defined by the 2N + 1 constant n × n-matrices D H/A k .Therefore, the hypothesis of locality-in-time allows the implementation of parametric estimation to determine noise spectra, rather than using non-parametric estimation as described in Sec.IV C for more general noises.Therefore noise spectroscopy methods for local-in-time stationary noises can thus reduce the amount of measurements required to reconstruct the noise spectrum.

C. Conditions for Markovian and non-Markovian noise processes
In this subsection, we state the conditions for the kernel operator D to represent a Markovian process.We show that non-Markovian, local-in-time processes can always be mapped to a Markovian process, which is described by the value of the field and its first N − 1 derivatives.We show that local-in-time processes are a generalization of Markovian processes and that those processes cannot generate long-time correlations.The probability distribution of B at a time t after a time t is shown to be based only on the state of the process at time t, if we consider the generalized process defined by the state of the joint probability distribution of B and its first N − 1 derivatives at time t.Therefore, local-in-time noises are generally non-Marovian and in our generalized framework appear as a natural extension of Markovian noises, as they maintain many of their properties while being able to model a greater variety of environments.They thus provide a natural model to study non-Markovian environments that does not contain long-time correlations.

Markovian process conditions
A stochastic process B is Markovian if the probability of the field B f at time t f starting from a value B 0 at t 0 is [79,[116][117][118] for any t f > t 0 , where P (B i , t i |B j , t j ) are the conditional probabilities that define the propagator of the process.A Markovian process is said to be memoryless, as the random variable "forgets" its previous state as it evolves [79,[116][117][118].Therefore, in general a stochastic process determined by the Kernel operator D as described in Eq. ( 7), is non-Markovian as it correlates the fluctuating field states at two distant times.Only local-in-time noise processes as defined in Sec.V A may be Markovian.We then only consider this case to evaluate the conditions required for the non-stationary noise process to be Markovian.
If the action A[B] of Eq. ( 24) contains derivatives of order N higher than 1, it implies a differentiability condition for the field paths so as the action does not diverge and they contribute to the propagator (See Appendix D).Therefore in this case B and its first N − 1 derivatives must be continuous functions.As Eq. ( 30) only imposes that the paths are continuous at t 1 to define a finite integral, if the action imposes more strict conditions on the possible paths, requiring continuity from their derivatives, the process of B will not be Markovian as the propagator requires information of different previous times.This implies in particular that the derivative of B is continuous, and therefore it contains information about the history of B. Then Eq. ( 30) cannot be satisfied as it does not include information about B encoded in its derivatives.Therefore, we found that the stochastic process that describes B is Markovian if and only if its Kernel operator is given by the local-in-time differential operator of the form where the derivatives of the paths are not continuous (see Appendix F for a full proof).In the frequency basis, the Kernel becomes following Eq.(27).Equations ( 31) and ( 32) are definitions of Markovian Gaussian processes within the presented framework that are equivalent to the conventional form of Eq. ( 30).
In the particular case of stationary noises, the most general n-dimensional Markovian noise spectrum is therefore according to Ec. (29).For the simplest case of one dimensional stationary noises, this means that the process that describes them is Markovian if and only if the noise spectrum is constant, i.e. white noise [N = 0 in Eq. ( 29)], or Lorentzian [N = 1 in Eq. ( 29)]. Figure 2 compares a Markovian field fluctuation for N = 1, where the field is not differentiable, and a non-Markovian field fluctuation with a continuous derivative for a one dimensional stationary noise.An important application of the noise spectrum form derived in Eqs. ( 27) and ( 33) is that they allow to determine when a noise process is Markovian or non-Markovian by experimentally measuring its spectrum with dynamical decoupling noise spectroscopy (Sec.IV C).

Generalized Markovian process for local-in-time non-stationary, non-Markovian noises
If the stochastic process is non-Markovian, but localin-time, it can still be described by a generalized Markovian process if all the derivatives of B are included in the stochastic process.We define a general conditional probability, or propagator, of the stochastic process based on the ordered set of all the derivatives of B of order lower than N , The Markovian condition of Eq. ( 30) is now satisfied for the stochastic process of {B (k) (t)}, as the equation imposes continuity on the first N − 1 derivatives of B. This means that both the action and the integral over {B (k) 1 (t)} impose that {B (k) (t)} are continuous, implying that the stochastic process {B (k) (t)} is Markovian (see the proof details in Appendix G).Therefore the family of fluctuating local-in-time non-stationary noises considered here imply that if B is not Markovian, the state of B is not only determined by the probability distribution of B, but by the joint probability distribution of B and its derivatives.This means that the information of the initial condition is encoded in the derivatives of B.
Since local-in-time noises can be described by the generalized Markovian process of {B (k) (t)}, its propagator P ({B 0 }, t 0 ) contains all the information about the stochastic process that describes the field paths B.Moreover, we found that the path integral framework allows to calculate the propagator of {B (k) } without actually requiring to perform path integrals.Instead, the propagator can be obtained by solving an ordinary linear differential equation with different types of boundary conditions.The full propagator expression and its derivation are given in the Appendix H.
In summary, the local-in-time stochastic process of B is Markovian if and only if the differential operator D(t) does not contain terms with k higher than 1.All other processes are non-Markovian.Yet, the generalized localin-time process of {B (k) (t)} is always Markovian, and it follows that the information of previous states of B is encoded in its derivatives when they are continuous.

The long time limit of local-in-time stationary noises is indistinguishable from a Markovian process
The spectral density S(ω) for local-in-time stationary noises of Eq. ( 29) is infinitely differentiable, as it is the inverse of a non-singular matrix polynomial, and therefore the integral of its derivatives is finite dω < ∞ for all k-th derivative order.Since the corresponding correlation functions are G ij (t) = dω 2π S ij (ω)e −iωt , one can see that and that they are bounded by the expression [119] |G Therefore, we obtain that lim |t|→∞

G(t)
t k = 0 for all k, implying that G(t) decays to zero exponentially (or faster) for long times t → ∞.Since S(ω) is the inverse of a matrix polynomial, we know that it is not entire when analytically continued to the complex plane of ω for N > 0. Therefore for N > 0, the correlation functions G(t) cannot decay faster than an exponential in the long time limit, so they must decays exactly as exponentials [120].Only for N = 0, the correlation function decays faster as its corresponding noise is a white noise.Therefore in the long time limit, every local-in-time stationary noise is indistinguishable by means of noise spectroscopy from a Markovian noise, as its self-correlation functions decay exponentially or are zero for long times, which coincides with the correlations functions derived from what we showed are Markovian noises.We consider an arbitrary differential operator D for a stationary local-in-time noise.In general, it is not possible to find a closed formula for the correlation functions G(t), since that would imply finding the roots of a polynomial of an arbitrarily high degree.However, it is possible to get information about the general behavior of G(t) at short times, by analyzing its representation in the frequency-basis, i.e. the noise spectrum S(ω).
In the large frequency ω limit, the noise spectrum is N ω −2N according to Eq. ( 29).Since D(ω) is positive definite, the spectral density S(ω) remains finite for all ω and the integral and are continuous [119].This means that for processes with N > 1, the correlation functions decay quadratically or slower around their maxima.Then as the correlation functions of each component of B satisfy they have a global maximum at t = 0, implying that if N > 1 they decay at least quadratically for short times.This agrees well with the expected behavior of correlation functions predicted by quantum mechanics [19,20,[121][122][123][124][125][126][127].
Therefore, every stationary noise process generated by a quantum mechanical process is not Markovian.Notice that the Markovian noise correlation functions for N ≤ 1 are a Dirac delta function that describes a white-noise spectrum (N = 0), and for a one dimensional noise with N = 1, we obtain a Lorentzian spectrum that describes an Ornstein-Uhlenbeck process where G(t) = 1 2D0τc e −|t|/τc , with τ c = D1 /D0.These Markovian noise processes disagree with what quantum mechanics predicts, that for short times G should decay at least quadratically.This is true for all Markovian processes, as Eq. ( 26) demands for the first derivative of G(t, t ) to be discontinuous at t = t when N = 1.

VI. IMPLEMENTATION OF THE FRAMEWORK ON PARADIGMATIC NON-STATIONARY NOISES
In this section, we show how to implement the presented path integral framework to two paradigmatic examples of non-stationary noises.We consider a noise determined from a quench on the environment and a noise that acts near to a point of time.Both examples shows some characteristic features that arise from our framework that distinguish non-stationary from stationary noise effects on the qubit-probe dephasing.

A. Quenched diffusion
We here consider a quenched environment described by a diffusion process of the fluctuating field that begin at an instant of time .This sets a paradigmatic model of typical environments that can be suddenly quenched to put them out of equilibrium, where excitations start spreading over a large number of degrees of freedom [13,50,56,64,74].The environment dynamics sensed by the qubit-probe can be represented with a generalized diffusion process, and possible examples can be encountered in spin ensembles coupled to single NV centers in diamond [94,128,129], macromolecular dynamics [130][131][132][133], spin diffusion on environments that becomes out of equilibrium [13,35,86,134,135], dynamics of spin and current fluctuations in a material at the nanoscale probed by magnetic noise sensors [136,137], and molecular diffusion out of equilibrium [63,[138][139][140][141].
As a simple model, in particular we consider that the fluctuating field is zero and at a given time suddenly starts to fluctuate, driven by a one dimensional Ornstein-Uhlenbeck diffusion process [47,48,93,94,129,136].Considering that the quench is at time t = 0, the fluctuating field is therefore confined to a point for t < 0. For t > 0 the noise process is described by a differential operator equal to a stationary one that gives a Lorentzian spectrum This quenched diffusion noise process is local-in-time and is modeled by the differential operator with where we considered D as a scalar Kernel operator.The action and the differential operator that describe the possible field paths are equivalent to the ones derived from a Schrödinger equation that describe an infinite potential wall at the position 0. Based on this analogy one can consider that x and D 0 (t) is mapped to the energy potential that describe the wall by replacing t with the position x.Therefore, the eigenmode basis that diagonalizes the Kernel operator D is where (Ω|Ω ) = δ(Ω − Ω ) and Ω ≥ 0 ∈ R. Following Eq. ( 18), the noise spectrum on the eigenmode basis of the non-stationary field fluctuations is (Fig. 3) and the generalized filter function is defined by Notice how the functional form of the non-stationary noise spectrum of Eq. ( 43) is equal to that of a stationary case for ω ≥ 0 [Eq.(39)] replacing Ω by ω.However the eigenmodes are different to the ones obtained in the stationary case.In this example, this means that the difference between the stationary and non-stationary cases is encoded in the filter function F .For a stationary Ornstein-Uhlenbeck process, F is obtained by performing the complex Fourier transform of f (t), but in the non-stationary process F is obtained by a real Fourier transform based on the sinusoidal component given by Eq. ( 44).

Manifestation of the quench on the noise correlation functions
The correlation function is the solution of the local-intime differential equation D(t)G(t, t ) = δ(t−t ).The correlation function satisfies G(t, t ) = G † (t , t), G(0, t ) = 0 due to the fluctuating field is 0 at t = 0 and it also satisfies lim t→∞ G(t, t ) = 0 for a fixed t as the correlation function of the Ornstein-Uhlenbeck process decays exponentially to 0. The solution for the differential equation is thus where τ c = D1 D0 is the correlation time and Θ(t) is the Heaviside function.
For long times after the quench t/τ c , t /τ c 1, we recover the correlation function for a Ornstein-Uhlenbeck stationary diffusion process In this case G(t, t ) is given by the Green function of the differential Kernel operator D 1 ∂ 2 t + D 0 acting over time dependent functions with t ∈ R instead of t > 0 as in the quenched case.Therefore, for times much longer than τ c , the environment forgets the boundary condition imposed by the quench.The non-stationary effects induced by the quench are thus modeled by the second term of Eq. ( 45).The quench effects are thus only manifested for times much lower than τ c with respect to the quench time t = 0, in contrast with the stationary effects that depend on the time correlation difference |t − t |.
The corresponding spectral density in the frequencybasis of Eq. ( 12) is the bispectrum where the first term is the stationary noise spectrum of Eq. ( 39), and the other term is due to the quench.Figure 3(a) shows this non-stationary bispectrum, where the stationary component is only manifested on the diagonal ω 1 = ω 2 .The noise eigenmodes [Eq.( 42)] and the nonstationary noise eigen-spectrum [Eq.(43)] are obtained by diagonalising this bispectrum.Figure 3(b) shows the corresponding diagonalized non-stationary spectrum S(Ω) of Eq. ( 43).

Manifestation of the quench on the qubit-probe dephasing
The difference in the qubit-probe decay induced by this quenched diffusion, between the non-stationary effects and its stationary counterpart is given by Therefore the non-stationary effects for this type of quenched noise always reduce the dephasing induced on the qubit-probe compared with a stationary process.We consider in particular the control modulation function of the typical continuous wave irradiation to show this manifestation of the non-stationary effects on the qubitprobe dephasing.We start the control at time t 0 after the quench and it acts during a time T .The argument of the dephasing given in Eq. ( 8) at time t 0 + T is where (f |G 0 |f ) is the dephasing obtained for the stationary case, and the second term is due to the quench -nonstationary-effects.This second term provides a dephasing term that oscillates with the control frequency ω as a function of the duration of the control modulation T .This oscillation is attenuated with the exponential decay e −T /τc , thus it disappears at long times after the quench.Then, if the control modulation duration is T τ c , we still have an extra term due to the quench effects Notice that this last term provides a constant term into the dephasing that contains information about the quench if t 0 τ c as it decays exponentially with the time t 0 when qubit-probe control started.This prediction thus gives a tool to probe the quench effects by monitoring the dephasing change as a function of t 0 , and gives a method to probe the self-correlation times induced by the quench.Moreover, if one monitor the decay rate of the qubit-probe as typically done in several noise spectroscopy approaches [35,40,45], one would obtain only the stationary decay rate given by Therefore, the effects of the quench are not manifested on the decay rate, but they are evidenced on a shift of the decaying signal given by the second term of Eq. ( 51).

B. Noise produced near to a point of time
As one of our results is that non-stationary spectrums can be discrete according to Eq. ( 18), we exploit an analogy with the Schrödinger equation to describe a paradigmatic example that manifests this discrete nature.A noise that contains these discrete features, is for example one that acts only near to an instant of time, a pulsed noise interaction.This is a paradigmatic model for a quantum probe that interacts with a noise during a finite duration of time.Examples of this can be moving charges or particles that pass near to quantum sensor [76,138,139,[142][143][144], forces or interactions detected by a moving cantilever or tip that contains the sensor [77,129,145], and biomedical applications as the detection of neuronal activity [75,76,78].
Every non-stationary noise described by a differential operator of the form where D 1 is a scalar, can be mapped to a quantummechanical problem with the Hamiltonian by replacing x → t, m → 1 2D1 and V (x) → D 0 (t).Therefore for every solvable quantum-mechanical Hamiltonian with positive eigenvalues, we obtain a solution for the noise spectrum and its eigenmodes of a Markovian localin-time, non-stationary Gaussian noise.The energy levels of the Hamiltonian must be positive so as the differential operator D(t) is positive definite, but every bounded from below Hamiltonian can be transformed into a positive definite one by adding a large enough constant C to H → H + C.
Based on this analogy, we describe the paradigmatic example of a pulsed noise with a one dimensional, nonstationary noise process that is local-in-time.We consider the noise described by a differential operator D that is mappable to the Hamiltonian of a quantum harmonic oscillator where D 0 is an additive constant that does not change the eigenvectors of the Hamiltonian H.The corresponding differential operator is where the necessary map is m → 1 2D1 , ω 0 → √ 4αD 1 and x → t.This stochastic process models a noise probed by the qubit-system that appears near to a point of time, where the fluctuating field paths are forced to be 0 for times |t| → ∞, since D 0 (|t| → ∞) → ∞.Therefore, the fluctuating fields are only allowed to deviate from 0 near to the local instant of time t = 0 (see blue dashed line in Fig. 4).The noise eigenmodes of D(t) are where H n are the Hermite polynomials with n = 0, 1, 2, ... in analogy with the eigenfunctions of the quantum harmonic oscillator [Fig.4(a)].The non-stationary noise spectrum expressed on its eigenmode basis is then discrete, given by as shown in Fig. 4(a).Since the differential operator D(t) is positive definite if and only if all its eigenvalues are positive, the model is well defined only for ω 0 > −2D 0 .This noise model sets a paradigmatic example of some of the differences between probing non-stationary and stationary noise spectra with dynamical decoupling noise spectroscopy.In this case, the natural control modulation would be f n (t) = (t|Ω n ) for n = 0, 1, 2, ... as shown in Fig. 4(a).Here the dephasing in the long time limit after applying these modulations will saturate to a constant value that provides the noise eigenvalues This is in contrast to the predicted exponential decay with a constant rate for stationary noises typically used for noise spectroscopy [35].Figure 4(b) shows a schematic representation of the qubit-probe signal decay when it is controlled by the modulation functions f n (t) = (t|Ω n ).This behaviour is universal for all noises with a discrete spectrum, therefore this example shows how noise spectroscopy must be done for such noises.

VII. SUMMARY AND CONCLUSIONS
We introduced a path integral framework for determining the dephasing on a quantum probe induced by nonstationary Gaussian noises.This type of noises models a fluctuating qubit-probe interaction with a quantum environment that is out-of-equilibrium under the so-called weak coupling approximation.We show that the noise generated by this non-stationary environment is completely determined by a bispectrum defined by the inverse of the Kernel operator that describes the probability density of the field paths.This complement recent results, where similar bispectra functions characterize stationary noises that are non-Gaussian [42,67].
The presented framework introduced a generalized noise spectrum for non-stationary environments, defined by the inverse of the eigenvalues of the kernel operator that determines the probability of the noise field paths.The noise eigenmodes define the proper basis to generalize a filter function derived from the control, whose overlap with the noise spectrum determines the qubit-probe dephasing.This results into an extension of the validity of the universal formula for the dephasing of open quantum systems that depends on the overlap between a noise spectral density and a qubit-control filter function.The main result of this generalization is that it allows to implement two important tools, already developed for stationary noises, to probe spectral properties and to mitigate decoherence effects of non-stationary environments.
We then considered a broad subclass of non-stationary noises, that we called local-in-time.We show they are described by a differential operator based on constraints to the derivatives of the fluctuating field paths.We also show how these constraints are reflected on the functional behavior of the non-stationary noise spectrums.Such subclass of the discussed non-stationary noises are the only ones that can be Markovian if the first derivative of the fluctuating fields is not continuous, e.g.white noises and stationary noises with Lorentzian noise spectra.Therefore, if the field derivative is continuous or the noise is not local-in-time, the noise process is non-Markovian.Remarkably, we found that local-in-time, non-stationary Gaussian noises that are non-Markovian, still can be described by a generalized Markovian noise process that includes the field and a finite number of its derivatives.This approach simplifies the noise description as it is fully determined by a differential equation and the constraints on its derivatives thus redusing the dimensionality for the characterization of the noise spectra.An important application for the derived forms of the noise spectrum is that they allow to determine when a noise process is Markovian or non-Markovian by experimentally measuring its spectrum with dynamical decoupling noise spectroscopy.In the particular case of stationary noises that are local-in-time, we found that the differential equation gives a noise spectra given by the inverse of matrix polynomials of the frequency modes.The polynomial coefficients associated to a power of the frequency are given by the constraint to the corresponding order of the fluctuating field derivative.This thus allow the implementation of parametric estimation methods to determine the noise spectra, rather than using non-parametric estimation that is much more complex and requires more available experimental time.
We have shown that in some cases local-in-time, nonstationary noises can be mapped to the Schrödinger equation.Thus every solvable 1 dimensional quantum mechanical Hamiltonian H for a single particle in one spatial dimension with eigenvalues bounded from below, generates a whole class of solvable noise probability distributions.We have used this map to apply the presented path integral framework to two paradigmatic non-stationary noises: a noise acting near to a point of time -a pulsed noise-by analogy to a quantum harmonic oscillator and a quench on the environment that suddenly starts an Ornstein-Uhlenbeck diffusion process.Both of these examples describe out-of-equilibrium environments.The pulsed noise is a paradigmatic model for a quantum probe interacting with a noise during a finite duration of time that manifests a discrete nature on the generalized nonstationary noise spectrum.In this case the noise spectrum is obtained by saturation of the qubit-probe dephasing rather than on its decay rate.The quenched diffusion noise sets a paradigmatic model for typical environments that can be suddenly quenched to put them out of equilibrium, where excitations start spreading over a large number of degrees of freedom.In this case we show some features on the spin dephasing and noise spectra that manifest the quench -non-stationary-effects, evidenced by a bispectrum and a reduced dephasing compared with a stationary noise.In particular, we show how the quench correlation can be probed by monitoring this dephasing.
The results presented here thus set a general framework for a large universal class of non-stationary -outof-equilibrium-noise sources of decoherence, allowing to probe and interpret noise spectral properties and time-correlations by a quantum probe.Thus they provide tools and insights for probing and understanding the dynamics of quantum information of out-ofequilibrium complex quantum systems via a quantum sensor [13,49,50,53,57,61,74,135,137,146].In particular, they can be useful for quantum sensing the dynamics of single, but complex, large molecules as proteins [133,[147][148][149] and neuronal activity [75,76,78] with potential applications in biology and medicine.At the same time the general framework sets an universal formula to allow finding optimal control for protecting against decoherence generated by more realistic environments, that at atomic scales produce non-stationaryout-of-equilibrium-noise fluctuations.This tool is very important for implementing quantum technologies that can span from memory storage to information processing in quantum devices [1][2][3][4]144].
where δ is de Dirac delta distribution and the boundary condition lim |t|→∞ G(t, t ) = 0 must be satisfied.The path integral is well defined if and only if D is a positive definite operator.For these operators, functions such that lim |t|→∞ B(t) = 0 do not contribute to the integral, as (B|D|B) diverges.
Since for any real anti-Hermitian operator D A , (B|D A |B) = 0, one can only consider integrals where D is a Hermitian operator without loss of generality as in Eq. (A1).This can be demonstrated by considering that an arbitrary D can be decomposed as with its Hermitian and anti-Hermitian parts, respectively.The value of the integral thus depends only on D H , and therefore we can consider D A = 0.In this article, we have considered J = if and B the noise fluctuating fields, therefore Appendix B: Parametrization of the noise eigenmode basis As described in Sec.IV A, Ω is a parameter that in general its physical meaning depends specifically on the non-stationary process, and on the parametrization of the functional behaviour S(Ω), as it is always possible to reparametrize the eigenmodes by a different parameter.Similarly, while the frequency ω seems a natural choice for stationary environments, different parameters can be used to describe the noise eigenmodes also in stationary systems.
Here, we show examples of possible reparametrization of the noise eigenmodes for stationary noises.Specifically, rather than using Ω = (ω, m) as discussed in Sec.
Another example is analogous to how the free particle Hamiltonian eigenstates can be indexed either by their momentum or by their energy and parity.In this case, the complex exponentials are replaced by trigonometric functions and the noise eigenmodes are |E, π, m).
, where the prefactors are again obtained from the normalization condition The most general action that describes a local-in-time, non-stationary noise process is of the form as in Eq. ( 23) of the main text.Since the only paths that contribute to the dephasing integral satisfy lim |t|→∞ B (k) (t) = 0 for all k ≤ N , we integrate by parts each of the action terms and find that where Ḋk,l (t) is the derivative of D k,l (t).Integrating by parts thus transforms a single term containing the derivatives of order k and l of B into two terms, one containing the derivatives of order k − 1 and l + 1 and another containing the ones of order k − 1 and l.Since l ≤ k, this integration by parts reduces the difference between the orders of the derivatives when applied to terms with k − l > 1.By repeating this procedure successively, one can express the action only containing terms with k = l and k = l + 1, where DH k (t) and DA k (t) are real matrices.Considering that the derivative and that ) the action can be written as and the real anti-Hermitian ones the action can be simplified to Defining then the Hermitian differential operator as we thus obtain as defined in the main text in Eq. (24).The operator D acts on the field function B(t) as where ← ∂ t = −∂ t over the space of functions considered, as implementing integration by parts we find that In this Appendix we demonstrate that the field paths and their first N − 1 derivatives must be continuous for local-in-time noise processes.We thus study the differentiability of the paths allowed by the action.Considering a particular B(t), let us assume that at some time t some of the first N − 1 derivatives of B(t) are not continuous at t = t.Then, the N order derivative B (N ) (t) = B for functions for which these limits exist and B The last term of this integral gives a positively divergent contribution that depends quadratically on the discontinuity of B(t), given by ∆B (k) , and the first N − 1 derivatives of the field at t = t.The discontinuities of higher derivatives do not appear in the action.Since the operator D(t) is definite positive, the total contribution of these discontinuous paths to the action must be infinite.Therefore the probability of discontinuous paths and paths with discontinuities in their first N − 1 derivatives, is zero, and do not contribute to the path integral.Thus we have concluded that the field paths and its first N − 1 derivatives must be continuous for local-in-time processes.
Appendix E: Discretization of local-in-time processes In this appendix we show an interpretation of the meaning of local-in-time processes by implementing a time discretization.We replace the continuous time variable t by the discrete sequence of times t j = j∆t.The action is thus where the field derivatives have been replaced using finite differences Ḃ(t j ) = B(tj+1)−B(tj−1) 2∆t with N the highest derivative order that contributes to the action.Within this discrete representation, the fluctuating field B can be defined by a vector B j = B(t j ) and the kernel operator D can be represented by the matrix D jj = D(t j , t j ).Therefore, the k-th derivative of the field B In this Appendix we analyze the conditions for a nonstationary noise to be non-Markovian.As we stated in the main text, noise processes that are not local-in-time cannot be Markovian since the action explicitly relates the value of the fluctuating field B at different times.A noise with N = 0 is the so-called white noise, and therefore it is Markovian.Therefore, we only consider in the following demonstration, local-in-time noises with N ≥ 1.In Appendix G we prove that the generalized noise process given by {B (k) (t)} is Markovian.As a particular case, if N = 1, then {B (k) (t)} ≡ {B (k) (t) : k = 0} = {B(t)}, therefore showing that processes with N = 1 are Markovian.In the Appendix D, we prove that the field paths that contribute to the path integral for local-intime noises are not only continuous, but are N − 1 times continuously differentiable.This property allows us to demonstrate that noises with N > 1 are not Markovian.
To do this, we consider the values of the field at two infinitesimally close times t 0 and t 1 to be B 0 and B 1 , respectively.According to this assumption, the field derivative Ḃ(t 1 ) = B1−B0 t1−t0 must be continuous.If we now consider the field value at a time t f , infinitesimally close and after t 1 , we know again that Ḃ will change continuously.Therefore, the mean value of the probability distribution P (B f , t f |B 1 , t 1 ; B 0 , t 0 ) is B 1 + Ḃ(t 1 ) (t f − t 1 ) = t1−t0 .This means that the state of the field at t f depends on the state B 0 and B 1 at times t 0 and t 1 respectively.Therefore, the probability distribution of the field depends on the state of the system at least at two previous times, thus

FIG. 1 :
FIG. 1: Representation of the qubit-probe interaction with fluctuating fields derived from the path integral framework.(a) Schematic representation of a qubit-probe interacting with the fluctuating fields Bi(t) that generate pure dephasing.The system-bath coupling strengths gi are dynamically controlled.The index i labels properties of the qubit-bath coupling network morphology.(b) Representation of the fluctuating fields paths Bi(t) that give a generalized Gaussian distribution that is sensed by the quantum probe as a function of time.The fluctuating fields are described in the time-basis |t).(c) Frequency-basis |ω) representation of the fluctuating fields correlation functions, which provides a bidimensional, non-stationary noise spectrum S(ω1, ω2) = (ω1|G|ω2).(d) Noise eigenmode Ω-basis representation of the non-stationary, generalized noise spectrum S(Ω) (red solid-line), obtained after diagonalization of the noise spectrum in the frequency-basis.A generalized filter function F (Ω) in the noise eigenmode Ω-basis is shown in green solid-line.Here, it mainly probes a single noise eigenmode Ω0.

FIG. 2 :
FIG. 2: Schematic comparison between a Markovian noise for N = 1 and a non-Markovian noise with N = 2 for one dimensional, local-in-time stationary noises with different constraints on the differential operator D. (a) Examples of single realizations of field fluctuations Bi(t) for both cases.The differential operator has constraints up to the first derivative for N = 1, where the field fluctuation is continuous but not differentiable.For N = 2, the field fluctuation and its derivative are continuous.(b) The corresponding noise spectrums S(ω).For N = 1 the spectral density is a Lorentzian and for N = 2 the inverse of a quartic polynomial, with power law tails ∝ ω −2 and ∝ ω −4 for large frequencies, respectively.
H/A k (t), which are 2N + 1 functions of R → R n×n .Conversely, a general non local-in-time process, requires estimating the Kernel D

4 .
Short time limit t → 0 of local-in-time stationary noises

FIG. 3 :
FIG. 3: Non-stationary noise spectrum of a one dimensional quenched diffusion process of the environmental field fluctuations, driven by a one dimensional Ornstein-Uhlenbeck stochastic process.(a) The absolute value of the corresponding spectral density S(ω1, ω2) in the frequency-basis, the bispectrum given by Eq. (47), is shown.Only the diagonal ω1 = ω2 contains the stationary component of the noise spectrum that is dominant at long times after the quench.As the stationary component is ∝ δ(ω1 − ω2), we have plotted it by replacing the δ(ω1 − ω2) function by a non-divergent gaussian function for presentation purposes.The inset shows a surface colored map of the bispectrum with contour lines in white color.(b) The corresponding generalized, non-stationary noise spectrum S(Ω), given in the noise eigenmode basis |Ω) that diagonalize the noise spectrum.

FIG. 4 : 1 Ωn+D 0
FIG. 4: Scheme for the characteristic features of a paradigmatic local-in-time, non-stationary noise that acts only near to a point of time and therefore produce a discrete, generalized noise spectrum.The variance of a one dimensional noise that appears near to a point of time is shown with a blue dashed line.(a) The generalized non-stationary noise spectrum S(Ωn) = 1 Ωn+D 0 in the noise eigenmode basis is shown with orange horizontal lines.The corresponding noise eigenmodes |Ωn) are shown in the time-basis fn(t) = (t|Ωn) for n = 0, 1, 2, ... with solid black lines.They also provide the natural control modulations to probe the noise eigenmodes.(b) Schematic representation of the qubit-probe signal decay when it is controlled by the modulation functions fn(t).They feature a signal saturation at long times given by exp {−(Ωn|G|Ωn)} = exp {−S(Ωn)}.
Appendix C: Derivation of the local-in-time differential operator D

∆B
(k) δ (N −1−k) (t − t), (D1)where the coefficients∆B (k) = lim t→t + B (k) (t) − lim t→t − B (k) (t) (D2) ) is finite.Then, terms like the following contribute to the actiondt B (N ) † (t)D H N (t)B (N ) (t) = dt B (k) j = j+k j =j−k c k ∆t k B jis a linear combination of the values that B takes at the 2k + 1 times closest to j, where the coefficients c k are independent of ∆t.Thus, a kernel operator corresponding to a local-in-time process has coefficients that satisfy D jj = 0 for |j −j | > N .Within this discrete picture, local-in-time processes are those whose kernel operator is N -diagonal, i.e. its matrix representation has non zero coefficients only in the first N central diagonals.In this representation only if N = 0, the kernel operator is diagonal in the time basis and therefore B(t) and B(t ) are uncorrelated for t = t .However for N ≥ 1 the field at the N -th closest times are correlated.Still, local-intime processes are such that long-time correlations do not exist.Appendix F: Non-Markovian proof for N > 1