Introduction

Quantum correlations (QCs) are essential in quantum information science1,2,3,4 and quantum computation5,6,7,8,9, as they are pivotal for understanding the inherent non-locality in quantum mechanics10,11,12. During quantum information processing, quantum systems are inevitably affected by interactions with their surrounding environment. These interactions result in quantum decoherence, which is key to comprehend the transition from quantum to classical behavior13,14,15,16,17,18. In order to grasp the environment induced-decoherence, the model of a single central spin interacting with an environment, known as the central spin model (CSM)19,20,21,22,23,24,25, has been extended to the notion of quantum phase transition15,26. Within the CSM framework, an environmental quantum spin system (ESS) interacting with a central spin (CS) or qubit can be either time-independent or time-dependent.

In the time-independent case, the environmental spin system starts in its lowest energy state, whereas the central spin/qubit is initially in a pure state. The overall connection between the qubit and the environment is structured so that the ESS’s initial ground state evolves along two distinct pathways, each governed by a different Hamiltonian. Although the qubit starts in a pure state, it has been shown that it loses almost all of its purity as the ESS approaches its quantum critical point15,26. Furthermore, in the context of time-dependent scenarios, as the ESS is gradually moved through its quantum critical point, there is a notable increase in decoherence. This amplification arises not only from the heightened susceptibility present near the critical point but also from the provoked excitations, suggesting an aspect that is dependent on the universality class. This phenomenon shows a remarkable similarity to the dynamics associated with defect formation seen in nonequilibrium phase transitions, as described by the Kibble-Zurek mechanism.27,28.

Nevertheless, there has been a limited focus on the investigation of stochastically driven ESSs characterized by noisy Hamiltonians, and the influence of quantum coherence within these systems is still largely unexamined.

Noise is a major obstacle in achieving goals in all quantum technologies, especially in the progression of quantum computing.29,30,31. In particular, by altering the outcome of a quantum dynamics through the disturbance that it introduces to the system parameters. Such disturbance can become significant which leads to information loss in qubits and a general deviation of the system from its intended state. Due to the conceptual and technical complexities in dealing with the system plus environment fully quantum mechanically, an alternative approach is to simply consider that the effect of the environment is to introduce classical noise in the system’s degrees of freedom32,33,34,35,36,37. In other words, noise arises as an efficient way of describing the evolution of systems interacting with environments or external driven fields. Surmounting this challenge, is vital for advancements in quantum computing, with a goal to attain greater reliability and accuracy.

In addition, in any real experiment, the simulation of the intended time-dependent Hamiltonian is inherently imperfect, and the presence of noisy fluctuations is inevitable. In essence, noise is pervasive and unavoidable in any physical system, exemplified by noise-induced heating that may result from amplitude variations in the lasers used to create the optical lattice38,39,40,41. Consequently, it is essential to comprehend the effects of noise on Hamiltonian evolution to accurately forecast experimental results and to develop advanced configurations that are robust against noise effects35,38,42,43,44,45,46.

The question addressed in this paper is: What are the effects of a noisy ESS on the decoherence of a central qubit when it is driven across the QCPs? Specifically, is there still a universal pattern in the dynamically induced decoherence, as measured by the decoherence factor (DF) of the CS? More precisely, can the decoherencies due to proximity to a critical point and noise be distinguished?

We demonstrate that decoherence factor decreases in the presence of both correlated (colored) and uncorrelated (white) Gaussian noises. This reduction exhibits exponential scaling with the system size, the square of noise intensity, and the noise correlation time at the critical points. Moreover, in the case of strong environment-qubit coupling (large ramp time scale), decoherence exhibits revivals in both noiseless and noisy scenarios. These revivals scale exponentially with the square of noise intensity. However, in the presence of fast colored noise (small correlation time), the revivals scale linearly with the noise correlation time, whereas in slow noise (large correlation time), they exhibit a power-law scaling with noise correlation time. Additionally, we find that the measure of non-Markovianity decreases with noise intensity and increases linearly with the noise correlation time.

Theoretical model

Fig. 1
figure 1

(a) The schematic diagram of spin qubit symmetrically coupled to the critical spin environment described as a transverse field Ising chain. When the magnetic field is large (\(|h(t)| > 1\)), (right and left panels), the environment is in a paramagnetic phase, where spins aligned with the field. In contrast, when the field is small (\(|h(t)| < 1\); (center panel), the spin chin enters a ferromagnetic phase, where spins ordered in x or \(-x\) direction. (b) The decoherence factor can be directly measured through a Ramsey experiment, which involves a \(\pi /2\) pulse sequence followed by a projective measurement in the Z-basis on the central qubit47,48,49,50.

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The full Hamiltonian which considers a qubit coupled to a driven transverse field Ising chain (Fig. 1), is expressed as23,24,25

$$\begin{aligned} \begin{aligned} \mathcal{H} = \mathcal{H}_E + \mathcal{H}_I + \mathcal{H}_q, \end{aligned} \end{aligned}$$
(1)

where

$$\begin{aligned} \mathcal{H}_E = \mathcal{H}_E (h(t)) = - \sum _{j=1}^N \Big (\sigma _j^{x}\sigma _{j+1}^{x} + h(t)\sigma _j^{z} \Big ), \end{aligned}$$

represents the time dependent transverse field Ising model in a ring configuration,

$$\begin{aligned} \mathcal{H}_I =\mathcal{H}_I (\delta ) = - \delta \sum _{j=1}^N \sigma _j^{z}\sigma ^z_0,\ \end{aligned}$$

describes the interaction between the surrounding spin ring and the central qubit, and

$$\begin{aligned} \mathcal{H}_q = \sigma _0^z, \end{aligned}$$

corresponds to the Hamiltonian of the qubit. Here, \(\sigma ^{\alpha = x, y, z}\) are Pauli matrices, and \(\delta\) represents the interaction strength between environment and the qubit. To search the effects of noise on the decoherence, we consider the noise added time dependent transverse field, i.e., \(h(t)= h_0(t)+S(t)\), in which the noiseless transverse field \(h_0(t)\) varying from an initial value \(h_i\) at time \(t_i\) to a final value \(h_0(t)\) at time t following the linear quench protocol \(h_0(t) = t/\tau _Q\), where \(\tau _Q\) is the ramp time scale, and S(t) is the stochastic noise. When the transverse field is time-independent and noiseless, \(h_0(t) = h\), the ground state of the model is in the ferromagnetic phase for \(|h| < 1\), while the system is in the paramagnetic phase for \(|h| > 1\), with the phases separated by equilibrium quantum critical points at \(h_c = \pm 1\)51.

Assuming the qubit is initially in a pure state at \(h(t_i)\)23,24,25,

$$\begin{aligned} |\phi (t_i)\rangle _{q}=c_u|\uparrow \rangle +c_d|\downarrow \rangle , \end{aligned}$$
Fig. 2
figure 2

Noiseless decoherence, as measured by the visibility or decoherence function during a quench, is analyzed for \(\delta = 0.01\) with varying ramp time scales \(\tau _Q\) and system sizes N: (a) For \(\tau _Q = 250\), nearly perfect revivals of decoherence observed between two critical points. However, decoherence is reduced when the magnetic field swept through the second critical point, and wiped out for very large system sizes. (b) For \(\tau _Q = 10\), there is a monotonic decay of decoherence between the critical points. This behavior is observed when \(\tau _Q\) exceeds the threshold value of \(\pi /(16\delta )\), indicating that \(\tau _Q\) is approximately around this threshold. (c) The decoherence behavior during the quench for \(\tau _Q = 1\) is illustrated, highlighting the differences in decoherence loss compared to the other time scales.

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with coefficients satisfying \(|c_u|^2 + |c_d|^2 = 1\), and the environment, E, is in the ground state denoted by \(|\varphi (t_i)\rangle _E\), the total wave function of the composite system at time \(t_i\) can then be written in the direct product form23,24,25

$$\begin{aligned} |\Psi (t_i)\rangle = |\phi (t_i)\rangle _q \otimes |\varphi (t_i)\rangle _E . \end{aligned}$$
(2)

It is straightforward to show that, the total wave function at an instant t is given by23,24,25

$$\begin{aligned} |\Psi (t)\rangle = c_d|\downarrow \rangle \otimes |\varphi ^{-}(t)\rangle + c_u|\uparrow \rangle \otimes |\varphi ^{+}(t)\rangle , \end{aligned}$$
(3)

where

$$\begin{aligned} |\varphi ^{\pm }(t)\rangle = \hat{T} \exp \left[ -i \int _{t_i}^t \mathcal{H}_E \Big ( h(t) \pm \delta \Big ) \, dt\right] |\varphi ^{\pm }(t_i)\rangle , \end{aligned}$$
(4)

and \(\hat{T}\) denotes the time-ordering operator. Since interaction Hamiltonian commutes with the central spin Hamiltonian, the basis \(\{|\uparrow \rangle , |\downarrow \rangle \}\) are stationary states. Thus, the evolution of the entire system simplifies to the dynamics of two Ising branches, each evolving in an effective magnetic field given by \(h_\textrm{eff}(t)=h(t) \pm \delta\). Consequently, the spin-dependent evolution of the environmental states is governed by23,24,25

$$\begin{aligned} i \frac{\partial }{\partial t} \left| \varphi ^\pm (t)\right\rangle = \mathcal{H}_E \Big ( h(t) \pm \delta \Big ) \left| \varphi ^\pm (t)\right\rangle . \end{aligned}$$
(5)

Given the qubit in the \({|\uparrow \rangle , |\downarrow \rangle }\) basis, the reduced density matrix is described by23,24,25

$$\begin{aligned} \begin{aligned} \rho _q(t)= \textrm{Tr}_E \Big [ |\psi (t)\rangle \langle \psi (t)| \Big ] = \left( \begin{array}{cc} |c_u|^{2} & c_u c_d^* d^*(t) \\ c_u^* c_d d(t) & |c_d|^{2} \end{array} \right) , \end{aligned} \end{aligned}$$
(6)

where \(d(t) = \langle \varphi ^+(t)|\varphi ^-(t)\rangle\) captures the coherency of the spin. We focus on studying its squared modulus, \(|d(t)|^2\), which is known as the decoherence factor, to analyze the time evolution of decoherence23,24,25:

$$\begin{aligned} D = |d(t)|^2 = |\langle \varphi ^+(t)|\varphi ^-(t)\rangle |^2. \end{aligned}$$
(7)

Not to be confused with the term noise, it is preferable to refer to this property as visibility, which measures the interference contrast. It is worth noting that the entanglement entropy, concurrence, and maximum quantum Fisher information are directly connected to the decoherence factor52,53,54. The qubit is fully decohered when \(D = 0\), but it stays in a pure superposition state when \(D = 1\). In the unique situation where \(c_u = c_d = 1/\sqrt{2}\), the density matrix assumes the subsequent straightforward form

$$\begin{aligned} \begin{aligned} \rho _q(t) = \frac{1}{2}\left( \begin{array}{cc} 1 & d^*(t) \\ d(t) & 1 \end{array} \right) = \frac{1}{2}\Big (\mathbbm {1} + \langle \sigma _0^x\rangle \sigma _0^x + \langle \sigma _0^y\rangle \sigma _0^y\Big ). \end{aligned} \end{aligned}$$
(8)

From this, it becomes evident that one can directly measure D by evaluating the Pauli x and y observables of the central spin. Specifically, \(d(t) = \langle \sigma _0^x \rangle + i \langle \sigma _0^y \rangle\). The standard experimental scheme to perform this measurement is known as the Ramsey measurement, which involves suitable pulse sequences followed by a projective measurement in the Z-basis47,48,49,50, as illustrated in Fig. 1b. In this scheme, the first pulse performs a \(\pi /2\)-rotation of the initial qubit state around the y-axis. The second pulse applies another \(\pi /2\)-rotation, but around either the x-axis or the y-axis, depending on whether \(\sigma _0^x\) or \(\sigma _0^y\) is to be measured. The outcome statistics are then read out in the Z-basis to determine the expectation values of \(\sigma _0^y\) or \(\sigma _0^x\).

Fig. 3
figure 3

The noisy decoherence during a ramp up the magnetic field in the presence of the white noise, for \(\delta =0.01\), \(N=500\) and different values of \(\tau _Q\) and noise intensity: (a) As seen, revivals of decoherence which appear between two critical points for the ramp time scale \(\tau _Q=250\), are still partially present in the presence of the white noise. (b) Monotonic decay of decoherence between the critical points for \(\tau _Q=10\) increases in the presence of the white noise. (c) Illustrates decay of the noisy decoherence during the quench for \(\tau _Q=1.\).

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Decoherence factor and time dependent Schrödinger equation

The behavior of the decoherence factor (visibility) can be determined by solving the time-dependent Schrödinger equation (Eq. 5). Through the application of Jordan-Wigner fermionization and use of Fourier transformation55,56, the Hamiltonian presented in Eq. (1) can be reformulated as the sum of N/2 non-interacting terms23,24,25:

$$\begin{aligned} H(t) = \sum _{k} H_{k}(t), \end{aligned}$$
(9)

with

$$\begin{aligned} \begin{aligned} H_{k}(t)= [-(h(t)\pm \delta -\cos k)](c_{k}^{\dagger }c_{k}+c_{-k}^{\dagger }c_{-k}) +\sin (k)(c_{k}^{\dagger }c_{-k}^{\dagger }+c_{k}c_{-k}), \end{aligned} \end{aligned}$$
(10)

where \(c_{k},~ (c_{k}^{\dagger })\) represents spinless fermionic annihilation (creation) operator, and the wave number k is given by \(k = (2m - 1)\pi / N\), with m ranging from 1 to N/2, and N denotes the total number of spins (or sites) in the chain (mathematical details of analysis can be found in Supplemental Material). The Bloch single-particle Hamiltonian \(H_{k}(t)\) can be expressed as23,24,25

$$\begin{aligned} H_{k}(t) = \left( \begin{array}{cc} h_{k}(t) & \Delta _{k} \\ \Delta _{k} & -h_{k}(t) \\ \end{array} \right) , \end{aligned}$$
(11)

where \(\Delta _{k} = \sin (k)\) and \(h_{k}(t) = -( h(t) \pm \delta - \cos (k))\). Thus, the time-dependent Schrödinger equation (Eq. (5)) with \(|\varphi _{k}^{\pm }(t)\rangle =(v^{\pm }_{k}, u^{\pm }_{k})^{T}\) can be expressed as

$$\begin{aligned} \begin{aligned} i \frac{d}{dt} v^{\pm }_{k} =&-(h(t) \pm \delta - \cos k) v^{\pm }_{k} + \sin k \, u^{\pm }_{k}, \\ i \frac{d}{dt} u^{\pm }_{k} =&(h(t) \pm \delta - \cos k) u^{\pm }_{k} + \sin k \, v^{\pm }_{k}. \end{aligned} \end{aligned}$$
(12)

Within this framework, one can derive that

$$\begin{aligned} D(t) = \prod _{k>0} F_k(t); \quad F_k(t) = \left| u_{k}^{+*}(t) u_{k}^-(t) + v_{k}^{+*}(t) v_{k}^-(t) \right| ^2, \end{aligned}$$
(13)

where \(F_k(t)\) captures the dynamics of decoherence from the perspective of momentum space23,24,25.

Fig. 4
figure 4

Scaling of the minimum decoherence at the critical point \(h_c=1\) versus \(N\xi ^2\) for the different ramp time scales: (a) \(\tau _Q=250\), (b) \(\tau _Q=10\), (c) \(\tau _Q=1\).

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It is worthy to note that, in the absence of noise \(S(t)=0\), it can be demonstrated that the coupled deferential equations in Eq. (12) are exactly solvable23,24,25,57,58,59,60. While in the presence of noise, the ensemble average of \(u_{k}^{\pm }(t)\) and \(v_{k}^{\pm }(t)\) can be calculated numerically using the master equation32,58,59,60,61,62. In the following we first review the dynamics of decoherence factor in the noiseless case and then we search the effects of noise on the dynamics of decoherence factor.

Noiseless decoherence factor

To study the dynamics of the decoherence, we initially prepare the system in its ground state at \(t_i\rightarrow -\infty\) (\(h_0(t) \ll -1\)), and then ramp up the magnetic field in such a way that the system crosses the ferromagnetic phase (\(-1< h_0(t) < 1\)), to enter the other paramagnetic phase (\(h_0(t) > 1\)). In such a case the quench field crosses the critical points where located at \(h_c = \pm 1\).

As the transverse field is ramped up, the responsive of environment to external field increases. This heightened sensitivity leads to enhanced decoherence, which is observed as a gradual reduction in the decoherence factor D, beyond the critical point; see Fig. 2. In the noiseless case, it has been shown that the decoherence factor in the paramagnetic phase is approximately given by23,24,25

$$\begin{aligned} D(t)\approx \exp \left( -\frac{N\delta ^2}{4h(t)^2(h(t)^2-1)}\right) . \end{aligned}$$
(14)

As the transverse field crosses the first critical point, \(h(t)>-1\), the adiabatic evolution breaks down which leading to acceleration of decoherence; appears as a substantial reduction in decoherence factor. As a result, the dynamics of decoherence influenced by the two effects: (i) the excitations in the environments due to the crossing the critical point, (ii) the perturbation which amplified by enhancement of environment’s sensitivity close to the quantum critical points58,59,60.

As the driven transverse field crosses the first critical point at \(h_c = -1\), decoherence either partially revives (Fig. 2a) or decays monotonically (Fig. 2b,c) in the region between two critical points. These behaviors arise from the non-adiabatic dynamics of the system near the critical point, as the band gap closes for \(k = \pi\) at \(h_c = -1\), where large k modes are excited. The large k modes, particularly those with \(k \sim \pi - \hat{k}\) with \(\hat{k} \sim \tau _Q^{-1/2}\) experience significant excitation23,24,25, while the small k modes, for which \(k \ll \pi - \hat{k}\), evolve adiabatically through the critical point. Additionally, partial revivals of decoherence between the critical points have been observed when the environment-qubit coupling is sufficiently strong, specifically when \(\delta \gg \pi /(16\tau _Q)\)23,24,25; see Fig. 2a. These revivals manifest within the magnetic field, h(t), domain with a period of \(\pi /(4\tau _Q\delta )\)23,24,25. In contrast, in the weak coupling regime where \(\delta \ll \pi /(16\tau _Q)\), the decoherence factor exhibits a monotonic decrease, as illustrated in Fig. 2b. Moreover, more complex decoherence dynamics arise when the driven field inverses polarity and takes the system through the second critical point at \(h_c = 1\). At this critical point, where the gap closing occurs at \(k = 0\) the modes with \(k \sim \hat{k}\) become excited.

In the next section we will study the effect of noise on the dynamics of the decoherence factor using the numerical solution of the exact master equation.

Fig. 5
figure 5

The noisy decoherence during a quench in the presence of colored noise, with parameters \(\delta =0.01\) and \(N=500\): (a) For \(\tau _Q=250\) and \(\xi =0.003\), revivals of decoherence observed between critical points in the ramp time scale \(\tau _Q=250\) are also partially visible in the presence of colored noise. (b) For \(\tau _Q=10\) and \(\xi =0.01\), there is a monotonic decay of decoherence between critical points, which becomes more pronounced with an increase in \(\tau _n\). (c) For \(\tau _Q=1\) and \(\xi =0.012\), the decay of noisy decoherence is shown for a quench with \(\tau _Q=1\).

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Impact of environmental noise: noisy dynamics

Noise is everywhere and unavoidable in any physical system. Specifically, when energy is moved into or out of a system in the lab, there will always be some fluctuations, “noise”, in this process. In addition, in the Ramsey interference scheme, the central qubit used as a noise spectrometer47,48,49,50. The noise is caused by a classical fluctuating field. In this section, we look at how noise affects the decoherence of a qubit that’s linked to a time-varying environmental spin system. To this end, we consider an added noise to the time dependent magnetic field

$$\begin{aligned} h(t) = h_{0}(t) + S(t) = \frac{t}{\tau _Q} + S(t), \end{aligned}$$
(15)

in the ramp interval \([t_i = h_i \tau _Q, ~ t_f = h(t) \tau _Q],\) where S(t) represents random fluctuations with vanishing mean, \(\langle S({t})\rangle =0\). This extra noisy term can resemble either a pure dephasing dynamics for the central spin. We assume the noise distribution is Gaussian with two-point correlations (Ornstein-Uhlenbeck process)

$$\begin{aligned} \langle S(t)S(t')\rangle =\frac{\xi ^{2}}{2\tau _n} e^{- \frac{ |{t}-{t}'| }{ \tau _n} }, \end{aligned}$$

where \(\xi\) characterizes the strength of the noise and \(\tau _n\) is the noise correlation time58,61,63,64,65. White noise is approximately equivalent to fast colored noise (\(\tau _n \rightarrow 0\)) with two-point correlations

$$\begin{aligned} \langle S(t)S(t')\rangle =\xi ^2 \delta (t-t'). \end{aligned}$$

The mean transition probabilities over the whole noise distribution S(t) are obtained numerically using the exact master equation58,61,63,64,65 for the averaged density matrix \(\rho _{k}(t)\) of the noisy Hamiltonian

$$\begin{aligned} \begin{aligned} H^{(\xi )}_{k}(t)=H^{(0)}_{k}(t)+S(t)H^{(1)}_{k}, \end{aligned} \end{aligned}$$
(16)

i.e.,

$$\begin{aligned} \dot{\rho }_k(t)=-i\left[ H^{(0)}_{k}(t),\rho _k(t)\right] -\frac{\xi ^{2}}{2\tau _n}\left[ H^{(1)}_{k},\int _{t_i}^{t}e^{-|t-s|/\tau _{n}}[H^{(1)}_{k},\rho _k(s)]ds\right] . \end{aligned}$$
(17)

where \(H^{(0)}_{k}(t)\) represents the noise-free Hamiltonian and \(H^{(1)}_{k} = -\sigma ^{z}\) denotes the noisy component58.

As described in Supplemental Material, by converting Eq. (17) into a pair of coupled differential equations, we numerically compute the mean values of \(|u_{k}^{\pm }(t)|^2\), \(|v_{k}^\pm (t)|^2\), \(u_{k}^{\pm }(t)v_{k}^{\pm *}(t)\), and \(u_{k}^{\pm *}(t)v_{k}^{\pm }(t)\) as ensemble averages over the noise distribution S(t). It is important to note that in the limit as \(\tau _n \rightarrow 0\), the above master equation simplifies to the white noise master equation

$$\begin{aligned} \dot{\rho }_k(t) = -i\left[ H^{(0)}_{k}(t), \rho _k(t)\right] - \frac{\xi ^2}{2} \left[ H^{(1)}_{k}, \left[ H^{(1)}_{k}, \rho _k(t) \right] \right] . \end{aligned}$$
(18)
Fig. 6
figure 6

Scaling of the maximums (revivals) of decoherence (\(D_{Max}\)) at the presence of the white noise and colored noise for \(\tau _Q=250\). (a) Linear scaling of \(\ln (D_{Max})\) with square of noise intensity \(\xi ^2\) corresponding to Fig. 3a. (b) Linear scaling of revivals (\(D_{Max}\)) for noise intensity \(\xi =0.003\) with noise correlation \(\tau _n\) for fast noise corresponding to Fig. 5a. (c) Scaling of \(\ln (D_{Max})\) versus \(\ln (\tau _n)\) for slow noise corresponding to Fig. 5a for noise intensity \(\xi =0.003\).

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Analysis in the presence of white noise

Our numerical analysis begins by examining the effects of white noise on the system, as illustrated in Fig. 3 for ramp time scales: (a) \(\tau _Q = 250\), (b) \(\tau _Q = 10\), and (c) \(\tau _Q = 1\). As depicted in Fig. 3a, when the environment-qubit coupling is strong, partial revivals are still observable in the presence of the white noise, and diminish by increasing the noise intensity \(\xi\). While, in the absence of noise, the revivals do not decrease by increasing the quench time t, the partial revivals decay by increasing the quench time in the presence of the noise. Decaying the revivals by increasing the quench time in the presence of the noise originates from the accumulation of noise-induced excitations during the evolution. Notably, the period of the revivals remains consistent across both noisy and noiseless cases. In scenarios with weak coupling between the environment and the qubit, as shown in Fig. 3b, noise exacerbates the monotonic decay of decoherence.

As demonstrated in Fig. 3c, when the ramp time scale decreases, indicating weaker coupling between the environment and the qubit, decoherence decreases and becomes less affected by noise. Furthermore, as shown in Fig. 3b,c, the finite-size decoherence exhibits a sharp decay at the critical points \(h_c = \pm 1\) for small ramp time scales, where the environment-qubit coupling is weak. In the case of strong coupling and large ramp time scales, the decay is observed specifically at \(h_c = -1\). While for strong environment-qubit coupling, the decoherence exhibits a maximum at \(h_c = 1\), as seen in Fig. 3a. In other words, the critical points are signaled by the finite-size decoherence. In the absence of noise, the decoherence exhibits exponential scaling with the size of the system at the critical point, i.e.,

$$\begin{aligned} D|_{h=\pm 1}\sim e^{-N}. \end{aligned}$$

In Fig. 4, we investigated the scaling behavior of the decoherence at the critical points in the presence of noise. We discovered that the decoherence exhibits exponential scaling with \(N \xi ^2\), such that

$$\begin{aligned} D|_{h = \pm 1} \sim e^{-N \xi ^2}. \end{aligned}$$

Analysis in the presence of colored noise

The decoherence during a quench in the presence of colored (correlated) noise is plotted in Fig. 5 for different values of the ramp time scale. As seen in Fig. 5a, when the environment-qubit coupling is strong enough, partial revivals are still present in the presence of colored noise, but they decrease as the noise correlation time \(\tau _n\) decreases. This means that decoherence is less affected by correlated noise than by white noise. Moreover, the period of the revivals remains the same in both noisy and noiseless cases. As shown in Fig. 5b, in the case of weak coupling between the environment and the qubit, noise enhances the monotonic decay of decoherence. As the ramp time scale get smaller, which is equivalent to weaker environment-qubit coupling, the coherency increases and affected less by noise as illustrated in Fig. 5c. Furthermore, it is clear that the critical points are signaled by the finite-size decoherence. In the presence of colored noise, we investigated the scaling behavior of the decoherence at the critical points. We found that the decoherence exhibits exponential scaling with \(N \xi ^2 / \tau _n\), specifically,

$$D|_{h = \pm 1} \sim e^{-\frac{ N \xi ^2 }{ \tau _n} }.$$

In Supplemental Material more details are given on the scaling of decoherence D.

Figure 6 illustrates the scaling of the revivals (the maximum of decoherence) in the presence of white and colored noise for \(\tau _Q = 250\). As seen in Fig. 6a, the maximum of the revivals scales exponentially with the square of the noise intensity. Additionally, the decreasing slope of the lines indicates the accumulation of noise-induced excitations during the evolution, which results in the decay of the revivals as the quench time t increases. The scaling of the revivals with the noise correlation time \(\tau _n\) is depicted in Fig. 6b and c for the noise intensity \(\xi = 0.003\). As observed, the maximum of the decoherence increases with the noise correlation time and scales linearly with \(\tau _n\) for fast noise (\(\tau _n \le 100\)), while it scales exponentially for slow noise (\(\tau _n \ge 250\)).

Fig. 7
figure 7

The measure of non-Markovianity for the white, as a function of the noise strength and the inset shows the measure of non-Markovianity for the color noises versus the correlation time. In the inset panel the dashed black line is the best linear fit to the data. Here, we have \(\xi =0.003\).

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Non-Markovianity

The revivals in decoherence of the central spin through the quench signals non-Markovianity of the dynamics. Here, to study the features of this behavior we quantify deviation of this dynamics from a Markovian one by computing the measure proposed in Ref.66.

The measure is defined based on the rate of change in the trace distance between two initially distinguishable states \(\rho _1(0)\) and \(\rho _2(0)\) for a given process that we show as \(\Phi (t)\). Indeed, for a Markovian process any two states become increasingly similar over time. Therefore, the measure is computed based on the deviation from this property as

$$\begin{aligned} \mathcal {N} = \int _{>0}\hspace{-2mm} dt\ \tfrac{d}{dt}\mathcal {D}[\Phi (t)\rho _1(0),\Phi (t)\rho _2(0)], \end{aligned}$$
(19)

where the integration is only performed for the time intervals that the integrand is positive. Here,

$$\mathcal {D}[\rho _1,\rho _2] = \frac{1}{2}\Vert \rho _1 - \rho _2 \Vert _1$$

is the trace distance with \(\Vert \cdots \Vert _1\) standing for the trace norm67. The above quantity must be maximized for different initial states. For the pure dephasing of a central spin that we are interested in this work the superposition state already is the optimal state.

The results for the measure of non-Markovianity are shown in Fig. 7 for both white and colored noise cases. In the case of white noise, we observe a monotonic decay in the non-Markovianity of the process as the noise strength increases. For large enough \(\xi\) values the process becomes fully Markovian. In order to understand the role of finite noise correlation time, we compute the measure in Eq. (19) with different \(\tau _n\) values when the noise strength is fixed to \(\xi =0.003\). Interestingly, the measure exhibits a linear growth with the noise correlation time.

Summary and conclusion

Characterizing noisy signals is crucial for understanding the dynamics of open quantum systems, enhancing our ability to control and engineer them effectively. This work investigated the impact of noisy environmental spin systems (ESSs) on the decoherence of the central qubit as a sensitive noise estimator compared to other methods. We specifically focus on how noise in a time-dependent external magnetic field influences the coherence dynamics of a central spin coupled to a spin chain, particularly as the ESS is driven across its quantum critical point (QCP).

By extending the central spin model to incorporate stochastic variations in the external magnetic field, we demonstrate that noise not only amplifies decoherence resulting from the nonequilibrium critical dynamics of the environment but also profoundly affects the system’s temporal evolution. Our numerical calculations reveal that decoherence exhibits exponential scaling at the critical points with both the square of the noise intensity and the noise correlation time. This contrasts with the noiseless case, where decoherence revivals occur when the chain-qubit coupling is sufficiently strong; however, these revivals diminish in the presence of noise, scaling exponentially with white noise intensity and linearly or a power law in the case of colored noise, depending on the correlation time. It should be emphasized that, both colored noise and white noise are classified as Gaussian noise, and a more comprehensive understanding could be achieved by exploring the effects of non-Gaussian noise on the decoherence factor.

Additionally, our exploration of non-Markovianity reveals that it decreases with the square of the noise intensity but increases linearly with the noise correlation time, highlighting the complex interplay between noise and memory effects in quantum systems. These findings underscore the importance of accounting for noise to model and predict quantum system behavior under realistic conditions accurately. They also offer new perspectives on the challenges and opportunities in quantum control, decoherence mitigation, and potential applications in noise spectroscopy of external signals47,48,49,50.

Last but not least, interference phenomena exemplified by the collapse and revival of the decoherence function serve as a direct evidence of entanglement dynamics and information flow between the central spin and the chain. But from a fundamental point of view the interference effect alone is not sufficient to rule out a classical description of the environment68; collapse and revival phenomena can indeed be generated by coupling the qubit to an engineered classical field69. Therefore, to definitively demonstrate the quantum nature of the system, a more fundamental approach is needed, similar to quantum-witness equality70 and the Leggett-Garg test50,69. Therefore, the present work anticipates a systematic exploration of entanglement dynamics and quantum coherence using stronger nonclassicality criterion for distinguishing quantum behavior from classical effects in various conditions.

The quick advancements in the realization of analog quantum simulators indicate that we might have the opportunity to experimentally verify our predictions. Noise-averaged measurements are anticipated to be entirely feasible with current experimental techniques and are expected to provide significant insights29,30,31,47,48,49. Ramped magnetic quenches, conducted in the presence of amplitude-controlled noise, have already been successfully implemented with trapped ions simulating the transverse-field XY chain71. Additionally, the foundational aspect of experimental investigation−detection and characterization of decoherence−is also established, as evidenced across various platforms for TFI-type chains with finite-range interactions, including trapped ions72,73,74, Rydberg atoms75, and NV centers76. These advancements, along with recent progress in quantum-circuit computations on NISQ devices77, suggest a promising avenue for exploring decoherence following noisy quenches in the nearest-neighbor interacting TFI chain discussed in this paper.