Objectivity in non-Markovian spin-boson model

TL;DR

This paper investigates how objectivity, a key feature of the classical world, emerges from quantum mechanics in a non-Markovian spin-boson model by analyzing the formation of Spectrum Broadcast Structures.

Contribution

It explores the emergence of Spectrum Broadcast Structures in a non-Markovian spin-boson model, linking quantum-to-classical transition mechanisms with non-Markovian dynamics.

Findings

Spectrum Broadcast Structures can form in non-Markovian regimes.

Non-Markovian effects influence the emergence of objectivity.

The model provides insights into quantum-to-classical transition processes.

Abstract

Objectivity constitutes one of the main features of the macroscopic classical world. An important aspect of the quantum-to-classical transition issue is to explain how such a property arises from the microscopic quantum world. Recently, within the framework of open quantum systems, there has been proposed such a mechanism in terms of the, so-called, Spectrum Broadcast Structures. These are multipartite quantum states of the system of interest and a part of its environment, assumed to be under an observation. This approach requires a departure from the standard open quantum systems methods, as the environment cannot be completely neglected. In the present work we study the emergence of such a state-structures in one of the canonical models of the condensed matter theory: Spin-boson model, describing the dynamics of a two-level system coupled to an environment made up by a large number of harmonic oscillators. We pay much attention to the behavior of the model in the non-Markovian regime, in order to provide a testbed to analyze how the non-Markovian nature of the evolution affects the surfacing of a spectrum broadcast structure.

Equations

$${\rho }_{S:fE}=\sum _i{p}_i|x_i⟩⟨x_i|\otimes {\rho }_i^{E_1}...\otimes {\rho }_i^{E_{fN}},$$

$${\rho }_i^{E_k}\perp {\rho }_{i^{\prime }}^{E_k}\text{ for every }{i}^{\prime }\ne i\text{ and }k=1,\dots ,fN.$$

$$H=H_S+H_E+H_{int},$$

$$H_S=\frac{1}{2}\Omega {\sigma }_z,H_E=\sum _i\left(\frac{p_i^2}{2m_i}+\frac{1}{2}{m}_i{\omega }_i^2{x}_i^2\right)$$

$$H_{int}={\sigma }_z\otimes \sum _i{g}_i\left(a_i+a_i^{†}\right).$$

$$[H_S,{\sigma }_z]=0.$$

$$H_{int}^I(t)={\sigma }_z\otimes \left(g_k{a}_k^{†}{e}^{i{\omega }_{k}t}+g_k{a}_k{e}^{-i{\omega }_{k}t}\right).$$

$$[H_{int}^I(t),H_{int}^I(t^{\prime })]=-2i\sum _k|g_k{|}^2\sin \left[{\omega }_k(t-t^{\prime })\right],$$

$$U(t)=$$

$$\equiv$$

$$H_k^n(t)\equiv (-1{)}^n\left(g_k{a}_k^{†}{e}^{i{\omega }_{k}t}+g_k{a}_k{e}^{-i{\omega }_{k}t}\right),$$

$$U_{E_k}(n,t)\equiv D_i\left([-1{]}^n{\alpha }_k(t)\right),$$

$${\alpha }_i(t)=2\frac{g_k}{{\omega }_k}(1-e^{i{\omega }_{k}t}),$$

$${\rho }_{S:fE}(t)={\text{Tr}}_{(1-f)E}\left[U(t){\rho }_{0S}\otimes \underset{k=1}{\overset{N}{⨂}}{\rho }_{0k}U(t{)}^{†}\right],$$

$${\rho }_{SE}(0)={\rho }_{0S}\otimes \underset{k=1}{\overset{N}{⨂}}{\rho }_{0k}.$$

$${\rho }_{S:fE}(t)=$$

$$+$$

$${\rho }_f^{nm}(t)\equiv \underset{k=1}{\overset{fN}{⨂}}{U}_{E_k}(n,t){\rho }_{0k}{U}_{E_k}(m,t{)}^{†}\equiv \underset{k=1}{\overset{fN}{⨂}}{\rho }_{nm}^{(k)}(t).$$

$${\Gamma }_{nm}(t)=\prod _{k\in (1-f)E}\text{Tr}\left[U_{E_k}(n,t){\rho }_{0k}{U}_{E_k}(m,t{)}^{†}\right]$$

$${\rho }_{nn}^{(k)}(t){\rho }_{mm}^{(k)}(t)=0,$$

$$B({\rho }_1,{\rho }_2)\equiv \text{Tr}\sqrt{\sqrt{{\rho }_1}{\rho }_2\sqrt{{\rho }_1}}.$$

$$B_{nm}^{\mathcal{M}}(t)$$

$$=\prod _{k\in \mathcal{M}}B\left[{\rho }_{nn}^{(k)}(t),{\rho }_{mm}^{(k)}(t)\right].$$

$$|{\Gamma }_{nm}(t)|\approx 0,B_{nm}^{\mathcal{M}}(t)\approx 0.$$

$${\rho }_{0E}=\frac{e^{-\beta H_E}}{Z_E}$$

$$\Gamma (t)=\exp \left[-2{\int }_{(1-f)E}d\omega \coth \left(\omega /2T\right)g(\omega ,t)\right],$$

$$g(\omega ,t)\equiv J(\omega )\frac{1-\cos \left(\omega t\right)}{{\omega }^2},$$

$$J(\omega )=2\sum _i{\left|g_i\right|}^2\delta \left(\omega -{\omega }_i\right).$$

$$B(t)=\exp \left[-2{\int }_{\mathcal{M}}d\omega \tanh \left(\omega /2T\right)g(\omega ,t)\right],$$

$${\dot{\rho }}_{S:fE}=$$

$$H_{S:fE}(t)\equiv \sum _n|n⟩⟨n|\otimes H_{fE}^n(t),H_{fE}^n(t)\equiv \underset{k=1}{\overset{fN}{⨂}}{H}_k^n(t)$$

$$\gamma (t)=\frac{\dot{\Gamma }(t)}{\Gamma (t)}.$$

$$\mathcal{N}=-{\int }_{\gamma <0}\gamma (t)dt.$$

Full text

I Introduction

Quantum mechanics is one of the most successful theories, correctly predicting a huge class of physical phenomena. Its validity remains confined to the microscopic regime, where such a theory provides a good explanation of the behavior of the constituents of matter. In contrast, there is no trace of quantum effects on macroscopic scales, fully ruled by classical physics Schlosshauer (2007, 2005); Zurek (2003). The rare counter examples involve typically unstable systems, and phenomena such as super-radiance Altland and Haake (2012), superfluorescence Haake et al. (1979), or spontaneous stimulated Raman scattering Walmsley and Raymer (1983), where the quantum fluctuations might become macroscopically enhanced. Another prominent counterexample concerns obviously Bose-Einstein condensation and superfluidity and superconductivity Pitaevskii and Stringari (2003). Despite these important but rather rare counterexamples, one natural question arises: How do the classical features of the macroscopic world emerge from the underlying quantum domain?

In particular, from everyday experience we are accustomed to perceive nature as objective: We all observe the same (modulo different reference frames) properties of an observed object, without disturbing it. This point of view has been fundamentally challenged by quantum mechanics, since the act of observation generally modifies a state of the observed system. So it is natural to wonder how the objective character of the classical theory can be derived from the (inherently subjective) quantum theory.

An important contribution to this fundamental problem has been given by the quantum Darwinism program Zurek (2009), which is a more realistic and elaborated form of the decoherence theory. It attributes objectification of information about a quantum system to an unavoidable interaction of the latter with its environment. The main breakthrough of this approach lies in the role played by the environment: It is no longer a mere source of noise and dissipation, but is recognized to be an information carrier as indeed most of our everyday observations are indirect. Moreover, the environment is considered to be divided into several different portions, $\{E_{1},...,E_{N}\}$, each representing degrees of freedom available for observation for a single observer. Inevitably, some portions will escape observation and are lost and hence, observed and unobserved portions of the environment deserve to be distinguished. A good example here is a visual observation of the same object by a group of people: Each person perceives a portion of the object’s photonic environment, given by the solid angle of the observation. But of course not all of the photons scattered by the object will be detected. If it happens that each of these portions contains the same information about the object and it can be extracted without any disturbance, then we may speak of a certain, operational form of objective existence of this information Zurek (2009).

One important step beyond the quantum Darwinism program has been accomplished in Horodecki et al. (2015); Korbicz et al. (2014) where the authors analyzed so-defined operational objectivity directly in terms of quantum states. Under certain assumptions, they have proven that a state of the system becomes objective if and only if a joint state of the system plus the observed portions of its environment, $\{E_{1},...,E_{fN}\}$, is of the following form, called spectrum broadcast structure (SBS):

[TABLE]

Here $\{|x_{i}\rangle\}$ is the so-called pointer basis of the central system to which it decoheres, $p_{i}$ are initial pointer probabilities, and $\rho^{E_{k}}_{i}$ are some states of the observed parts of the environment with mutually orthogonal supports for different pointer index $i$. Simply put, the reason why this state structure corresponds to a form of objectivity is the following. The mutual orthogonality of the supports of $\rho^{E_{k}}_{i}$ means those states are one-shot perfectly distinguishable. Hence, by performing the right measurements (projections on the orthogonal supports of $\rho^{E_{k}}_{i}$), each of the observers extracts the same information about the system – the value of the index $i$, enumerating the possible states the system can be in. On average (after forgetting the results) this extraction does not disturb neither the other observers nor the central system, as the whole state in Eq. (1) stays unchanged. The implication in the other direction – that demanding objectivity in the above sense (under certain assumptions) uniquely leads to the structure in Eq. (1) is much more elaborate and can be found in Horodecki et al. (2015). The central points of the reasoning are Bohr’s definition of non-disturbance Wiseman and, so called, strong objectivity: The only correlation between the portions of the environment is common information about the system. Let us stress the conceptual importance of the SBS states Eq. (1): They put a rather philosophical notion of objectivity into a concrete form of a multipartite quantum state. A form that in principle can be checked in concrete models.

Let us note that Eq. (1) is an idealized structure and an approach to it by real-life states has been analyzed in Mironowicz et al. (2017). The quantities which control this approach are: i) the decoherence factor due to the unobserved part of the environment and ii) the fidelities of the states $\rho^{E_{k}}_{i}$ for different $i$’s Korbicz et al. (2014). Spectrum broadcast structures have been so far found in a number of paradigmatic for the open quantum systems theory models, including: the illuminated sphere model Korbicz et al. (2014), massive quantum Brownian motion Tuziemski and Korbicz (2015a, b, 2016), and the spin-spin model Mironowicz et al. (2017). Recently it has also been shown that SBSs are typical for von Neumann measurements with measuring apparatuses composed of a large number of degrees of freedom Korbicz et al. (2016). This motivates further their studies as an important bridge in the quantum-to-classical transition.

The main purpose of the current paper is to investigate the objectification processes through the SBS formation in another canonical model of decoherence – the spin-boson model. It consists of a two-level central system interacting with a large reservoir of bosonic modes Weiss (2008); Leggett et al. (1987); Schlosshauer (2007); Breuer and Petruccione (2007); Ferialdi (2017). The model plays an important role in quantum computing, as well as in experiments on macroscopic quantum coherence, for instance in those aimed to analyze the role of quantum coherence in biological systems. An important part of the paper is devoted to explore the behavior of the model in the non-Markovian regime. By non-Markovianity we mean the presence of memory effects making the evolution of the central system strongly dependent on its past history Gardiner and Zoller (2004); Breuer and Petruccione (2007). This situation constitutes a rule rather than an exception, especially in the low-temperature regime, or when the interaction between the central system and the surrounding degrees of freedom gets sufficiently strong. It is then a natural question to ask if and how the non-Markovianity affects objectification processes in this model. This analysis is one of the main goals of the present present. We would like to stress that there are many different definitions of non-Markovianity (see the recent review Breuer et al. (2016) or Addis et al. (2014)). Here we use the definition of Hall et al. (2014) based on non-positivity of decoherence rates.

The paper is organized as follows. In Sec. (II) we introduce the spin-boson model. In Sec. (III) we study the partially reduced state $\rho_{S:fE}$, which describes the system plus a fraction of its environment. We are focused on checking weather the partially reduced state approaches the SBS form in Eq. (1), which can be studied by calculating: (i) the decoherence factor induced by the unobserved part of the environment and (ii) the fidelity Fuchs and van de Graaf (1999) for different values of the central spin of the fragments of the observed part of the environment. Derivation of the fidelity as a function of the physical parameters of the model, such as the temperature, the coupling strength etc., is an original result of our paper. In Sec. (IV) we move our analysis towards the non-Markovian regime. In particular, we look for the range of the model parameters defining the non-Markovian behavior. Such issues have been extensively studied before (see Addis et al. (2014); Breuer et al. (2016)), however our approach requires a distinction between the observed and the unobserved environment, which needs to be taken into account in the evaluation of non-Markovianity. This extension is also an original outcome of the present paper. Finally, in Sec. (V) we focus on the analysis of an influence of the non-Markovianity on the emergence of a SBS. A similar problem has also been treated in Galve et al. (2016) in the context of the quantum Brownian motion model. Based on our analysis, we conclude that there is no direct connection between the non-Markovianity and the objectification processes in the spin-boson model.

II Introduction to Spin-Boson Model

The spin-boson model is described by the following Hamiltonian:

[TABLE]

where:

[TABLE]

are, respectively, the self-Hamiltonian of the central system and the environment. The former is represented by a two-level system while the latter is represented by a set of uncoupled harmonic oscillators. In Eq. (4) we set $\hbar=1$, and hereafter we work with these units. The interaction Hamiltonian is given by the expression:

[TABLE]

It is important to stress that the above model is not the most general one, since the interaction Hamiltonian in Eq. (5) does not lead to dissipation processes because:

[TABLE]

In fact, in the literature this model is commonly called the pure dephasing spin-boson model, although we still refer to it as spin-boson for brevity. In realistic systems the time scale for decoherence is typically many orders of magnitude shorter than the timescale of the energy exchange. Thus, our model can be regarded as a good representation of such rapid decoherence processes during which the energy dissipation is negligible.

Because of Eq. (6), the self-Hamiltonian $H_{S}$ can be effectively neglected. Passing to the interaction picture, the interaction Hamiltonian takes the following form:

[TABLE]

Since for two arbitrary instances of time $t,t^{\prime}$, the commutator $[H_{int}(t),H_{int}(t^{\prime})]$ is a c number:

[TABLE]

the evolution can be easily solved, using e.g. the Magnus series:

[TABLE]

where $\xi(t)$ is a global phase factor, which we drop out as it is not important for our considerations,

[TABLE]

and $n=\pm 1$ are the eigenvalues of $\sigma_{z}$, while the evolution of the environment is governed by:

[TABLE]

where $D(\alpha)$ is a displacement operator.

III Structure of a partially reduced state

In this section we investigate the structure of a partially reduced state, describing the central spin and a fraction of its oscillatory environment. We derive tools that allow us to conclude if in the course of evolution the partially reduced state approaches the SBS form, defined in Eq. (1). We assume that the environment consists of $N$ oscillators, $fN$ ($0<f<1$) out of which are under a potential observation and constitute what we call the observed fraction $fE$ of the environment, while the rest is assumed to be lost and is the unobserved fraction $(1-f)E$. This promotion of a part of the environment from a mere source of noise to an information carrier is the key point of the current approach, inherited from the quantum Darwinism idea Zurek (2009), and a novelty in comparison with the traditional point of view in open quantum systems, where environment is always treated as a set of unobserved and uncontrollable degrees of freedom .

The partially reduced state is obtained by simply tracing out the non-observed fragment of the environment:

[TABLE]

where as customary we assumed a fully product initial state for the global system:

[TABLE]

Eq. (13) can be expressed as follows:

[TABLE]

where $c^{nm}_{0S}\equiv\langle n|\rho_{0S}|m\rangle$, and:

[TABLE]

The quantity:

[TABLE]

represents the decoherence factor between the state $|n\rangle$ and $|m\rangle$ of the central system due to the unobserved fraction of the environment $(1-f)E$.

From Eq. (15) one sees that one necessary condition to approach an SBS is given by the usual decoherence condition: $\Gamma_{nm}(t)=0$. However, it is not sufficient. One has also to check whether the information deposited in the environment during the decoherence can be perfectly read out, i.e., if the system-dependent states of the fragments of the environment have non-overlapping supports Korbicz et al. (2014):

[TABLE]

and hence are perfectly one-shot distinguishable. Among different measures of distinguishability, the most suitable turns out to be the generalized overlap (also known as Uhlmann’s fidelity Uhlmann (1976)):

[TABLE]

One cannot expect Eq. (18) to hold at the level of single fragments. To the contrary, since each of the unitaries $U_{E_{k}}(n;t)$ weakly depends on the parameter $n$, the states $\rho^{(k)}_{nn}(t)$ are almost identical for different $n$’s. However, it can happen that by grouping subsystems of the observed part $fE$ into larger fractions, called macrofractions $\mathcal{M}$, one can approach the perfect distinguishability in Eq. (18) at the level of macrofraction states $\rho_{n}^{\mathcal{M}}(t)\equiv\bigotimes_{k\in\mathcal{M}}\rho^{(k)}_{nn}(t)$ Korbicz et al. (2014). Generalized overlap is well suited for such tests due to its factorization with the tensor product:

[TABLE]

We stress that the measure of distinguishability we introduce refers to macrofractions of the observed part of the environment.

Summarizing, the formation of SBS in Eq. (1) is equivalent to Mironowicz et al. (2017):

[TABLE]

We will refer to the decoherence factor and the fidelity as to indicator functions. As it is well known, for the thermal environment:

[TABLE]

where $\beta=1/T$ and $Z_{E}$ is the partition function, the decoherence factor can be written in the analytical form Breuer and Petruccione (2007); Schlosshauer (2007):

[TABLE]

where:

[TABLE]

and the integral is performed over the frequency range related to the unobserved part of the environment $(1-f)E$. The quantity $J(\omega)$ is the spectral density containing the information of the coupling with the environment Breuer and Petruccione (2007); Schlosshauer (2007):

[TABLE]

We postpone to next Section further discussion of the spectral density as it plays a fundamental role in defining Markovian and non-Markovian behavior.

The expression for the fidelity in the present model constitutes in turn the result:

[TABLE]

where the integral is performed over the frequency range associated with a macrofraction $\mathcal{M}$. The derivation is presented in Appendix B for the case of the initial thermal state of the environment (see Tuziemski and Korbicz (2015b)). The most important difference between the fidelity Eq. (26) and the decoherence Eq. (23) lies in the dependence on the temperature. In particular, when $T\rightarrow 0$ both quantities approach the same form, and the emergence of a SBS reduces just to decoherence. This is in agreement with the fact that at low temperature the state of the fragments of the environment becomes pure.

IV Non-Markovian environments

One of the main purposes of the current paper is to investigate if and how non-Markovianity affects the emergence of objective properties in the pure-dephasing spin-boson model. As presented in the introduction, the technical tools we use for checking for objectification are SBS states (1) – their formation implies objectification of a state of the central spin as it becomes redundantly stored in the bosonic environment in many perfect copies. Yet more technically, we argued in the previous section that the above state structures arise when both the decoherence factor and the appropriate fidelity vanish. These quantities, as well as the amount of non-Markovianity, depend on the system parameters: the temperature and the couplings as summarized by the spectral density. We proceed thus by manipulating appropriately these parameters in order to increase non-Markovianity, and to analyze how decoherence and fidelity react.

First, we have to carefully introduce a measure of non-Markovianity, adapted to our specific scenario. We will use the measure from Hall et al. (2014) but based on the dynamics of the partially reduced state, i.e. the state of the central spin and the observed part of the environment. The evolution of the latter [see (15)] can be expressed in terms of the following local-in-time master equation (see Appendix A for the details):

[TABLE]

where:

[TABLE]

and

[TABLE]

The structure of Eq. (IV) allows one to identify $\gamma(t)$ as the only non-zero canonical decoherence rate for the considered problem Hall et al. (2014): Its positivity guarantees that the evolution is completely positive or “time-dependent Markovian” Rivas et al. (2010); Hall et al. (2014). Accordingly, if $\gamma(t)$ becomes strictly negative the evolution is non-Markovian. To quantify the amount of non-Markovianity in a given interval of time we will adopt the measure introduced in Hall et al. (2014):

[TABLE]

The above expression is formally similar to definitions of non-Markovianity based on the reduced dynamics of the spin only Rivas et al. (2010); Addis et al. (2014); Hall et al. (2014). However, as we are interested in the structure of the partially reduced state, in the present case $\gamma(t)$ incorporates non-unitary effects caused by the non-observed fraction of the environment. To the best of our knowledge such a situation has not been studied so far in the context of non-Markovianity measures.

In the high-temperature limit, $T\gg s\Lambda$, Eq. (30) becomes:

[TABLE]

Let us now specify the spectral density. We will follow the usual approach and choose Schlosshauer (2007); Breuer and Petruccione (2007):

[TABLE]

where $\Lambda$ is the cutoff frequency and $s$ is the Ohmicity parameter. The case $s=1$ corresponds to pure Ohmic, whereas $s<1$ to sub- and $s>1$ corresponds to super-Ohmic regimes. The spectral density provides an information about the coupling of the two-level system with the environment. We note that he integral in Eq. (31) is performed over the frequencies belonging to the unobserved part of the environment. We will use the spectral density in Eq. (32) in two basic ways to define the observed and unobserved parts of the environment: (i) Each part is so large that the full spectral density applies to it (we call it uncut ); and (ii) Each part is defined by some frequency range of Eq. (32) only (cut case).

Let us first briefly recall known facts about the connection between the non-Markovianity and the uncut spectral density (see e.g. Haikka et al. (2013)). Fig. 1 shows the plot of the measure in Eq. (30) as a function of the Ohmicity parameter $s$ for different values of the cutoff $\Lambda$ scaled to the thermal frequency $\Omega_{T}=k_{B}T/\hbar$. The degree of non-Markovianity monotonically increases with $s$ for $s>3$, as for $s<3$ the dynamics is purely Markovian, i.e. $\mathcal{N}=0$. Moreover, in Addis et al. (2014), where the measure in Eq. (30) has been calculated in the zero-temperature limit, it has been proved that the threshold at $s=3$, distinguishing the Markovian regime from the non-Markovian one, holds even for other measures.

Fig. 1 also provides the dependence on the cutoff $\Lambda$ of the present measure, at fixed $s$. Precisely, it monotonically decreases as $\Lambda$ grows, whatever values of the Ohmicity parameter we consider. This result could also be inferred by looking to the self-correlation function for the environment, decaying exponentially as $1/\Lambda$, as discussed in Breuer and Petruccione (2007). Finally, it is easy to infer, considering Eq. (31), that in the high-temperature limit non-Markovianity decreases linearly as $T$ grows, provided $s>3$.

So far we have described the unobserved part of the environment by means of the whole spectral density in Eq. (32). Now, we consider a more realistic case in which such a spectral density represents the whole environment, while its observed and unobserved part are constituted just by sets of oscillators related to different frequency ranges, separated by a cut $\beta$. Here, the $fE$ is represented by the oscillators with frequency $\omega\in\left[0,\beta\right]$, while $(1-f)E$ is constituted by the complement $\omega\in\left[0,\beta\right]^{c}$. This situation corresponds to that sketched in Fig. 2 once one sets $\alpha=0$.

An interesting question is how the measure in Eq. (30) depends on the cut $\beta$, distinguishing the observed and unobserved part of the environment. This behavior is presented in Fig. 3.

In general, the dependence on the cut does not show a monotonic trend, but there is a monotone decrease for large values of $\beta$, i.e. at the tight edge of the frequency domain. The distinction between observed and unobserved environment may be implemented also in an another manner. We introduce two cuts, rather than just one. The observed part of the environment is constituted by the oscillators with frequency in the range $\omega\in\left[\alpha\equiv\beta-\Delta,\beta\right]$, while the unobserved one is related to the complementary set $\left[\alpha,\beta\right]^{c}$. This is sketched in Fig. 2. We calculated the non-Markovianity measure as a function of $\beta$ and various interval widths $\Delta$ for $s=4$ (Fig. 3). We see that the degree of non-Markovianity depends on the position of the cuts in a non-monotonic way.

We remark on an important feature of the plot in Fig.  3 for $s=3$ for the single cut case: The value of the measure is non-zero (see Haikka et al. (2013)). Thus, the introduction of the frequency cut changes the behavior from Markovian to non-Markovian. One can show it also for more than one cut. We conclude that the typical framework of quantum Darwinism, entailing a distinction into the observed and the unobserved parts of the environment, can lead to a certain amount of non-Markovianity, as certified by the measure (30). We make one last remark concerning the implementation of the cuts. We used here sharp cuts defined by the step function. To exclude any artifacts connected with that, we also performed the parallel analysis using soft cuts (modeled by the $\tanh x$ function) and found that there is no quantitative difference between the two implementations. Thus, the non-Markovianity induced by the cuts does not depend on the their sharpness.

V Formation of Spectrum Broadcast Structures in Non-Markovian environments

We now move to the main study of the current paper: the study of the connection between Markovianity and emergence of an objective value of the central spin in the sense of SBS formation. We first study the full, uncut spectral density in Eq. (32) and then introduce the cuts, defining the observed and unobserved parts of the environment.

V.1 Case I: Uncut spectral density

In the first part of the discussion, we consider both observed and unobserved environment to be so large as to include the whole frequency domai; namely, we start with the simple case in which there are no cuts in the spectral density. In this case, the integrals in Eqs. (23) and (26), defining, respectively, the decoherence and the fidelity, are both performed for $\omega\in[0,\infty)$, with the spectral density given by Eq. (32) – note that $|\Gamma(t)|\leq B(t)$ because for $T>0$, $\tanh\left[\omega/(2T)\right]>\coth\left[\omega/(2T)\right]$. The integrals can be performed analytically. For the integer values of $s$, the results are presented below with both the decoherence and the fidelity factors decomposed into a vacuum and thermal parts:

[TABLE]

where:

[TABLE]

and:

[TABLE]

where $B_{vac}(t)=|\Gamma_{vac}(t)|$ as for pure states decoherence and fidelity factors become equal, and:

[TABLE]

In the above formulas $c.c.$ stands for the complex conjugate, $\wp(z)$ is the Euler gamma function, and $\Psi^{m}(z)$ is the so-called polygamma function defined as NIS :

[TABLE]

Generalization to the non-integer $s$ is presented in Appendix C.

From our point of view, the most interesting regime is the intermediate temperature one:

[TABLE]

since for low temperatures the decoherence and fidelity factors become identical modulo the macrofraction sizes. For high temperatures $T\gg s\Lambda$; on the other hand, by approximating the hyperbolic functions involved in the formulas one sees that the decoherence process is very strong but fidelity will be close to $1$, meaning that the environmental states become almost indistinguishable and hence no information about the state of the central system is stored in the environment – it is too noisy. Unfortunately the above functions are too complicated for an analytical analysis for the temperature regime given by Eq. (39) and we resort here to a numerical analysis.

In Fig. 4 and 5 we plot the minus logarithms of the indicator functions (the decoherence and the fidelity) for $T=\Lambda$. Their high values for times larger than the inverse cutoff indicate that the partially reduced state quite quickly approaches SBS. We also see that the increase of the Ohmicity $s$ results in higher asymptotic values so that the higher $s$, the closer the partially reduced state is to SBS. Moreover, in Fig. 1 we see that the degree of non-Markovianity also grows with $s$. Thus, by looking to the dependence on the Ohmicity parameter, non-Markovianity favors, rather than hinders, the emergence of SBS and, as a result, objectivity. One possible insight for that is that bigger $s$ corresponds to stronger coupling between the system and environment, as can be seen from Eq. (32).

The degree of non-Markovianity may be changed also by tuning other parameters, e.g. the cutoff $\Lambda$. Fig. 1 shows that non-Markovianity increases when $\Lambda\rightarrow 0$. In Fig. 6 we plotted minus logarithms of the asymptotic values of decoherence and fidelity, respectively, as a function of the cutoff and the temperature. We note that decoherence gets weaker as $\Lambda\rightarrow 0$, i.e., when the non-Markovianity increases, while the fidelity shows the opposite behavior and in this case there seems to be no straightforward connection between the non-Markovianity and the SBS formation.

The value of the non-Markovianity measure depends also on temperature $T$. In Sec. IV we pointed out that in the high-temperature regime non-Markovianity decreases linearly with $T$, provided that $s>3$. However, the indicator functions depend on the temperature in opposite ways: With growing temperature decoherence gets stronger, while the states of the environment become harder to distinguish due to the higher thermal noise. In this case there is also no clear connection between the degree of non-Markovianity and the formation of SBS.

Summarizing the uncut spectral density case, when looking at the Ohmicity parameter $s$ alone, it seems that the stronger non-Markovianity enhances the formation of SBS. However, taking into account the other parameters $\Lambda,T$ does not support this claim. To the contrary, it seems that the non-Markovianity does not have a direct influence on the process of SBS formation in the considered model. The most important parameter is the Ohmicity $s$ that controls coupling strength between the system and the environment.

V.2 Case II: Cut spectral density

We consider now the situation in which the whole environment is modeled through the spectral density in Eq. (32), while the observed part $fE$ and the unobserved one $(1-f)E$ are represented only by its different fragments. We begin with a single cut case: The observed and the unobserved environments are defined in the frequency domain by a single cut located at some frequency $\beta$; see Fig. 2 with $\alpha=0$. Unlike in the previous case of the uncut environment, the analytical solution is not possible and we have to resort to a numerical analysis straight from the beginning. In Fig. 7 we plotted the minus logarithms of the decoherence factor and the fidelity as a function of time. We note the presence of oscillations in the time evolution of both functions. These oscillations indicate a non-Markovian behavior since they constitute non-monotonicity areas. They occur even for $s=3$, when the behavior is pure Markovian for the uncut case Haikka et al. (2013). This means that the presence of the cut turns on non-Markovianity effects, as we have already shown in Fig. 3.

Fig. 7 also shows that after a very long time ($\Lambda t\gg 1$) both decoherence and fidelity approach constant values. In Fig. 8 we plot these asymptotic values as functions of the frequency cut $\beta$. This plot shows an interesting reciprocal behavior: While decoherence gets stronger when $\beta$ decreases, state distinguishability gets weaker. This is in agreement with the fact that decoherence is related to the unobserved environment while distinguishability is related to the observed one. When $\beta$ increases, the observed environment enlarges, so the state distinguishability gets better, while the unobserved environment shrinks which negatively affects the decoherence process. In Fig. 8 we also plot the value of the non-Markovianity measure as a function of the frequency cut $\beta$. In the lower part of the frequency domain, the growth of $\beta$ corresponds to both an increase of the non-Markovian effects and an approach to SBS. Past the maximum at approximately $\beta/\Lambda=4$, both the non-Markovianity and the fidelity to SBS decrease (the latter due to the decreased decoherence). Thus, in this particular case it is possible to link non-Markovianity with the approach to objectivity.

Finally, we investigate the two cuts case, i.e., when the observed part of the environment corresponds to a frequency window and the unobserved parts correspond to the rest of the spectrum. The results, presented in Fig. 9, show that moving the frequency window towards mid frequencies leads to decrease of decoherence and increase of distinguishability. When the cuts are placed around $\alpha=3\Lambda,\beta=5\Lambda$, the partially reduced state is a good approximation to SBS. On the other hand, shifting the frequency window towards the upper part of the frequency domain results in increase of decoherence and decrease of distinguishability. The value of the non-Markovianity measure, also plotted in Fig. 9, is a complicated function of the frequency window location. Thus, in this case, there is no straightforward connection between non-Markovianity and the approach to objectivity.

VI Conclusions

We studied the emergence of a SBS in a paradigmatic model of an open quantum system: the pure dephasing spin-boson model. SBS represents the structure of the quantum states which encode objective property after the interaction with the environment. We showed that this structure arises in the spin-boson model as a result of the temporal evolution. In this case what becomes objective is the state of the central spin, which, after SBS formation, is perceived by many observers as a classical bit with values $\pm 1/2$. This is an original result of our paper, which enforces the reliability of SBS to describe in terms of state the emergence of objectivity of classical theory starting by the underlying quantum domain.

A large part of the paper has been devoted to the analysis of if and how the presence of non-Markovian effects influences the SBS formation. This task has been already investigated in Giorgi et al. (2015) for a qubit in a spin environment and Galve et al. (2016) in the context of quantum Brownian motion, using quantum Darwinism rather than a more fundamental SBS approach. We discuss a particular situation where non-Markovianity favors the formation of a SBS. This is the case in which we control non-Markovianity degree by tuning the degree of super Ohmicity. Our insight is that, rather than non-Markovianity, there are other physical properties that are decisive in the process of emergence of a SBS, for instance strength of the coupling.

We also showed that in the framework of quantum Darwinism, where environment is divided into observed and unobserved parts, there is a certain amount of non-Markovianity caused by this “environment cutting”.

The analysis of the emergence of the SBS for the present model opens also the possibility to study objectivity from the experimental point of view. In fact, the pure dephasing spin-boson model admits several practical counterparts. In Cirone et al. (2009); Haikka et al. (2011) it has been shown that an experimental realization of the spin-boson model may be obtained by means of an impurity in an ultracold gas. In this context the degrees of freedom of the gas play the role of the oscillators in Eq. (4), while the two-level system may be constructed putting the impurity in a double-well trap potential. In Liu et al. (2011), instead, a realization of the pure dephasing spin-boson model with photons has been presented: The two polarization states correspond the two-level open system while the bosonic modes of the environment are represented by the frequency degree of freedom of the photon which is coupled to the system via an interaction induced by a birefringent material.

Acknowledgements.

Insightful discussion with Paweł Horodecki, Manabendra Nath Bera, Vincenzo D’Ambrosio, Arnau Riera, Jan Wehr, Bogna Bylicka and Dariusz Chruściński are gratefully acknowledged. This work has been funded by a scholarship from the Programa Másters d’Excel-léncia of the Fundació Catalunya-La Pedrera, ERC Advanced Grant OSYRIS, EU IP SIQS, EU PRO QUIC, EU STREP EQuaM (FP7/2007-2013, No. 323714), Fundació Cellex, the Spanish MINECO (SEVERO OCHOA GRANT SEV-2015-0522, FOQUS FIS2013-46768, FISICATEAMO FIS2016-79508-P), and the Generalitat de Catalunya (SGR 874 and CERCA/Program). The work was made possible through the support of grant from the John Templeton Foundation (JT and JKK). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. JT acknowledges support of the Polish National Science Center by means of project no. 2015/16/T/ST2/00354 for the PhD thesis.

Appendix A Canonical form of a time-local Master equation describing evolution of the partially reduced state

In this appendix we show that evolution of the partially reduced state [Eq. (15] of the main text) can be rewritten in the canonical form of a time-local master equation Hall et al. (2014):

[TABLE]

where for sake of the simplicity explicit time dependence of the partially reduced density matrix $\rho_{S:fE}$ was omitted and time derivation was denoted by $\dot{\rho}$.

Using Eq. (15) of the main text we can write:

[TABLE]

where:

[TABLE]

with, real for initial thermal states, factor:

[TABLE]

and (as in the main text):

[TABLE]

The first step in derivation is calculation of $\dot{\rho}_{S:fE}$. We find:

[TABLE]

where (c.f. Eq. (10) of the main text):

[TABLE]

Finally, defining:

[TABLE]

we can recast Eq. (A) as:

[TABLE]

After identification $L(t)\equiv\sigma_{z}\otimes I_{fE}$ we arrive at Eq. (40). Quantity $\gamma(t)$ is called the canonical decoherence rate. If this rate is positive at all times, then the evolution over any time interval is completely positive Rivas et al. (2010); Hall et al. (2014). On this basis Hall et al. (2014) the following definition of non-Markovian evolution was introduced.

*Definition. A time-local master equation is Markovian, at some given time, if and only if the canonical decoherence rates are positive. Correspondingly, the evolution is non-Markovian if one or more of the canonical decoherence rates is strictly negative. *

According to the definition, the integral Hall et al. (2014):

[TABLE]

can be used to measure the total amount of non-Markovianity for the considered evolution.

Appendix B Fidelity in the spin-boson model

In this appendix we present a detailed derivation of the expression for the fidelity. The quantity we want to evaluate is:

[TABLE]

denoting a single-subsystem overlap. Dropping the explicit dependence on the index $k$ we obtain:

[TABLE]

where we have pulled the extreme left and right unitaries out of both the square roots and used the cyclic property of the trace to cancel them out. The free evolutions $e^{-inE_{k}t}$ cancel out as both unitary operators under the square root are Hermitian conjugates of each other. Thus, modulo phase factors:

[TABLE]

where $D(\alpha)$ represents the displacement operator. Next, assuming all the initial states $\rho_{0k}$ are thermal with the same temperature, we use the corresponding $P$-representation for the middle $\rho_{0}$ under the square root in (52):

[TABLE]

where $\bar{n}=1/(e^{\beta\omega}-1)$, $\beta\equiv 1/T$. Indicating the Hermitian operator under the square root in (52) by $\tilde{B}_{t}$, we obtain:

[TABLE]

The next step is to calculate explicitly the square root in the equation above. For this aim we expand $\rho_{0}$ in the Fock basis:

[TABLE]

Replacing it in Eq. (B) we have:

[TABLE]

with:

[TABLE]

Using the Fock basis $|j\rangle$ representation of coherent states one may get the explicit scalar product $\langle j|\gamma+\eta_{t}\rangle$. Accordingly Eq. (57) gets:

[TABLE]

We now show that this equation is formally equivalent to that of a thermal state introduced in Eq. (54). For this aim, we underline that we are interested in the square root of the operator $\tilde{B}_{t}$, rather than in itself. Therefore, there is a freedom of rotating $\tilde{B}_{t}$ by a unitary operator, and in particular a displacement one:

[TABLE]

In particular we find:

[TABLE]

where we have omitted the irrelevant phase factor arising from the action of the displacement. We replace Eq. (B) with Eq. (59). Dropping displacement operators due to Eq. (60) and introducing the variable:

[TABLE]

one obtains:

[TABLE]

In order to calculate explicitly the square root we recall the Fock expansion in Eq. (56). Finally, we get:

[TABLE]

and generalize it to the fidelity over all macrofractions:

[TABLE]

The expression in Eq. (26) follows by taking the limit of continuum spectrum in the above equation.

Appendix C Analytical expressions for decoherence factor and fidelity

When both observed and unobserved fragments of environment can be described in terms of the full spectral density, the decoherence factor and fidelity of the environmental sates are given by:

[TABLE]

In what follows we assume that $s$ is an integer number such that $s>1$ (the case for s=1 needs to be treated separately). Integrals in Eqs. (66) and (67) can be expressed in terms of Hurwitz zeta function NIS :

[TABLE]

what leads to the following expressions for decoherence factor:

[TABLE]

where $\wp(s-1)$ is the Euler gamma function and $c.c$ denotes complex conjugate. Similarly we find:

[TABLE]

For integer $s\geq 2$, using the following relation between Hurwitz theta function and polygamma function NIS :

[TABLE]

one reach Eqs. (V.1,V.1) presented in the main text.