# Spin-Boson Model as A Simulator of Non-Markovian Multiphoton Jaynes-Cummings Models

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## Abstract

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## 1. Introduction

## 2. The Spin-Boson Model

## 3. Analogue Simulation of Multiphoton Spin-Boson Interactions

#### 3.1. Reaction Coordinate Mapping

#### 3.2. Structured Environments

## 4. Examples and Numerical Simulations

#### 4.1. Dissipationless Multiphoton Jaynes-Cummings Models

#### 4.2. Dissipative Multiphoton Jaynes-Cummings Models

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Reaction Coordinate Mapping

## Appendix B. Derivation of ${H}_{b}$ and ${H}_{\mathrm{n}}$

## References

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**Figure 1.**(

**a**) Spin-boson model in the customary star configuration, where each of the circles corresponds to a harmonic oscillator of the environment with frequency ${\omega}_{k}$ interacting with the spin through ${\sigma}_{x}{f}_{k}({c}_{k}+{c}_{k}^{\u2020})$, before the reaction coordinate mapping. In (

**b**), we show an underdamped spin-boson spectral density ${J}_{\mathrm{SB}}(\omega )$, peaked at ${\omega}_{0}$ (cf. Equation (8)). Upon the reaction coordinate mapping, a collective degree of freedom is included into the system, which in turn interacts with the residual environment, as sketched in (

**c**) (see the main text for further details). For an underdamped ${J}_{\mathrm{SB}}(\omega )$, ${J}_{\mathrm{RC}}(\omega )$ adopts a Markovian form, as depicted in (

**b**). Such interaction with a collective coordinate can be exploited to realize Hamiltonians containing multiphoton interaction terms, as indicated in (

**c**) and explained in detail in Section 3. For structured environments, one can still rearrange the original environment using more collective coordinates into the augmented system ${S}^{\prime}$, where each of them interacts now with its own residual environment, as sketched in (

**d**) (see Section 3.2 for further details).

**Figure 2.**Dynamics of the simulated multiphoton Jaynes-Cummings models, $n=2$ (top) and $n=3$ (bottom). In Panels (

**a**) and (

**c**), we show the targeted dynamics (solid lines) and the one obtained using the spin-boson Hamiltonian (points) for $\u2329{a}^{\u2020}a\u232a$ and $\u2329{\sigma}_{z}\u232a$, as indicated in the plots and as a function of the time rescaled by ${\tau}_{n}$ (Equation (31)). In Panels (

**b**) and (

**d**), we plot the infidelity $1-F(t)$ between the ideal ${\rho}_{\mathrm{nJCM}}(t)$ state and its approximated one $\mathsf{\Phi}{\rho}_{\mathrm{S}+\mathrm{RC}}(t){\mathsf{\Phi}}^{\u2020}$ for different conditions, namely in (

**b**) for different temperatures (or mean occupation number ${n}^{\mathrm{th}}$) and in (

**d**) for different values of ${\u03f5}_{0}/\mathsf{\Omega}$. See Section 4.1 for further details regarding the parameters and states considered in the simulation. JCM, Jaynes-Cummings model.

**Figure 3.**Non-Markovian dynamics for a 2JCM and its simulation using a spin-boson model ${H}_{{\mathrm{S}}^{\prime}}$. In Panel (

**a**), we show the dynamics for the expectation values $\langle {a}_{i}^{\u2020}{a}_{i}\rangle $ with $i=1,2$ and $\langle {\sigma}_{z}\rangle $ for the target 2JCM model (solid lines) and its reconstructed values using ${H}_{{\mathrm{S}}^{\prime}}$ (points). The considered initial state reads as ${\rho}_{{\mathrm{S}}^{\prime}}(0)=\left|-\right.\u232a\left.\u2329-\right|\otimes {\rho}_{{\mathrm{RC}}_{1}}^{\mathrm{th}}\otimes {\rho}_{{\mathrm{RC}}_{2}}^{\mathrm{th}}$, with $\beta $ very large such that ${\rho}^{\mathrm{th}}\approx \left|0\right.\u232a\left.\u23290\right|$. In (

**b**), we plot the time-derivative of the trace distance, $\sigma (t)$, after tracing out the second bosonic mode and considering the initial states $\left|e\right.\u232a$ and $\left|g\right.\u232a$ for the spin in ${H}_{{\mathrm{S}}^{\prime}}$, while both reaction coordinates find themselves in their vacuum. Clearly, $\sigma (t)>0$ during certain intervals, revealing the non-Markovianity introduced due to the interaction with the second mode. Panel (

**c**) shows the evolution of purity for the state upon tracing the second mode, $\mathrm{Tr}[{\rho}_{\mathrm{S}+{\mathrm{RC}}_{1}}^{2}(t)]$ and for the reduced state of the spin, $\mathrm{Tr}[{\rho}_{\mathrm{S}}^{2}(t)]$, for the same case shown in (

**a**). In Panel (

**d**), we compare the infidelity $1-F(t)$ between the ideal state and the simulated one using ${H}_{{\mathrm{S}}^{\prime}}$ for the three different initial states employed here. We refer to Section 4.1 for further details regarding the parameters and states considered in the simulation.

**Figure 4.**Dynamics of a dissipative 2JCM using a spin-boson model. In Panel (

**a**), we show the dynamics of the expectation values of $\u2329{a}^{\u2020}a\u232a$ and $\u2329{\sigma}_{z}\u232a$, as in Figure 2, for the dissipative 2JCM (solid lines) and its simulation using the spin-boson model (points), for $\mathsf{\Gamma}/\tilde{\nu}=2\xb7{10}^{-1}$ and ${\rho}_{\mathrm{S}+\mathrm{RC}}(0)=\left|-\right.\u232a\left.\u2329-\right|\otimes {\rho}_{\mathrm{RC}}^{\mathrm{th}}$ with ${n}^{\mathrm{th}}={10}^{-3}$. For the same case, we also show in (

**b**) the evolution of the purities for the spin state $\mathrm{Tr}[{\rho}_{\mathrm{S}}^{2}(t)]$ and for the total state $\mathrm{Tr}[{\rho}_{\mathrm{S}+\mathrm{RC}}^{2}(t)]$. In (

**c**), we compare the different behaviour as $\mathsf{\Gamma}/\tilde{\nu}$ varies for the von Neumann entropy of the reduced spin state, ${S}_{\mathrm{vN}}({\rho}_{\mathrm{S}}(t))$. The values of $\mathsf{\Gamma}/\tilde{\nu}$ are indicated close to each curve. Finally, the state infidelity $1-F(t)$ between the targeted ${\rho}_{2\mathrm{JCM}}$ and its approximate simulation, $\mathsf{\Phi}{\rho}_{\mathrm{S}+\mathrm{RC}}(t){\mathsf{\Phi}}^{\u2020}$, is plotted in Panel (

**d**) for different $\mathsf{\Gamma}/\tilde{\nu}$. See the main text for further details on the parameters employed for the simulation.

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**MDPI and ACS Style**

Puebla, R.; Zicari, G.; Arrazola, I.; Solano, E.; Paternostro, M.; Casanova, J. Spin-Boson Model as A Simulator of Non-Markovian Multiphoton Jaynes-Cummings Models. *Symmetry* **2019**, *11*, 695.
https://doi.org/10.3390/sym11050695

**AMA Style**

Puebla R, Zicari G, Arrazola I, Solano E, Paternostro M, Casanova J. Spin-Boson Model as A Simulator of Non-Markovian Multiphoton Jaynes-Cummings Models. *Symmetry*. 2019; 11(5):695.
https://doi.org/10.3390/sym11050695

**Chicago/Turabian Style**

Puebla, Ricardo, Giorgio Zicari, Iñigo Arrazola, Enrique Solano, Mauro Paternostro, and Jorge Casanova. 2019. "Spin-Boson Model as A Simulator of Non-Markovian Multiphoton Jaynes-Cummings Models" *Symmetry* 11, no. 5: 695.
https://doi.org/10.3390/sym11050695