The task now consists of bringing the spin-boson Hamiltonian

${H}_{\mathrm{SB}}$ into the form of a

n-photon model, i.e, into a model containing interaction terms of the form

${\sigma}^{\pm}{a}^{n}$ and

${\sigma}^{\pm}{({a}^{\u2020})}^{n}$. For that, one could perform the approximate mapping used in [

43,

44] directly onto

${H}_{\mathrm{SB}}$. This would require the selection of a particular bosonic mode out of the environment with frequency

${\omega}_{q}$ to now play the role of

a in the interaction with the spin (

${c}_{q}\to a$), while treating the rest of

${c}_{k\ne q}$ as a residual environment. Here, however, we resort to a more sophisticated procedure, based on the so-called reaction coordinate (RC) mapping [

22,

23,

24,

25,

26,

27,

28], which consists of rearranging the environment degrees of freedom, such that a small number of collective coordinates can be included in the Hamiltonian part, which in turn interact with the residual environment. In certain cases, the open-quantum system description of the augmented system is considerably simplified with respect to the original system plus environment. Clearly, if the spin-boson model involves just a discrete number of modes, the reaction-coordinate procedure then trivially retrieves the original discrete environment.

#### 3.1. Reaction Coordinate Mapping

In the following, we summarize how to make use of the RC mapping for a spin-boson model, which has been studied previously in different works [

24,

25], while referring to

Appendix A and References [

22,

23,

24,

25,

26,

27,

28] for further details of the calculations and of the RC mapping.

We shall start by defining a collective mode or reaction coordinate, described by the annihilation and creation operators

a and

${a}^{\u2020}$, such that:

while the residual environmental degrees of freedom transform into

${b}_{k}$ and

${b}_{k}^{\u2020}$, requiring that the latter appear in a normal form in the Hamiltonian. In this manner, the original spin-boson Hamiltonian adopts the form of

${H}_{\mathrm{SB}}={H}_{\mathrm{S}+\mathrm{RC}}+{H}_{\mathrm{RC}-{\mathrm{E}}^{\prime}}+{H}_{{\mathrm{E}}^{\prime}}$, where the former is given by:

and the other two terms are

${H}_{\mathrm{RC}-{\mathrm{E}}^{\prime}}+{H}_{{\mathrm{E}}^{\prime}}=(a+{a}^{\u2020}){\sum}_{k}{g}_{k}({b}_{k}+{b}_{k}^{\u2020})+{(a+{a}^{\u2020})}^{2}{\sum}_{k}\frac{{g}_{k}^{2}}{{\omega}_{k}}+{\sum}_{k}{\omega}_{k}{b}_{k}^{\u2020}{b}_{k}$. The reaction coordinate map is completed upon the identification of the parameters

$\lambda $,

$\mathsf{\Omega}$, and

${g}_{k}$ or, thus,

${J}_{\mathrm{RC}}(\omega )={\sum}_{k}{g}_{k}^{2}\delta (\omega -{\omega}_{k})$. For certain cases, such mapping allow for an exact relation between the original and transformed parameters [

28]. Indeed, considering an underdamped spin-boson spectral density in the initial spin-boson model,

one can show that the resulting spectral density for the residual environment interacting with the reaction coordinate reads as:

provided

$\mathsf{\Lambda}/\omega \gg 1$ and where the parameters are related according to

$\gamma =\mathsf{\Gamma}/(2\pi {\omega}_{0})$,

$\mathsf{\Omega}={\omega}_{0}$, and

$\lambda =\sqrt{\pi \alpha {\omega}_{0}/2}$ (see

Appendix A or [

22,

23,

24,

28] for further details of this derivation). Here, the frequency

${\omega}_{0}$ in

${J}_{\mathrm{SB}}(\omega )$ denotes the position at which the spectral density features a maximum, while

$\mathsf{\Gamma}$ and

$\alpha $ account for its width and strength, respectively. For

${J}_{\mathrm{RC}}(\omega )$, the coupling strength is given by

$\gamma $. In this manner, by augmenting the system incorporating a collective mode, the original spin-boson model with

${J}_{\mathrm{SB}}(\omega )$ is transformed into a spin plus reaction coordinate, which now in turn interacts with a Markovian environment, where the standard Born-Markov approximations can be performed [

45]. Indeed, the master equation governing the dynamics of the augmented system, spin plus reaction coordinate, reads as (see

Appendix A for the details of the calculation, which closely follows [

24]):

with

$x=a+{a}^{\u2020}$, while the quantities

$\chi $ and

$\mathsf{\Theta}$ define the rates affecting the reaction coordinate. They are defined as:

where

${x}_{jk}=\left.\u2329{\varphi}_{j}\right|x\left|{\varphi}_{k}\right.\u232a$,

${H}_{\mathrm{S}+\mathrm{RC}}\left|{\varphi}_{j}\right.\u232a={\phi}_{j}\left|{\varphi}_{j}\right.\u232a$, and

${\xi}_{jk}={\phi}_{j}-{\phi}_{k}$.

Having obtained the reaction coordinate Hamiltonian, we undertake the transformation of

${H}_{\mathrm{S}+\mathrm{RC}}$, and thus, of Equation (

10), to achieve a model that comprises spin-multiphoton interaction terms. For that purpose, we will introduce two auxiliary Hamiltonians

${H}_{a}$ and

${H}_{b}$, which will arise in the intermediate steps by moving into a suitable interaction picture and transforming them accordingly. The first step consists indeed of moving to a rotating frame in which

${H}_{\mathrm{S}+\mathrm{RC}}\equiv {H}_{a,1}^{I}$ where

${H}_{a}={H}_{a,0}+{H}_{a,1}$ with

${H}_{a,0}=-{\Delta}_{0}/2{\sigma}_{x}$. In this manner, we find:

while Equation (

10) transforms into:

where

$\widehat{\chi}={U}_{a,0}\chi {U}_{a,0}^{\u2020}$ and

$\widehat{\mathsf{\Theta}}={U}_{a,0}\mathsf{\Theta}{U}_{a,0}^{\u2020}$, such that

${U}_{x}=\mathcal{T}{e}^{-i{\int}_{0}^{t}ds{H}_{x}(s)}$ is the time-evolution operator of a Hamiltonian

${H}_{x}$. Then, we perform a further transformation using the unitary operator

$T(\alpha )$, defined as

$T(\alpha )=1/\sqrt{2}\left[{D}^{\u2020}(\alpha )\left(\left|e\right.\u232a\left.\u2329e\right|-\left|g\right.\u232a\left.\u2329e\right|\right)+D(\alpha )\left(\left|g\right.\u232a\left.\u2329g\right|+\left|e\right.\u232a\left.\u2329g\right|\right)\right]$ with

$D(\alpha )={e}^{\alpha {a}^{\u2020}-{\alpha}^{*}a}$ the standard displacement operator. Hence,

${H}_{b}\equiv {T}^{\u2020}(-\lambda /\mathsf{\Omega}){H}_{a}T(-\lambda /\mathsf{\Omega})$ such that

${\rho}_{b}={T}^{\u2020}{\rho}_{a}T$, which leads to (see

Appendix B for further details):

where the Hamiltonian

${H}_{b}$ can be written as:

Hence, the dissipator acting on

${\rho}_{b}$ has the same form as in Equation (

14), but with transformed operators, namely

${T}^{\u2020}xT$,

${T}^{\u2020}\widehat{\chi}T$, and

${T}^{\u2020}\widehat{\mathsf{\Theta}}T$, where

$T\equiv T(-\lambda /\mathsf{\Omega})$. Finally, by moving to an interaction picture with respect to

${H}_{b,0}=(\mathsf{\Omega}-\tilde{\nu}){a}^{\u2020}a-\tilde{\omega}{\sigma}_{z}/2$ and expanding the exponential in Equation (

16) (the latter requires that

$|2\lambda /\mathsf{\Omega}|\sqrt{\langle {(a+{a}^{\u2020})}^{2}\rangle}\ll 1$ for truncating the exponential to a finite number of terms), we arrive at a Hamiltonian containing multiphoton interaction terms. The latter condition is commonly known as the Lamb-Dicke regime. In addition, we consider the driving frequencies to be

${\Delta}_{j}=\pm {n}_{j}(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$ with

$|\mathsf{\Omega}-\tilde{\nu}|\gg {\u03f5}_{j}/2$, so that one can safely perform a rotating-wave approximation keeping only those terms that are resonant, i.e., time independent (see

Appendix B for further details of the calculation). Note that, as

${H}_{b}$ is similar to the Hamiltonian describing an optical trapped ion under the action of lasers driving vibrational sidebands [

48], the procedure to obtain Jaynes-Cummings or quantum Rabi models is analogous to those cases [

35,

49,

50]. In this manner, we can approximate

${H}_{b,1}^{I}\equiv {U}_{b,0}^{\u2020}{H}_{b,1}{U}_{b,0}\approx {H}_{\mathrm{n}}$, where

${H}_{\mathrm{n}}$ contains the aimed at multiphoton interactions,

Note that the sets

r and

b encompass the terms with amplitude

${\u03f5}_{j}$ driving red- and blue-sidebands, that is those terms in Equation (

5) with frequency

${\Delta}_{j\in r}=+{n}_{j}(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$ and

${\Delta}_{j\in b}=-{n}_{j}(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$. Each of these drivings will contribute with a multiphoton interaction, either

${\sigma}^{+}{a}^{{n}_{j}}+\mathrm{H}.\mathrm{c}.$ for

$j\in r$ or

${\sigma}^{-}{a}^{{n}_{j}}+\mathrm{H}.\mathrm{c}.$ for

$j\in b$, which produce transitions between the states

$\left|m\right.\u232a\left|g\right.\u232a\leftrightarrow \left|m\mp {n}_{j}\right.\u232a\left|e\right.\u232a$. We stress that for a time-independent spin-boson model, as given in Equations (

1)–(

4) (or equivalently with

${n}_{d}=0$ in

${H}_{S,d}$ as given in Equation (

5), one obtains a single

n-photon [anti]-Jaynes-Cummings interaction term,

${\sigma}^{+}{a}^{n}+\mathrm{H}.\mathrm{c}.$ [

${\sigma}^{+}{(-{a}^{\u2020})}^{n}+\mathrm{H}.\mathrm{c}.$], by choosing

${\Delta}_{0}=n(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$ [

${\Delta}_{0}=-n(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$] in the original spin-boson Hamiltonian

${H}_{\mathrm{SB}}$. Thus, one needs the knowledge of the relevant bosonic frequency

$\mathsf{\Omega}$ to simulate multiphoton interaction terms properly.

In order to show how the dissipative part transforms, it is advisable to introduce the time-dependent unitary operator:

Then, one can see that, defining

$\tilde{\chi}=\mathsf{\Phi}\chi {\mathsf{\Phi}}^{\u2020}$,

$\tilde{\mathsf{\Theta}}=\mathsf{\Phi}\mathsf{\Theta}{\mathsf{\Phi}}^{\u2020}$ and

$\tilde{x}=\mathsf{\Phi}(a+{a}^{\u2020}){\mathsf{\Phi}}^{\u2020}$, the resulting master equation for

${\rho}_{\mathrm{n}}(t)$ is:

where the state

${\rho}_{\mathrm{n}}(t)$ of the multiphoton model is related to the original spin-boson upon the reaction coordinate mapping,

${\rho}_{\mathrm{S}+\mathrm{RC}}(t)$, through a unitary transformation:

From the previous expression, it follows that the purity of the total state

${\rho}_{\mathrm{S}+\mathrm{RC}}$ and that of

${\rho}_{\mathrm{n}}$ are approximately equal. Moreover, the reduced spin state in the different frameworks are related according to

${\mathrm{Tr}}_{\mathrm{B}}[{\rho}_{\mathrm{SB}}(t)]={\mathrm{Tr}}_{\mathrm{RC}}[{\rho}_{\mathrm{S}+\mathrm{RC}}(t)]\approx {\mathrm{Tr}}_{\mathrm{RC}}[{\mathsf{\Phi}}^{\u2020}{\rho}_{\mathrm{n}}(t)\mathsf{\Phi}]$, where

${\mathrm{Tr}}_{\mathrm{B}}[\xb7]$ and

${\mathrm{Tr}}_{\mathrm{RC}}[\xb7]$ denote the trace over the environment degrees of freedom and reaction coordinate, respectively. In this manner, having access to the spin degree of freedom, one can have access to the dissipative spin dynamics dictated by the master Equation (

19) under a multiphoton Hamiltonian

${H}_{\mathrm{n}}$, given in Equation (

17), whose parameters can be tuned. In addition, we remark that the initial state at

${t}_{0}=0$ in the multiphoton frame is related to that of the spin-boson model as

${\rho}_{\mathrm{n}}(0)={T}^{\u2020}{\rho}_{\mathrm{S}+\mathrm{RC}}(0)T$.

At this stage, a few comments regarding the validity of Equation (

20) are in order. While the steps performed from

${H}_{\mathrm{S}+\mathrm{RC}}$ to

${H}_{b}$ are exact,

${H}_{\mathrm{n}}$ is attained in an approximate manner. The good functioning of the simulation depends on how these approximations are met. That is, Equation (

20) holds within the Lamb-Dicke regime

$|2\lambda /\mathsf{\Omega}|\sqrt{\langle {(a+{a}^{\u2020})}^{2}\rangle}\ll 1$ and for parameters satisfying

$|\mathsf{\Omega}-\tilde{\nu}|\gg {\u03f5}_{j}/2\phantom{\rule{4pt}{0ex}}\forall j$, so that one can perform a rotating-wave approximation. As a consequence, this approximation also sets a constraint on the total duration for a good simulation (see

Appendix B). Note that, as the parameters

$\lambda $ and

$\mathsf{\Omega}$ are directly related to the original spin-boson spectral density, these conditions set constraints onto the accessible parameters, as well as on the temperature of the environment. Furthermore, in order to observe coherent multiphoton dynamics, the noise rates in Equation (

19) must be small compared to the parameters involved in

${H}_{\mathrm{n}}$. For the considered shape of

${J}_{\mathrm{SB}}(\omega )$, this translates into

$\mathsf{\Gamma}\ll \tilde{\nu},{\tilde{g}}_{n}$, where

${\tilde{g}}_{n}={\u03f5}_{0}{(2\lambda )}^{n}/(2{\mathsf{\Omega}}^{n}n!)$ for an

${n}_{d}=0$ and

${\Delta}_{0}=\pm n(\tilde{\nu}-\mathsf{\Omega})-\tilde{\omega}$ (cf. Equation (

17).

Finally, we comment that the previous scheme can be carried out beyond the Lamb-Dicke regime [

44]. Admittedly, when the Lamb-Dicke approximation does not hold, the Hamiltonian

${H}_{\mathrm{n}}$ is no longer a good approximation to the dynamics. In this case, the Hamiltonian

${H}_{\mathrm{n}}$ must be replaced by a suitable nonlinear Jaynes-Cummings or quantum Rabi model, whose coupling constants crucially depend on the Fock-state occupation number in a nonlinear fashion [

51,

52,

53,

54]. These nonlinear, yet multiphoton Hamiltonians appear then as a good approximation to

${H}_{b}$, and thus to

${H}_{\mathrm{SB}}$ whenever

$|2\lambda /\mathsf{\Omega}|\sqrt{\langle {(a+{a}^{\u2020})}^{2}\rangle}\ll 1$ is not fulfilled, as recently shown in [

44]. In this article, however, we will constrain ourselves to parameters within the Lamb-Dicke regime.

#### 3.2. Structured Environments

As previously mentioned, the simulation of multiphoton spin-boson interactions is not restricted to a determined form of

${J}_{\mathrm{SB}}(\omega )$. Here, we show the derivation of the procedure to obtain an effective multiphoton Hamiltonian when the initial spin-boson model features a more complicated interaction with the environment. For simplicity, we consider that

${J}_{\mathrm{SB}}(\omega )$ can be split in two parts,

${J}_{\mathrm{SB}}(\omega )={J}_{\mathrm{SB},1}(\omega )+{J}_{\mathrm{SB},2}(\omega )$, although its generalization to more is straightforward. The first contribution,

${J}_{\mathrm{SB},1}(\omega )$, is considered here to be suitable for the realization of multiphoton interactions as described in

Section 3.1. In addition, we will work under the assumption that the environment degrees of freedom corresponding to

${J}_{\mathrm{SB},2}(\omega )$ can be treated and simplified using again a collective or reaction coordinate, as sketched in

Figure 1c.

As discussed previously, we identify a collective coordinate for each of the contributions to the spectral density

${J}_{\mathrm{SB}}(\omega )$. In this manner, we augment the system to include both reaction coordinates, denoted here by

${\mathrm{S}}^{\prime}=\mathrm{S}+{\mathrm{RC}}_{1}+{\mathrm{RC}}_{2}$. Hence, its Hamiltonian is given by:

where

${H}_{\mathrm{S},\mathrm{d}}$ is the original spin Hamiltonian, which may contain spin rotations, introduced in Equation (

5), while the subscripts denote the corresponding reaction coordinate. The parameters

${\lambda}_{i}$ and

${\mathsf{\Omega}}_{i}$ are determined by the spectral density

${J}_{\mathrm{SB},\mathrm{i}}(\omega )$. The dynamics of the augmented system

${\mathrm{S}}^{\prime}$ is governed by the following master equation:

where

${x}_{i}={a}_{i}+{a}_{i}^{\u2020}$ for

$i=1,2$, and

${\chi}_{i}$ and

${\mathsf{\Theta}}_{i}$ are defined in analogy to Equations (

11) and (

12).

In order to find a suitable transformation to realize multiphoton interaction terms from

${H}_{{\mathrm{S}}^{\prime}}$, we proceed in a similar manner as for a single reaction coordinate. That is, we first move to a rotating frame where

${H}_{{\mathrm{S}}^{\prime}}\equiv {H}_{a,1}^{I}$, with

${H}_{a}={H}_{a,0}+{H}_{a,1}$ and

${H}_{a,0}=-{\Delta}_{0}/2{\sigma}_{x}$. Therefore, the transformed Hamiltonian reads as:

The next step is to perform the transformation using the unitary operator

$T(\alpha )$. As previously mentioned, we consider that the first reaction coordinate is suitable for the quantum simulation of multiphoton interaction terms, due to the form of its spectral density. This argument enables one to choose

$\alpha \equiv -{\lambda}_{1}/{\mathsf{\Omega}}_{1}$, hence

${H}_{b}\equiv {T}^{\u2020}(-{\lambda}_{1}/{\mathsf{\Omega}}_{1}){H}_{a}T(-{\lambda}_{1}/{\mathsf{\Omega}}_{1})$. This transformation acts trivially on the second reaction coordinate, but it does affect the coupling between the latter and the spin. Finally, if we move to an interaction picture with respect to

${H}_{b,0}=({\mathsf{\Omega}}_{1}-{\tilde{\nu}}_{1}){a}_{1}^{\u2020}{a}_{1}-\tilde{\omega}{\sigma}_{z}/2$, we obtain the Hamiltonian

${H}_{\mathrm{n},2}\approx {H}_{b,1}^{I}\equiv {U}_{b,0}^{\u2020}{H}_{b,1}{U}_{b,0}$,

where we have considered

${\Delta}_{j}=\pm {n}_{j}(\tilde{\nu}-{\mathsf{\Omega}}_{1})-\tilde{\omega}$ and assumed the Lamb-Dicke regime

$|{\lambda}_{1}/{\mathsf{\Omega}}_{1}|\sqrt{\langle {({a}_{1}+{a}_{1}^{\u2020})}^{2}\rangle}\ll 1$, and

$|{\mathsf{\Omega}}_{1}-\tilde{\nu}|\gg {\u03f5}_{j}/2$ to perform a rotating-wave approximation. Note that, while the multiphoton terms are identical to those of

${H}_{\mathrm{n}}$ in Equation (

17), the second reaction coordinate interacts with the spin degree of freedom. Indeed, depending on the parameters of

${H}_{\mathrm{n},2}$, the effect of such an interaction may turn effectively into non-Markovian effects for the reduced state of the spin and first reaction coordinate,

${\rho}_{\mathrm{n}}={\mathrm{Tr}}_{2}[{\rho}_{\mathrm{n},2}]$. The final master equation governing the dynamics of

${\rho}_{\mathrm{n},2}$ is:

where the operators involved are defined as in the case involving a single reaction coordinate (cf. Equation (

19)). It is worth stressing that the relation between the states given in Equation (

20) still holds. From the previous derivation, one can observe that the extension to more collective coordinates is straightforward.