Abstract
We establish an approximate equivalence between a generalised quantum Rabi model and its nth order counterparts, where spinboson interactions are nonlinear as they comprise a simultaneous exchange of n bosonic excitations. Although there exists no unitary transformation between these models, we demonstrate their equivalence to a good approximation in a wide range of parameters. This shows that nonlinear spinboson couplings, i.e., nth order quantum Rabi models, are accessible to quantum systems with only linear coupling between boson and spin modes by simply adding spin rotations and after an appropriate transformation. Furthermore, our result prompts novel approximate analytical solutions to the dynamics of the quantum Rabi model in the ultrastrong coupling regime improving previous approaches.
Introduction
The quantum Rabi model (QRM) lies not only at the heart of our understanding of lightmatter interaction,^{1} but is also of importance in diverse fields of research.^{2} The Rabi model was primarily proposed to describe a nuclear spin interacting with classical radiation,^{3,4} whose quantised version only appeared two decades later.^{5} This contemplates a scenario which is of great generality as it encompasses two of the most basic, yet essential, ingredients in quantum physics, namely, a twolevel system and a bosonic mode. Indeed, this model emerges in disparate settings, ranging from ion traps^{6,7} to circuit or cavity QED,^{8,9} quantum optomechanical systems,^{10} colourcentres in membranes,^{11} and cold atoms.^{12}
Even though the QRM has been exhaustively investigated in the last decades, a number of recent findings has brought it again into the research spotlight. Among them we can mention its integrability,^{13} the existence of a distinctive behaviour in the deep strong coupling regime,^{14} or the emergence of a quantum phase transition.^{15,16,17,18} Closely related to the QRM, we find the nth order QRM (nQRM) which differs from the QRM in that the nQRM comprises nboson exchange interaction terms because of the presence of a nonlinear spinboson coupling. This generalisation of the QRM has recently attracted attention, mainly in its secondorder form (2QRM) as it shows striking phenomena such as spectral collapse,^{19,20,21} due to its relevance in preparing nonclassical states of light in quantum optics^{22,23} and regarding its solvability.^{24,25,26} These studies have also been extended to a mixed QRM comprising both oneboson and twoboson interaction terms, which appears in the context of circuit QED.^{27,28,29} Furthermore, solutions to this mixed QRM have recently been found,^{30} and it has also been reported that this model displays quantum phase transitions.^{31} Due to these compelling physical properties, the coherent control of nth order quantum Rabi models could open new avenues to develop different fields as quantum computing or quantum simulations. In addition, because of their different spectra, it is worth noting that there is no unitary map between the QRM and the nQRM with n > 1.
In this article, we demonstrate the existence of a connection, i.e., an approximate equivalence, among a family of Hamiltonians comprising nth order boson interaction terms, where the standard QRM or the 2QRM appear as special cases. As a proof of concept, we show how the dynamics of a 2QRM and a 3QRM can be captured without having access to the required nonlinear twophoton and threephoton interactions, and after an appropriate transformation of a linear QRM that includes spin driving terms, i.e., spin rotations. The latter is dubbed here as generalised QRM (gQRM). In this manner we can argue that, a quantum system that contains a linear spinboson coupling but lacks of nonlinear interactions suffices for the simulation of models where nonlinear terms are crucial. Our method works as follows: The dynamics of a state ψ〉 evolving under a nQRM (the targeted dynamics) can be successfully retrieved from a gQRM (the starting point of our method) by (i) evolving a transformed initial state Tψ〉 under gQRM during a time t and (ii) measuring customary spin and boson observables of the gQRM. We will demonstrate that the latter corresponds to expectation values of observables of the state \(\left {\psi (t)} \right\rangle\) evolved under the nonlinear nQRM (see Fig. 1 for a scheme of the method). Indeed, as creating nboson interactions is considered challenging in many quantum platforms, our method opens new avenues for their inspection. It is worth stressing that this reported method fundamentally differs from previous works where resonant multiboson effective Hamiltonians were obtained, either via amplitude modulation as used in circuit QED,^{32,33,34} or via adiabatic passage.^{35,36} In these works, effective multiboson exchange terms do not comprise nonlinear spinboson couplings and hold only in a very limited parameter regime and/or for particular states. Certainly, in this article we report an approximate equivalence among nQRMs which holds in a large range of parameters and grants a large tunability to explore their physics, as well as it unveils a fundamental relation between these models. Moreover, we present a potentially scalable platform,^{37} a microwavedriven trapped ion setting,^{38,39,40,41} in which nQRMs are unattainable without resorting to our approximate equivalence, which highlights the applicability of our method. Finally, we use our theory to analyse the standard QRM and find that our method provides, in addition, approximate analytical solutions that surpass in accuracy previous approaches in the ultrastrong coupling regime.^{42,43,44,45,46,47}
Results
Description of the approximate equivalence
We begin with the following general Hamiltonian (later we will demonstrate its connection with the gQRM that only contains linear spinboson interactions and represents the starting point of our approximate equivalence)
whose first two terms correspond to a bosonic mode of frequency ν and a twolevel system with a frequency splitting ω, described by the usual annihilation (creation) operator a (a^{†}) and spin\({\textstyle{1 \over 2}}\) Pauli matrices \(\vec \sigma = \left( {\sigma _x,\sigma _y,\sigma _z} \right)\). Both subsystems interact through a set of coupling terms with amplitude Ω/2 and parameter η, considered here equal ∀j, and α_{j} being a time dependent phase. The Hamiltonian H_{s} is central for our theory, as sketched in Fig. 1, and establishes an approximate map between gQRM dynamics with those of the nQRM. We perform a unitary transformation on H_{s} to find \(H_T = T(i\eta /2)H_{\mathrm{s}}T^\dagger (i\eta /2)\), where \(T(\beta ) = 1/\sqrt 2 \left[ {{\cal D}(\beta )\left( {\left e \right\rangle \left\langle g \right + \left g \right\rangle \left\langle g \right} \right) + {\cal D}^\dagger (\beta )\left( {\left e \right\rangle \left\langle e \right  \left g \right\rangle \left\langle e \right} \right)} \right]\) with \(\sigma _z = \left e \right\rangle \left\langle e \right  \left g \right\rangle \left\langle g \right\) and \({\cal D}(\beta ) = e^{\beta a^\dagger  \beta ^ \ast a}\) is the displacement operator. Note that this transformation has been used in the context of trapped ions to derive the eigenstates of a system that comprises a laser interacting with a trapped ion, and for fast implementations of the QRM.^{48,49} Now, choosing time dependent phases, α_{j} = (ω + δ_{j})t, and moving to a rotating frame with respect to H_{T,0} = −(ω + δ_{1})σ_{x}/2, the resulting Hamiltonian, H_{gQRM}, reads (for more details see Methods section)
with \(p = i(a^\dagger  a)\) and \({\cal U}_{T,0} = e^{  itH_{T,0}}\). The previous Hamiltonian is the one of the gQRM, where its last term can be viewed as a classical driving acting on the system, i.e. this is the term leading to spin rotations. In particular, we note that H_{gQRM} adopts the form of a standard QRM in the case of having δ_{j} = 0 ∀j.
On the other hand, the Hamiltonian H_{s} in Eq. (1) can be brought into the form of a H_{nQRM} by properly choosing α_{j} and in a suitable interaction picture. More specifically, by defining H_{s} = H_{s,0} + H_{s,1} with \(H_{{\mathrm{s}},0} = (\nu  \tilde \nu )a^\dagger a + (\omega  \tilde \omega )\sigma _z/2\) and considering two interaction terms (i.e., j = 1,2) such that \(\delta _{1,2} = \mp n\nu  \tilde \omega \pm n\tilde \nu\) (recall that α_{j} = (ω + δ_{j})t and thus α_{j} and δ_{j} are related) with \(\tilde \omega > 0\) and \(\tilde \nu > 0\), we find that \(H_{{\mathrm{s}},1}^I = e^{itH_{s,0}}H_{s,1}e^{  itH_{s,0}}\) approximately leads to
with ϕ_{n} = nπ/2 and g_{n} = η^{n}Ω/(2 n!). The validity of Eq. (3) is ensured when \({\mathrm{\Omega }} \ll \nu\), \(\tilde \omega + n\tilde \nu  \ll n\nu\) together with \(\eta \sqrt {\left\langle {(a + a^\dagger )^2} \right\rangle } \ll 1\) to safely perform a rotating wave approximation (RWA) in the joint Hilbert space involving spin and bosonic degrees of freedom. In this respect, an expression of the leading order error committed by our scheme can be found in section I of Supplementary Information for further explanation and details of the calculation. The simulated nQRM can be brought into strong or ultrastrong coupling regimes as the parameters \(\tilde \omega\) and \(\tilde \nu\) can be tuned to frequencies comparable to g_{n}.
In this manner, having access to H_{gQRM} that includes only a linear spinboson interaction, enables the exploration of a nQRM with nonlinear spinboson coupling (n > 1), whose physics is fundamentally different. For example, the most exotic hallmarks of the twophoton QRM (2QRM), are that the spectrum becomes a continuum for \(g_2 = \tilde \nu /2\) regardless of \(\tilde \omega\), and for \(g_2 > \tilde \nu /2\) the Hamiltonian is not longer lower bounded.^{19,20,25,50} The gQRM lacks these features, and it is therefore not obvious that the physics of H_{2QRM} can be accessed from H_{gQRM}. Moreover, the H_{gQRM} allows to simulate more exotic scenarios like combined nQRM and mQRM (see section II in Supplementary Information for further explanation and details of the calculation).
Based on the previous transformations one can find the following expression among operators that establishes a relation between the gQRM and nQRM dynamics, which is the central result of this article (see Methods for a more detailed derivation):
Here, \({\cal U}_{{\mathrm{gQRM}}}\) and \({\cal U}_{{\mathrm{nQRM}}}\) are the propagators of the gQRM and nQRM respectively, \(\Gamma (t) = {\cal U}_{s,0}^\dagger T^\dagger (i\eta /2){\cal U}_{T,0}\) with \({\cal U}_{s,0} = e^{  itH_{s,0}}\), and the approximate character of Eq. (4) is only a consequence of the RWA performed to achieve H_{nQRM} from H_{s}. Hence, an initial state ψ_{nQRM}(0)〉 after an evolution time t under H_{nQRM} can be approximated as ψ_{nQRM}(t)〉 ≈ Γ(t)ψ_{gQRM}(t)〉 with the initial state ψ_{gQRM}(0)〉 = T(iη/2)ψ_{nQRM}(0)〉.
Remarkably, while the dynamics under the gQRM occurs in a typical time 1/(ην), see Eq. (2), the simulated nQRM (Eq. (3)) involves parameters that are much smaller than ν since they satisfy the previously commented conditions \(\Omega \ll \nu\), \(\tilde \omega + n\tilde \nu  \ll n\nu\), and g_{n} = η^{n}Ω/(2 n!). As a consequence, a long evolution time of gQRM is required to effectively reconstruct the dynamics of nQRM.
Finally, our theory is completed with a mapping for the observables. As it can be derived from Eq. (4) (see Methods), the expectation value of an observable O_{nQRM}, i.e., an observable of the nQRM, corresponds to evaluate \(O_{{\mathrm{gQRM}}} = {\mathrm{\Gamma }}^\dagger (t)O_{{\mathrm{nQRM}}}{\mathrm{\Gamma }}(t)\) in the gQRM. Because Γ(t) involves bosonic displacement and spin rotations, O_{gQRM} may be in general intricate. Yet, for two relevant observables in nQRM, σ_{z} and a^{†}a, the mapping leads to simple operators, namely, σ_{z} transforms into −σ_{x} and a^{†}a into \(a^\dagger a  \eta /2p\sigma _x + \eta ^2/4\) (see Methods). Interestingly, it still possible to obtain good approximations for other observables by truncating bosonic operators. Indeed, \(e^{  \eta ^2/2}\left[ {\sigma _{z,y}{\mathrm{cos}}((\tilde \omega + \delta _1)t) \mp \sigma _{y,z}{\mathrm{sin}}((\tilde \omega + \delta _1)t)} \right]\) turns to be a good approximation of σ_{x,y} in the gQRM frame (see Sec. III in Supplementary Information for further explanation and details of the calculation) which allows to recover the full qubit dynamics of nQRM.
Approximate equivalence among gQRM and 2 and 3QRM
To numerically confirm our approximate equivalence, in Fig. 2 we show the results of the simulated dynamics of a 2QRM and a 3QRM using a gQRM for a certain set of parameters and initial states ψ(0)_{2QRM}〉 = 2〉↑〉_{x} and \(\left {\psi (0)_{{\mathrm{3}}QRM}} \right\rangle = \left( {\left 0 \right\rangle + \left 1 \right\rangle } \right)/\sqrt 2 \left \uparrow \right\rangle _x\), with \(\left { \uparrow ( \downarrow )} \right\rangle _x = \left( {\left e \right\rangle \pm \left g \right\rangle } \right)/\sqrt 2\). In addition, in Fig. 2c we show that the targeted σ_{x} of a nQRM is retrieved by means of the previously mentioned bosonic truncation of σ_{x} in the gQRM frame, i.e., \(e^{  \eta ^2/2}\left[ {\sigma _z\cos((\tilde \omega + \delta _1)t)  \sigma _y\sin((\tilde \omega + \delta _1)t)} \right]_{}^{}\). Furthermore, in order to quantify the agreement among these models and the validity of the previous theory, we compute the fidelity between the ideal quantum state of the nQRM and the approximated state evolved in the gQRM and properly transformed with Γ(t), that is, \(F_{{\mathrm{g}},n}(t) = \left\langle {\psi _{{\mathrm{gQRM}}}(t)} \right{\mathrm{\Gamma }}^\dagger (t)\left {\psi _{{\mathrm{nQRM}}}(t)} \right\rangle\). The computed fidelities of the considered cases are well above 0.99, showing the good agreement among these two models. Note that although H_{3QRM} could present truncation problems for g_{3} ≠ 0 (see ref. ^{51}), these do not affect the dynamics for the particular case plotted in Fig. 2. Indeed, for the chosen parameters and initial state, the dynamics during the considered evolution takes place in a constrained region of the Hilbert space and thus it does not show Fock space truncation problems (see section IV in Supplementary Information for further explanation and details of the calculation] for further details). It is however worth stressing that this is not the general case, because the 3QRM is not bounded from below. Therefore, the number of excitations can grow very fast and, as a consequence, the simulation of the 3QRM relying on the approximate equivalence will break down since \(\eta \sqrt {\left\langle {\left( {a + a^\dagger } \right)^2} \right\rangle } \ll 1\) is not longer satisfied. It is important to mention that our approximate equivalence is, in addition, not restricted to small times. The latter assertion is corroborated in Fig. 2 where the propagators for the 2QRM and 3QRM for the final time \(t_f = 2\pi \frac{4}{{\tilde \nu }}\) (values of \(\tilde \nu\) in the caption) are \({\mathrm{exp}}\left\{ {  i\pi \left[ {\frac{{\tilde \nu }}{{g_2}}a^\dagger a + \frac{{\tilde \omega }}{{2g_2}}\sigma _z  \sigma _x\left( {a^2 + (a^\dagger )^2} \right)} \right]} \right\}\) and \({\mathrm{exp}}\left\{ {  i2\pi /5\left[ {\frac{{\tilde \nu }}{{g_3}}a^\dagger a + \frac{{\tilde \omega }}{{2g_3}}\sigma _z + \sigma _y\left( {a^3 + (a^\dagger )^3} \right)} \right]} \right\}\) respectively. Note that in both previous cases the coupling terms are multiplied by a phase π and 2π/5 respectively. We furthermore stress that these phases (π and 2π/5) can be increased without deteriorating the achieved fidelities by simply choosing a larger value for ν. As previously commented, this is indeed possible since the approximate character of our method appears when we equal H_{s} to H_{nQRM}, whose performance is enhanced for large values of ν (see Methods).
Application for microwave driven ions
The proofofconcept of our method can be illustrated in a microwave driven ions platform. Note that the developed theory may be relevant in other systems as circuit QED.^{9} A microwavedriven trapped ion in a magnetic field gradient is described by (for more details see ref. ^{38,39,40,41})
where ω is the qubit energy splitting with a value that depends on the ion species. For example, for ^{171}Yb^{+}, we have ω ≈ 12.4 GHz^{52} plus a factor γB_{z} with γ ≈ 1.4 MHz/G that depends on the applied static magnetic field B_{z}. The coupling parameter Δ determines the rate of the spinboson coupling, while the last term corresponds to the action of microwave radiation on the system.^{53} In this setup the spinboson coupling is restricted to be linear, and therefore our theory appears as an alternative to introduce higherorder boson couplings in the dynamics. In order to take Eq. (5) into the form of Eq. (2), and subsequently (via the mapping T) into the general expression in Eq. (1), we define \(\omega = \delta _1 + \tilde \omega _{}^{}\) and move to a rotating frame with respect to the term \(\frac{{\tilde \omega }}{2}\sigma _z\). Considering two drivings such that φ_{1,2} = π, \(\omega _1 = \tilde \omega\) and \(\omega _2 = \tilde \omega  (\delta _2  \delta _1)\) and after eliminating terms that rotate at frequencies on the order of GHz, we find
which equals H_{gQRM} after a basis change, that is, \(e^{  i\frac{\pi }{4}\sigma _y}e^{  i\frac{\pi }{2}a^\dagger a}H_{{\mathrm{MW}}}^Ie^{i\frac{\pi }{2}a^\dagger a}e^{i\frac{\pi }{4}\sigma _y} = H_{{\mathrm{gQRM}}},\) where H_{gQRM} is given in Eq. (2) with η = 2Δ/ν. Hence, it is possible to use a microwavedriven ion to simulate models with nonlinear spinboson couplings (see section V in Supplementary Information for further explanation and details of the calculation] for more details concerning the implementation in this setup).
Approximate analytical solution for the QRM
Finding a solution to the QRM has been subject of a longstanding debate, which still attracts considerable attention.^{13,25,54,55} Based on our theory, we obtain a simple expression for the timeevolution propagator and expectation values of the QRM. The general expression given in Eq. (2) adopts the form of a standard QRM with a unique driving and δ_{1} = 0,
which, applying our method, approximately corresponds to \(H_{{\mathrm{aux}}} = {\mathrm{\Omega }}/2\sigma _x[1  \eta ^2(a^\dagger a + 1/2)]\). Indeed, from H_{s} = H_{s,0} + H_{s,1} with \(H_{s,0} = \nu a^\dagger a + \omega \sigma _z/2\) (setting \(\tilde \omega = \tilde \nu = 0\)), we obtain now \({\cal U}_{s,0}^\dagger H_{s,1}{\cal U}_{s,0} \approx H_{{\mathrm{aux}}}\) instead of H_{nQRM}, and where fast oscillating terms have been neglected performing a RWA, requiring again \(\eta \sqrt {\left\langle {(a + a^\dagger )^2} \right\rangle } \ll 1\), and only considering resonant terms up to η^{2} (see Sec. IV in Supplementary Information for further explanation and details of the calculation). As a consequence, the following analysis does not apply to the deepstrong coupling regime,^{14} found here when η ≥ 2. Hence, the propagator for the QRM is approximated as
which is expected to hold even in the ultrastrong coupling regime of the QRM, although restricted to the condition \({\mathrm{\Omega }} \ll \nu\). Because H_{aux} has a simple form, the time evolution can be analytically solved, with an initial state \(\left {\psi _{{\mathrm{aux}}}(0)} \right\rangle = T^\dagger (i\eta /2)\left {\psi _{{\mathrm{QRM}}}(0)} \right\rangle\). Indeed, \(H_{{\mathrm{aux}}} = \mathop {\sum}\nolimits_{n, \pm } E_n^ \pm \left {\varphi _n^ \pm } \right\rangle \left\langle {\varphi _n^ \pm } \right\) with \(\left {\varphi _n^ \pm } \right\rangle = \left n \right\rangle \left { \uparrow ( \downarrow )} \right\rangle _x\) and \(E_n^ \pm = \pm {\mathrm{\Omega }}/2(1  \eta ^2(n + 1/2))\). Now, employing the map between the two models, Eq. (8), we obtain the relation between observables. For example, a^{†}a in the QRM translates to \(a^\dagger a + \eta ^2/4 + \eta /2(x\sigma _z{\mathrm{sin}}\nu t  p\sigma _z{\mathrm{cos}}\nu t)\) in H_{aux} (see section VII in Supplementary Information for further explanation and details of the calculation). In addition, we show that our method improves the typical BlochSiegert (BS) approximation^{43,44} and the generalised RWA (GRWA) of the QRM^{45,46,47} in a particular parameter regime. The former, i.e., the BS, is found as e^{−S}H_{QRM}e^{S} ≈ H_{BS}, with \(H_{{\mathrm{BS}}} = (\nu + \tilde g{\mathrm{\Lambda }}\sigma _z)a^\dagger a + ({\mathrm{\Omega }} + \tilde g{\mathrm{\Lambda }})/2\sigma _z  \tilde g(ia^\dagger \sigma ^   ia\sigma ^ + )\), where the antiHermitian operator is given by \(S = i{\mathrm{\Lambda }}(a^\dagger \sigma _ + + a\sigma _  )  \xi \sigma _z(a^2  (a^\dagger )^2)\), with parameters \(\Lambda = \tilde g/(\nu + {\mathrm{\Omega }})\), \(\xi = \tilde g{\mathrm{\Lambda }}/(2\nu )\) and \(\tilde g = \eta \nu /2\) (see ref. ^{43,44}). The GRWA of the QRM is attained in a similar manner, but with \(S = \tilde g/\nu \chi \sigma _z(a^\dagger  a)\) such that e^{−S}H_{QRM}e^{S} ≈ H_{GRWA}, where H_{GRWA} has a JaynesCummings form with modified parameters, see ref. ^{45,46,47} and section VIII in the Supplementary Information for further explanation and details of the calculation] for furtherdetails. In Fig. 3 we compute the overlap between timeevolved states for these three approaches (our approximate solution, the BS approximation, and the GRWA) and the QRM. The approximate solution reproduces correctly the time evolution of the QRM as the coupling enters in the nonperturbative ultrastrong regime, \(\tilde g/\nu = 0.2\) (see ref. ^{44}) with a fidelity F_{QRM,aux} > 0.99, while approximations H_{BS} and H_{GRWA} fail as their fidelities drop significantly. For smaller couplings these approaches lead to similar high fidelities (see Fig. 3a).
Discussion
We have presented a connection, i.e., an approximate equivalence, among a family of Hamiltonians, including the QRM and its higher order counterparts (nQRM) comprising a nonlinear interaction term that involves the simultaneous exchange of n bosonic excitations with the spin–qubit, such as the twophoton QRM. In particular, the standard QRM including spin driving terms, i.e., the gQRM, allows us to retrieve the nQRM dynamics with very high fidelities. This theoretical framework shows that nQRMs can be accessed even in the absence of the required nonlinear spinboson exchange terms, as illustrated with a microwavedriven trapped ion. Therefore, we find that this fundamental model, the gQRM, approximately contains the dynamics of all other nth order models. Moreover, we have derived an approximate solution to the dynamics of the QRM even in the ultrastrong coupling regime which surpasses in accuracy previous approximate solutions. In this manner, we have defined a general theoretical frame for the study and understanding of this family of fundamental Hamiltonians and their associated dynamics, which may open new avenues in quantum computing and simulation.
Methods
Transformation between H _{s}, H _{gQRM}, and H _{nQRM}
The Hamiltonian H_{s}, given in Eq. (1), after the unitary transformation \(H_T = T(i\eta /2)H_sT^\dagger (i\eta /2)\), adopts the following form
which becomes H_{gQRM} in the rotating frame with respect to H_{T,0} = −(ω + δ_{1})σ_{x}/2, namely, \(H_{{\mathrm{gQRM}}} \equiv {\cal U}_{T,0}^\dagger (t)(H_T  H_{T,0}){\cal U}_{T,0}(t)\) as given in Eq. (2). For simplicity, we constrain ourselves to the case in which Ω_{j} ≡ Ω ∀j, although the procedure can be easily extended to a more general scenario. On the other hand, H_{s} leads to the desired nQRM when moving to an interaction picture with respect to \(H_{s,0} = (\nu  \tilde \nu )a^\dagger a + (\omega  \tilde \omega )/2\sigma _z\) with H_{s} = H_{s,0} + H_{s,1} Then, the interacting part of H_{s} can be written as
with \(a(t) = ae^{  i(\nu  \tilde \nu )t}\), \(a^\dagger (t) = a^\dagger e^{i(\nu  \tilde \nu )t}\), and \({\cal U}_{s,0}(t,t_0)\) the timeevolution operator associated to H_{s,0} such that t′ = t − t_{0}. Then, expanding the exponential, considering that \({\mathrm{\Omega }} \ll \nu\) and \(\eta \sqrt {\left\langle {(a + a^\dagger )^2} \right\rangle } \ll 1\), and that \(\delta _{1,2} = \mp n\nu  \tilde \omega \pm n\tilde \nu\) with \(\tilde \omega + n\tilde \nu  \ll n\nu\), one can perform a rotating wave approximation just keeping those terms resonant with σ^{+}a^{n} and σ^{−}a^{n}. In general,
with g_{n} = η^{n}Ω/(2 n!) and ϕ_{n} = nπ/2. Hence, it is possible to achieve a H_{nQRM} from H_{s}. Note however that the corresponding attained coupling g_{n} becomes smaller for increasing n, as it is proportional to η^{n}/n!. In particular, for n = 2, \(H_{s,1}^I\) can be approximated as
Note that, while the Hamiltonians H_{s} and H_{gQRM} are related through a unitary transformation, the achievement of a nphoton QRM, H_{nQRM}, from H_{s} requires of certain relations between parameters, such as \({\mathrm{\Omega }} \ll \nu\), \(\tilde \omega + n\tilde \nu  \ll n\nu\) and \(\eta \sqrt {\left\langle {(a + a^\dagger )^2} \right\rangle } \ll 1\) to safely perform the rotating wave approximation. In addition, it is worth stressing that the equivalence to a good approximation is not restricted to H_{nQRM} and H_{gQRM}. For example, a H_{gQRM} can lead into a more complex Hamiltonian, such as one comprising both nQRM and mQRM interaction terms (see Supplementary Information for further explanation and details of the calculation).
Transformations of observables and states
Here we show the derivation of the Eq. (4) which is a central result of this article. Having established the transformations that connect H_{gQRM} with H_{s}, and H_{nQRM} with H_{s} we can relate them in terms of the timeevolution operators,
where \({\cal U}_{x,1}^I\) denotes the timeevolution propagator of H_{x,1} in an interaction picture with respect to H_{x,0} such that H_{x} = H_{s,0} + H_{s,1}. Note that we have dropped the explicit time dependence for the sake of readability (see previous Eqs. (9–11) for the specific transformations). Then, combining the Eqs. (13–15), we arrive to
which is the Eq. (4), \({\cal U}_{{\mathrm{gQRM}}} = {\mathrm{\Gamma }}^\dagger (t){\cal U}_{{\mathrm{nQRM}}}T^\dagger (i\eta /2)\) with \({\mathrm{\Gamma }}(t) = {\cal U}_{s,0}^\dagger T^\dagger (i\eta /2){\cal U}_{T,0}\). Then,
with the relation between initial states ψ_{gQRM}(0)〉 = T(iη/2)ψ_{nQRM}(0)〉. Finally, from Eq. (17) it is straightforward to obtain the observable that must be measured in the gQRM frame in order to retrieve O_{nQRM} of the nQRM, i.e., \(O_{{\mathrm{gQRM}}} = {\mathrm{\Gamma }}^\dagger (t)O_{{\mathrm{nQRM}}}{\mathrm{\Gamma }}(t)\). Explicitly, Γ(t) reads
and thus, for O_{nQRM} = σ_{z} and a^{†}a the transformation leads to
while for other observables, like σ_{x} and σ_{y}, a more intricate expression is attained,
as it involves qubit and bosonic operators due to the presence of the displacement operator \({\cal D}(\beta )\). However, because the condition \(\eta \sqrt {\left\langle {(a + a^\dagger )^2} \right\rangle } \ll 1\) is required to guarantee a good realisation of H_{nQRM} and so that of Eq. (4), the previous expression can be well approximated by truncating \({\cal D}(\beta )\). Indeed, in our case \({\cal D}(i\eta )\) can be approximated up to third order as
In general, we can approximate the observable (σ_{j})_{gQRM} by truncating at order M, that is,
where the terms \((\sigma _j)_{{\mathrm{gQRM}}}^{(n)}\) for j = x,y and can be calculated from Eqs. (20–22). In particular, for _{σx,y} and for n = 0,
Note that measuring \((\sigma _{x,y})_{{\mathrm{gQRM}}}^{(M)}\) would require measurements of observables in the gQRM of the form \(\sigma _{y,z}(a^M + (a^\dagger )^M)\) as well as \(\sigma _{y,z}(a^\dagger )^na^m\) with n + m = M and n ≥ m (see Supplementary Information for further explanation and details of the calculation). Remarkably, for the considered cases here, the zeroth order approximation already reproduces reasonably well the expectation value of σ_{x,y} of a nQRM. Therefore, having access to qubit observables in gQRM, σ_{x,y,z}, allows to reconstruct the full qubit dynamics of a nQRM. Note that Eqs. (24) and (25) correspond to the expressions given in Results, which for σ_{x} is plotted in Fig. 2c for the simulation of a 2QRM and 3QRM.
Data availability
The data files used to prepare the figures shown in the manuscript are available from the first corresponding author upon request.
References
 1.
Scully, M. O. & Zubairy, M. S. Quantum Optics. (Cambridge University Press, Cambridge, 1997).
 2.
Braak, D., Chen, Q.H., Batchelor, M. T. & Solano, E. Semiclassical and quantum Rabi models: in celebration of 80 years. J. Phys. A 49, 300301 (2016).
 3.
Rabi, I. I. On the process of space quantization. Phys. Rev. 49, 324–328 (1936).
 4.
Rabi, I. I. Space quantization in a gyrating magnetic field. Phys. Rev. 51, 652–654 (1937).
 5.
Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89–109 (1963).
 6.
Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281–324 (2003).
 7.
Häffner, H., Roos, C. F. & Blatt, R. Quantum computing with trapped ions. Phys. Rep. 469, 155–203 (2008).
 8.
Haroche, S. & Raimond, J.M. Exploring the Quantum: Atoms, Cavities, and Photons. (Oxford University Press, Oxford, 2006).
 9.
Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. Science 339, 1169–1174 (2013).
 10.
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).
 11.
Abdi, M., Hwang, M.J., Aghtar, M. & Plenio, M. B. Spinmechanical scheme with color centers in hexagonal boron nitride membranes. Phys. Rev. Lett. 119, 233602 (2017).
 12.
Schneeweiss, P., Dareau, A. & Sayrin, C. Coldatom based implementation of the quantum Rabi model. https://arxiv.org/abs/1706.07781 (2017).
 13.
Braak, D. Integrability of the Rabi model. Phys. Rev. Lett. 107, 100401 (2011).
 14.
Casanova, J., Romero, G., Lizuain, I., GarcaRipoll, J. J. & Solano, E. Deep strong coupling regime of the JaynesCummings model. Phys. Rev. Lett. 105, 263603 (2010).
 15.
Hwang, M.J., Puebla, R. & Plenio, M. B. Quantum phase transition and universal dynamics in the Rabi model. Phys. Rev. Lett. 115, 180404 (2015).
 16.
Puebla, R., Hwang, M.J. & Plenio, M. B. Excitedstate quantum phase transition in the Rabi model. Phys. Rev. A 94, 023835 (2016).
 17.
Puebla, R., Hwang, M.J., Casanova, J. & Plenio, M. B. Probing the dynamics of a superradiant Quantum phase transition with a single trapped ion. Phys. Rev. Lett. 118, 073001 (2017).
 18.
Puebla, R., Casanova, J. & Plenio, M. B. A robust scheme for the implementation of the quantum Rabi model in trapped ions. New J. Phys. 18, 113039 (2016).
 19.
Felicetti, S. et al. Spectral collapse via twophonon interactions in trapped ions. Phys. Rev. A 92, 033817 (2015).
 20.
Duan, L., Xie, Y.F., Braak, D. & Chen, Q.H. Twophoton Rabi model: analytic solutions and spectral collapse. J. Phys. A 49, 464002 (2016).
 21.
Puebla, R., Hwang, M.J., Casanova, J. & Plenio, M. B. Protected ultrastrong coupling regime of the twophoton quantum Rabi model with trapped ions. Phys. Rev. A 95, 063844 (2017).
 22.
Brune, M., Raimond, J. M., Goy, P., Davidovich, L. & Haroche, S. Realization of a twophoton maser oscillator. Phys. Rev. Lett. 59, 1899–1902 (1987).
 23.
Toor, A. H. & Zubairy, M. S. Validity of the effective Hamiltonian in the twophoton atomfield interaction. Phys. Rev. A 45, 4951–4959 (1992).
 24.
Travĕnec, I. Solvability of the twophoton Rabi Hamiltonian. Phys. Rev. A 85, 043805 (2012).
 25.
Chen, Q.H., Wang, C., He, S., Liu, T. & Wang, K.L. Exact solvability of the quantum Rabi model using Bogoliubov operators. Phys. Rev. A 86, 023822 (2012).
 26.
Cui, S., Cao, J.P., Fan, H. & Amico, L. Exact analysis of the spectral properties of the anisotropic twobosons Rabi model. J. Phys. A 50, 204001 (2017).
 27.
Bertet, P. et al. Dephasing of a superconducting qubit induced by photon noise. Phys. Rev. Lett. 95, 257002 (2005).
 28.
Bertet, P., Chiorescu, I., Harmans, C. J. P. M. & Mooij, J. E. Dephasing of a fluxqubit coupled to a harmonic oscillator. https://arxiv.org/abs/condmat/0507290.
 29.
Felicetti, S., Rossatto, D. Z., Rico, E., Solano, E. & FornDíaz, P. Twophoton quantum Rabi model with superconducting circuits. Phys. Rev. A 97, 013851 (2018).
 30.
Duan, L., Xie, Y.F. & Chen, Q.H. Solutions to the mixed quantum Rabi model. https://arxiv.org/abs/1807.02676.
 31.
Ying, Z.J., Cong, L. & Sun, X.M. Quantum phase transition and spontaneous symmetry breaking in a nonlinear quantum Rabi model. https://arxiv.org/abs/1804.08128.
 32.
Strand, J. D. et al. Firstorder sideband transitions with fluxdriven asymmetric transmon qubits. Phys. Rev. B 87, 220505 (2013).
 33.
Allman, M. S. et al. Tunable resonant and nonresonant interactions between a phase qubit and LC resonator. Phys. Rev. Lett. 112, 123601 (2014).
 34.
Lü, Z., Zhao, C. & Zheng, H. Quantum dynamics of twophoton quantum Rabi model. J. Phys. A 50, 074002 (2017).
 35.
Ma, K. K. W. & Law, C. K. Threephoton resonance and adiabatic passage in the largedetuning Rabi model. Phys. Rev. A 92, 023842 (2015).
 36.
Garziano, L. et al. Multiphoton quantum Rabi oscillations in ultrastrong cavity QED. Phys. Rev. A 92, 063830 (2015).
 37.
Lekitsch, B. et al. Blueprint for a microwave trapped ion quantum computer. Sci. Adv. http://advances.sciencemag.org/content/3/2/e1601540 (2017).
 38.
Mintert, F. & Wunderlich, C. Iontrap Quantum logic using longwavelength radiation. Phys. Rev. Lett. 87, 257904 (2001).
 39.
Timoney, N. et al. Quantum gates and memory using microwavedressed states. Nature 476, 185–188 (2011).
 40.
Weidt, S. et al. Trappedion Quantum logic with global radiation fields. Phys. Rev. Lett. 117, 220501 (2016).
 41.
Piltz, Ch., Sriarunothai, T., Ivanov, S. S., Wölk, S. & Wunderlich, C. Versatile microwavedriven trapped ion spin system for quantum information processing. Sci. Adv. http://advances.sciencemag.org/content/2/7/e1600093 (2016).
 42.
FornDaz, P. et al. Observation of the BlochSiegert shift in a qubitoscillator system in the ultrastrong coupling regime. Phys. Rev. Lett. 105, 237001 (2010).
 43.
Beaudoin, F., Gambetta, J. M. & Blais, A. Dissipation and ultrastrong coupling in circuit QED. Phys. Rev. A 84, 043832 (2011).
 44.
Rossatto, D. Z., VillasBôas, C. J., Sanz, M. & Solano, E. Spectral classification of coupling regimes in the quantum Rabi model. Phys. Rev. A 96, 013849 (2017).
 45.
Feranchuk, I. D., Komarov, L. I. & Ulyanenkov, A. P. Twolevel system in a onemode quantum field: numerical solution on the basis of the operator method. J. Phys. A 29, 4035 (1996).
 46.
Irish, E. K. Generalized rotatingwave approximation for arbitrarily large coupling. Phys. Rev. Lett. 99, 173601 (2007).
 47.
Gan, C. J. & Zheng, H. Dynamics of a twolevel system coupled to a quantum oscillator: transformed rotatingwave approximation. Eur. Phys. J. D. 59, 473–478 (2010).
 48.
MoyaCessa, H., Jonathan, D. & Knight, P. L. A family of exact eigenstates for a single trapped ion interacting with a laser field. J. Mod. Opt. 50, 265–273 (2003).
 49.
MoyaCessa, H. Fast Quantum Rabi model with trapped ions. Sci. Rep. 6, 38961 (2016).
 50.
Peng, J., Ren, Z., Guo, G., Ju, G. & Guo, X. Exact solutions of the generalized twophoton and twoqubit Rabi models. Eur. Phys. J. D. 67, 162 (2013).
 51.
Lo, C. F., Liu, K. L. & Ng, K. M. The multiquantum JaynesCummings model with the counterrotating terms. Europhys. Lett. 42, 1 (1998).
 52.
Olmschenk, S. et al. Manipulation and detection of a trapped Yb^{+} hyperfine qubit. Phys. Rev. A 76, 052314 (2007).
 53.
Arrazola, I. et al. Pulsed dynamical decoupling for fast and robust twoqubit gates on trapped ions. Phys. Rev. A 97, 052312 (2018).
 54.
Zhong, H., Xie, Q., Batchelor, M. T. & Lee, C. Analytical eigenstates for the quantum Rabi model. J. Phys. A 46, 415302 (2013).
 55.
Batchelor, M. T. & Zhou, H.Q. Integrability versus exact solvability in the quantum Rabi and Dicke models. Phys. Rev. A 91, 053808 (2015).
Acknowledgements
This work was supported by the ERC Synergy grant BioQ, the EU STREP project EQUAM. The authors acknowledge support by the state of BadenWürttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 40/4671 FUGG. J. C. acknowledges Universität Ulm for a Forschungsbonus and support by the Juan de la Cierva grant IJCI201629681. H. M.C. thanks the Alexander von Humboldt Foundation for support. R. P. acknowledges DfESFI Investigator Programme (grant 15/IA/2864).
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J. C. and R. P. have contributed equally to this work. J.C. and R.P. conceived the idea and develop the theory with inputs from H. M.C. and M. B. P. All authors contributed to the writing of the manuscript.
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Casanova, J., Puebla, R., MoyaCessa, H. et al. Connecting nth order generalised quantum Rabi models: Emergence of nonlinear spinboson coupling via spin rotations. npj Quantum Inf 4, 47 (2018). https://doi.org/10.1038/s4153401800969
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