A Comparison of Quantum and Semiclassical Rabi Models Near Multiphoton Resonances in the Presence of Parametric Modulation

Abstract

We compare the semiclassical and quantum predictions for the unitary dynamics of a two-level atom interacting with a single-mode electromagnetic field under parametric modulation of the atomic parameters in the regime of multiphoton atom–field resonances. We derive approximate analytic solutions for the semiclassical Rabi model when the atomic transition frequency and the atom–field coupling strength undergo harmonic external modulations. These solutions are compared to the predictions of the quantum Rabi model, which we solve numerically for an initial coherent state with a large average photon number (on the order of ${10}^4$), in the regime of three-photon resonance. We show that, for short enough times and sufficiently intense coherent states, the semiclassical dynamics agrees quite well with the quantum dynamics, although it inevitably fails at longer times due to the absence of collapse–revival behavior. Furthermore, we describe how the field state evolves throughout the interaction, presenting numerical results for the average photon number, entropies (related to atom–field entanglement), and other quantities characterizing the photon number statistics of the electromagnetic field.

1. Introduction

The semiclassical Rabi model (SRM) [1,2,3] plays a fundamental role in understanding light–matter interactions, particularly in regimes where the electromagnetic field can be treated classically while the atomic system retains its quantum nature [4]. It provides essential insights into the dynamics of two-level systems (qubits) driven by external fields, capturing key phenomena such as Rabi oscillations, Bloch–Siegert shift, Autler–Townes effects, coherent control, dynamical Stark shift, nonlinear optical susceptibilities, the pulse-area theorem, multiphoton resonances, parametric modulations, and others [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Despite its apparent simplicity, the SRM forms the basis for experimental implementations in areas ranging from laser–atom interactions to quantum control protocols [10,22,23]. However, since the model neglects the quantum nature of light, it cannot fully describe qubit dynamics at all times, nor does it provide the correct physical interpretation of qubit excitation as a photon absorption process [24,25]. These limitations are overcome by the quantum Rabi model (QRM) [26,27,28], in which the quantum nature of light is fully incorporated, allowing resonances and transitions induced by parametric modulations to be consistently explained from a quantum perspective [24,29,30,31,32,33,34,35,36].

Although not commonly recognized, both the SRM [11,12,13,14,17] and the QRM [29,32,33,36] intrinsically exhibit multiphoton resonances, whereby the atom can be (almost) completely excited from the ground state if its transition frequency is nearly equal to an odd multiple of the field frequency. These resonances are significantly narrower than the standard one-photon resonance; as a result, the associated oscillation periods are considerably longer, which renders the system dynamics more strongly affected by dissipation. Moreover, recent studies have demonstrated that parametric modulation of qubit parameters [37,38,39,40,41,42,43,44,45] can be exploited to generate photons from a vacuum [30,31,34,46,47], engineer effective light–matter interactions [48], create nonclassical states of light with metrological utility [49], induce qubit oscillations in the dispersive regime [30,38,39], implement quantum thermal cycles [50], and others. Applications of external modulation near multiphoton resonances have already been explored in both the quantum and semiclassical regimes [21,32,33,34]. However, the transition from the quantum to the semiclassical regime—which takes place when the average photon number of the initial coherent state becomes large enough for the quantum dynamics to approach the semiclassical dynamics—has not been investigated in this context (although such a transition for the one-photon resonance was recently analyzed in Refs. [24,25]). This raises several open questions, including how accurately the SRM captures qubit dynamics for coherent states of large but finite amplitude, what degree of atom–field entanglement persists in the semiclassical regime, what are the intrinsic limitations on qubit purity, and how initial coherent states are modified due to the atomic excitation and parametric modulations.

In this paper, we address these questions through a combination of analytical calculations and numerical simulations. The first goal is to provide a closed analytical description of the SRM under parametric modulation of system parameters, focusing on the regime of multiphoton resonances. From a practical standpoint, this involves investigating how (almost) complete oscillations of the qubit excitation probability can arise in the far-dispersive regime when the qubit transition frequency and the external modulation frequencies are properly tuned. In particular, we consider both separate and simultaneous modulations of the qubit transition frequency and the qubit–field coupling parameter, and analyze in detail the first-order and second-order effects with respect to the modulation amplitudes. The second goal is to provide a full characterization of the qubit dynamics within the QRM, highlighting the differences between the semiclassical and quantum predictions for the excitation probability. Finally, we study how qubit excitation and de-excitation influence the state of the electromagnetic field. To this end, we assume that the cavity field is initially prepared in a coherent state of relatively large amplitude and examine the evolution of the average photon number, the von Neumann and linear entropies, and other quantities related to photon-number statistics. We show that, for relatively short times and sufficiently intense coherent states, semiclassical and quantum predictions coincide, whereas for relatively longer times, both quantitative and qualitative discrepancies emerge. Moreover, the numerical simulations yield solid estimates of the qubit purity during the evolution, directly linked to the degree of atom–field entanglement in this bipartite, lossless system. We note that current circuit QED (quantum electrodynamical) systems can achieve dissipation rates lower than ${10}^{-6}\nu$, with $\nu$ being the cavity field frequency [51]. Therefore, within the time intervals t that we consider in this paper, $\nu t\sim {10}^4$, the impact of dissipation is not expected to be significant.

The paper is organized as follows. Section 2 presents the approximate analytic solution of the SRM in the presence of external modulations, when the conditions for a multiphoton resonance are met. In Section 3, we consider the QRM under the three-photon resonance condition and analytically show how resonant external modulations can affect the system dynamics for an initial coherent state with relatively large average photon number. Section 4 presents the numerical results for the system dynamics according to the QRM, considering initial coherent states with average photon numbers equal to $5\times {10}^3$ and $3\times {10}^4$, and compares the predictions of the SRM and QRM. Finally, Section 5 contains the conclusions. In Appendix A, we present closed analytic expressions for some transition rates relevant to the current study.

2. Semiclassical Rabi Model

Let us consider the semiclassical Rabi Hamiltonian [1,2] with externally prescribed qubit transition frequency $\Omega \left(t\right)={\Omega }_0+{\epsilon }_{\Omega }sin\left({\eta }_{\Omega }t\right)$, and the atom–field coupling strength $g\left(t\right)=g_0+{\epsilon }_{g}sin\left({\eta }_{g}t\right)$:

$${\widehat{H}}_S\left(t\right)={\displaystyle \frac{\Omega \left(t\right)}{2}}{\widehat{\sigma }}_z+g\left(t\right)\left(E_1+E_{-1}\right){\widehat{\sigma }}_x,$$

(1)

where ${\widehat{\sigma }}_i$, $i=x,y,z$, are the Pauli matrices and we set the reduced Planck constant $\hslash =1$. $g_0$ and ${\Omega }_0$ are the initial (and average) values of g and $\Omega$, respectively; ${\epsilon }_g$ and ${\epsilon }_{\Omega }$ denote the modulation amplitudes of g and $\Omega$, while ${\eta }_g$ and ${\eta }_{\Omega }$ denote the modulation frequencies of g and $\Omega$, respectively. We use the short-hand notation

$$E_{k+X}\equiv exp\left[it\left(k\nu +X\right)\right],$$

(2)

where k is an integer and X is an arbitrary real parameter. In the interaction picture defined by the operator ${\widehat{U}}_1=exp\left(-i\nu t{\widehat{\sigma }}_z/2\right)$, the Hamiltonian becomes

$${\widehat{H}}_I\left(t\right)={\widehat{H}}_{\mathrm{ef}}+{\widehat{H}}_t\left(t\right),$$

(3)

$${\widehat{H}}_{\mathrm{ef}}={\displaystyle \frac{{\Omega }_0-\nu }{2}}{\widehat{\sigma }}_z+g_0{\widehat{\sigma }}_x,$$

(4)

$${\widehat{H}}_t\left(t\right)={\displaystyle \frac{{ϵ}_z\left(t\right)}{2}}{\widehat{\sigma }}_z+{ϵ}_{+}\left(t\right){\widehat{\sigma }}_{+}+{ϵ}_{-}\left(t\right){\widehat{\sigma }}_{-},$$

(5)

where ${\widehat{\sigma }}_{+}={\widehat{\sigma }}_{-}^{†}=|e\rangle \langle g|$ is the ladder operator, the symbol † denotes the Hermitian conjugate, $|g\rangle$ and $|e\rangle$ denote the ground and the excited states of the atom, and we define the time-dependent functions

$${ϵ}_z\left(t\right)={\epsilon }_{\Omega }sin\left({\eta }_{\Omega }t\right)\mathrm{and}{ϵ}_{+}\left(t\right)={ϵ}_{-}^{*}\left(t\right)={\epsilon }_{g}sin\left({\eta }_{g}t\right)+g\left(t\right)e^{2i\nu t},$$

(6)

where the asterisk denotes the complex conjugate.

The time-independent part ${\widehat{H}}_{\mathrm{ef}}$ (4) of the interaction-picture Hamiltonian (3) has the eigenenergies $\pm R/2$ with the corresponding eigenstates

$$|{\Phi }_{+}\rangle ={\tilde{R}}_{+}^{-1/2}\left({\tilde{R}}_{+}|g\rangle +{\tilde{g}}_0|e\rangle \right) \text{and} |{\Phi }_{-}\rangle ={\tilde{R}}_{-}^{-1/2}\left({\tilde{R}}_{-}|g\rangle -{\tilde{g}}_0|e\rangle \right),$$

(7)

where $R=\sqrt{4g_0^2+{(\nu -{\Omega }_0)}^2}$, ${\tilde{g}}_0=g_0/R$, and ${\tilde{R}}_{\pm }=\left(1\pm \tilde{\Delta }\right)/2$ with $\tilde{\Delta }=(\nu -{\Omega }_0)/R$.

2.1. Jacobi–Anger Expansion

Several analytical approaches have been developed in the past to obtain a closed approximate description of the system dynamics [11,12,13,14,15,16,17,18]. Here, following the method described in Refs. [19,20,21], we expand the wavefunction corresponding to the Hamiltonian ${\widehat{H}}_I\left(t\right)$ (3) to

$$|{\psi }_1\rangle =A_{+,1}\left(t\right)e^{i{T}_t/2}{e}^{-itR/2}|{\Phi }_{+}\rangle +A_{-,1}\left(t\right)e^{-i{T}_t/2}{e}^{itR/2}|{\Phi }_{-}\rangle ,$$

(8)

where

$$\begin{array}{ccc}\hfill T_t& =& \phi -Y_{0}cos\left(2\nu t-\pi /2\right)-Y_{1}cos\left({\eta }_{\Omega }t\right)\hfill \\ & & +Y_{2}cos\left({\eta }_{g}t\right)+Y_{3}cos\left[\left({\eta }_g+2\nu \right)t\right]+Y_{4}cos\left[\left({\eta }_g-2\nu \right)t\right]\hfill \end{array}$$

(9)

with

$$Y_0=g_0{\displaystyle \frac{2{\tilde{g}}_0}{\nu }},Y_1={\epsilon }_{\Omega }{\displaystyle \frac{\tilde{\Delta }}{{\eta }_{\Omega }}},Y_2={\epsilon }_g{\displaystyle \frac{4{\tilde{g}}_0}{{\eta }_g}},Y_3={\epsilon }_g{\displaystyle \frac{2{\tilde{g}}_0}{{\eta }_g+2\nu }},Y_4={\epsilon }_g{\displaystyle \frac{2{\tilde{g}}_0}{{\eta }_g-2\nu }}$$

(10)

and $\phi =Y_1-Y_2-Y_3-Y_4$. For a common initial condition, $|{\psi }_1\left(0\right)\rangle =|g\rangle ,$ the coefficients $A_{+,1}\left(0\right)={\tilde{R}}_{+}^{1/2}$ and $A_{-,1}\left(0\right)={\tilde{R}}_{-}^{1/2}$.

The exact system dynamics can be obtained by solving the pair of equations

$${\dot{A}}_{-,1}=-iQ\left(t\right)A_{+,1}\mathrm{and}{\dot{A}}_{+,1}=-iQ{\left(t\right)}^{*}{A}_{-,1},$$

(11)

where the dot on top denotes the time derivative, with the time-dependent function

$$Q\left(t\right)=e^{-itR}{e}^{i{T}_t}\left[{\tilde{R}}_{-}{ϵ}_{+}^{*}\left(t\right)-{\tilde{g}}_0{ϵ}_z\left(t\right)-{\tilde{R}}_{+}{ϵ}_{+}\left(t\right)\right].$$

(12)

Using the Jacobi–Anger expansion

$$e^{izcos\theta }=J_0\left(z\right)+\sum _{n=1}^{\infty }{i}^n{J}_n\left(z\right)\left(e^{i\theta n}+e^{-i\theta n}\right),$$

(13)

where $J_0$ and $J_n$ are the Bessel functions of the first kind, one can recast $Q\left(t\right)$ to

$$Q\left(t\right)=\sum _{j=1}^{\infty }{p}_j{e}^{it{f}_j}\equiv Q_f\left(t\right)+Q_s\left(t\right),$$

(14)

where $Q_s\left(t\right)$ denotes the “slowly” varying terms (for which $|f_j|\lesssim |p_j|$) and $Q_f\left(t\right)$—the “rapidly” varying terms (for which $\left|f_i\right|\gg \left|p_i\right|$). All the constant coefficients $p_j$ and the corresponding frequencies $f_j$ can be derived after straightforward but lengthy manipulations.

So far, all the calculations are exact. However, under the experimentally feasible conditions, $|Y_k|\ll 1\forall k$, $Q\left(t\right)$ can be expanded in orders of ${\epsilon }_g$ and ${\epsilon }_{\Omega }$; to abbreviate the notation, under the “kth-order terms in $\epsilon$” we understand the terms proportional to ${\epsilon }_g^k$, ${\epsilon }_{\Omega }^k$, and ${\epsilon }_g^n{\epsilon }_{\Omega }^{k-n}$. It is advantageous to first consider the “rapidly” varying terms $Q_f\left(t\right)$, which to the zeroth-order in $\epsilon$ read

$$\begin{array}{ccc}\hfill Q_f^{\left(0\right)}& =& {\Xi }_{-R}{E}_{-R}+{\Xi }_{-2-R}{E}_{-2-R}+{\Xi }_{-4-R}{E}_{-4-R}\hfill \\ & & +{\Xi }_{-6-R}{E}_{-6-R}+{\Xi }_{-8-R}{E}_{-8-R}+\dots ,\hfill \end{array}$$

(15)

where the time-independent coefficients are:

$${\Xi }_{-R}=-g_0{e}^{i\varphi }{J}_1^0{J}_0^1{J}_0^2{J}_0^3{J}_0^4,$$

(16)

$${\Xi }_{-2-R}=e^{i\varphi }{g}_0{J}_0^1{J}_0^2{J}_0^3{J}_0^4\left(J_0^0{\tilde{R}}_{-}-J_2^0{\tilde{R}}_{+}\right),$$

(17)

$${\Xi }_{-4-R}=e^{i\varphi }{g}_0{J}_0^1{J}_0^2{J}_0^3{J}_0^4\left(J_1^0{\tilde{R}}_{-}-J_3^0{\tilde{R}}_{+}\right),$$

(18)

$${\Xi }_{-6-R}=e^{i\varphi }{g}_0{\tilde{R}}_{-}{J}_2^0{J}_0^1{J}_0^2{J}_0^3{J}_0^4,$$

(19)

and

$${\Xi }_{-8-R}=e^{i\varphi }{g}_0{\tilde{R}}_{-}{J}_3^0{J}_0^1{J}_0^2{J}_0^3{J}_0^4,$$

(20)

with $J_i^k\equiv J_i\left(Y_k\right)$.

Table 1. Some values of the zeroth-order coefficients (16)–(20) for different values of ${\Omega }_0$ and $g_0=0.1\nu$.

Coefficient	${\mathbf{\Omega }}_0=1.5\mathit{\nu }$	${\mathbf{\Omega }}_0=2\mathit{\nu }$	${\mathbf{\Omega }}_0=2.5\mathit{\nu }$	${\mathbf{\Omega }}_0=3\mathit{\nu }$	${\mathbf{\Omega }}_0=3.5\mathit{\nu }$
$\left|{\Xi }_{-R}\right|$	$1.9\times {10}^{-3}$	$9.8\times {10}^{-4}$	$6.6\times {10}^{-4}$	$5\times {10}^{-4}$	$4\times {10}^{-4}$
$\left|{\Xi }_{-2-R}\right|$	$9.6\times {10}^{-2}$	$9.9\times {10}^{-2}$	${10}^{-1}$	${10}^{-1}$	${10}^{-1}$
$\left|{\Xi }_{-6-R}\right|$	$1.7\times {10}^{-5}$	$4.8\times {10}^{-6}$	$2.2\times {10}^{-6}$	$1.2\times {10}^{-6}$	$7.9\times {10}^{-7}$
$\left|{\Xi }_{-8-R}\right|$	${10}^{-7}$	$1.6\times {10}^{-8}$	$4.8\times {10}^{-9}$	$2\times {10}^{-9}$	$1.1\times {10}^{-9}$

presents the values of the coefficients (16)–(20) for $g_0=0.1\nu$. One finds that, at least for ${\Omega }_0>1.5\nu$, $\left|{\Xi }_{-2-R}\right|$ is at least by two orders of magnitude larger than the other coefficients.

2.2. Coarse-Grained Approximation

Let us solve Equation (11) assuming that $Q_s=0$ and $Q_f\left(t\right)={\sum }_{i=1}^K{p}_i{e}^{it{f}_i}$, where K is some integer, $\left|p_i\right|\ll \left|f_i\right|$, and $\left|p_j\right|\ll \left|p_1\right|\forall j>1$ (as occurs in the present study for ${\Omega }_0>1.5\nu$, where $p_1={\Xi }_{-2-R}$ and $f_1=-2\nu -R$). One obtains the second-order differential equations

$${\ddot{A}}_{-,1}-{\displaystyle \frac{{\dot{Q}}_f}{Q_f}}{\dot{A}}_{-,1}+{\left|Q_f\right|}^2{A}_{-,1}=0$$

(21)

and

$${\ddot{A}}_{+,1}-{\left({\displaystyle \frac{{\dot{Q}}_f}{Q_f}}\right)}^{*}{\dot{A}}_{+,1}+{\left|Q_f\right|}^2{A}_{+,1}=0,$$

(22)

where

$${\displaystyle \frac{{\dot{Q}}_f}{Q_f}}\approx i\left[f_1+\sum _{j=2}^K\left(f_j-f_1\right){\displaystyle \frac{p_j}{p_1}}{e}^{it\left(f_j-f_1\right)}-\sum _{j,k=2}^K{f}_k{\displaystyle \frac{p_k}{p_1}}{\displaystyle \frac{p_j}{p_1}}{e}^{it\left(f_k+f_j-2f_1\right)}\right]$$

(23)

and

$${\left|Q_f\right|}^2=\sum _{j,k=1}^K{p}_j{p}_k^{*}{e}^{it\left(f_j-f_k\right)}.$$

(24)

Under the conditions $\left|f_j+f_k-2f_1\right|\gtrsim \left|f_j\right|,\left|f_k\right|$ for $j,k>1$ and $\left|f_j-f_{k\ne j}\right|\gtrsim \left|f_1\right|\forall j$, one can apply the coarse-grained approximation by assuming that $A_{\pm ,1}$ changes negligibly on the time scale $\tau =2\pi /\left|f_1\right|$ and averaging the dynamics over $\tau$:

$${\ddot{A}}_{-,1}-{\dot{A}}_{-,1}{\displaystyle \frac{1}{\tau }}{\int }_t^{t+\tau }d{t}^{\prime }{\displaystyle \frac{{\dot{Q}}_f\left(t^{\prime }\right)}{Q_f\left(t^{\prime }\right)}}+A_{-,1}{\displaystyle \frac{1}{\tau }}{\int }_t^{t+\tau }d{t}^{\prime }{\left|Q_f\left(t^{\prime }\right)\right|}^2\approx 0.$$

(25)

Under the assumptions made just above,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\tau }}{\int }_t^{t+\tau }d{t}^{\prime }{\displaystyle \frac{{\dot{Q}}_f\left(t^{\prime }\right)}{Q_f\left(t^{\prime }\right)}}& \approx & -{\displaystyle \frac{\left|f_1\right|}{2\pi }}\sum _{j,k=2}^K{\displaystyle \frac{f_k}{f_k+f_j-2f_1}}{\displaystyle \frac{p_k{p}_j}{p_1^2}}{e}^{it\left(f_k+f_j-2f_1\right)}\left(e^{i\tau \left(f_k+f_j\right)}-1\right)\hfill \\ & & +i{f}_1+{\displaystyle \frac{\left|f_1\right|}{2\pi }}\sum _{j=2}^K{\displaystyle \frac{p_j}{p_1}}{e}^{it\left(f_j-f_1\right)}\left(e^{i\tau f_j}-1\right)\approx i{f}_1\hfill \end{array}$$

(26)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\tau }}{\int }_t^{t+\tau }d{t}^{\prime }{\left|Q_f\left(t^{\prime }\right)\right|}^2& =& \sum _{j=1}^K{\left|p_j\right|}^2-i{\displaystyle \frac{\left|f_1\right|}{2\pi }}\sum _{j\ne k}^K{\displaystyle \frac{p_j{p}_k^{*}}{f_j-f_k}}{e}^{it\left(f_j-f_k\right)}\left(e^{i\tau \left(f_j-f_k\right)}-1\right)\hfill \\ & \approx & \sum _{j=1}^K{\left|p_j\right|}^2.\hfill \end{array}$$

(27)

Thus, under the coarse-grained approximation, Equations (21) and (22) become

$${\ddot{A}}_{\pm ,1}\pm i{f}_1{\dot{A}}_{\pm ,1}+\sum _{j=1}^K{\left|p_j\right|}^2{A}_{\pm ,1}\approx 0$$

(28)

and the slowly varying solutions read

$$A_{\pm ,1}=A_{\pm ,2}{e}^{\mp it\delta },$$

(29)

where $A_{\pm ,2}$ are some constants and the frequency shift

$$\delta =\mathrm{sign}\left(f_1\right)\times {\displaystyle \frac{\left|f_1\right|-\sqrt{f_1^2+4{\sum }_{j=1}^K{\left|p_j\right|}^2}}{2}}$$

(30)

is defined due to the “rapidly” oscillating terms.

Based on the values from Table 1, in this paper, we keep only the terms ${\Xi }_{-R}$, ${\Xi }_{-2-R}$ and ${\Xi }_{-4-R}$ in Equation (30), neglecting all the other (insignificant) contributions.

To conclude, in this Section it is shown that the “rapidly” oscillating terms lead to the effective frequency shift (30) and the approximate slowly varying solution is given by Equation (29). For a detailed discussion about the validity of the coarse-grained approximation, see Ref. [20].

2.3. Approximate Description Under Resonances

Now, let us return to Equations (11) and consider the ansatz

$$A_{\pm ,1}\left(t\right)=A_{\pm ,2}\left(t\right)e^{\mp it\delta },$$

(31)

where $A_{\pm ,2}\left(t\right)$ are relatively slowly varying coefficients (on the characteristic timescale $\tau$). As discussed in Section 2.2, under the coarse-grained approximation, this ansatz naturally eliminates the “rapidly oscillating part” $Q_f\left(t\right)$ (which merely amounts to introducing the frequency shift $\delta$). So, one is left with the equations

$${\dot{A}}_{-,2}=-iq\left(t\right)A_{+,2}\mathrm{and}{\dot{A}}_{+,2}=-iq{\left(t\right)}^{*}{A}_{-,2},$$

(32)

where

$$q\left(t\right)\equiv e^{-2i\delta t}{Q}_s\left(t\right)=\sum _{k=0}^{\infty }{q}^{\left(k\right)}\left(t\right)$$

(33)

and $q^{\left(k\right)}$ denotes the slowly oscillating kth-order term.

Defining $r\equiv R+2\delta$, the zeroth-order terms read

$$q^{\left(0\right)}\left(t\right)=\sum _{k=1}^{\infty }\prime {\Xi }_{2k-r}{E}_{2k-r},$$

(34)

where the prime in the sum indicates that only the terms with $|2\nu k-r|\lesssim |{\Xi }_{2k-r}|$ are to be retained; other rapidly oscillating terms must be treated as part of $Q_f\left(t\right)$ and can be included into the frequency shift $\delta$ (30). The first four coefficients in Equation (34) are (approximately):

$${\Xi }_{2-r}=e^{i\varphi }{g}_0{J}_0^1{J}_0^2{J}_0^3{J}_0^4\left(J_2^0{\tilde{R}}_{-}-J_0^0{\tilde{R}}_{+}\right)$$

(35)

$${\Xi }_{4-r}=e^{i\varphi }{g}_0{J}_0^1{J}_0^2{J}_0^3{J}_0^4\left(J_1^0{\tilde{R}}_{+}-J_3^0{\tilde{R}}_{-}\right),$$

(36)

$${\Xi }_{6-r}=-e^{i\varphi }{g}_0{\tilde{R}}_{+}{J}_2^0{J}_0^1{J}_0^2{J}_0^3{J}_0^4,$$

(37)

and

$${\Xi }_{8-r}=e^{i\varphi }{g}_0{\tilde{R}}_{+}{J}_3^0{J}_0^1{J}_0^2{J}_0^3{J}_0^4.$$

(38)

The terms under sum in Equation (34) correspond to the multiphoton atom–field resonances, $2k\nu -r=0$, which occur when the atomic transition frequency matches the $\left(2k+1\right)$-photon resonance:

$${\Omega }_0=\nu +2\sqrt{{\left(k\nu -\delta \right)}^2-g_0^2}\approx \left(2k+1\right)\nu -2\delta -{\displaystyle \frac{g_0^2}{k\nu }}.$$

(39)

This means that when ${\Omega }_0\approx \left(2k+1\right)\nu$, the atom exhibits Rabi oscillations without any external perturbation due to the “counter-rotating terms”$\left(E_1{\sigma }_{+}+E_{-1}{\sigma }_{-}\right)$ in the semiclassical Rabi model, as can be seen from Equations (32) and (34) or from other approaches [11,12,13,14,17].

The first-order terms in Equation (33) are

$$q^{\left(1\right)}\left(t\right)=\sum _{\eta ={\eta }_{\Omega },{\eta }_g}\left(\underset{k=0}{\overset{\infty }{{\sum }^{\prime }}}{\Xi }_{-2k-r,\eta }{E}_{-2k-r+\eta }+\underset{k=1}{\overset{\infty }{{\sum }^{\prime }}}\sum _{j=\pm 1}{\Xi }_{2k-r,j\eta }{E}_{2k-r+j\eta }\right),$$

(40)

where the prime in the sum reminds that only the slowly oscillating terms are to be retained; for relatively small values of k, the coefficients are given in Appendix A.1.

Analogously, one can calculate the higher-order terms $q^{\left(k\right)}\left(t\right)$. As an instructive example, we give the second-order terms dependent on the product ${\epsilon }_{\Omega }{\epsilon }_g$ (i.e., when both the modulations are imposed simultaneously):

$$\begin{array}{ccc}\hfill q^{\left(2\right)}\left(t\right)& =& {\Xi }_{-r,{\eta }_{\Omega }+{\eta }_g}{E}_{-r+{\eta }_{\Omega }+{\eta }_g}+{\Xi }_{-r,{\eta }_{\Omega }-{\eta }_g}{E}_{-r+{\eta }_{\Omega }-{\eta }_g}\hfill \\ & & +{\Xi }_{-r,-{\eta }_{\Omega }+{\eta }_g}{E}_{-r-{\eta }_{\Omega }+{\eta }_g}+{\Xi }_{-2-r,{\eta }_{\Omega }+{\eta }_g}{E}_{-2-r+{\eta }_g+{\eta }_{\Omega }}\hfill \\ & & +{\Xi }_{-2-r,{\eta }_{\Omega }-{\eta }_g}{E}_{-2-r+{\eta }_{\Omega }-{\eta }_g}+{\Xi }_{-2-r,-{\eta }_{\Omega }+{\eta }_g}{E}_{-2-r-{\eta }_{\Omega }+{\eta }_g}+\dots ,\hfill \end{array}$$

(41)

where multiple dots denote weaker contributions than those shown and the coefficients of Equation (41) are given in Appendix A.2. From Equation (41), one finds that resonances can also take place when the combinations ${\eta }_{\Omega }\pm {\eta }_g$ match certain values.

Under exact resonance(s), the solutions to Equation (32) are

$$A_{-,2}=A_{-}\left(0\right)cos\left(\left|\Xi \right|t\right)-i{e}^{i\xi }{A}_{+}\left(0\right)sin\left(\left|\Xi \right|t\right)$$

(42)

and

$$A_{+,2}=A_{+}\left(0\right)cos\left(\left|\Xi \right|t\right)-i{e}^{-i\xi }{A}_{-}\left(0\right)sin\left(\left|\Xi \right|t\right),$$

(43)

where $\Xi \equiv {\sum }_k{\Xi }_k=\left|\Xi \right|e^{i\xi }$, and ${\Xi }_k$ stands for the coefficient for which the subindex of the E-exponents in Equations (34), (40), and/or (41) is zero. For the atom prepared initially in the ground state, the probability of the excited state reads

$$P_e\left(t\right)={\left|\langle e|{\psi }_1\left(t\right)\rangle \right|}^2={\tilde{g}}_0^2{\left|e^{i{T}_t}{e}^{-itr}{\tilde{R}}_{+}^{-1/2}{A}_{+,2}-{\tilde{R}}_{-}^{-1/2}{A}_{-,2}\right|}^2.$$

(44)

Therefore, $\left|\Xi \right|$ expresses the total transition rate, which depends on the precise matching of the atomic transition frequency to a multiphoton resonance condition, as well as the adjustment in the modulation frequencies to the system resonances.

Equations (42) and (43) fully describe the system dynamics only under exact resonances; in the general case, Equations (32) must be solved numerically, for example, using the Runge–Kutta method. For the approximate analytic solution of the SRM in the presence of Markovian dissipation, see Refs. [19,20,21].

3. Quantum Rabi Model

The quantum Rabi Hamiltonian [27,28] reads

$${\widehat{H}}_Q=\nu \widehat{N}+\Omega \left(t\right){\widehat{\sigma }}_e+\check{g}\left(t\right)(\widehat{a}+{\widehat{a}}^{†})({\widehat{\sigma }}_{+}+{\widehat{\sigma }}_{-})={\widehat{H}}_0+2\sum _{k=\Omega ,g}{\widehat{W}}^{\left(k\right)}sin\left({\eta }_{k}t\right),$$

(45)

where ${\widehat{\sigma }}_e=|e\rangle \langle e|$, ${\widehat{H}}_0=\nu \widehat{N}+{\Omega }_0{\widehat{\sigma }}_e+{\check{g}}_0\left(\widehat{a}+{\widehat{a}}^{†}\right)\left({\widehat{\sigma }}_{+}+{\widehat{\sigma }}_{-}\right)$ is the standard time-independent Rabi Hamiltonian with bare parameters, and we define

$${\widehat{W}}^{\left(\Omega \right)}={\displaystyle \frac{{\epsilon }_{\Omega }}{2}}{\widehat{\sigma }}_e\mathrm{and}{\widehat{W}}^{\left(g\right)}={\displaystyle \frac{{\check{\epsilon }}_g}{2}}(\widehat{a}+{\widehat{a}}^{†})({\widehat{\sigma }}_{+}+{\widehat{\sigma }}_{-}).$$

(46)

Here, $\widehat{a}$ and ${\widehat{a}}^{†}$ are the annihilation and creation operators of the cavity field, $\widehat{N}={\widehat{a}}^{†}\widehat{a}$ is the photon number operator, and $\check{g}\left(t\right)={\check{g}}_0+{\check{\epsilon }}_{g}sin\left({\eta }_{g}t\right)$ is the atom–field coupling parameter in the quantum regime, the latter indicated by the inverted hat. While numerous unexpected phenomena may occur in the ultrastrong coupling regime [52,53,54,55,56,57], here, we focus on the less demanding weak coupling regime, ${\check{g}}_0\ll \nu$, which can be readily achieved in many circuit QED setups [37,52,53].

Let us show how the quantum Rabi Hamiltonian is related to the semiclassical Hamiltonian (1). In the interaction picture generated by the unitary operator $exp\left(-i\nu \widehat{N}t\right)$, the Hamiltonian (45) reads

$${\widehat{H}}_{\mathrm{int}}=\Omega \left(t\right){\widehat{\sigma }}_e+\check{g}\left(t\right)(\widehat{a}{e}^{-i\nu t}+{\widehat{a}}^{†}{e}^{i\nu t})({\widehat{\sigma }}_{+}+{\widehat{\sigma }}_{-}).$$

(47)

For the cavity field prepared initially in the coherent state $|\alpha \rangle$, with $\alpha \gg 1$, let us assume that the cavity field is not altered during the evolution. Then, the total density operator is ${\widehat{\rho }}_{\mathrm{tot}}\left(t\right)=\widehat{\rho }\left(t\right)\otimes |\alpha \rangle \langle \alpha |$, where $\widehat{\rho }$ is the qubit’s density operator, and the qubit dynamics is governed by the Hamiltonian

$${\widehat{H}}_q={\mathrm{Tr}}_f\left({\widehat{H}}_{\mathrm{int}}\otimes |\alpha \rangle \langle \alpha |\right)=\Omega \left(t\right){\widehat{\sigma }}_e+\alpha \check{g}\left(t\right)(e^{-i\nu t}+e^{i\nu t})({\widehat{\sigma }}_{+}+{\widehat{\sigma }}_{-}),$$

(48)

where ${\mathrm{Tr}}_f$ denotes the partial trace over the field (without loss of generality, we assume that $\alpha$ is real). By subtracting the classical term $\Omega \left(t\right)/2$, one recovers the semiclassical Hamiltonian (1) with the identification $g\left(t\right)=\alpha \check{g}\left(t\right)$, which relates the quantum and semiclassical atom–field coupling parameters. For a rigorous analysis of the transition from the QRM to SRM, see Refs. [4,56].

3.1. Dressed-States Expansion

Near the multiphoton resonances, when ${\Omega }_0\approx K\nu$, ${\check{g}}_0\ll \nu$ and K is a small enough odd integer, the eigenvalues and eigenstates of ${\widehat{H}}_0$ can be found by numerical diagonalization (here, we use the routine EVCRG from the International Mathematics and Statistics Library (IMSL) for FORTRAN [58]). We write the two eigenstates, whose eigenergies ${\lambda }_{\pm ,n}$ are close to $\nu n$, as $|{\lambda }_{\pm ,n}\rangle ={\sum }_l\left({\varrho }_{\pm ,n}^{\left(2l\right)}|g,n+2l\rangle +{\tilde{\varrho }}_{\pm ,n}^{\left(2l-1\right)}|e,n+2l-1\rangle \right)$, where ${\varrho }_{\pm ,n}^{\left(l\right)}$ are the probability amplitudes and l runs over all the possible values (although, as shown just below, in the case considered here, it is sufficient to consider $l\in \left[-5,5\right]$).

Figure 1 shows the eigenenergies and the coefficients of the eigenstates for the parameters $\Omega$, $\check{g}$, and $\alpha$ corresponding to the three-photon resonance of the SRM. Figure 1a shows (${\lambda }_{\pm ,n}/\nu -n$) and the photon number probability distribution $p_n=e^{-{\alpha }^2}{\alpha }^{2n}/n!$ for the coherent state $|\alpha \rangle$. As discussed in Ref. [21], when the difference ($|{\lambda }_{+,n}\rangle -|{\lambda }_{-,n}\rangle$) is constant for all the relevant values of n (i.e., within the “hump” of the photon number distribution), the QRM and SRM provide the same results for $P_e$ (44) for initial times. One can see that, for the parameters used, the SRM does not accurately reproduce the results of the QRM even for the initial times (as also is attested by Figure 3). Figure 1b shows the coefficients of $|{\lambda }_{+,n}\rangle$ with the largest absolute values, and the curves correspond to ten different coefficients, as indicated. Figure 1c shows the same for $|{\lambda }_{-,n}\rangle$.

From Figure 1b,c, one can find that, near the three-photon resonance, the non-normalized eigenstates (“dressed-states”) for $n\gg 1$ are (approximately) symmetric and asymmetric combinations of the states $|g,n\rangle$ and $|e,n-3\rangle$:

$$\begin{array}{ccc}\hfill |S_n\rangle & \approx & {\mu }_n|g,n\rangle +\sqrt{1-{\mu }_n^2}|e,n-3\rangle -{\varsigma }_n|e,n-1\rangle +{\varpi }_n|g,n-2\rangle +\dots \hfill \\ \hfill \mathrm{and}\\ \hfill |A_n\rangle & \approx & \sqrt{1-{\mu }_n^2}|g,n\rangle -{\mu }_n|e,n-3\rangle -{\varpi }_n|e,n-1\rangle -{\varsigma }_n|g,n-2\rangle +\dots ,\hfill \end{array}$$

where the coefficients ${\mu }_n$, ${\varsigma }_n$, ${\varpi }_n$, and others are to be found numerically [21]. Moreover, in the quite close vicinity of the three-photon resonance, ${\mu }_n\approx 2^{-1/2}$, while ${\varsigma }_n$, ${\varpi }_n$, and other coefficients are at least by two orders of magnitude smaller. Similar conclusions hold for higher-order resonances.

Let us expand the wavefunction corresponding to the Hamiltonian (45) as

$$|\psi \left(t\right)\rangle =\sum _n\left(c_n^{\left(S\right)}\left(t\right)e^{-it{S}_n}|S_n\rangle +c_n^{\left(A\right)}\left(t\right)e^{-it{A}_n}|A_n\rangle \right),$$

(49)

where the sum is taken over all the Hilbert space, and $c_n^{\left(S\right)}$ and $c_n^{\left(A\right)}$ are the time-dependent probability amplitudes of the symmetric and asymmetric dressed-states, respectively. For the initial state $|\psi \left(0\right)\rangle =\sum c_n|g,n\rangle$:

$$c_n^{\left(S\right)}\left(0\right)\approx c_n{\mu }_n\mathrm{and}{c}_n^{\left(A\right)}\left(0\right)\approx c_n\sqrt{1-{\mu }_n^2}.$$

(50)

From the Schrödinger equation, one obtains the differential equations

$$\begin{array}{ccc}\hfill {\dot{c}}_m^{\left(S\right)}& =& -2i\sum _{k=\Omega ,g}sin{\eta }_{k}t[W_{S_m,S_m}^{\left(k\right)}{c}_m^{\left(S\right)}+W_{S_m,A_m}^{\left(k\right)}{e}^{-it\left(A_m-S_m\right)}{c}_m^{\left(A\right)}\hfill \\ & & +\sum _{n\ne m}\left(W_{S_m,S_n}^{\left(k\right)}{e}^{-it\left(S_n-S_m\right)}{c}_n^{\left(S\right)}+W_{S_m,A_n}^{\left(k\right)}{e}^{-it\left(A_n-S_m\right)}{c}_n^{\left(A\right)}\right)]\hfill \end{array}$$

(51)

and

$$\begin{array}{ccc}\hfill {\dot{c}}_m^{\left(A\right)}& =& -2i\sum _{k=\Omega ,g}sin\left({\eta }_{k}t\right)[W_{A_m,A_m}^{\left(k\right)}{c}_m^{\left(A\right)}+W_{A_m,S_m}^{\left(k\right)}{e}^{-it\left(S_m-A_m\right)}{c}_m^{\left(S\right)}\hfill \\ & & +\sum _{n\ne m}\left(W_{A_m,S_n}^{\left(k\right)}{e}^{-it\left(S_n-A_m\right)}{c}_n^{\left(S\right)}+W_{A_m,A_n}^{\left(k\right)}{e}^{-it\left(A_n-A_m\right)}{c}_n^{\left(A\right)}\right)],\hfill \end{array}$$

(52)

where we define the matrix elements between the dressed-states $|D\rangle$ and $|B\rangle$ as

$$W_{D,B}^{\left(k\right)}\equiv \langle D|{\widehat{W}}^{\left(k\right)}|B\rangle ,k=\Omega ,g.$$

(53)

To qualitatively estimate the dynamics described by Equations (51) and (52), let us define new probability amplitudes $C_m^{\left(S\right)}$ and $C_m^{\left(A\right)}$:

$$C_m^{\left(S\right)}\left(t\right)=c_m^{\left(S\right)}\left(t\right)exp\left(-2i\sum _{p=\Omega ,g}{\displaystyle \frac{cos\left({\eta }_{p}t\right)-1}{{\eta }_p}}{W}_{S_m,S_m}^{\left(p\right)}\right)$$

(54)

and

$$C_m^{\left(A\right)}\left(t\right)=c_m^{\left(A\right)}\left(t\right)exp\left(-2i\sum _{p=\Omega ,g}{\displaystyle \frac{cos\left({\eta }_{p}t\right)-1}{{\eta }_p}}{W}_{A_m,A_m}^{\left(p\right)}\right).$$

(55)

Now, one has to solve the equations

$$\begin{array}{ccc}\hfill {\dot{C}}_m^{\left(S\right)}& =& -2i\sum _{k=\Omega ,g}{\sum }_n^{\prime }sin\left({\eta }_{k}t\right)\left\{W_{S_m,S_n}^{\left(k\right)}exp\left[2i\Lambda \left(S_n,S_m\right)\right]e^{-it\left(S_n-S_m\right)}{C}_n^{\left(S\right)}\right.\hfill \\ & & \left.+W_{S_m,A_n}^{\left(k\right)}exp\left[2i\Lambda \left(A_n,S_m\right)\right]e^{-it\left(A_n-S_m\right)}{C}_n^{\left(A\right)}\right\}\hfill \end{array}$$

(56)

and

$$\begin{array}{ccc}\hfill {\dot{C}}_m^{\left(A\right)}& =& -2i\sum _{k=\Omega ,g}{\sum }_n^{\prime }sin\left({\eta }_{k}t\right)\left\{W_{A_m,S_n}^{\left(k\right)}exp\left[2i\Lambda \left(S_n,A_m\right)\right]e^{-it\left(S_n-A_m\right)}{C}_n^{\left(S\right)}\right.\hfill \\ & & \left.+W_{A_m,A_n}^{\left(k\right)}exp\left[2i\Lambda \left(A_n,A_m\right)\right]e^{-it\left(A_n-A_m\right)}{C}_n^{\left(A\right)}\right\},\hfill \end{array}$$

(57)

where the prime indicates that the sum is taken over all the coefficients different from the coefficient on the left-hand side of the equation, and we define

$$\Lambda \left(D,B\right)\equiv \sum _{p=\Omega ,g}{\displaystyle \frac{cos\left({\eta }_{p}t\right)-1}{{\eta }_p}}\left(W_{D,D}^{\left(p\right)}-W_{B,B}^{\left(p\right)}\right).$$

(58)

If $|W_{D,D}^{\left(p\right)}|\ll {\eta }_p$ for all the relevant states $|D\rangle$ and modulation frequencies ${\eta }_p$, one can expand the exponentials containing $\Lambda$ in the power series and rewrite Equations (56) and (57) in the generic form

$${\dot{C}}_m=\sum _{n\ne m}\left({\theta }_{m,n,1}{e}^{i{\chi }_{m,n,1}t}+{\theta }_{m,n,2}{e}^{i{\chi }_{m,n,2}t}+\dots \right)C_n,$$

(59)

where $C_n$ stands for the probability amplitudes and ${\theta }_{m,n,l}$ and ${\chi }_{m,n,l}$ are some constant coefficients (whose exact form is not significant at this point). As discussed in Section 2, the major contribution to the dynamics comes from the “slowly” varying terms, for which $\left|{\chi }_{m,n,l}\right|\lesssim \left|{\theta }_{m,n,l}\right|$. Therefore, in the case considered, the relevant figure of merit is the ratio between the modulation frequency detuning from the resonant transition frequency and the corresponding transition rate. Hence, for the first-order effects, the figure of merit reads

$$f_k\left(D,B\right)=\left|{\displaystyle \frac{\left|D-B\right|-{\eta }_k}{W_{D,B}^{\left(k\right)}}}\right|,$$

(60)

where D and B are the eigenenergies of, respectively, the $|D\rangle$ and $|B\rangle$ states within the set $\left\{\right|S_n\rangle ,|A_m\rangle \}$. In broad terms, the modulation frequency ${\eta }_k=\left|D-B\right|$ causes the transition between the states $\left\{\right|D\rangle ,|B\rangle \}$ whenever $f_k\left(D,B\right)\lesssim 1$. For higher-order effects, one merely needs to replace $W_{D,B}^{\left(k\right)}$ by the appropriate coefficient in Equation (60).

In Figure 2, we consider the initial state $|\psi \left(0\right)\rangle =|g\rangle \otimes |\alpha \rangle$, with the atom in the ground state and the cavity in the coherent state, where we set $\alpha ={10}^3$ and use the parameters ${\Omega }_0$, ${\epsilon }_{\Omega }$, ${\check{\epsilon }}_g$, ${\eta }_{\Omega }$ and ${\eta }_g$ of the Hamiltonian ${\widehat{H}}_Q$ (45) as indicated. Figure 2a shows the initial values of $c_n=e^{-{\alpha }^2/2}{\alpha }^n/\sqrt{n!}$ along with $c_n^{\left(S\right)}$ and $c_n^{\left(A\right)}$ from Equation (50). One sees that, for $n>{\alpha }^2$, $c_n^{\left(S\right)}>c_n^{\left(A\right)}$, while for $n<{\alpha }^2$, $c_n^{\left(A\right)}>c_n^{\left(S\right)}$. Figure 2b presents the figures of merit (60) for pairs of different type dressed-states and Figure 2c presents the figures of merit (60) for pairs of same type dressed-states One sees that, for $n>{\alpha }^2$, one has $f_{\Omega }\left(S_n,A_{n+2}\right)<f_{\Omega }\left(A_n,S_{n+2}\right)$ and $f_g\left(S_n,A_{n+4}\right)<f_g\left(A_n,S_{n+4}\right)$, signifying that, for such values of n, the transitions $|S_n\rangle \to |A_{n+2}\rangle$ and $|S_n\rangle \to |A_{n+4}\rangle$ are more favorable than $|A_n\rangle \to |S_{n+2}\rangle$ and $|A_n\rangle \to |S_{n+4}\rangle$. In addition, right for such values of n, $c_n^{\left(S\right)}>c_{n+2}^{\left(A\right)},c_{n+4}^{\left(A\right)}$. For $n<{\alpha }^2$, there is the opposite scenario. Therefore, the external modulations with frequencies ${\eta }_{\Omega }=2\nu$ and ${\eta }_g=4\nu$ induce transitions that increase the number of photons compared to the unperturbed case. Since $f_{\Omega }\left(S_n,S_{n+2}\right)\approx f_{\Omega }\left(A_n,A_{n+2}\right)$ and $f_g\left(S_n,S_{n+4}\right)\approx f_g\left(A_n,A_{n+4}\right)$, while $c_n^{\left(S,A\right)}\approx c_{n+2}^{\left(S,A\right)}\approx c_{n+4}^{\left(S,A\right)}$ (as seen in Figure 2a,c), the transitions between the states $|A_n\rangle \to |A_{n+2}\rangle$, $|S_n\rangle \to |S_{n+2}\rangle$, $|A_n\rangle \to |A_{n+4}\rangle$, and $|S_n\rangle \to |S_{n+4}\rangle$ do not contribute significantly to the dynamics in this case.

Hence, Figure 2 provides a qualitative explanation of how resonant external modulations may alter the system dynamics for the initial coherent state with a relatively large amplitude. For the initial vacuum state, the dynamics of the photon generation due to parametric modulation has been extensively studied earlier in Refs. [30,31,34,46,47].

3.2. Approximate Equations for First-Order and Second-Order Effects

Let us now find the approximate relations that describe the system dynamics during the first-order and second-order resonances. For illustration purposes, let us first assume that ${\eta }_{\Omega }=2\nu$ and ${\eta }_g=0$. Neglecting the rapidly oscillating terms, for which $f_{\Omega }\left(D,B\right)\gg 1$, up to the first order in ${\epsilon }_{\Omega }$, one obtains the following approximate formulas:

$$\begin{array}{ccc}\hfill {\dot{C}}_m^{\left(S\right)}& \simeq & W_{S_m,S_{m-2}}^{\left(\Omega \right)}{e}^{it\left(\left|S_m-S_{m-2}\right|-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(S\right)}-W_{S_m,S_{m+2}}^{\left(\Omega \right)}{e}^{-it\left(\left|S_{m+2}-S_m\right|-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(S\right)}\hfill \\ & & +W_{S_m,A_{m-2}}^{\left(\Omega \right)}{e}^{it\left(\left|S_m-A_{m-2}\right|-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(A\right)}-W_{S_m,A_{m+2}}^{\left(\Omega \right)}{e}^{-it\left(\left|A_{m+2}-S_m\right|-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(A\right)}\hfill \end{array}$$

(61)

and

$$\begin{array}{ccc}\hfill {\dot{C}}_{A,m}& \simeq & W_{A_m,S_{m-2}}^{\left(\Omega \right)}{e}^{it\left(\left|A_m-S_{m-2}\right|-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(S\right)}-W_{A_m,S_{m+2}}^{\left(\Omega \right)}{e}^{-it\left(\left|S_{m+2}-A_m\right|-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(S\right)}\hfill \\ & & +W_{A_m,A_{m-2}}^{\left(\Omega \right)}{e}^{it\left(\left|A_m-A_{m-2}\right|-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(A\right)}-W_{A_m,A_{m+2}}^{\left(\Omega \right)}{e}^{-it\left(\left|A_{m+2}-A_m\right|-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(A\right)}.\hfill \end{array}$$

(62)

If ${\eta }_g\ne 0$, one just need to add the terms proportional to $W_{D,B}^{\left(g\right)}$ on the right-hand side (r.h.s.) of Equations (61) and (62). For instance, the additional modulation with ${\eta }_g=4\nu$ contributes with terms proportional to $C_{m\pm 4}^{\left(S\right)}$ and $C_{m\pm 4}^{\left(A\right)}$ on the r.h.s. of Equations (61) and (62), the latter corresponding to the coupling between the pairs of states $\left\{\right|S_m\rangle ,|A_m\rangle \}$ and $\left\{\right|S_{m\pm 4}\rangle ,|A_{m\pm 4}\rangle \}$.

To illustarte the second-order effects, let us consider the coupling among the states $\left\{|S_m\rangle ,|A_m\rangle ,|S_{m\pm 2}\rangle ,|A_{m\pm 2}\rangle \right\}$ for ${\eta }_{\Omega }+{\eta }_g=2\nu$. One finds:

$$\begin{array}{ccc}\hfill i{\dot{C}}_m^{\left(S\right)}& \simeq & \left[{\displaystyle \frac{W_{S_m,S_{m+2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{S_{m+2},S_{m+2}}^{\left(g\right)}-W_{S_m,S_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{S_m,S_{m+2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{S_{m+2},S_{m+2}}^{\left(\Omega \right)}-W_{S_m,S_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{-it\left(\left|S_{m+2}-S_m\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(S\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{S_m,A_{m+2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{A_{m+2},A_{m+2}}^{\left(g\right)}-W_{S_m,S_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{S_m,A_{m+2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{A_{m+2},A_{m+2}}^{\left(\Omega \right)}-W_{S_m,S_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{-it\left(\left|A_{m+2}-S_m\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_{m+2}^{\left(A\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{S_m,S_{m-2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{S_{m-2},S_{m-2}}^{\left(g\right)}-W_{S_m,S_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{S_m,S_{m-2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{S_{m-2},S_{m-2}}^{\left(\Omega \right)}-W_{S_m,S_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{it\left(\left|S_m-S_{m-2}\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(S\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{S_m,A_{m-2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{A_{m-2},A_{m-2}}^{\left(g\right)}-W_{S_m,S_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{S_m,A_{m-2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{A_{m-2},A_{m-2}}^{\left(\Omega \right)}-W_{S_m,S_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{it\left(\left|S_m-A_{m-2}\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_{m-2}^{\left(A\right)}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill i{\dot{C}}_m^{\left(A\right)}& \simeq & \left[{\displaystyle \frac{W_{A_m,S_{m+2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{S_{m+2},S_{m+2}}^{\left(g\right)}-W_{A_m,A_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{A_m,S_{m+2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{S_{m+2},S_{m+2}}^{\left(\Omega \right)}-W_{A_m,A_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{-it\left(\left|S_{m+2}-A_m\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_n^{\left(S\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{A_m,A_{m+2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{A_{m+2},A_{m+2}}^{\left(g\right)}-W_{A_m,A_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{A_m,A_{m+2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{A_{m+2},A_{m+2}}^{\left(\Omega \right)}-W_{A_m,A_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{-it\left(\left|A_{m+2}-A_m\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_n^{\left(A\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{A_m,S_{m-2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{S_{m-2},S_{m-2}}^{\left(g\right)}-W_{A_m,A_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{A_m,S_{m-2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{S_{m-2},S_{m-2}}^{\left(\Omega \right)}-W_{A_m,A_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{it\left(\left|A_m-S_{m-2}\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_n^{\left(S\right)}\hfill \\ & & +\left[{\displaystyle \frac{W_{A_m,A_{m-2}}^{\left(\Omega \right)}}{{\eta }_g}}\left(W_{A_{m-2},A_{m-2}}^{\left(g\right)}-W_{A_m,A_m}^{\left(g\right)}\right)+{\displaystyle \frac{W_{A_m,A_{m-2}}^{\left(g\right)}}{{\eta }_{\Omega }}}\left(W_{A_{m-2},A_{m-2}}^{\left(\Omega \right)}-W_{A_m,A_m}^{\left(\Omega \right)}\right)\right]\hfill \\ & & \times e^{it\left(\left|A_m-A_{m-2}\right|-{\eta }_g-{\eta }_{\Omega }\right)}{C}_n^{\left(A\right)}.\hfill \end{array}$$

The transition rates depend on the product ${\epsilon }_{\Omega }{\epsilon }_g$ so, under the realistic scenario of modulations with quite small amplitudes, the second-order effects are significantly weaker than those of the first-order [46].

4. Numeric Results

In this Section, we analyze the system dynamics for the initial state $|g\rangle \otimes |\alpha \rangle$ and two values of the initial average photon number: ${\alpha }^2=5\times {10}^3$ and ${\alpha }^2=3\times {10}^4$. To compare the predictions of the QRM in the semiclassical regime (i.e., for quite large values of $\alpha$) to the those of the SRM, we solve numerically the Schrödinger equation corresponding to Equation (45) using the Runge–Kutta–Verner fifth-order and sixth-order method (here, we use the routine IVPRK from the IMSL FORTRAN library [58]). We assume the truncated subspace

$$|\psi \left(t\right)\rangle =\sum _{n=N_{min}}^{N_{max}}\left(A_n|g,n\rangle +B_n|e,n\rangle \right),$$

(63)

where $N_{min}$ and $N_{max}$ are such that, for the initial coherent state, $\langle n|\alpha \rangle <{10}^{-18}$ for $n\notin \left[N_{min},N_{max}\right]$. Since we treat a dissipationless case with the exchange of a relatively small number of excitations between the atom and the field, Equation (63) serves as a suitable approximation for the purposes of this study.

4.1. Three-Photon Qubit Excitation Without Modulation

In Figure 3, we consider the three-photon atomic excitation in the absence of any external modulation, considering the parameters $\check{g}\left(t\right)$ and ${\Omega }_0$, corresponding to the exact three-photon resonance in the SRM (the comparison between the SRM and QRM for the standard one-photon resonance has been described in Refs. [24,25]). To illustrate the transition from the quantum to semiclassical behavior, we consider two values of the initial average photon number: ${\alpha }^2=5\times {10}^3$ and $3\times {10}^4$. Figure 3a shows the behavior of the atomic excitation probability, $P_e\left(t\right)=\mathrm{Tr}\left(|e\rangle \langle e|{\widehat{\rho }}_{\mathrm{tot}}\right)$, where ${\widehat{\rho }}_{\mathrm{tot}}=|\psi \left(t\right)\rangle \langle \psi \left(t\right)|$ is the density matrix of the total system. The black line depicts the analytic Formula (44) for the SRM. We verified that, in all cases studied in this paper, the analytic result by the Formula (44) is indistinguishable from the numeric solution of the Hamiltonian (1) so the numeric solution of the SRM is not visible. One finds that, for ${\alpha }^2=5\times {10}^3$, the classical and quantum cases behave quite differently, while for ${\alpha }^2=3\times {10}^4$ the behavior is almost identical, at least for the initial times. For long enough times, inevitably, the behavior of quantum and semiclassical solutions differ due to the collapse–revival phenomenon [22,59], which occurs more rapidly for smaller $\alpha$ values and is not captured by the SRM.

In Figure 3b–f, we analyze quantities existing only in the QRM framework. Figure 3b shows the variation in the average photon number, $〈N\left(t\right)〉-〈N\left(0\right)〉$, where $〈n〉=\mathrm{Tr}\left(\widehat{N}{\widehat{\rho }}_{\mathrm{tot}}\right)$, and illustrates that the atomic excitation is accompanied by a decrease in the average number of photons in the cavity: for ${\alpha }^2=3\times {10}^4$, the average number of photons decreases by almost three photons, while for ${\alpha }^2=5\times {10}^3$, the variation is smaller (of the order of $-2.5$) because not all the Fock states undergo transitions. To study the atom–field entanglement, Figure 3c,d presents the behavior of the von Neumann entropy, $S=-〈ln{\widehat{\rho }}_{\mathrm{sub}}〉$, and the linear entropy, $S_L=1-\mathrm{Tr}\left({\widehat{\rho }}_{\mathrm{sub}}^2\right)$ (trivially related to the purity $\mathrm{Tr}\left({\widehat{\rho }}_{\mathrm{sub}}^2\right)$). Here, ${\widehat{\rho }}_{\mathrm{sub}}$ is the reduced density operator of either of the subsystems, since, from the Schmidt decomposition for pure states, the subsystems have the same eigenvalues. Nonzero values of the entropies S or $S_L$ attest the atom–field entanglement. The excitation of the atom leads to the atom–field entanglement, and, for more intense (more “classical”) coherent states, the degree of entanglement is smaller for a given instant of time. Furthermore, when the atom returns to the ground state due to the re-emission of photons, the entropies only exhibit local minima, but do not vanish. Indeed, the entropies increase with time, undergoing oscillations, attaining the allowed maximum values $S=ln2$ and $S_L=1/2$ for large enough times (for the time interval in Figure 3, this only occurs for ${\alpha }^2=5\times {10}^3$), indicating the formation of maximally entangled states. Let us note that the approach considered neglects the damping and dephasing mechanisms, so the numeric results for large times, ${\gamma }_{max}t\gtrsim 1$ (where ${\gamma }_{max}$ denotes the largest of the damping and dephasing rates), are not experimentally significative.

To better understand how the atom alters the cavity field state, in Figure 3e and Figure 3f, we show, respectively, the probability of the system to be found in the state $|g,\alpha (t)\rangle$, $P_{\mathrm{in}}={\left|\langle \psi \left(t\right)|g,\alpha \left(t\right)\rangle \right|}^2$, and the probability of the cavity field to be found in the coherent state $\left|\alpha \right(t)\rangle$, $P_{|\alpha \rangle }=\mathrm{Tr}\left(\right|\alpha \left(t\right)\rangle \langle \alpha \left(t\right)\left|{\widehat{\rho }}_{\mathrm{tot}}\right)$, where $\alpha \left(t\right)=\alpha e^{-i\nu t}$ denotes the amplitude of the original coherent state $|\alpha \rangle$ evolving in the absence of atom–field interaction. From Figure 3e, one finds that, at the moments of the maximum excitation of the qubit, the probability $P_{\mathrm{in}}$ goes to zero, signifying that the system state is indeed driven away from the initial state. Moreover, $P_{|\alpha \rangle }$ is seen to decrease with time, exhibiting oscillations with the periodicity of the qubit oscillations. The decay of $P_{|\alpha \rangle }$ is considerably slower for larger values of $\alpha$, pointing that, for sufficiently intense coherent states, the assumption of the SRM that the field remains in the original coherent state becomes partially justified for initial times.

Figure 4 illustrates how the photon number distribution is modified due to the interaction with the atom. Figure 4a shows the photon number distribution of the original field state, $p_n\left(0\right)=e^{-{\alpha }^2}{\alpha }^{2n}/n!$, for ${\alpha }^2=3\times {10}^4$. Figure 4b–h show the modification of the photon number probability distribution, $\Delta p_n=p_n\left(t_{*}\right)-p_n\left(0\right)$, at the instant of time $\nu t_{*}=18,910$ where $p_n\left(t\right)=\mathrm{Tr}\left(|n\rangle \langle n|{\widehat{\rho }}_{\mathrm{tot}}\left(t\right)\right)$. One finds that the atomic excitation is accompanied by a relativlely small decrease in the photon number probability for $n>{\alpha }^2$ along with a similar increase for $n<{\alpha }^2$, what explains the decrease in the average number of photons.

4.2. Modulation: First-Order Effects

In Figure 5, we consider the three-photon atomic excitation using the same parameters as in Figure 3 supplemented by the $\Omega$-modulation. The period of the atomic Rabi oscillations decreases, and the dynamics is quite well described by the QRM already for ${\alpha }^2=5\times {10}^3$. However, now the photon number behavior is different, because the external modulation activates the additional transitions $|{\phi }_n\rangle \to |{\phi }_{n+2}\rangle$ (where $\phi$ stands for S and A), as one finds from Figure 5b.

In Figure 6, we add the g-modulation to the scenario of Figure 5. The qubit oscillations become even faster now, and the photon dynamics is different again, similar to that observed in Figure 5b, since the additional transitions $|{\phi }_n\rangle \to |{\phi }_{n+4}\rangle$ are being activated in this case.

4.3. Modulation: Second-Order Effects

Now, let us consider the second-order transitions taking place during the simultaneous $\Omega$- and g-modulations.

In Figure 7, we set the parameters ${\epsilon }_{\Omega }=0.02{\Omega }_0$, ${\epsilon }_g=-0.02g_0$, ${\eta }_{\Omega }=0.4\nu$, and ${\eta }_g=1.6\nu$ (with the other parameters of Figure 3 kept unchanged) so that to induce the transition $|{\phi }_n\rangle \to |{\phi }_{n+2}\rangle$ via the resonance ${\eta }_g+{\eta }_{\Omega }=2\nu$. In Figure 8, we use the same parameters as in Figure 7, except ${\eta }_g=2.4\nu$, set to induce to the resonance ${\eta }_g-{\eta }_{\Omega }=2\nu$. In both cases—of Figure 7 and Figure 8—the dynamics is qualitatively alike to that to conclude from Figure 5.

Finally, in Figure 9 and Figure 10, we set the parameters ${\epsilon }_{\Omega }=0.04{\Omega }_0$ and ${\epsilon }_g=-0.04g_0$ in order to induce weaker transitions $|{\phi }_n\rangle \to |{\phi }_{n+4}\rangle$ (other parameters being kept unchanged). In addition, in Figure 9, the parameters ${\eta }_{\Omega }=2.3\nu$ and ${\eta }_g=1.7\nu$, and in Figure 10, the parameters ${\eta }_{\Omega }=2.3\nu$ and ${\eta }_g=6.3\nu$ are changed so that to match the resonances ${\eta }_g\pm {\eta }_{\Omega }=4\nu =2\nu +r$. The behaviors observed in Figure 9 and Figure 10 are nearly identical, apart from negligibly small difference in the total transition rate $\Xi$ in Equations (42) and (43).

Note that, in all the cases considered here, the approximate analytic solution for $P_e$ in the SRM framework shows exceptionally good agreement with the predictions of the QRM, provided that ${\alpha }^2\gtrsim 3\times {10}^4$ and the evolution time $t\lesssim 3\times {10}^{-4}{\nu }^{-1}$.

5. Conclusions

In this paper, we considered the lossless semiclassical and quantum Rabi models in the regime of multiphoton resonances and external harmonic perturbations of atomic parameters. We considered the modulation of the atomic transition frequency, the atom–field coupling strength, or the simultaneous modulation of both. In particular, we studied the less common “second-order resonances”, where the sum of or difference in different modulation frequencies matches a resonance condition of the total system.

We obtained approximate analytical solutions for the SRM, which exhibited exceptionally good agreement with exact numerical results. For the QRM, we explained how the external modulation modifies the system dynamics near multiphoton resonances for coherent states with quite large amplitudes by analyzing the transition rates among the dressed-states.

As the main result of this paper, we numerically solved the QRM for initial coherent states with relatively large average photon numbers ($5\times {10}^3$ and $3\times {10}^4$) and exemplified how the behavior of the qubit population according to the QRM approaches the semiclassical picture, when the amplitude of the coherent state increases. However, at comparably longer times, the predictions of the QRM and SRM are found to start inevitably differ qualitatively due to the characteristic collapse–revival phenomenon in the quantum model.

Finally, we have illustrated how the cavity field is modified due to the interaction with the atom. We presented the behavior of the average photon number, von Neumann and linear entropies (which quantify the degree of entanglement), the probabilities of finding the system and the cavity field alone in their original states, and the deviation in the photon number probability distribution from the initial one. We showed that, after a complete oscillation of the atomic excitation probability, the cavity field does not return to its previous state and the atom–field entanglement grows as a function of time, attaining the allowed maximum value at long enough times.

We believe that this study contributes to a better understanding of the actual dynamics occurring during the interaction of an atom with a strong (i.e., “classical”) monochromatic coherent field. Although the semiclassical approach accurately describes the qubit dynamics at initial times, the quantum treatment explicitly reveals how photons are absorbed and re-emitted and how the cavity field state is continuously modified due to light–matter entanglement. Finally, the results obtained show that the semiclassical model inevitably fails after a characteristic time interval, which depends on the initial average photon number and the atom–field coupling parameter.