Effective and efficient resonant transitions in periodically modulated quantum systems

We analyse periodically modulated quantum systems with $SU(2)$ and $SU(1,1)$ symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.


I. INTRODUCTION
Effective transitions in time-dependent quantum systems have been extensively studied since the classical paper [1], later generalized in [2] and widely applied for the description of atomic dynamics in external fields [3]- [14]; and in more involved periodically perturbed quantum systems [15]- [21]. Effective transitions are described by operators that: i) be-  [1], [2] and the parametric quantum oscillator [22]. Even in these simplest systems, where the Hamiltonian is a linear form on the su(2) and su(1, 1) Lie algebras, it turns out that the general expressions for the effective interaction constants and the frequency shifts in the vicinity of each resonance, are not easy to obtain.
The situation becomes even more complicated when two coupled quantum systems are subjected to time-dependent periodic perturbations e.g., as in the quantum Rabi model with modulated coupling and/ or frequency. In these types of models, CR terms (in the absence of external fields) are responsible for several physical effects such as: multiphoton atom-field interactions in the Rabi model [23,24], an improvement of a qubit photodetector readout [25], the excitation of several atoms by a single photon [23,26], and several other effective processes now experimentally achievable in solid state circuit QED setups [27][28][29].
An additional periodic excitation makes the situation even richer, leading to e.g., the enhancement of CR interactions in the Rabi model [14], the generation of specific non-classical photon states [15], the emergence of non-linear spin-boson couplings [16], the appearance of quantum to classical phase-transitions [17], lasing with a single atom [18], simulation of the anisotropic Rabi model [19], or the dynamical Casimir effect [20].
One of the possibilities to construct a perturbation theory that unveils the effective resonant interactions, is the Lie transformation method [30]. Such a method consists in orderby-order elimination of CR terms by a specific set of transformations, which particular form directly follows from the algebraic structure of the original Hamiltonian [23]. The advantage of this approach consists in a rather simple and systematic procedure for obtaining the general form of effective resonance terms and the order of the corresponding effective coupling constants.
The aim of the present paper is to provide a systematic approach to the analysis of effective resonant transitions in quantum systems obeying the SU(2), SU (1,1) and H (1) symmetries with periodically modulated frequencies and/or coupling constants. We construct the Lie-type all-order perturbation theory allowing to determine the order of every possible resonance that may emerge in the effective Hamiltonians. We consider both single and coupled quantum systems and determine the efficient resonant transitions emerging as a combination of the time modulation and intrinsic non-linearities, especially relevant in interacting systems.
In Sec.II we outline the Lie transformation method in an example of a single periodically perturbed su(2)/su(1, 1) system and provide not only the order of the effective resonance terms [31] but the explicit iterative procedure for determining the effective interaction constants both for the weak and strong modulation. In Sec.III we analyze coupled timedependent quantum systems and discuss types of efficient resonances proper to different symmetries of the interacting systems.

II. SINGLE PERIODICALLY MODULATED QUANTUM SYSTEM
A. General settings Let us consider a quantum system described by the following time-dependent Hamiltonian (the case where only the frequency of the system is modulated is considered in Sec. 2.4), where the operators X ± , X 0 , satisfy the following commutation relations the signs ± correspond to the su(2) and su(1, 1) algebras respectively. In the interaction picture the Hamiltonian takes the form where the terms ∼ e i(ω−ν)t X + +h.c. become time independent if ω = ν and correspond to the principal resonance, while the counter-rotating (CR) terms ∼ cos (νt) X 0 and e i(ω+ν)t X + + h.c. rapidly oscillate for any relation between the frequencies.
The CR terms are neglected in the zero-order approximation when g 1 ∼ g 0 ≪ ω, ν (RWA) and the Hamiltonian (3)-(4) acquires a simple form In the opposite limit, g 1 ≪ g 0 ∼ ω, ν the zero-order approximation gives the diagonal characterized by a trivial dynamics. The CR terms in (3)-(4) lead to the emergence of non-trivial resonant transitions, not explicitly present in the original Hamiltonian. It is well known that in the case of H(1) symmetry, X + = a † , X − = a, X 0 = a † a, no resonances additional to the principal one ∼ ae i(ω−ν)t appear.
In order to develop the Lie-perturbation theory, that allows to describe all possible effective resonances both in the limits g 0,1 ≪ ω, ν and g 1 ≪ g 0 ∼ ω, ν, we introduce the Euclidean algebra operators E, E † , E 0 , obeying the commutation relations where E † E = EE † = I. Then, the Hamiltonian (1) can be put in a one-to-one correspondence with the following time-independent (Floquet) form [31] where the Euclidean operators (E, E † , E 0 ) describe a "classical" field interacting with the X-system. The time-dependent Hamiltonian (1) is recovered from (4) by transforming it to the frame rotating with the "classical frequency" ν, with a subsequent averaging over the eigenstates of E, E † operators (phase-states), E |φ = e −iφ |φ , where E 0 |n = n |n , and setting the initial phase φ = 0 without loss of generality. The CR We start with the most complicated limit g 0 ∼ g 1 ≪ ω, ν, when the contributions of diagonal and non-diagonal CR terms in (3)-(4) are of the same order. The CR terms appearing in the Floquet Hamiltonian (6) can be removed order -by -order by applying a set of Lie-type transformations according to the general scheme [23], [30] as shown in Appendix A.
The common feature of all of these transformations (small rotations) is their form where ε k ≪ 1 are some appropriate small parameters, under the condition that the Hamiltonian, which is transformed by (8), should contain the term Z The resonance expansion is obtained as a power series of the small parameters ε k and only contains terms that become time-independent in appropriate reference frames.
The resonance expansion (9) contains all possible effective resonant transitions that may emerge in (6) and indicates that such transitions happen only at (k + 1)ν =ω, where the case k = 0 corresponds to the principal resonance present in the original Hamiltonian.
In principle, the effective Hamiltonian, describing the system excitation in the vicinity of every particular resonance, should still be obtained from the resonance expansion (9) by removing all of the other resonances. However, as is proven in Appendix A, the elimination of all terms in (9) that become non-resonant under the condition ω ≈ (k + 1)ν does not change the leading order of the effective interaction constants g 1 ǫ ±k , thus the effective Hamiltonian has the form The effective X-system frequencyω ±(k+1) includes small shifts that should be taken into account up to the order of the coupling constant g 1 ǫ ±k , which determines the width of the corresponding resonance.
The evolution operator corresponding to the effective Hamiltonian (13) under the resonance conditionω ±(k+1) = (k + 1)ν is and can be disentangled in the standard way. Using (14), the evolution of any observable can be computed without returning to the time-dependent frame. This is achieved by transforming the corresponding X-system operator with (14) and averaging the result over the phase states (7). Strictly speaking, the evolution operator should still be transformed with all the transformations of the form (8) used for removing non-resonant terms in order to obtain the effective Hamiltonian (13). Nevertheless, since the transformations (8) are time independent, they lead only to small modifications of amplitudes and can be neglected in the first approximation. For instance, the evolution of X 0 operator in the resonancẽ ω ±(k+1) = (k + 1)ν can be easily found using (14), The frequency shifts for the lowest resonances can be easily found by a direct application of the transformations given in Appendix A. In order to obtainω ±(k+1) for the highest order resonances the following procedure can be applied: the effective Hamiltonian (13) where such that under the condition ω ≈ (k + 1)ν Taking into account the form of the perturbative action of transformations of the type (15) on the Hamiltonian (6), as discussed in Appendix A, we realize that every coefficient in (16) can be expanded in a series on some small parameters to be determined where the operators A ± and B ± can be easily found. Expanding A ± α ± , α † ± , β ± , β † ± and B ± α ± , α † ± , β ± , β † ± in series according to (17) and equaling to H ±(k+1) up to ε (k) one can, in principle, determine all needed x (m) ±j , m ≤ k and eventually findω ±(k+1) . However, such a procedure, although systematic, faces significant numerical difficulties and in practice is not very efficient.

Semi-classical Rabi model
The semi-classical Rabi model describes the evolution of an S-spin system in a periodic field and the corresponding time-dependent Hamiltonian has the form where S z,± are generators of the 2S + 1 dimensional representation of the su(2) algebra, The Floquet form of (18) is where E 0 , E † , E are the generators of the Euclidian algebra (5). The Hamiltonian (18) corresponds to g 0 = 0 in (4), so that h +(2k+1) = ǫ +(2k+1) = 0, and thus only odd resonances in (9) survive, where ǫ +2k are given in (10) andω is the shifted atomic frequency. In the vicinity of the resonance ω ≈ (2k + 1)ν, the effective Hamiltonian takes the form Interactions ) and effective couplings g +(2k+1) , k = 0, 1, 2 for the semiclassical Rabi model in terms of the small parameter ε = g/ω.
D. Effective Hamiltonian, The situation is less involved in the limit g 1 ≪ g 0 ∼ ω, ν if the expansion of the effective coupling constants is restricted by the leading order in the expansion of the effective coupling TABLE II: The frequency shifts δω −(k+1) =ω −(k+1) −ω +O(ε (k+1) ) and effective couplings g −(k+1) , k = 0, 1, 2 for the parametric quantum oscillator in terms of the small parameter ε = g/ω.
constants. Applying the transformation where ǫ = 2g 0 /ν, to the Hamiltonian (6) the following expression is obtained where J k (ǫ) are the Bessel functions. Removing all CR terms in (20) in the weak interaction limit g 1 ≪ ω, ν, results in the following resonance expansion where ε k = g 1 /(ω + kν) and In the vecinity of each resonance ω ≈ mν, the effective Hamiltonian takes the form where the frequency corrections, appear as a result of eliminating all the other transitions in (21).

E. Modulated quantum system with intensity dependent coupling
Our approach can be easily extended to Hamiltonians non-linear on the algebra generators when only the frequency of the system is modulated. Let us consider the following where f (X 0 ) is a function of the "diagonal" operator X 0 , in the strong modulation limit, ωγ ν. The interaction Hamiltonian in (26) describes a wide class of quantum optical systems as atom-photon interactions, parametric processes, etc [37]. It is clear, that only assisted transitions, i.e. induced by the external field, can be generated by (25) due to the presence of the term ωX 0 .
It is easy to find that the effective Hamiltonian in the vecinity of m-th resonace, ω ≈ mν, m < M, has the form whereĨ m (ǫ) = It is worth noting that in the weak modulation limit, ǫ ε ≪ 1, only the first resonance ω ≈ ν survives in the non-linear case, since the effective coupling is of order of the intensity dependent shift, Observe, that in the particular case, f (X 0 ) = 1, the resonant expansion for linear Hamiltonians is recovered. For instance, for the Dicke model in the strong modulation limit, ω 0 γ ≫ g, the resonace expansion has the following form

III. TWO PERIODICALLY MODULATED COUPLED QUANTUM SYSTEMS
The application of Lie transformations in order to determine the effective interaction constants, corresponding to effective resonances emerging in the case of two coupled and periodically modulated systems, becomes a quite involved task. We will analyze the situation where the coupling between the systems is significantly smaller than the bare frequencies of both systems. Thus, for consistency, all CR terms in the interaction Hamiltonian, appearing even in the absence of time-dependence, should be taken into account.
Let us consider two interacting quantum systems X and Y in dipole approximation, where the frequency of one of those is harmonically modulated. The corresponding Hamiltonian is where the operators describing X or Y systems can be from su (2), su(1, 1) or h(1) algebras.
The commutation relations have the following generic form where φ z (Z 0 ) = Z + Z − is a second degree polynomial for su(2) and su(1, 1) algebras, and is a first degree polynomial for the Heisenberg-Weyl algebra h(1); the discrete derivative is defined as where n ∈ Z.
The resonance expansion in the limit of strong modulation, ωγ ν and weak coupling, g ≪ ω 0,1 is obtained in Appendix B and has a generic form The intensity dependent frequency shift K(X 0 , Y 0 ) explicitly given in (B5)-(B8), leads to inhibition of higher-order transitions in X and Y systems. For the considered symmetries h(1), su(2) and su(1, 1), the effective interacion Hamiltonian H int has the following structure nk are given in Appendix B.
The form of the intensity dependent shift K(X 0 , Y 0 ) (B5)-(B8) depends on the degree of the polynomials φ x (X 0 ) and φ y (Y 0 ): i) both X and Y systems are described by the h(1) algebra. In this case K(X 0 , Y 0 ) is a linear form on X 0 , Y 0 .
ii) one of the systems is described by h(1) and another by su(2)/su(1, 1) algebra. In this case the leading term in K(X 0 , Y 0 ) is a second degree polynomial on X 0 and Y 0 , and the first correction is of a third degree one.
iii) The leading term in K(X 0 , Y 0 ) is a third degree polynomial if both systems have su(2)/su(1, 1) symmetry.

A. Modulated quantum parametric amplifier
Let us start with a non-degenerated parametric quantum amplifier with modulated interaction constant [35], described by In this case X + = a † , X − = a, X 0 = a † a, Y + = b † , Y − = b, Y 0 = b † b and no intensity dependent shift (B5)-(B8) appears in the resonant expansion since φ(a † a) = a † a and ∇φ(a † a) = 1. The resonance expansion (38) is reduced to the following where I(ǫ) is defined in (31) with ω = ω a + ω b and where the values of the summation index satisfying |ω a ± ω b | = nν are excluded.
For instance, in the case 2ω b ≈ ν y ω a = ω b /2, the effective Hamiltonian where describes the b-mode squeezing.
In Fig. 2 we compare the exact evolution of the average photon number in the b-mode, starting with the initial vacuum state |0 a , 0 b with the results of analytical calculations using the effective Hamiltonian (42),

Dicke model with modulated frequency
The dynamics of the quantum Dicke model, describing the interaction between an effective S-spin system and a single mode of a quantized field with harmonically modulated atomic frequency [14,15,18,20,33] is governed by the following Hamiltonian, where g ≪ ω 0,1 and, 0 < ǫ = ω 0 γ/ν < 1, which corresponds to (36)  the form: where ε (k) ∼ ε k , ε ∼ g/l.c.(ω, ν) ≪ 1, are some homogeneous polynomials on the Bessel functions J k (ǫ), 0 < ǫ = ω 0 γ/ν 1 (B9). In particular, i) the dynamic Stark shift term ∼ ε (1) k (ǫ)a † aS z suppressess all transitions between the field and the atomic system with an exchange of more than one excitation; ii) the atomic Kerr term ∼ ε (1) k (ǫ)S 2 z does not allow to efficiently absorb more than one excitation by the atomic system; iii) the field Kerr term ∼ ε (3) k (ǫ)(a † a) 2 makes the generation of more than four photons by the quantum field inefficient. Thus, the resonance expansion containing only efficient transitions takes the form

Non-symmetric excitation of an atomic system in a vacuum field
The resonant expansion (46) reveals the existence of effective processes consisting in the excitation of two atoms in the symmetric configuration, described by However, this process is rapidly suppressed by the atomic Kerr term ∼ S 2 z , which is of the first order on the small parameters. Thus, the symmetry of the atomic system should be broken in order to render the two-atom excitation process efficient.
Let us consider the following generalization of the Hamiltonian (45) to the two-atom case, which corresponding Flouquet form is and ω c = ω 1 + ω 2 with g 0 = g 1 = g 2 = 1.
For simplicity, we also assume that ǫ = g 0 /ν ≪ 1. In this case we obtain a resonant expansion, which up to the second order on the small parameters is given in Appendix B 1, Eq. (B10). Instead, the Kerr term Eq. (B10) contains the spin exchange operator s −1 s +2 + h.c., which can be taken out of resonance under appropriate frequency conditions.
For instance, choosing Imposing appropriate conditions on the frequencies all of the first order transitions in (B10) can be removed thus arriving at the following effective Hamiltonian for the initial vacuum field mode , and ε 2i = g i /ω c . The effective interaction constant is g ef f = ǫ (g 1 ε 12 + g 2 ε 11 − g 1 δ 12 − g 2 δ 11 ) .

IV. CONCLUSIONS
Even the simplest periodically modulated quantum systems exhibit a rich resonance structure captured by the expansion (9). This resonance expansion is obtained by a specific Lie-type perturbation theory where coupling constant is small with respect to the bare sys-tem´s frequencies both for weak and strong modulation amplitude. In the framework of this approach the order of each resonance, which determines the width of the related transition, and consequently the Rabi frequency of corresponding oscillations can be found. In case of single modulated linear systems we have been able to obtain the principal contribution to the effective interaction constant corresponding to each resonant term. Here we obtain the resonance expansion corresponding to the Flouquet Hamiltonian (6) by removing CR terms with adequate small Lie transformations and keeping only the principal order on g 0,1 ≪ ω, ν.
The CR term g 1 E † X + + h.c. can be exactly eliminated by the transformation where and T (x) = tan(x) for su(2) case, [X + , X − ] = 2X 0 , and T (x) = tanh(x) for the case where ∆ ±1 = 1 ± 4g 2 1 /(ω + ν) 2 , and ω ±1 = (ω + ν)∆ ±1 . The elimination of the CR term ∼ X + + X − only produces corrections to the terms already present in (A3), and thus can be neglected, since we are interested only in the principal order of the effective interaction constants. On the contrary, the elimination of the CR term ∼ E †2 X + + h.c. leads to the appearance of ∼ E 3 X 0 + h.c., ∼ E †4 X + + h.c. and ∼ E †5 X + + h.c., and in addition to the modification of the coefficient of ∼ E †3 X + + h.c..
An important observation should be made here about the order of f ±k and ε ±k where ε (k) , is a homogeneous polynomial of order k on some small parameters ε j ∼ g 0,1 /l.c(ω, ν) ≪ 1, being l.c.(ω, ν) a linear combination of ω and ν, where n j 1 + · · · + n js = k, and c j 1 ,...,js are real numbers.
Oncethe principal orders of CR terms ∼ E †k X + + h.c. are removed we arrive at the following form whereω is the system's modified frequency. The couplings h ±k are obtained from the following recurrence relations for k = 2, . . ., and for k = 1, . . .. The CR terms of the form ∼ E k X 0 + h.c commute with each other and can be removed altogether with the transformation obtaining the expansion The term EX + e ± k=1 δ ±k E †k + h.c does not contribute to the principal order of the effective coupling constants and can be neglected. Then, using the standard expansion where B k (a ±1 , a ±2 , . . .) are complete Bell polynomials [32], we finally obtain the required resonance expansion, which contains only the resonant terms i.e., terms that become timeindependent under appropriate relations between the frequencies ω and ν, where ǫ ±0 = 1, for m = 1, . . . and m = k + 1, where to the expansion (A7). This results in the following effective Hamiltonian describing the resonant transitionω ≈ (k + 1)ν implicitly present in the Hamiltonian (6), where∆ ±m = 1 ± 4g 2 1 ǫ 2 ±m−1 /(ω − mν) 2 . It can observed that the modified frequencyω is changed, but the principal order of the coupling constant corresponding to the resonant term ∼ E k+1 X + + h.c. in (A8) remains the same as in (A7). All of the other terms (A9)-(A10) generate contributions of smaller order.
First, by applying the transformation (27), with ǫ = ω 0 γ/ν 1 we obtain Now we consequtively apply the set of transformations where ε k = g/(ω 0 + ω 1 + kν), to the Hamiltonain (B2) in order to remove CR terms J k (ǫ) E †k X + Y + + h.c. in the weak coupling limit, g ≪ ω 0,1 . The transformed Hamiltonain contains, in addition to the resonant terms, CR contributions of the form: ∼ After eliminating all those CR terms we eventually arrive at a resonance expansion that contains a diagonal contribution K(X 0 , Y 0 ) as an important ingredient. The operator K(X 0 , Y 0 )depends non-linearly on X 0 and Y 0 , except for the case when both X and Y systems are described by h(1) algebra and can be interpreted as an intensity-dependent frequency shift. Up to third order on small parameters ε k ≪ 1, it has the form The intensity dependent frequency shift (B5)-(B8) automatically suppresses higher-order transitions leading to the excitation of X and Y systems. Since we consider only h(1), su (2) and su(1, 1) algebras, the maximum degree of Φ(X 0 , Y 0 ) on X 0 and Y 0 is four. Thus, the resonance expansion that includes only possible efficient transitions takes the form where the effective interacion Hamiltonian H int has the following structure x φ x (X 0 + 1)] + h.c.
Higher orders of the interaction Hamiltonians contain higher discrete derivatives of the structural functions φ x (X 0 ) and φ y (Y 0 ). This in particular, allows to determine all possible resonances when both X and Y systems are described by h(1) algebras.

Non-symmetric excitation of an atomic system in a vacuum
Applying an elimination procedure similar to that described in Appendix A to the Hamiltonian (48) we arrive at the following resonance expansion up to the second order on ǫ (ω i + g i ε 1i ) a † as zi − g 2 ε 11 (s −1 s +2 + h.c.) g i ε 1i ε 2i a 3 s +i + h.c.