Population trapping in a pair of periodically driven Rydberg atoms

We study the population trapping extensively in a periodically driven Rydberg dimer. The periodic modulation of the atom-light detuning effectively suppresses the Rabi couplings and, together with Rydberg-Rydberg interactions, leads to the state-dependent population trapping. We identify a simple, yet a general scheme to determine population trapping regions using driving induced resonances, the Floquet spectrum, and the inverse participation ratio. Contrary to the single atom case, we show that the population trapping in the two-atom setup may not necessarily be associated with level crossings in the Floquet spectrum. Further, we discuss under what criteria population trapping can be related to dynamical stabilization, taking specific initial states which include both product and the maximally entangled Bell states. The behavior of the entangled states is further characterized by the bipartite entanglement entropy.


I. INTRODUCTION
Periodic driving emerged as a tool to coherently manipulate the states of quantum systems. Consequently, Floquet systems exhibit a wide variety of unique phenomena related to non-equilbirium dynamics and many-body physics [1][2][3][4][5][6][7][8]. One such phenomenon, the dynamical stabilization, has been a subject of study in both classical and quantum mechanical systems. Dynamical stabilization is the stabilization of an otherwise dynamically unstable configuration of a system by periodically varying the system parameters in time. It has been first demonstrated using a classical pendulum, by Kapitza [9]. By periodically moving the point of suspension with high frequency, it is possible to stabilize the pendulum in its inverted position. In the quantum world, a closely analogous phenomenon to the Kapitza pendulum is the population trapping in a two-level atom [10][11][12]. The population can be trapped for a substantial time in an initial quantum state by periodically varying the atom-field detuning in time, even in cases where the state would otherwise evolve instantly into another state due to the Rabi coupling. Effectively, the periodic modulation may suppress the Rabi coupling depending on the modulation amplitude and frequency, leading to dynamical stabilization of the initial state. Dynamical stabilization has various applications for instance, in ion-trapping [13], mass spectrometers, and particle synchrotrons [14].
Other quantum phenomena related to population trapping are coherent destruction of tunneling in a double-well potential [15][16][17], the localization of a moving charged particle under the action of a time-periodic electric field [18,19], or the localization of a wavepacket in a periodic lattice due to periodic shaking of the lattice [20][21][22][23] or modulating the inter-particle interactions [24]. In interacting quantum gases, a Kapitza or a dynamically stabilized state has different manifestations, for instance, stabilizing a Bose-Einstein condensate [25] or a bright soliton [26,27] against collapse, freezing spin mixing dynamics in spinor condensates [28][29][30], stabilizing a classically unstable phase (π-mode) in a bosonic Josephson junction [31], or giving rise to unconventional or-dered phases that have no equilibrium counterparts [32]. Additionally, dynamical stabilization has been used to control the superfluid-Mott insulator quantum phase transition of bosons in an optical lattice [22].
Currently, ultracold Rydberg atoms are emerging as a promising platform for probing quantum many-body phenomena and implementing quantum information protocols [33,34]. The Rydberg blockade, in which strong Rydberg-Rydberg interactions (RRIs) suppress simultaneous excitation of two Rydberg atoms within a finite volume [35][36][37][38], and the breaking of the blockade (anti-blockade) [39][40][41] are of central utility for these applications. For two atoms, it has been predicted that through modulation induced resonances, one can engineer the parameter space for both Rydberg-blockade and anti-blockade [42][43][44]. Not only that, periodically driven Rydberg gases may provide control over the blockade and anti-blockade regimes, but also find applications such as to implement robust quantum gates [44][45][46] and accelerating the formation of dissipative entangled steady states [47]. To realize periodic driving in a Rydberg chain, one can modulate the light field that couples the ground to the Rydberg state. Another method is to apply additional radio-frequency or microwave fields that provide off-resonant couplings to other Rydberg states. Both approaches give rise to sidebands either in the driving field or in the atomic levels [48,49]. Experiments with interacting Rydberg atoms in oscillating electric fields [50] have, so far, mainly explored dipole-dipole interactions via Förster resonances [51][52][53][54]. Also, the dynamical stabilization of thermal Rydberg atoms against ionization, exposed to periodic kicks, has been a subject of intense study in the past, especially in classical-quantum correspondence [55,56]. In the latter case, the RRIs were not relevant.
In this paper, we study the population trapping comprehensively in a pair of periodically driven interacting Rydberg atoms. The population trapping may or may not indicate the dynamical stabilization. Only if the initial state is dynamically unstable in the absence of periodic driving can the population trapping be interpreted as the dynamical stabilization. The two-atom setup is one of the most common scenarios in Rydberg atom experiments [37,[57][58][59][60][61][62][63][64][65][66][67][68], and can be easily re-alizable using optical tweezers or microscopic optical traps [60]. The two-atom setup also constitutes the basic building block for quantum simulations and quantum information protocols [34]. We show that the presence of RRIs leads to statedependent population trapping in a two-atom setup. In particular, we look at how a specific set of states, including both product and maximally entangled Bell states, can be dynamically stabilized for significantly long periods of time. The product states we consider include both atoms are either in ground or Rydberg states. In a Rydberg setup, the Bell states have been demonstrated experimentally using various techniques [57,58,64,[69][70][71]. We identify a simple scheme for locating population trapping regions for any initial state, relying on driving induced resonances and the Floquet spectrum. We also introduce inverse participation ratio (IPR), calculated from the overlap of the initial state with the Floquet states, as an indicator of population trapping. Contrary to the previous conception from the single atom case, the population trapping or the dynamical stabilization in the two-atom setup is not necessarily related to the level crossings in the Floquet spectrum.
The paper is structured as follows. In Sec. II, we discuss the physical setup, the Hamiltonians including an effective time-independent one in the high-frequency limit, and techniques which we employ to study the emergence of Kapitza or dynamically stabilized states. The population trapping and the dynamical stabilization in a single two-level atom and the scheme for identifying dynamical stabilization is discussed in Sec. III. In Sec. IV, we extend the scheme to the two atom setup, and in particular, discuss the population trapping in both product and entangled states, including the driving induced resonances, the Floquet spectrum. Finally, we summarize in Sec. V.

II. SETUP, MODEL, AND TECHNIQUES
We consider a chain of two two-level atoms, in which the electronic ground state |g is coupled to a Rydberg state |e via a light field, the frequency of which is varied periodically in time t. The system is described in the frozen gas limit, after the rotating wave and dipole approximations, by the timedependent Hamiltonian ( = 1): whereσ ab = |a b| with a, b ∈ {e, g} includes both transition and projection operators,σ x =σ eg +σ ge , Ω is the Rabi frequency, ∆(t) = ∆ 0 + δ sin ωt is the time-dependent detuning with modulation amplitude δ > 0 and the modulation frequency ω. The Rydberg excited atoms interact via strong van der Waals interactions, V 0 = C 6 /r 6 , where C 6 is the interaction coefficient, and r is the separation between two Rydberg excitations [60]. The exact dynamics of the system is obtained by numerically solving the Schrödinger equation: i∂ψ(t)/∂t =Ĥ(t)ψ(t). To gain an insight, especially at high frequency (ω), we move to a rotating frame: ee ] with f (t) = (δ/ω) cos ωt − ∆ 0 t. The new Hamiltonian, H (t) =ÛĤÛ † − i ÛU † , after using the Jacobi-Anger expansion exp(±iz cos ωt) = ∞ m=−∞ J m (z) exp(±im[ωt + π/2]), is [42] where J m (α) is the mth order Bessel function with α = δ/ω and g m (t) = exp[i(mω−∆ 0 )t]. Comparing Eq. (1) with Eq. (2), we can see that the periodic detuning has effectively modified the Rabi coupling, thereby affecting the excitation dynamics. Further, using e ±iV 0 k jσ k ee t = k j σ k ee (e ±itV 0 − 1) + I , where I is the identity operator, we rewrite the Hamiltonian in Eq. (2) aŝ ee describes the correlated Rabi coupling [42,72]. The correlated Rabi process is analogous to the density assisted inter-band tunneling or density-dependent hopping for atoms in optical lattices [73,74]. Then, we obtain the effective time-independent Hamiltonian or the zeroth-order Floquet Hamiltonian [8,24,75] in the high-frequency limit as, H eff = 1/T T 0 dtĤ (t) where T = 2π/ω, and we get, , which expose the resonances in the driven system. Note that the effective Hamiltonian in Eq. (4) describes the long time dynamics in the limit ω {∆ 0 , V 0 }. The first term in Eq. (4) is a sum of singleparticle Hamiltonians, which drives the |g − |e transition, whereas the last term depends on the RRIs.
Floquet Theory.-According to the Floquet theorem, the time evolution operator associated with a time-periodic HamiltonianĤ(t) isÛ(t) = P(t)e −iĤ F t , where the Floquet HamiltonianĤ F is defined through the evolution operator over a full period T = 2π/ω, i.e.,Û(T ) = e −iĤ F T [8,15,[76][77][78]. The unitary operatorP(t) =P(t + T ) has the same periodicity as that of the Hamiltonian, and it becomes an identity operator at the instants t n = nT where n = 0, 1, 2, .... Further, we can write,Û(T ) = e −iĤ F T = k e −iθ k |φ k φ k |, where the Floquet states {|φ k (t) } are the eigenstates of the HamiltonianĤ F , and they form a complete set of square-integrable states. The Floquet mode |φ k (t) has the same periodicity as that of the HamiltonianĤ(t), and the quasi-energy k = θ k /T is defined up to a multiple of ω. Then, a general state of the system can be written as where the time-independent co-efficient c k gives the probability amplitude for finding the system in the Floquet state |φ k (t) .
It is worth mentioning that the population in the Floquet states remains preserved even if the actual state of the system or the Hamiltonian is changing over time. In that spirit, if the initial state coincides with one of the Floquet eigenstates, the population trapping takes place. The Floquet quasi-energies k and the eigenvectors are calculated numerically by obtaining the eigenvalues, λ k = exp(−i k T ) of the operatorÛ(T ) [79,80]. Further, to characterize the behavior of Rydberg excitation dynamics we define the inverse participation ratio (IPR), where p k = | φ k |I | 2 , is the projection of the initial state |I on the Floquet state |φ k and N is the number of atoms. If the initial state coincides with one of the Floquet states, IPR vanishes. Since, the Floquet state doesn't evolve in time, Π |I N = 0 may indicate the dynamical stabilization of the state |I . In that spirit, a smaller value of Π |I N indicates a slower transition rate from the state |I to other states.

III. A TWO-LEVEL ATOM (N = 1)
In the following, we briefly review the population trapping in a periodically driven single two-level atom. In particular, we discuss the criteria under which the population trapping can be identified as dynamical stabilization. For N = 1, the Hamiltonian in Eq. (2) takes the simplest form [10][11][12], In the high-frequency limit (ω Ω), the terms satisfying the resonance condition, n 1 ω = ∆ 0 , where n 1 = 0, 1, 2, ... becomes the most relevant in the summation of Eq. (7). The latter can also be seen from Eq. (4), in which for N = 1, only the first term with A m exits, and the poles of A m (ω) provide us the resonance criteria. Once the resonance condition is met, the dynamics shows coherent Rabi oscillations of the population between |g and |e . As shown in Figs. 1(a) and 1(b), resonances can be identified as either avoided crossings in the Floquet spectrum or peaks in the IPR (Π |g 1 ). At those peaks (Π |g 1 = 1), the Floquet states become an equal superposition of |g and |e . Far away from the avoided crossings or resonances, i.e., for ∆ 0 n 1 ω and ∆ 0 Ω, the periodic driving is ineffective. In that case, the Floquet states approximately become the eigenstates of the undriven Hamiltonian, H(t = 0), which are either |g or |e with a weak mixing be-tween them. Due to this, Π |g 1 decays to almost zero between the resonances.
Once the resonance condition n 1 ω = ∆ 0 is satisfied, the effective Rabi coupling between the states |g and |e is proportional to J n 1 (α). Therefore, at the Bessel zeros [J n 1 (α) = 0], the dynamics freezes and leads to population trapping. This can be further verified by looking at the quasi-energies k as a function of α at the resonances [see Fig. 1(c) for the case ∆ 0 = Ω]. The quasi-energies or the energy gap between them oscillates as a function of α, and crossings occur at the zeros of the Bessel function, J 1 (α), as shown in Fig. 1(d) for ∆ 0 = ω [80]. At those crossings, the Floquet states become purely |g and |e , which results in a vanishing Π |g 1 or Π |e 1 . Since the Floquet states do not evolve in time we have the population trapping in the states, |g or |e . Thus, a vanishing IPR (Π |I 1 ) at the driving induced resonances indicates the freezing of the state, |I . Now, we raise the following question: under which conditions do the population trapping in |I signifies dynamical stabilization of |I ? To answer this, we take the example of the classical Kapitza pendulum [9]. If the initial state |I is dynamically unstable in the absence of periodic driving (δ = 0), and can only be stabilized by the periodic driving, we term the resulting population trapping as the dynamical stabilization of |I . It is easy to see that the above situation holds only when n 1 = 0 for ω Ω. For n 1 0, in the high-frequency limit, the resonance condition demands a large value of ∆ 0 . Such a large value of ∆ 0 (off-resonant coupling) makes |g and |e stable even in the absence of periodic driving. Therefore, population trappings for n 1 > 0 cannot be interpreted as dynamical stabilization. In other words, the population trapping at the primary resonance (n 1 = 0), i.e., when J 0 (α) = 0 for ∆ 0 = 0 results in the phenomenon of dynamical stabilization. The results for the latter case are shown in Figs. 1(e) (quasi-energies) and 1(f) (IPR).
More extensive results for the IPR (Π |g 1 ) are summarized in Fig. 2. In the α − ∆ 0 plane, Π |g 1 exhibits pearl-chains along α axis at the resonances n 1 ω = ∆ 0 . The local minima along the chains provide the values of α at which population trapping takes place [or J n (α) = 0], and those along α at ∆ 0 = 0 are the points of dynamical stabilization. Between the stripes (along ∆ 0 axis), Π |g 1 vanishes due to the far off-resonant driving of the atom, as discussed above. Note that, in the case of a single two-level atom, the effect of a finite ω is apparent only for sufficiently small ω for which the crossings in Floquet energies start to deviates slightly from the Bessel zeros.
In short, we have seen that, by tuning the amplitude of modulation, the avoided crossings of the Floquet quasi-energies at the resonances of a periodically driven two-level atom become actual level crossings. At those crossing points, the population dynamics freezes, and also the IPR vanishes. Population trapping at the primary resonance is identified as the dynamical stabilization. Thus, we have a scheme to identify population trapping (including dynamical stabilization) of any initial state in two steps. First, identify resonances in which the initial state is involved, and second, vary the amplitude of modulation satisfying the resonance condition. In this section, we extend the above analysis to two interacting Rydberg atoms and discuss how RRIs affect the population trapping or dynamical stabilization. In particular, we uncover the conditions at which the states |gg , |ee , |+ = (|eg + |ge )/ √ 2, and |B = (|gg + |ee )/ √ 2 are dynamically stabilized. The first two states are product states, and the last two are the maximally entangled Bell states. If we restrict the dynamics to the symmetric states, we can truncate the basis to {|gg , |+ , |ee }. In this basis, the off-diagonal matrix elements ofĤ in Eq. (3) provide the time-dependent coupling strengths for |gg ↔ |+ and |+ ↔ |ee transitions, and they are respectively, and in general, Ω 1 Ω 2 . As a first step towards analyzing the population trapping or the dynamical stabilization, we discuss the resonances in the two-atom driven setup.

A. Resonances
At high ω, the most relevant terms in Eqs. (8) and (9) give us the resonance criteria n 1 ω = ∆ 0 (R1) and n 2 ω = ∆ 0 − V 0 (R2), and they are associated with the transitions |gg ↔ |+ and |+ ↔ |ee , respectively. The same resonance criteria are also obtained from the poles of the functions A m (ω) and B m (ω) in Eq. (4), respectively. For sufficiently large values of |V 0 − nω| with n = 0, ±1, ±2, ..., the resonances of the types R1 and R2 can be well separated along the ∆ 0 axis. On the other hand, the two resonance criteria can be simultaneously satisfied by taking V 0 = nω which gives n 1 = n 2 + n. Assuming the two types of resonances are not overlapping, and if the condition for R1 is fulfilled, we have the effective (time averaged) Rabi couplings, Ω 1 ≈ ΩJ n 1 (α)/ √ 2 and Ω 2 ≈ 0, for |gg ↔ |+ and |+ ↔ |ee transitions, respectively. Therefore, for the initial state |I = |gg , the system exhibits Rabi oscillations between |gg and |+ states [see Fig. 3(a) for n 1 = 1], which corresponds to the dynamics under the Rydberg blockade. In contrast, if |I = |ee , the dynamics freezes as shown in Fig. 3(b). The latter is expected, since the state |ee is far off-resonant from |+ due to large V 0 and hence, the periodic driving is non relevant. If the condition for R2 is satisfied, we have Ω 1 ≈ 0 and Ω 2 ≈ ΩJ n 2 (α)/ √ 2 which leads to the Rabi oscillations between |ee and |+ states and hardly any dynamics if the initial state is |gg , as shown in Figs. 3(c) and 3(d) for n 2 = −1, respectively. Apart from the resonances R1 and R2, there exists a third one n 3 ω = 2∆ 0 − V 0 (R3), which is not directly visible from Eqs. (8) and (9), but can be revealed using adiabatic impulse approximation [43]. R3 leads to resonant transitions between |gg and |ee .
In Figs. 4(b) and 4(c), we show the IPR (Π |I 2 ) as a function of ∆ 0 for the initial states |gg and |ee , respectively. The value of other parameters is the same as in Fig. 3. The peaks in Fig. 4(b) correspond to the resonances R1 and R3, labeled by n 1 and n 3 , respectively. Similarly, the peaks in Fig. 4(c) correspond to the resonances R2 and R3, labeled by n 2 and n 3 , respectively. As expected, the R3 resonances (marked by n 3 ) are very narrow in nature since |gg and |ee are not directly coupled. Between the resonant peaks, Π |I 2 vanishes due to the off-resonant driving as mentioned above. Each of these resonances can also be identified by the avoided crossings in the quasi-energies as shown in Fig. 4(a). To calculate k , we used the basis {|gg , |eg , |ge , |ee } and therefore we have four levels in the quasi-energy spectrum as shown in Fig. 4(a).
B. Dynamical stabilization of product states: |gg and |ee R1.-To identify the regions of dynamical stabilization we choose the primary resonance in each of R1, R2 and R3, i.e., n j∈1,2,3 = 0 and vary the amplitude of modulation, or equivalently α by keeping ω constant. First, we consider the resonance R1 with n 1 = 0 (∆ 0 = 0), for which the system exhibits Rabi oscillations between |gg and |+ states. Based on the results of the single atom case discussed in Sec. III, for the non-interacting case (V 0 = 0), the dynamical stabilization arises at the zeroes of the J 0 (α) at which the quasi-energies cross [dashed lines in Fig. 5(a)]. Since we have eliminated the asymmetric state |− = (|eg − |ge )/ √ 2 from the dynamics, there are only three relevant quasi-energy eigenvalues. The color bar in Fig. 5 quantifies the probability density of |gg in each of the Floquet states. A finite V 0 partially lifts the degeneracy of k at the crossings [see solid lines in Fig. 5(a)]. For small RRIs (V 0 Ω), the two types of resonances R1 and R2 are not well separated energetically and all the three states (|gg , |+ , |ee ) participate in the dynamics for any initial state. Therefore, we need to address the dynamical stabilization of both |gg and |ee when RRI is small.
For ∆ 0 = n 1 ω and V 0 /ω 1, we can expandĤ e f f in Eq. (4), in powers of V 0 /ω, and we get [42], Equation (10) implies that in the infinite-frequency limit (V 0 /ω → 0), the population trapping occurs at the zeros of the Bessel function J n 1 (α) irrespective of the initial state. In the particular case of n 1 = 0, we have the dynamical stabilization. For non-zero, but small values of V 0 /ω, the dominant interaction dependence comes from the second and third terms in Eq. (10), which are linear in V 0 /ω. For n 1 = 0, the third term in Eq. (10) vanishes, which means that the DS occurs at J 0 = 0. To verify this, we look at the IPRs, Π |gg 2 and Π |ee 2 as a function of α, see Figs. 5(d) and 5(e) for V 0 = 0.2Ω and ∆ 0 = 0 (green dashed lines). As expected, they both vanish when J 0 (α) = 0, indicating the dynamical stabilization of |gg and |ee . When n 1 0, and for α such that J n 1 (α) = 0, the third term in Eq. (10) also becomes vanishingly small and can be safely ignored. That means, for small values of V 0 /ω with R1 being satisfied, the population trapping always occurs at the zeros of the Bessel function J n 1 (α). The corrections from the terms involvingX in Eq. (10) may introduce a tiny shift in the value of α at which the DS occurs, especially for the case, |I = |ee . It can also be seen from Fig. 5(a) that the value of α for which the crossings in the Floquet spectrum occur is hardly affected by V 0 .
Keeping n 1 = 0 and as V 0 increases (excluding V 0 = nω where n is a non-zero positive integer), one quasi-energy level [top one in Figs. 5(a) and 5(b)] moves away from the other two, and eventually becomes purely |ee in the blockade regime (V 0 ≥ Ω), for any value of α [see Fig. 5(b)]. At that stage, the two lowest Floquet modes shown in Fig. 5(b) become superposition of |gg and |+ states, except at the level crossings. At the crossings, which occur at J 0 (α) = 0, the two Floquet states become purely |gg and |+ states. Note that, in the blockade regime, the state |ee is dynamically stable even in the absence of periodic driving. This makes Π |ee 2 ∼ 0 independent of α [see Fig. 5(e) for V 0 = 5Ω]. Therefore at the crossings of quasi-energy levels shown in Fig. 5(b), we have the dynamical stabilization of |gg . The latter can also be seen from Π |gg 2 , which vanishes at the crossings for sufficiently large V 0 as shown in Fig. 5(d). Π |gg 2 = 1 indicates the regimes of Rydberg blockade for which we have an effective two-level system consisting of |gg and |+ states.
R2.-Now we analyze the dynamical stabilization of |gg and |ee assuming the resonance criteria for R2 being satisfied, i.e., for n 2 ω = ∆ 0 −V 0 and that of R1 is not fulfilled. When the condition for R2 is met, the system exhibits Rabi oscillations between |+ and |ee states. Here, we restrict the analysis to the case for which n 2 = 0, i.e., for ∆ 0 = V 0 and look at how V 0 affects the dynamical stabilization of |ee and |gg . For small RRIs (V 0 Ω), the blockade is absent, and we need to address the dynamical stabilization of both |gg and |ee . Following the discussions we had for R1, it is easy to see that for V 0 Ω, and keeping V 0 = ∆ 0 , the dynamical stabilization of the states |ee and |gg is provided by the condition, J 0 (α) = 0. As V 0 (or equivalently ∆ 0 ) increases, the state |gg completely decouples from the dynamics (except when ∆ 0 = V 0 = nω). In that case, we need only to consider the dynamical stabilization of |ee , which is again provided by J 0 (α) = 0. In the particular case, when ∆ 0 = V 0 = nω, (both R1 and R2 are satisfied) the freezing of |gg is provided by J n (α) = 0, and the dynamical stabilization of the state |ee is given by J 0 (α) = 0. These results are identical to that for the case of R1 discussed above with ∆ 0 = 0, V 0 = nω, except that the role of |ee and |gg are interchanged. Therefore, Figs. 6(a) and 6(b) equivalently show Π |ee 2 and Π |gg 2 for V 0 = ∆ 0 , respectively. R3.-Now, we consider the case in which the resonance condition, R3: n 3 ω = 2∆ 0 − V 0 is satisfied whereas R1 and R2 are not satisfied. As mentioned earlier, the resonance condition for R3 cannot be extracted directly from the Hamiltonian in Eq. (3) or Eqs. (8) and (9) for the Rabi couplings, and hence, they do not provide us any hint on the dynamical stabilization. When R3 is satisfied, the system exhibits Rabi oscillations between |gg and |ee . Note that, for V 0 Ω, the resonances R1, R2, and R3 are not well separated, and all three states (|gg , |+ , |ee ) are relevant in the dynamics which leads to the population transfer between |gg and |ee via |+ state. For large values of V 0 , R3 gets well isolated from R1 and R2 along the ∆ 0 -axis. In that case, the population in |+ becomes negligible for sufficiently large values of V 0 /ω, except when ∆ 0 = nω. For small values of both RRIs and detuning compared to the driving frequency, i.e., for ∆ 0 /ω 1 and V 0 /ω 1 the effective Hamiltonian in Eq. (4) can be approximated to, To investigate the dynamical stabilization, we take n 3 = 0, i.e., for 2∆ 0 = V 0 . It can be seen from Eq. (11) that the dynamical stabilization of both |gg and |ee is provided by the zeros of J 0 (α). When n 3 = 0, the second term withX in Eq. (11) vanishes, and the dynamics is determined by the first term, which is proportional to J 0 (α). This result has been further verified by numerical calculations of the Schrödinger equation, using the crossings in the Floquet spectrum and IPR [see Fig. 7(a)]. We notice that, as V 0 increases, the dynamical stabilization demands both higher driving frequencies (ω) and larger modulation amplitudes (α). As shown in Fig. 7(a), for V 0 = 0.01Ω, we get the IPR identical to that of the noninteracting case as shown in Fig. 1(f The same as in (a), but for different ω and V 0 = 6Ω. In (c), we show the dynamics for the initial state |gg assuming R1 and R2 are met (n 1 = n 3 = 0) at the first root of J 0 (α), ω = 15Ω and V 0 = 6Ω. In (d), we show the same as in (c), except that the initial state is |ee and for the resonances R2 and R3, i.e. for n 2 = n 3 = 0.
creasing V 0 makes the minima broader, and in particular, those at small values of α get lifted from zero. That means, increasing V 0 /ω destroys dynamical stabilization at small values of α as seen for V 0 = 0.2Ω and V 0 = 1Ω in Fig. 7(a). In Fig. 7(b), we show IPR at a sufficiently large value of RRIs (V 0 = 6Ω) and for different ω, and we see that the sharp minima with vanishing IPR have disappeared completely and become smooth minima. These results can be understood from Eqs. (8) and (9). For sufficiently large V 0 , satisfying resonance condition 2∆ 0 = V 0 does not select a single Bessel function for the Rabi couplings, which hinders the dynamical stabilization. This strong dependence of V 0 on the dynamical stabilization under R3 resonance, is in high contrast with that of R1 and R2. To show that explicitly, we look at the dynamics at the first Bessel zero of J 0 (α) for the three resonances R1, R2, and R3 for sufficiently large V 0 [see Figs. 7(c) and 7(d)]. In Fig. 7(c), we show the dynamics for the initial state |gg , satisfying resonances R1 and R3, and in Fig. 7(c), the dynamics is shown for the initial state |ee with resonance conditions R2 and R3.
In both cases, we observe population dynamics in the initial state only for R3, as expected.

C. Dynamical stabilization of maximally entangled Bell states
In the following, we consider the dynamical stabilization of two class of Bell states: |+ and |B = (|gg + |ee )/ √ 2, and they both are maximally entangled two-qubit states. We also use the bipartite entanglement entropy to characterize the correlation or entanglement between the qubits. To quantify it, we label the qubits as A and B, and the entanglement entropy of subsystem A is obtained as S A = −Tr(ρ A log 2 ρ A ) = − k λ k log 2 λ k , where ρ A is the reduced density matrix of the subsystem A and λ k are the eigenvalues of ρ A . Both |+ and |B 2 have S A = 1, and under dynamical stabilization, we expect S A also to be stabilizing over time.
|+ state.-The state |+ is involved in two resonances: R1 and R2. For V 0 Ω, the resonances R1 and R2 are not entirely separable. The latter implies that the population from |+ state transfers almost equally to both |gg and |ee states for V 0 Ω. Following Eq. (10) for V 0 /ω 1, we can see that dynamical stabilization of |+ occurs when J 0 (α) = 0. For sufficiently large V0 (except when V 0 = nω), the resonances R1 and R2 can be well isolated from each other, and the dynamical stabilization of |+ is still determined by the zeros of J 0 (α) if either R1 or R2 is satisfied. When the condition for R1 is met, the Rydberg blockade, which prevents any transi-tion to |ee , helps to stabilize the |+ state dynamically when J 0 (α) = 0. On the other hand, the resonance condition R2 demands a large detuning, which prevents any population transfer from |+ to |gg , and that helps the dynamical stabilization in state |+ . Note that, when |+ is dynamically stabilized, one of the Floquet modes overlap completely with |+ , as we have discussed in Sec. IV B.
Keeping n 1 = 0 (R1 condition is satisfied) and for V 0 = nω with n being a non-zero integer, both R1 and R2 are satisfied simultaneously, and the dynamical stabilization of |+ requires both J 0 (α) = 0 and J −n (α) = 0. The latter criteria can never be satisfied with n 0, which prevents the dynamical stabilization of |+ when both R1 and R2 are satisfied simultaneously. This is in high contrast to the case of |gg as we have discussed in Sec. IV B. The above results are summarized in Fig. 8(a), in which we show the IPR, Π |+ 2 as a function of α and V 0 , in which the broken horizontal stripes correspond to the regions of dynamical stabilization of |+ state. The regions with Π |+ 2 = 1 indicate the blockade dynamics and those with Π |gg 2 = 2 indicate that all three states are very relevant in the dynamics. As expected, keeping the R2 condition with V 0 = ∆ 0 , also provides us the same results, and the only difference is that the regions with Π |+ 2 = 1 indicate the Rabi oscillations between |+ and |ee . Further, the general behavior of the time evolution of the entanglement entropy for the initial state |+ is shown in Fig. 7(b) for distinct IPR. As seen in Fig. 7(b), when Π |+ 2 = 0, we hardly find any dynamics in S A , which indicates that the correlation between the two atoms are preserved under the periodic driving. For the case in which Π |+ 2 = 1, the entropy S A undergoes periodic oscillation, and in this particular case, due to the Rabi oscillations between the states |+ and |gg .
|B state.-To discuss the dynamical stabilization of the Bell state |B , we need to consider the resonances, which includes either |gg or |ee , or both are included. We only comment on the latter case in which both |gg and |ee are involved in the resonance, and that happens when either R3 is satisfied or R1 and R2 are met simultaneously. As already mentioned, when the primary resonance of R3 is met (2∆ 0 = V 0 ), the system exhibits Rabi oscillations between |gg and |ee via |+ . Moreover, for large V 0 , the population in |+ can be neglected. In the latter case, |B becomes the eigenstate of the non-driven Hamiltonian and the question of dynamical stabilization become irrelavant. For small RRIs and V 0 /ω 1, the dynamical stabilization is given by the roots of J 0 (α), which can be easily seen from Eq. (11). On the other hand, satisfying R1 and R2 conditions simultaneously requires two different Bessel functions vanish at the same value of α, which is never possible, ruling out the possibility of dynamical stabilization of |B .

V. SUMMARY
In summary, we have studied the Dynamical stabilization of both product and entangled states in a Rydberg atom pair. The presence of Rydberg-Rydberg interactions leads to statedependent population trapping. As we have shown, unlike in the case of a single two-level atom, the population trapping or dynamical stabilization in two interacting Rydberg atoms may not be accompanied by level crossings in the Floquet spectrum. We have discussed the dynamical stabilization of a few selected states, including both product and entangled Bell states. The latter case offers a way to preserve entanglement or correlation between two qubits for sufficiently long times, with limitations arising only from the decoherent processes. The results we have discussed here on population trapping or dynamical stabilization are valid for a pair of any interacting two-level systems.
Our studies immediately raise the question of population trapping or dynamical stabilization in extended systems, i.e., beyond a pair of atoms. For instance, the effect of population trapping on the bipartite and tripartite entanglement of W-states and GHZ-states in a three atom setup can be studied. As the number of qubits or atoms increases, the Floquet spectrum's complexity also increases, which makes the scenario more intriguing.