Chaos and PT -symmetry breaking transitions in a driven, nonlinear dimer with balanced gain and loss
aa r X i v : . [ n li n . C D ] J u l Chaos and PT -symmetry breaking transitions in a driven, nonlinear dimer withbalanced gain and loss Shiguang Rong ∗ Department of Physics, Hunan University of Science & Technology, Xiangtan 411201, China andDepartment of Physics, Indiana University Purdue University Indianapolis (IUPUI), Indiana 46202, USA
Qiongtao Xie † College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
Yogesh N. Joglekar
Department of physics, Indiana University Purdue University Indianapolis (IUPUI), Indiana 46202, USA
Dynamics of a simple system, such as a two-state (dimer) model, are dramatically changed in thepresence of interactions and external driving, and the resultant unitary dynamics show both regularand chaotic regions. We investigate the non-unitary dynamics of such a dimer in the presence ofbalanced gain and loss for the two states, i.e. a PT symmetric dimer. We find that at low and highdriving frequencies, the PT -symmetric dimer motion continues to be regular, and the system is inthe PT -symmetric state. On that other hand, for intermediate driving frequency, the system showschaotic motion, and is usually in the PT -symmetry broken state. Our results elucidate the interplaybetween the PT -symmetry breaking transitions and regular-chaotic transitions in an experimentallyaccessible toy model. I. INTRODUCTION
In recent years, a special class of the non-hermitianHamiltonians that are invariant under combined opera-tions of parity and time-reversal ( PT ) has attracted ex-tensive interest. Starting from the seminal, theoreticalworks of Bender and co-workers three decades ago [1], ithas now become clear that open classical systems that arefaithfully described by PT -symmetric effective Hamilto-nians are of great experimental interest [2]. In quantummechanics, the requirement of a Hermitian Hamiltonianguarantees the existence of real eigenvalues and a com-plete set of orthogonal eigenvectors, and thus ensuresprobability conservation [3]. However, Bender and co-workers showed that many non-Hermitian Hamiltonianspossess entirely real spectra when the non-Hermiticityis below an energy scale determined by the Hermitianpart of the Hamiltonian; the spectrum changes into com-plex conjugate pairs when the gain-loss strength exceedsthis threshold, called the PT -symmetry breaking thresh-old [4, 5].This transition from purely real to complex-conjugate spectrum is called PT -symmetry breakingtransition.After the discovery of PT symmetric continuumHamiltonians [1], initial efforts were focused on devel-oping a self-consistent quantum theory, i.e. a com-plex extension of quantum mechanics [4], where a new,Hamiltonian-dependent inner product is defined to makethe eigenfunctions of the non-Hermitian Hamiltonian or-thogonal. These efforts led to significant insights intomathematical properties of pseudo-Hermitian operators ∗ Electronic mail: rong [email protected] † Electronic mail: [email protected] with real spectra that are self-adjoint with respect to anon-standard inner product [6–8]. Although the com-plex extension of quantum mechanics based upon non-Hermitian, PT symmetric Hamiltonians [4] is most likelyis not a fundamental theory [9], classical systems with PT -symmetric effective Hamiltonians have been widelyrealized. This mapping between Hamiltonians and classi-cal systems is primarily based on the equivalence betweenthe Schr¨odinger equation for a non-relativistic particleand the Maxwell equation for the slowly varying envelopeof the electric field in the paraxial approximation [10, 11].Resultant experimental examples include optical cou-plers [12, 13], microwave billiards [14], large-scale tem-poral lattices [15], microring single-mode lasers [16, 17],and coupled resonators [18]. Even in the non-interacting,linear regime, these systems show a wide variety of fasci-nating behaviors [19, 20] that are absent in their Hermi-tian counterparts. When a static, PT -symmetric Hamil-tonian is replaced by a time-periodic one, the result is arich phase diagram of PT symmetric and PT broken re-gions that are determined by the strength of the gain-lossterm and the frequency of its temporal modulation [21–23]. In particular, in the neighborhood of specific mod-ulation frequencies, the PT threshold is driven down tozero, thus facilitating the PT breaking transitions at van-ishingly small non-Hermiticity.The theoretical studies of non-interacting Hamiltoni-ans have been extended to the nonlinear systems [24, 25]such as dipolar Bose-Einstein condensates [26], and non-linear optical [27, 28] and optomechanical [29] structures.They predict that solitons in the strongly coupled, PT symmetric systems are stable [30, 31], nonlinear quan-tum Zeno effects can be observed [32], and chaos in the PT symmetric systems [33]. Experimental studies haveobserved optical solitons in PT symmetric synthetic lat-tices [34].In this paper, we present the dynamics of a PT sym-metric, nonlinear dimer under periodic driving field. Atlow modulation frequency, the structure of the instan-taneous fixed points is analyzed. At intermediate fre-quency, we find that the dimer exhibits chaotic behaviorand PT symmetry broken phase. The condition for theoccurrence of chaos is obtained by means of the Mel-nikov method. At high modulation frequencies, we findthat the driving force strongly renormalizes the dimercoupling constant and therefore strongly modifies its dy-namical behavior. II. DRIVEN DIMER WITH STATIC GAIN ANDLOSS
Consider a PT -symmetric dimer described by the non-linear Schr¨odinger equation (¯ h = 1) i dψ dt = − ν ψ −
12 [ ε ( t ) + iγ + λ | ψ | − | ψ | | ψ | + | ψ | ] ψ , (1) i dψ dt = − ν ψ + 12 [ ε ( t ) + iγ + λ | ψ | − | ψ | | ψ | + | ψ | ] ψ . (2)Here ν is the coupling between the two sites of the dimerthat, in the absence of all other terms, leads to Rabioscillations. We use it to set the frequency and timescale in the rest of the paper. γ > ε ( t ) isan external, Floquet drive characterized by an amplitude A and frequency ω , i.e. ε ( t ) = A sin( ωt ) = − ε ( − t ) , and λ is the antisymmetric nonlinearity that is proportionalto the dimer polarization. When λ = 0, this systemshows a sequence of PT -symmetric transitions when thedriving frequency ω is changed at a fixed value of drivingstrength A [35–37]. It is convenient to cast Eqs.(1)-(2) ina matrix form by defining an effective, state-dependentHamiltonian as i∂ t | ψ ( t ) i = H eff ( t ) | ψ ( t ) i , where the 2 × H eff ( t ) = − ν σ x −
12 [ ε ( t ) + iγ − λZ ( t )] σ z , (3)where σ x , σ z are the standard Pauli matrices, and Z ( t ) ≡h ψ ( t ) | σ z | ψ ( t ) i / h ψ ( t ) | ψ ( t ) i is the time-dependent polar-ization of the dimer. Note that, by definition, the dimerpolarization is a bounded, real function, − ≤ Z ( t ) ≤ H eff commutes withthe PT operator where P = σ x exchanges the first siteof the dimer with the second, and T : t → − t, i → − i isthe time-reversal operation. Therefore, the driven, non-linear, gain-loss dimer model is PT symmetric.Due to the presence of the gain and loss terms ± iγ ,the evolution generated by the time-dependent effectiveHamiltonian H eff is not unitary. The norm of the dimerwavefunction n ( t ) ≡ h ψ ( t ) | ψ ( t ) i = | ψ ( t ) | + | ψ ( t ) | is not conserved and depends on the dimer polarization,i.e. dn/dt = − γn ( t ) Z ( t ) = 0. Therefore, starting from anormalized initial state | ψ (0) i ) on the Bloch sphere, thetime-evolved state does not remain confined to it. Weseparate this motion of | ψ ( t ) i into the dynamics of itsnorm n ( t ) and its projection onto the Bloch sphere atevery instance of time, and consider a new, scaled state | ψ ′ ( t ) i ≡ | ψ ( t ) i p h ψ ( t ) | ψ ( t ) i = | ψ ( t ) i p n ( t ) , (4)that is normalized at all times and satisfies a differentialequation i∂ t | ψ ′ ( t ) i = H ′ eff | ψ ′ ( t ) i with a scaled effectiveHamiltonian H ′ eff given by H ′ eff = + iγ Z ( t ) − ν σ x −
12 [ ε ( t ) + iγ − λZ ( t )] σ z , (5)We would like to emphasize that although H ′ eff is notHermitian, it conserves the norm of the state, i.e. ∂ t h ψ ′ ( t ) | ψ ′ ( t ) i = 0. The equation of motion for thescaled state | ψ ′ ( t ) i = ( ψ ′ , ψ ′ ) T is simplified by express-ing it in terms of the polarization Z ( t ) and two phases0 ≤ θ ( t ) , θ ( t ) ≤ π , ψ ′ , ( t ) ≡ r ± Z ( t )2 e iθ , ( t ) . (6)The Schr¨odinger equation i∂ t | ψ ′ ( t ) i = H ′ eff ( t ) | ψ ′ ( t ) i forthe driven, nonlinear, PT -symmetric dimer then be-comes ∂ t Z = − ν p − Z sin θ − γ (1 − Z ) , (7) ∂ t θ = − ε ( t ) + λZ + ν Z √ − Z cos θ, (8)where θ ( t ) ≡ θ ( t ) − θ ( t ) is the phase difference be-tween the wavefunction weights on the two sites. In thefollowing section, we investigate the properties of thesetwo equations for a sinusoidal drive ε ( t ) = A sin( ωt ) as afunction of the amplitude A and the driving frequency ω across the entire frequency range. III. LOW-FREQUENCY DRIVING: ω/ν ≪ In the static-driving limit, we consider the instanta-neous fixed points of Eqs.(7)-(8). The steady-state dimerpolarization Z f satisfies the equation( γ + λ ) Z f − ελZ f + ( ν + ε − γ − λ ) Z f +2 ελZ f − ε = 0 . (9)In the absence of an external drive, ε = 0, Eq.(9) be-comes a biquadratic and its fixed points are analyticallyobtained, Z f = ( , , ± s − ν γ + λ ) . (10) t z (a) t θ (b) −0.4 −0.2 0 0.2 0.4−0.2−0.100.1 z θ (c) −0.1 −0.05 0 0.05 0.1−0.2−0.100.1 z θ (d) t (e) | ψ | | ψ | nnZ FIG. 1. Regular behavior of a PT symmetric dimer with γ/ν = 0 . λ/ν = 2, driven by an externalforce with moderate amplitude A/ν = 0 . ω/ν = 0 .
1. Temporal evolution of the dimer polarization Z ( t )(a) and phase difference θ ( t ) (b) shows oscillatory behavior with two distinct frequency components ν and ω = ν/
10. Thephase-space portrait (c) and the Poincare section (d) in the Z − θ plane show that the dimer evolution is regular. (e) Thetemporal evolution of norm of the state n ( t ), the wavefunction weights on the two sites | ψ , | , and their difference showsperiodic behavior that is characteristic of a PT symmetric phase. The corresponding, doubly-degenerate, steady-statephase difference values θ f are given bycos θ f = (cid:26) − λν , − λ √ λ + ν (cid:27) . (11)This static-limit analysis can be extended to the case ε = 0 in a straight forward manner. At low frequen-cies, we numerically obtain the temporal evolution of thedimer by solving Eqs.(7)-(8) with given initial conditions.Figure 1 shows typical results for a dimer with gain-lossstrength γ/ν = 0 . λ/ν = 2,driven by a moderate strength, low-frequency externalforce with A/ν = 0 . ω/ν = 0 .
1. The dimer isinitially in a symmetric state, | ψ (0) i = (1 , T / √ Z (0) = 0 = θ (0). Figure 1a,b show thatthe dimer polarization Z ( t ) and phase difference θ ( t ) os-cillate periodically with two dominant frequency com-ponents, namely ν and ω . Figure 1c,d are the phase-space portrait and Poincare sections of the correspondingtime evolution. They show that the dimer has a regu-lar motion when driven at low frequencies. In Fig. 1e,we plot the norm n ( t ) of the state vector | ψ ( t ) i , theweights | ψ , ( t ) | on the two sites, and their difference h ψ ( t ) | σ z | ψ ( t ) i = n ( t ) Z ( t ). The norm n ( t ) shows oscilla-tory behavior that is a hallmark of the PT symmetricphase, with two frequency scales, as do the other quanti-ties. These results show that in the low driving frequencyregime, the driven, nonlinear, PT symmetric dimer is inthe PT symmetric phase and has regular, non-chaoticdynamics. Due to strong nonlinearity in this system, an exhaus-tive or analytical investigation of the PT symmetricphase diagram as a function of the four dimensionlessparameters, i.e. the amplitude of the drive A/ν , thefrequency of the drive ω/ν , the strength of nonlinearity λ/ν , and the gain-loss strength γ/ν is virtually impossi-ble. Therefore, in this work, we focus primarily on theregime with strong nonlinearity and moderate externaldrive. We note that in the linear case ( λ = 0), this char-acterization can be analytically carried out [35, 36], andin the static case, a variety of integrable PT symmetricdimer models have been studied in the literature [38–41]. IV. HIGH-FREQUENCY REGIME: ω/ν ≫ When the external drive frequency is much larger thanthe Rabi frequency of the dimer, we can separate thedynamics of | ψ ′ ( t ) i into the high-frequency contributionand a slowly varying field, i.e. | ψ ′ ( t ) i = exp " ± i σ z Z tπ/ dt ′ ε ( t ′ ) | ϕ ( t ) i . (12)Note that the dimer polarization is solely determined bythe slowly varying field, Z ( t ) = h ψ ′ | σ z | ψ ′ i = h ϕ | σ z | ϕ i .The equation of motion for the slowly varying field isgiven by i∂ t | ϕ ( t ) i = iγ Z | ϕ ( t ) i −
12 [ iγ − λZ ] σ z | ϕ ( t ) i t z (a) t θ (b) −0.05 0 0.05−0.4−0.200.2 z θ (c) −0.04 −0.02 0 0.02 0.04−0.4−0.200.2 z θ (d) t (e) | ψ | | ψ | nnZ FIG. 2. Quasi-static, regular motion of the PT symmetric dimer with γ/ν = 0 . λ/ν = 2, drivenby an external force with moderate amplitude A/ν = 0 . ω/ν = 10. The effective coupling amplitude isessentially equal to the bare coupling, i.e. ν eff /ν = J ( A/ω ) = 1. The temporal evolution of the dimer polarization Z ( t ) (a)and phase difference θ ( t ) (b) shows oscillatory behavior with two frequencies ν and ω = 10 ν . The phase space portrait (c) andthe Poincare section (d) in the Z − θ plane show that the dimer dynamics is regular. (e) The temporal evolution of norm of thestate n ( t ), the wavefunction weights on the two sites | ψ , | , and their difference shows periodic behavior that is characteristicof a PT symmetric phase. −
12 [ ν ( t ) σ + + ν ∗ ( t ) σ − ] | ϕ ( t ) i , (13)where ν ( t ) = ν exp( iA cos ωt/ω ) is the complex couplingamplitude and σ ± = ( σ x ± iσ y ) /
2. In the high-frequencylimit, ignoring the higher-order Bessel functions in theexpansion of the exponential-cosine gives the followingeffective, time-independent Hamiltonian for the ϕ field, H ′ s = + iγ Z − νJ ( A/ω )2 σ x −
12 [ iγ − λZ ] σ z . (14)Comparison of Eq.(14) with Eq.(5) shows that, to firstapproximation, the high-frequency Hamiltonian behaveslike a nonlinear dimer with no drive ( ε = 0) and a smaller,effective coupling ν → ν eff = νJ ( A/ω ). Since ν eff canbe made arbitrarily small or driven to zero by appropri-ate choice of A/ω , the system can be driven from PT symmetric phase to PT broken phase and back [35].In Fig. 2 we show the typical results for the tempo-ral evolution of such a dimer with the same initial stateas in Fig. 1, but a high driving frequency ω/ν = 10.We note that for these parameters, the effective cou-pling is essentially equal to the bare coupling, i.e. ν eff = νJ ( A/ω ) = ν . Figure 2a, b show that the dimer po-larization and phase differences both evolve periodicallywith two distinct frequencies, and Fig. 2c, d show thatthe dynamics are regular when the dimer is driven bya high-frequency field. The temporal evolution of thenorm of the state n ( t ), on-site weights, and their differ-ences, Fig. 2e, show oscillatory behavior consistent withthe PT symmetric phase. In particular, the smallness of oscillations in Fig. 2e shows that the system is deep inthe PT symmetric phase, or almost Hermitian. V. INTERMEDIATE-FREQUENCY DRIVING
At last, we consider the most interesting case, namelythat of intermediate modulation frequencies ω ∼ ν .When there is no external drive or gain/loss potential,the resulting system of equations ∂ t Z ( t ) = − ν p − Z sin θ ≡ f , (15) ∂ t θ ( t ) = λZ + ν Z √ − Z cos θ ≡ f , (16)is integrable [42]. The equations (15)-(16) can be writtenin the Hamiltonian form [43] with Hamiltonian H = λ Z − ν p − Z cos θ. (17)By introducing an effective potential V ( Z ) = Z ( λ / − λH/ λ Z /
8, we find that the field Z ( t ) satisfies New-ton’s second law, i.e. ∂ tt Z = − ∂V /∂Z , and its totalenergy is given by E = 12 (cid:18) ∂Z∂t (cid:19) + V ( Z ) = 12 (cid:0) ν − H (cid:1) . (18)The separatrix solution of Eqs.(15)-(16), i.e. the E = 0case, is given by Z s ( t ) = 2 aλ sech( at ) , (19) ω / νγ / ν A / ν (a) drive frequency ω / ν c hao s t h r e s ho l d A c / ν Regular regionChaotic region(b)
FIG. 3. Chaos threshold amplitude A c as a function of the gain-loss strength γ and drive frequency ω , for a PT symmetricdimer with strong nonlinearity, λ/ν = 2. (a) A c ( γ, ω ) shows a linear-in- γ dependence, Eq.(25), and a marked minimum atintermediate frequencies ω/ν ∼
1. (b) A cut at γ/ν = 0 . sin [ θ s ( t )] = λ a sech ( at )tanh ( at ) λ − a sech ( at ) , (20)where a = √ νλ − ν . When this exact solution is subjectto a periodic perturbation, its stability analysis is carriedout via Melnikov method [44]. We start with the equationfor the two-component vector z ≡ ( Z, θ ) T , ∂ t z ( t ) = f ( t ) + ǫ g ( t ) , (21)where f ≡ ( f , f ) T , ǫ ≪ g ( t ) ≡ ( g , g ) T with g ( t ) = − γ p − Z /ǫ, (22) g ( t ) = − A sin ωt/ǫ. (23)The Melnikov function M ( t ) can be analytically calcu-lated from Eqs.(19)-(20), and gives M ( t ) = − Z ∞−∞ f ( t, t ) ∧ g ( t, t ) dt, = − ( γF + F A cos ωt ) /ǫ. (24)where f ∧ g ≡ ( f g − f g ), the two auxiliary func-tions are F = π ( λ − ν )(4 λ − ν ) / λ and F =(2 πω/λ )sech (cid:0) πω/ √ λν − ν (cid:1) , and t is the initial time.The condition for the onset of classical chaos is given by M ( t ) = 0, or, equivalently, the drive amplitude mustexceed the chaos threshold, i.e. A ≥ A c where A c = γ F ( λ, ν ) F ( ω, λ, ν ) . (25)Figure 3a shows the dimensionless chaos threshold am-plitude A c /ν as a function of the gain-loss strength γ andthe drive frequency ω , at a strong nonlinearity λ/ν = 2. In the absence of gain and loss, γ = 0, the chaos thresh-old is zero, meaning the nonlinear, driven dimer is al-ways in the chaotic region. As the gain-loss strength isincreased, the chaos threshold is large at both high andlow frequencies, but is suppressed at intermediate driv-ing frequencies. Figure 3b is a vertical cut at γ/ν = 0 . γ dependence of the chaos threshold, seenin Fig. 3a, is a reflection of Eq.(25). t z (a) t θ −0.2 −0.1 0 0.1 0.2−0.4−0.200.2 z θ (c) −0.2 −0.1 0 0.1 0.2−0.4−0.200.2 z θ (d)(b) t (e) | ψ | | ψ | nnZ FIG. 4. Regular, periodic motion of a PT dimer with asmall gain-loss strength, γ/ν = 0 . λ/ν = 2, driven by a field with an intermediate frequency ω/ν = 1, but a small amplitude A/ν = 0 .
2. The temporalevolution of Z ( t ) (a) and θ ( t ) shows periodic behavior withmultiple, closely space frequency components. The phase-space portrait (c) and the Poincare section (d) show that thedimer has a regular motion although it is close to boundarywith the chaotic region. (e) The norm of the state n ( t ), on-site weights, and their difference show oscillatory behaviorindicative of the PT symmetric phase. To demonstrate regular and chaotic behavior at inter-mediate driving frequencies ( ω/ν = 1), we consider the PT symmetric dimer with a small and large drive ampli- t z (a) t θ (b) −1 −0.5 0 0.5 1−70−60−50−40−30−20 z θ (c) −1 −0.5 0 0.5 1−70−60−50−40−30−20 z θ (d) t | ψ | | ψ | nnZ (e) FIG. 5. Chaotic behavior of a PT dimer with a small gain-loss strength, γ/ν = 0 . λ/ν = 2, driven bya field with an intermediate frequency ω/ν = 1 and a large amplitude A/ν = 1 . Z ( t ) (a) and θ ( t ) show aperiodic, oscillatorybehavior. The phase-space portrait (c) and the Poincare section (d) show that the dimer is in the chaotic regime. (e) the normof the state n ( t ), on-site weights, and their difference show increasing-in-time behavior indicative of a PT symmetry brokenphase. tudes respectively, and obtain its time evolution with thesame initial conditions as before. Thus, λ/ν = 2 , γ/ν =0 . Z (0) = 0 = θ (0). Figure 4 shows the resultsfor a small A/ν = 0 .
2, when the dimer is in the regu-lar region, but close to the boundary with the chaoticregion (Fig. 3b). Panels (a)-(b) show that the dimer po-larization and phase difference have an oscillatory be-havior with closely spaced multiple frequencies ∼ ν . Thephase-space portrait, Fig. 4c, and the Poincare section,Fig. 4d, show the regular motion of the dimer. Figure 4eshows that the norm of the state n ( t ), the on-site weights | ψ , ( t ) | , and their difference all oscillate, signaling a PT symmetric phase.When the same dimer is driven with a stronger ampli-tude, A/ν = 1 .
5, it is in the chaotic regime. Figure 5shows the results for its temporal dynamics. The polar-ization Z ( t ), Fig. 5a, and phase difference θ ( t ), Fig. 5b,do not show periodic behavior. The phase-space portrait,Fig. 5c, and the Poincare section, Fig. 5d, clearly showthat the dimer is in deep, chaotic regime. Fig. 5e showsthat the norm of the state n ( t ), on-site weights | ψ , ( t ) | ,and their difference oscillate in an aperiodic manner and,on average, increase with time. This increase shows thatthe dimer is in the PT symmetry broken phase. VI. CHAOS VS. PT SYMMETRY BREAKING
Numerical results from Sec. III, IV, V suggest a closecorrelation between regular dynamics and the PT sym-metric phase, or chaotic dynamics and the PT symme-try broken phase. While chaos can be inferred from the phase-space portraits in the Z − θ plane, the definitionof PT symmetry breaking in the absence of a static orFloquet Hamiltonian requires care. A key consequenceof real spectrum of a non-Hermitian, PT symmetric sys-tem is bounded, oscillatory, non-unitary time evolutionfor the norm n ( t ) of a state; a complex-conjugate spec-trum, on the other hand, necessarily leads to unbounded-in-time dynamics for the norm. We use this criterion todefine whether the dimer is in the PT symmetric phaseor a PT symmetry broken phase.For comparing the region of PT symmetry breakingand chaos, we set λ/ν = 2 and use the same initial stateto obtain the PT symmetry breaking transition, thresh-old drive strength A P T /ν , Fig. 6a, and the chaos thresh-old drive strength A c /ν , Fig. 6b, while restricting to driveamplitudes 0 ≤ A/ν ≤
6; if the threshold is not foundin this range, we assign it a zero value. The similaritybetween the two plots enforces the strong correlation be-tween chaotic regime and PT -symmetry broken regimein this system. The primary exception is the Hermitiancase, i.e. γ = 0, where the system is always in the PT symmetric phase, but can be driven to chaotic domainfrom a regular domain by increasing the drive amplitude.A detailed understanding of the differences between thechaotic/regular domains and PT -symmetric/ PT -brokenregions remains a topic of future work. VII. CONCLUSION
We have investigated the dynamics of a PT symmet-ric, nonlinear dimer under a periodic driving field. We drive frequency ω / ν ga i n − l o ss s t r eng t h γ / ν (a) A PT drive frequency ω / ν n − l o ss s t r eng t h γ / ν (b) A c FIG. 6. The thresholds of PT symmetry breaking and chaos for A/ν in interval [0,6] which are represented by color. (a)Thethreshold of PT symmetry breaking; (b) The threshold of chaotic region. The zero value means that it can’t find PT symmetrybreaking or chaos when A/ν in interval [0,6]. have shown that while the motion remains regular andthe dimer remains in the PT symmetric phase at lowand high driving frequencies, both chaotic and regulardimer behavior emerges at intermediate driving frequen-cies. It is worth noting that the PT -symmetry brokenregions are strongly correlated with the chaotic region inthe parameter space.Controlling the dynamics through external fields is asubject of great interest, particularly so in open systemswith balanced gain and loss. Our results provide new in-sights into the interplay between PT -symmetry breakingtransitions, that are caused by increasing the gain-lossstrength, and the interaction-driven chaotic-to-regulartransitions. They also raise the following key question:are these two, seemingly distinct transitions independent or not? How does one characterize the regions where thetransitions are correlated or anti-correlated? A detailedinvestigation of these questions will deepen our under-standing of interacting PT symmetric systems. ACKNOWLEDGMENTS