NMR studies of quantum chaos in a two-qubit kicked top
V R Krithika, V S Anjusha, Udaysinh T. Bhosale, T. S. Mahesh
NNMR studies of quantum chaos in a two-qubit kicked top
V R Krithika, V S Anjusha, Udaysinh T. Bhosale, and T. S. Mahesh ∗ Department of Physics and NMR Research Center,Indian Institute of Science Education and Research, Pune 411008, India
Quantum chaotic kicked top model is implemented experimentally in a two qubit system com-prising of a pair of spin-1/2 nuclei using Nuclear Magnetic Resonance techniques. The essentialnonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were re-alized using RF pulses. After a variable number of kicks, quantum state tomography was employedto reconstruct the single-qubit density matrices using which we could extract various measures suchas von Neumann entropies, Husimi distributions, and Lyapunov exponents. These measures en-abled the study of correspondence with classical phase space as well as to probe distinct features ofquantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top.
Keywords: Chaos, kicked top, entanglement, von Neumann entropy
I. INTRODUCTION
Classical chaos is an extensively studied field in physicstheoretically and experimentally. Classically chaotic sys-tems are deterministic systems which show sensitivity toinitial conditions, rendering the long-time predictions un-certain [1]. Chaos has far-reaching applications not justin physics, but in many diverse fields like biology, chem-istry, engineering, etc. [1–3].The correspondence principle states that classical me-chanics is a limiting case of quantum mechanics, in whichcase, there must be some signatures of chaos in the quan-tum regime. A direct extension of chaos to quantummechanics is however not straightforward since (i) quan-tum dynamics is governed by the Schr¨odinger equation,which is linear and preserves the overlap of states, and(ii) we can not define trajectories for quantum systemsdue to the constraint imposed by the uncertainty princi-ple in precisely locating a point in the phase space of thesystem. A major focus of the field of quantum chaos isto understand the correspondence between quantum andclassical evolutions in chaotic systems, and it has beena subject of theoretical as well as experimental interest[4–14].Study of quantum chaos is not only important fromthe perspective of understanding fundamental physics,but also for applications in building operable quantumcomputers since it was shown that the presence of quan-tum chaos in a system can affect the functionality of aquantum computer [15]. Since classical measures of chaoscannot be extended to the quantum domain, quantumchaos has to be quantified using inherent quantum me-chanical properties. Signatures of quantum chaos havebeen studied using various quantities like entanglement[16–20], Lyapunov exponents and Husimi probability dis-tributions [21], the dynamics of quantum discord [22],level statistics of chaotic Hamiltonians [23, 24], the dy-namics of open quantum systems undergoing continuous ∗ [email protected] quantum measurement [25], etc. The kicked top model isa classic example for studying chaos. It shows regular tochaotic behavior as a function of a parameter, has beenstudied theoretically [16–19, 23, 26–29], and has been re-alized experimentally in various systems like laser-cooledcesium atoms [8] and superconducting circuits [11]. Re-cently, the kicked top consisting of just two qubits, whichis in a deep quantum regime, has also been studied the-oretically in detail [18, 30]. For two qubits the model isexactly solvable and the same is shown to hold valid forthree and four qubits as well [28]. In this work, we inves-tigate quantum chaos in a two-qubit system formed by apair of spin-1/2 nuclei using Nuclear Magnetic Resonance(NMR) techniques. NMR has been a successful testbedto understand quantum correlations and implement vari-ous quantum information processing tasks [31, 32]. NMRoffers advantages in terms of long coherence times, pre-cise controllability of quantum dynamics, and efficientmeasurement of output states. We study quantum kickedtop (QKT) using spin-spin interaction between two nu-clear spins as the nonlinear evolution and intermittentRF pulses as linear kicks. After a variable number ofkicks, we characterize the final state via quantum statetomography (QST). Signatures of the corresponding clas-sical phase space are found in the time averaged vonNeumann entropy. Further analysis using Lyapunov ex-ponents and Husimi probability distributions also revealgood classical-quantum correspondence.The paper is organized as follows. Sec. II introducesthe theory of kicked top model. The NMR implementa-tion, results of the experiments, and their analysis alongwith numerical simulations are presented in Sec. III andfinal conclusions are given in Sec. IV. II. QUANTUM KICKED TOP
We now describe using a pair of qubits to simulatea QKT [8, 23] described by the piece-wise Hamiltonianconsisting of periodic x -kicks of width ∆ and strength p separated by nonlinear evolutions each of an interval a r X i v : . [ qu a n t - ph ] O c t FIG. 1. The linear kicks and nonlinear evolutions for simu-lating a QKT using two qubits. τ (cid:29) ∆ (see Fig. 1) H ( t ) = p J x , for t ∈ (cid:20) nτ − ∆2 , nτ + ∆2 (cid:21) and, H ( t ) = k jτ J z otherwise . (1)Here, J = [ J x , J y , J z ] is the total angular momentumvector and [ nτ − ∆2 , nτ + ∆2 ] describes the time lapse of the n th kick. The value of (cid:126) has been set to 1. The nonlinearterm describes a torsion about the z axis wherein k is thechaoticity parameter. Here, j is the total spin-quantumnumber. The advantage of this model is that, for a given j it corresponds to 2 j number of qubits and thus variousquantum correlations can be studied [22]. In the caseof two-qubits considered here, j = 1. Further, we set p ∆ = π/ U kick = e − i π J x , U NL = e − i kJ z j & U QKT = U NL U kick . (2)The overall unitary U QKT is applied repeatedly to re-alize the desired number of kicks. In the Heisenberg pic-ture, the evolution of angular momentum operator forany time step is given by [19] J (cid:48) = U † QKT J U QKT . (3)The x and y components of J can be recast in the formof raising and lowering operators as J x = ( J + + J − ) / J y = ( J + − J − ) / i which can then be studied in J z eigenbasis {| m (cid:105)} following the ladder equations J + | m (cid:105) = c m | m + 1 (cid:105) and J − | m (cid:105) = d m | m − (cid:105) . First let us consider the evolution of J + component since J − will simply be its Hermitian conjugate (H.c): J (cid:48) + = U † QKT J + U QKT = U † kick U † NL J + U NL U kick . (4)Computing the action of U NL on the operator in | m (cid:105) ba- sis, (cid:104) m | U † NL J + U NL | n (cid:105) = (cid:104) m | e i k j J z J + e − i k j J z | n (cid:105) = exp (cid:26) i k j ( m − n ) (cid:27) (cid:104) m | J + | n (cid:105) = exp (cid:26) i k j ( m − n ) (cid:27) c n δ m,n +1 = (cid:40) e i kj ( n + ) c n if m = n + 1 , (cid:104) m | J + e i kj ( J z + ) | n (cid:105) , (5)so that U † NL J + U NL = J + e i kj ( J z + ) . (6)Next, the kick Floquet unitary has to be applied on theabove operator. The action of kick unitary is to bringabout a clockwise rotation about the x axis by an angleof π /2 giving U † kick ( J x , J y , J z ) U kick = ( J x , − J z , J y ), sothat J (cid:48) + = U † QKT J + U QKT = U † kick J + e i kj ( J z + ) U kick = ( J x − iJ z ) e i kj ( J y + ) . (7)The post-iteration transverse components of the angularmomentum are thus, J (cid:48) x = J (cid:48) + + J (cid:48)− (cid:104) ( J x − iJ z ) e i kj ( J y + ) + H . c (cid:105) and J (cid:48) y = J (cid:48) + − J (cid:48)− i = 12 i (cid:104) ( J x − iJ z ) e i kj ( J y + ) − H . c (cid:105) . (8)For J z operator, the non-linear Floquet unitary bringsabout no change since it commutes with J z . The onlyevolution of J z is caused by π /2 rotation about x axisgiving U † kick J z U kick = J y , so that J (cid:48) z = J y . (9)In the next section, we will study the classical limit ofthe kicked top. Classical limit of kicked top
It is insightful to first look into the classical features ofthe kicked top in the semiclassical limit i.e., j → ∞ . Ex-pressing X = J x /j , Y = J y /j , and Z = J z /j , one obtains[ X, Y ] = iZ/j , which vanishes in the large j limit.Underthis classical limit, Eqs. 8 and 9 lead to the iterative map[19, 33] X (cid:48) = X cos( kY ) + Z sin( kY ) Y (cid:48) = X sin( kY ) − Z cos( kY ) Z (cid:48) = Y. (10) / (a) k = 0.5 (b) k = 2.5 / / (c) k = - 2.5 / (d) k = 2 +2.5 FIG. 2. Classical trajectories of the kicked top in | θ, φ (cid:105) phasespace for various values of chaoticity parameter k as indicated.The points chosen for detailed analysis are marked by blackdots. These components can be parametrized in terms ofthe polar coordinates ( θ, φ ) as X = sin θ cos φ , Y =sin θ sin φ , and Z = cos θ . As the value of the chaotic-ity parameter k increases, the phase-space undergoes atransition from a regular to a combination of regular andchaotic regions before becoming predominantly chaoticfor large values of k . The classical phase space is shownfor different values of k in Fig.2. The trivial fixed points( θ, φ ) = ( π/ ,
0) and ( π/ , π ) can be seen in Fig.2(a)which becomes unstable at k = 2. At k = 2 new fixedpoints are born and they move away as k is increased asshown in Fig. 2(b). For large value of k > III. QKT WITH A PAIR OF NMR QUBITS
Consider a pair of qubits with spin angular-momentumoperators I and I respectively. By denoting the total z -component J z = I z + I z , we obtain the nonlinear term J z = / / I z I z . Dropping identities which onlyintroduce global phases, we may realize the nonlinear dy-namics using the bilinear term, which naturally occurs ina pair of weakly coupled on-resonant heteronuclear NMRqubits. In a doubly rotating frame, the Hamiltonian isgiven by H I = 2 π I I z I z , (11)where I is the indirect spin-spin interaction strength.Comparing the above Hamiltonian with the second termof Eq. 1, we obtain k = 2 π I τ .In our experiments, the pair of qubits was formed by F and P spins of sodium fluorophosphate dissolvedin D O (5.3 mg in 600 µ l). All experiments wereperformed on a 500 MHz Bruker NMR spectrometerat ambient temperatures and on-resonant conditions. P PPS Initial State Kicked Top Detection F PFG θ 𝑦 φ 𝑧 𝜋2 𝑥 𝜋 𝑥
4J 14J τ τ TomoG G 𝜋2 𝑦 θ 𝑦 φ 𝑧 𝜋6 𝑥 𝜋3 −𝑦 FIG. 3. NMR pulse sequence for simulating a QKT in a two-qubit system. Here G and G correspond to pulsed-field-gradients. The indirect spin-spin coupling I = 868 Hz. Theexperiments consisted of two parts, i.e., preparation ofthe initial state ( θ , φ ), followed by simulating a QKTas illustrated in Fig. 1.In NMR systems, owing to the low nuclear polarizationat an ambient temperature T and a typical Zeeman field B , the initial thermal equilibrium state ρ = (cid:15) ρ. is highly mixed with the low purity factor (cid:15) ∼ − . Theuniform background population represented by identityremains invariant under the unitary evolution, while thetraceless deviation density matrix ˜ ρ = I z + I z evolvesand captures all the interesting dynamics.In the following we utilize {| (cid:105) , | (cid:105)} eigenbasis of I z as the computational basis. We first prepare the | (cid:105) pseudopure state (PPS) by transforming ˜ ρ into I z + I z +2 I z I z using a pair of pulses followed by a pulsed fieldgradient (PFG) as shown in Fig. III [34].Subsequently, a θ y rotation followed by a φ z rotationas shown in Fig. III initialize each of the qubits along aspin coherent state, | θ, φ (cid:105) = cos ( θ/ | (cid:105) + e iφ sin ( θ/ | (cid:105) , (12)on the Bloch sphere, analogous to the classical case. Thelatter pulses for different φ angles were generated by anoptimal control technique [35]. The resulting state is ρ θ,φ ≈ (cid:16) − (cid:15) (cid:17) (cid:15) | θ, φ (cid:105)(cid:104) θ, φ | . (13)We now apply kicks via radio-frequency ( π/ x pulseswith Hamiltonian H rf = π
2∆ ( I x + I x ) , (14)where the pulse duration ∆ (cid:28) τ = k/ (2 π J), the dura-
Kick number A v e r age F i de li t y FIG. 4. Average fidelity of the experimental states for variouskick-numbers. The errorbars indicate one standard deviationof distribution. tion of nonlinear evolution corresponding to the chaotic-ity parameter k (see Fig.1). Thus in our experiment, U kick = exp( − iH rf ∆), U NL = exp( − iH J τ ), and U QKT = U NL U kick (see Eq. 2)We applied U QKT for up to n times and estimated the F reduced density operator ρ n = Tr P (cid:104) U n QKT ρ θ,φ U n † QKT (cid:105) using single-qubit pure-phase QST. It consists of follow-ing three NMR experiments: (i) A PFG to destroy all thecoherences followed by ( π/ y pulse to obtain the diago-nal elements; (ii) ( π/ − y pulse followed by a PFG and( π/ y pulse to obtain real part of off-diagonal coherenceelement; (iii) ( π/ − x pulse followed by PFG and ( π/ y pulse to obtain the imaginary part of the off-diagonalcoherence element. This way one obtains a pure-phaseNMR signal which can be easily quantified without anyfurther numerical processing. We estimated the fidelity F (˜ ρ n , ˜ ρ th n ) = Tr (cid:2) ˜ ρ n ˜ ρ th n (cid:3)(cid:112) Tr [˜ ρ n ] Tr [(˜ ρ th n ) ] (15)of the experimental deviation state ˜ ρ n with the theoret-ical deviation state ˜ ρ th n for all initialization points ( θ, φ )and for all k values. The average fidelity versus kick num-ber displayed in Fig. 4 indicates high fidelities of above0.95 upto six kicks and above 0.8 upto 8 kicks. A. Probing quantum chaos viavon Neumann entropy
It has been observed that a kicked top in a state corre-sponding to a classically chaotic region results in a higherentanglement production [19]. Since the degree of entan-glement can be quantified by the von Neumann entropy S ( ρ n ) = − (cid:88) λ ± (cid:54) =0 λ ± log λ ± (16)of the reduced density operator ρ n with eigenvalues λ ± =(1 ± (cid:15)α ) / ± α are eigenvalues of the traceless devi-ation part. Since in low purity conditions, the von Neu- mann entropy is close to unity and displays very low con-trast between regular and chaotic regions, we define ann th -kick order parameter s n = 1 − n (cid:80) nm =1 S ( ρ m ) (cid:15) (17)which extracts information from the deviation part andhence is a convenient measure of chaos.We carried out four sets of experiments for chaoticityparameter k ∈ { . , . , π − . , π + 2 . } . In each case,we performed experimental QST and estimated the or-der parameter s n for the number n of kicks ranging from1 to 8. The contours in Fig. 5 display the experimen-tal order parameter s n for various values of n as well as k . The color background is provided to compare the ex-perimental contours with numerically simulated values oforder parameter. In each case, we have also calculatedthe root-mean-square (RMS) deviation δ between the ex-perimental and the simulated values. There appears tobe a general agreement between the experimental and thesimulated values.For one kick at k = 0 . s > .
6, while for other k val-ues, we observe similar patterns with a pair of highlyordered regular islands. Gradually, with larger numberof kicks, the order parameter settles to a characteristicpattern that resembles the classical phase-space exceptfor k = 2 π + 2 .
5. Ultimately, we see domains of reg-ular islands corresponding to high order parameter forall k values. As expected, we observe overall high orderparameter for the lowest k value. On the other hand,for high k values, unlike the classical case which showshighly chaotic phase-space, in the quantum scenario, theregular islands survive. This is due to the periodicityof the order parameter w.r.t. chaoticity parameter, i.e., s ( k ) = s (mod( k, π )). This is evident from the similaritybetween the contours of column 2 and 4 in Fig. 5 as wellas from the reflection symmetry between the columns 2(or 4) and 3. The periodicity of entropy distribution asa function of chaoticity parameter k and the number ofqubits has been theoretically studied in detail in [30]. B. Husimi probability distribution
Although the von Neumann entropy captures themixedness of the reduced density operator, it is not sen-sitive to its angular location on the Bloch sphere. TheHusimi probability function measures overlap of the state ρ n at any time with Bloch vectors {| θ, φ (cid:105)} on the phasespace and is given byQ n ( θ, φ, t ) = 1 π (cid:104) θ, φ | ρ n | θ, φ (cid:105) (18)While states initialized to regular regions are expected tobe localized at all times, those initialized at other regionsexplore more of the Bloch sphere. Higher the degree of . . . . . . . . (a1) s , = 0.1501/61/31/22/35/61 / . . . . . . . . . . (a2) s , = 0.1401/61/31/22/35/61 / . . . . . . . . . (a3) s , = 0.1301/61/31/22/35/61 / . . . . . . . . . . (a4) s , = 0.1301/61/31/22/35/61 / . . . . . . . . . . (a5) s , = 0.1301/61/31/22/35/61 / . . . . . . . . . . . (a6) s , = 0.1401/61/31/22/35/61 / . . . . . . . . . . . . . (a7) s , = 0.1501/61/31/22/35/61 / . . . . . . . . . . . . . (a8) s , = 0.150 1/2 1 3/2 2 /01/61/31/22/35/61 / . . . . . . . . . . . (b1) s , = 0.046 . . . . . . . . . . . . . . . . (b2) s , = 0.027 . . . . . . . . . . . . . . . . . . . . (b3) s , = 0.029 . . . . . . . . . . . . . . . (b4) s , = 0.036 . . . . . . . . . . . (b5) s , = 0.029 . . . . . . . . . . . (b6) s , = 0.036 . . . . . . . . . . . . . . (b7) s , = 0.033 . . . . . . . . . . . . . . . . (b8) s , = 0.0311/2 1 3/2 2 / . . . . . . . . . . . . (c1) s , = 0.045 . . . . . . . . . . . . . . . . (c2) s , = 0.035 . . . . . . . . . . . . . . . . . . . . . (c3) s , = 0.03 . . . . . . . . . . . . . . . . . . . . . . (c4) s , = 0.027 . . . . . . . . . . . . (c5) s , = 0.025 . . . . . . . . . (c6) s , = 0.028 . . . . . . . . . . (c7) s , = 0.028 . . . . . . . . . . . . . . (c8) s , = 0.031/2 1 3/2 2 / . . . . . . . . . . . . (d1) s , = 0.045 . . . . . . . . . . . . . . . . (d2) s , = 0.021 . . . . . . . . . . . . . . . . . . . (d3) s , = 0.029 . . . . . . . . . . . . . . . . . . . (d4) s , = 0.034 . . . . . . . . . . . (d5) s , = 0.026 . . . . . . . . . . . (d6) s , = 0.033 . . . . . . . . . . . . . (d7) s , = 0.033 . . . . . . . . . . . . . . . (d8) s , = 0.0391/2 1 3/2 2 / 0.20.30.40.50.60.7 FIG. 5. Contours represent experimental order parameter averaged over n -kicks ( s n ) for chaoticity parameter k = 0 . k = 2 . k = 2 π − . k = 2 π +2 . δ ) between the experimental and simulated values are shown in each case. FIG. 6. Experimental (mesh-grids) and simulated (color background) Husimi probability distributions (in units of 1 /π ) forcertain k values and initial states (as marked in Fig. 2) for various number of kicks. chaos, more is the spreading of the state. The Husimidistributions for select values of k and initial states areshown in Fig. 6. Here the mesh-grid lines represent theexperimental distribution while the colour backgroundrepresents the numerically simulated distribution. Wesee that the state | π/ , π (cid:105) for k = 0 . / (a) k=0.5; = /2; = (b) k=2.5; = /3; =7 /6 / (c) k=2.5; = /3; =4 /3 (d) k=2 -2.5; = /6; =4 /3 FIG. 7. Distribution of maxima neighborhood of Husimiprobability function on the Bloch sphere. Blue dots repre-sent simulated data and red dots represent experimental datafor the values of k and initial angles mentioned therein. ability distribution. As shown in Fig. 7, the maxima re-gion for k = 0 . k , the maxima regions spread out onthe phase space. Interestingly, the mismatch between ex-periment and simulated data increases with increasing k ,implying the sensitivity of the system dynamics to initialconditions and experimental imperfections. C. Lyapunov exponents
The Lyapunov exponent is a measure of chaos thatdetermines whether the trajectories of two initially veryclose points diverge or converge over time. In the classicalphase space, the Euclidean distance is usually used as thedistance measure. For nearby quantum initial states ρ (1)0 and ρ (2)0 , we may instead use the fidelity measure d m =1 − F ( ρ (1) m , ρ (2) m ) (see Eq. 15) to characterize the distanceafter m kicks. The discrete time Lyapunov exponent after n kicks is defined as λ ( n ) = 1 n n (cid:88) m =1 log d m d m − , (19)and its asymptotic limit lim n →∞ λ ( n ) being positive is con-sidered as a witness for chaoticity. A system initialized ina regular region is characterized by a negative Laypunovexponent and therefore a pair of nearby trajectories ul-timately converge. On the other hand, trajectories of apair of nearby initial states corresponding to a positiveLyapunov exponent diverge over time, and hence lead toa chaotic behavior. Fig. 8 displays experimentally ex-tracted Lyapunov exponents for certain pairs of nearbyinitial states after various number of kicks. For compari- -2024 (a) k=0.5; ( /2, ); ( /2, 5 /6) c = -3.3E-4; q = -1.0E-4 10 -2024 (b) k=2.5; ( /3, 7 /6); ( /3, ) c = -3.8E-4; q = -2.5E-410 No of kicks-2024 (c) k=2.5; ( /3, 4 /3); ( /3, 7 /6) c = -5.3E-4; q = -1.1E-3 10 No of kicks-2024 (d) k=2 -2.5; ( /6, 4 /3); ( /6, 7 /6) c = 1.7E-4; q = 3.5E-4900 950 1000-1E-3-5E-40 900 950 1000-4E-3-2E-302E-3900 950 1000-4E-3-2E-30 900 950 1000-4E-3-2E-302E-3 FIG. 8. Experimental Lyapunov exponents for up to 8 kicksfor certain k values and pairs of initial states (mentioned inthe titles) are shown by red circles. The simulated Lyapunovexponents with classical dynamics (black dashed line) andwith quantum dynamics (blue solid line) for up to 1000 kicksare shown for comparison. Means of the last 100 simulatedexponents (zoomed in the insets) are also mentioned for clas-sical ( λ c ) as well as quantum ( λ q ) cases to understand theasymptotic behavior. son, we have also plotted simulated Lyapunov exponentsfor the corresponding classical as well as quantum dy-namics for up to 1000 kicks which help us to evaluatethe asymptotic behavior. The last 100 exponents in eachcase are zoomed in the insets. The discrete time Lya-punov exponents in Fig. 8(a), which corresponds to themost ordered region, remain close to zero at all times,and slowly converge to a negative value. The means λ c and λ q of the last 100 classical and quantum exponentsrespectively, are also negative indicating the regular dy-namics. Although Lyapunov exponents of Fig. 8 (b) and(c) show relatively large fluctuations, they too convergetowards zero over larger number of kicks. It can seen thatfor Fig. 8(a), (b), and (c), which correspond to regularregions (see Fig. 2), both classical and quantum expo-nents are predominantly negative and accordingly theirrespective means λ c and λ q also are negative. However,for Fig. 8(d) which corresponds to a chaotic region (seeFig. 2), both classical and quantum exponents are pre-dominantly positive as reflected in their positive means. IV. CONCLUSIONS
In this work, we have experimentally studied quantumsignatures of classical chaos on a two-qubit NMR system using a kicked top model. We characterized the dynamicsvia three distinct ways:(i) Correspondence to classical phase-space was stud-ied using order-parameter profiles extracted from thevon-Neumann entropy. These profiles not only showeda good correspondence with the classical phase-space forlower chaoticity parameters, but also showed the inher-ent periodicity and symmetry in the quantum dynamicsfor larger values of the chaoticity parameter. It is in-teresting to see such signatures in the NMR case wherethe quantum state purity is well below the threshold forentanglement.(ii) The localization/delocalization of the quantumstates on the Bloch sphere was characterized via Husimiprobability distribution. They also showed temporalperiodicity that is characteristic of the quantum sys-tem. We observed the localization of the profiles for lowchaoticity conditions and significant delocalization oth-erwise. In addition, it also highlighted the sensitivity ofthe distribution to experimental imperfections particu-larly at higher values of chaoticity parameter.(iii) Finally we characterized the asymptotic behaviorof the nearby trajectories via Lyapunov exponents. Theexperimentally extracted discrete time exponents gradu-ally decayed towards zero. However, those correspond-ing to highly periodic region settled to negative valuesclearly indicating non-chaotic behavior. The simulatedexponents for large number of kicks could clearly distin-guish the periodic region from the chaotic region.The system considered here, being only two qubits, isdeeply embedded in the quantum regime, but the marksof quantum chaos are nonetheless interesting. NMRtestbed should facilitate the possibility of extending suchstudies with higher number of qubits. Further investi-gation of other quantum correlation measures such asdiscord, negativity, etc. will help better understand thebridge between chaos in classical and quantum systems.
ACKNOWLEDGMENTS
We acknowledge useful discussions with Prof. San-thanam, Sudheer Kumar, Deepak Khurana, and SohamPal. This work was partly supported by DST/SJF/PSA-03/2012-13 and CSIR 03(1345)/16/EMR-II. UTB ac-knowledges the funding received from Department of Sci-ence and Technology, India under the scheme Science andEngineering Research Board (SERB) National Post Doc-toral Fellowship (NPDF) file Number PDF/2015/00050. [1] E. Ott,
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