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exponential sensitivity to small perturbations is a defining property of classical chaos . determining whether quantum systems exhibit the same property is important for understanding both quantum chaos as such and its role in the foundations of statistical mechanics .
many studies have been looking for static or dynamic manifestations of sensitivity to small perturbations in quantum systems that have chaotic classical limit @xcite .
there are conflicting opinions about whether nonintegrable quantum systems can exhibit exponential sensitivity to perturbations .
the often - mentioned argument in favour of the absence of exponential sensitivity is that quantum mechanics is intrinsically linear , and , therefore , quantum amplitudes , which define all measurable properties , are not subject to nonlinear dynamics , which , in turn is required for the onset of classical chaos . moreover , in bound quantum systems , the discreteness of the energy spectrum makes the evolution of any observable periodic or quasi - periodic in contrast to the random behavior generated by the lyapunov instability in the classical domain .
a related argument is that the heisenberg uncertainty relation does not allow one to define phase space trajectories , which , in turn , makes the notion of diverging phase space trajectories meaningless .    on the other hand
, it is widely believed that the classical behavior should be restored from the quantum laws of motion in the limit of large quantum numbers .
if a quantum system has a chaotic classical limit , and if its initial wave function is a very narrow wavepacket , then the subsequent evolution of this wavepacket should follow the classical trajectory for a finite time interval and hence exhibit hypersensitivity to the perturbations of initial conditions .
the late - time recurrences exhibited by the wavefunction due to the discreteness of the energy levels do not contradict the possible initial instability .
this is similar to the situation in a bounded classical chaotic system where the local lyapunov instability in the phase space does not prevent two initially diverging trajectories from coming arbitrary close to each other at a later time @xcite . for a macroscopic number of particles , astronomical time scales
will not be sufficient to observe the above recurrences - quantum or classical . from a somewhat different perspective
, it can also be argued that the linearity of quantum mechanics should suppress sensitivity to small perturbations only as much as the linearity of liouville s equation does for classical systems @xcite .
more discussion of related issues can be found , for example , in ref .
@xcite .    particularly difficult in the above respect
are nonintegrable macroscopic systems of spins 1/2 , where macroscopic observables , such as the total magnetization , are expected to behave classically , but the individual microscopic constituents , i.e. spins 1/2 , are as far from the classical limit as a quantum system can only be .
it is natural in the context of this difficulty to adopt the following approach .
one first substitutes spins 1/2 with classical spins and identifies the signatures of microscopic chaos in the behaviour of a macroscopic observable of the resulting classical system .
then one looks at the behavior of the same observable for the original spin-1/2 system to see whether it exhibits similar signature .
the task of identifying macroscopic signatures of microscopic chaos even for purely classical systems has been a long - standing challenge for the statistical physics community .
one prominent approach was to try to extract the value of kolmogorov - sinai entropy ( the sum of all positive lyapunov exponents of the system ) from the behaviour of a macroscopic observable @xcite .
this effort did not bring a conclusive outcome , because the true value of the kolmogorov - sinai entropy for macroscopic systems is not accessible either numerically or experimentally@xcite .
another approach was to look for the the exponential tails of relaxation functions@xcite and power spectra@xcite as signatures of microscopic chaos .
it led to a number of experimentally verified predictions@xcite .
however , the difficulty of this approach is that , even though the above tails are indeed generic for chaotic many - particle systems , no quantitative connection is known between the exponential decay constants of these tails and the primary characteristics of chaos , namely , lyapunov exponents of the system .
recently , we were able to make progress on the above agenda by first investigating the lyapunov spectra of classical spin lattices@xcite and then identifying an experimentally feasible manipulation of a classical spin system that would be able to access system s largest lyapunov exponent@xcite .
this manipulation is a weakly perturbed time reversal known in statistical physics as `` loschmidt echo '' @xcite and in nuclear magnetic resonance ( nmr ) as  magic echo"@xcite .
the outcome of our analysis was that loschmidt echoes in classical spin systems exhibit exponential sensitivity to small perturbations of perfect time reversal .
the constant characterising this sensitivity is equal to twice the value of the largest lyapunov exponent of the system . at the same time
, we have shown that the corresponding spin-1/2 systems are not exponentially sensitive to small perturbations even in the macroscopic limit , and , therefore , lyapunov exponents can not be defined for them .
the above conclusions were supported by numerical simulations of loschmidt echoes for spin-1/2 and for classical spin systems .
the question then arises how the transition from non - exponential to exponential sensitivity of loschmidt echoes proceeds .
it is known@xcite that the classical limit of quantum spin systems can be obtained by increasing the quantum spin number with the proper normalization of spin operators .
as far as the loschmidt echoes are concerned , our qualitative reasoning in ref .
@xcite was that spin-1/2 systems are not exponentially sensitive to small perturbations , because one can not slightly perturb an individual spin projection : any small perturbation creates a superposition of completely unperturbed state and a strongly perturbed state where the spin projection changes by 1 , which means that the spin 1/2 flips entirely . from the above perspective
, a larger quantum spin @xmath1 has @xmath2 possible values , which means that , if the spin projection is perturbed by 1 , then there is still a range of values for the difference between unperturbed and perturbed evolution to grow before reaching the maximum value of the order of @xmath3 .
therefore , we expect that the range of nearly exponential sensitivity of loschmidt echoes for systems of macroscopic number of spins @xmath1 to be of the order of @xmath4 .
the goal of the present article is to verify numerically the presence of the above nearly exponential growth numerically .
( 88 , 107 ) ( 3 , 55 ) ( 3 , 0 ) ( 2 , 50)(b ) ( 2 , 100)(a )    in the following , we compute loschmidt echoes for quantum ( @xmath5 ) and classical spin chains consisting of 6 spins with periodic boundary conditions .
the spins interact with the nearest - neighbor ( nn ) hamiltonian @xmath6 where @xmath7 represents either the quantum operator of the @xmath8 ( @xmath9 , @xmath10 or @xmath11 ) projection of a quantum spin on @xmath12 lattice site or the corresponding projection of a vector of length 1 representing a classical spin .
we take @xmath13 and @xmath14 for the classical hamiltonian , while for the quantum hamiltonian we divide these values by the factor @xmath15 to match the characteristic timescale of the dynamics with the classical counterpart .
our loschmidt echo manipulation is similar to nmr magic echo @xcite .
the system is initially slightly polarized in the @xmath9-direction in the vicinity of the infinite temperature equilibrium and then allowed to evolve under the action of the hamiltonian ( [ h ] ) for a certain time @xmath16 . at time @xmath16 , the sign of the hamiltonian is reversed , and the system evolves under the action of the reversed hamiltonian for another time interval @xmath16 .
afterwards , the value of the total magnetization in the @xmath9- direction @xmath17 is registered .    in real experiments ,
the reversal of the sign of the hamiltonian is not perfect @xcite . in our simulations , we intentionally introduce violations of the perfect time reversal by two methods :    \(a ) we apply small random rotations to the spins of the system at the moment of the hamiltonian reversal . specifically , each spin is rotated around the @xmath11-axis by an angle randomly selected from the interval @xmath18 $ ] .
\(b ) we add a small random - field perturbation @xmath19 to the hamiltonian @xmath20 during the backward evolution .
the fields @xmath21 are randomly selected from the interval @xmath22 $ ] .
we characterize the loschmidt echo response by function @xmath23 where @xmath24 and @xmath25 represent the averages computed with respect to the probability distributions or density matrices for the initial and the final states of the system respectively .
it was shown in ref .
@xcite that for sufficiently small perturbations in chaotic classical spin systems , there is a regime of exponential departure from the perfect time reversal : @xmath26 where @xmath27 is the maximum lyapunov exponent of the system . the above regime sets in after time @xmath16 of the order of @xmath28 required to suppress contributions from smaller lyapunov exponents
. the smallness of the initial perturbations should be such that @xmath29 in the regime exponential growth .
eventually , as @xmath16 increases , the system leaves the regime of small perturbations and enters the saturation regime characterized by @xmath30 .    for quantum systems ,
the initial perturbation should be large enough to ( ideally ) make the overlap of perturbed and unperturbed wave functions equal to zero , but , at the same time , small enough to avoid entering the saturation regime immediately .
the former restriction is of particular concern for finite - size numerical simulations : together with the latter one , it leaves a rather limited growth range for the regime @xmath29 . for macroscopic systems ,
the former restriction is consistent with virtually any physically realizable small perturbations of either external fields or the interaction hamiltonian .
we computed @xmath31 for the ring of 6 classical spins and the ring of 6 quantum spins @xmath32 described above . in both cases ,
the initial nonequilibrium polarization along the @xmath9-axis was equal to 5% of the maximum polarization . for the classical system , we averaged over an ensemble of @xmath33 different initial conditions for both perturbations ( a ) and ( b ) .
for the quantum system , it was sufficient , based on the recent results on quantum typicality@xcite , to use one initial quantum superposition state .
the equations of motion in both the classical and the quantum cases are propagated numerically by a fourth - order runge - kutta algorithm . in the classical case ,
@xmath34 was computed following the technique presented in ref.@xcite and was found to be equal to 0.28 .
both the calculations of @xmath34 and the classical simulations of loschmidt echoes were done for zero total energy and zero total polarization along the @xmath11-axis .
in fig . 1 , we present the evolution of @xmath35 for both the quantum and the classical spin systems for two types of perturbations ( a ) and ( b ) , together with the fit of the form @xmath36 , where @xmath37 is a fitting parameter , and @xmath34 is the directly computed lyapunov exponent .
we observe that , in the quantum system , @xmath35 exhibits a region of nearly exponential growth by about a factor of 10 , which is consistent with our expectation that ( i ) this range exists , and ( ii ) the growth factor is limited by @xmath38 .
we further observe that in the regime of nearly exponential growth for the spin-@xmath32 chain , the growth rate is consistent with the rate for the corresponding classical spin system , which is , in turn , equal to @xmath39 with accuracy of about 15 percent .
( this small discrepancy requires further investigation . )    for comparison , we include in fig .
1(a ) the loschmidt echo for a spin-1/2 chain .
we define the term `` nearly exponential growth '' as the growth of @xmath35 that can be well fitted by an exponential function over at least a factor of 10 .
with such a definition , the above chain of spins 1/2 exhibits no range of nearly exponential growth .
the existence of the region of nearly exponential growth of @xmath35 for a large - spin quantum system indicates that the experiments to determine effective lyapunov exponents in materials with large quantum spins may be feasible .
we have shown numerically , to as much extent as direct simulations allow , that systems of large quantum spins can produce signatures of exponential sensitivity to small perturbations similar to that of classical chaotic systems with positive lyapunov exponents .
this result is consistent with the expectation that classical behavior is reproduced by increasing spin quantum number @xmath1 .
it also indicates that , in solids containing large quantum spins , it is realistic to observe a behavior similar to chaotic lyapunov instabilities with the help of nmr or other experimental techniques .
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we investigate the sensitivity of nonintegrable large - spin quantum lattices to small perturbations with a particular focus on the time reversal experiments known in statistical physics as `` loschmidt echoes '' and in nuclear magnetic resonance ( nmr ) as `` magic echoes . '' 
 our numerical simulations of quantum spin-@xmath0 clusters indicate that there is a regime , where loschmidt echoes exhibit nearly exponential sensitivity to small perturbations with characteristic constant approximately equal to twice the value of the largest lyapunov exponent of the corresponding classical spin clusters . 
 the above theoretical results are verifiable by nmr experiments on solids containing large - spin nuclei .