Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems
The exponential growth of the out-of-time-ordered correlator (OTOC) has been proposed as a quantum signature of classical chaos. The growth rate is expected to coincide with the classical Lyapunov exponent. This quantum-classical correspondence has been corroborated for the kicked rotor and the stadium billiard, which are one-body chaotic systems. The conjecture has not yet been validated for realistic systems with interactions. We make progress in this direction by studying the OTOC in the Dicke model, where two-level atoms cooperatively interact with a quantized radiation field. For parameters where the model is chaotic in the classical limit, the OTOC increases exponentially in time with a rate that closely follows the classical Lyapunov exponent.
Quantum chaos tries to bridge quantum and classical mechanics. The search for quantum signatures of classical chaos has ranged from level statistics Haake (1991); Stöckmann (2006) and the structure of the eigenstates Chirikov (1985); Flambaum et al. (1999) to the exponential increase of complexity Peres (1996); Borgonovi et al. and the exponential decay of the overlap of two wave packets Jalabert and Pastawski (2001); Cucchietti et al. (2002); Gorin et al. (2006). Recently, the pursuit of exponential instabilities in the quantum domain has been revived by the conjecture of a bound on the rate growth of the out-of-time-ordered correlator (OTOC) Maldacena and Stanford (2016); Maldacena et al. (2016). First introduced in the context of superconductivity Lar , the OTOC is now presented as a measure of quantum chaos, with its growth rate being associated with the classical Lyapunov exponent. The OTOC is not only a theoretical quantity, but has also been measured experimentally via nuclear magnetic resonance techniques Gärttner et al. (2017); Li et al. (2017); Wei et al. (2018).
The correspondence between the OTOC growth rate and the classical Lyapunov exponent has been explicitly shown in two cases of one-body chaotic systems, the kicked-rotor Rozenbaum et al. (2017) and, after a first unsuccessful attempt Hashimoto et al. (2017), the stadium billiard Rozenbaum et al. . For interacting many-body systems, while exponential behaviors for the OTOC have been found for the Sachdev-Ye-Kitaev model Maldacena and Stanford (2016); Bagrets et al. (2017) and for the Bose-Hubbard model Bohrdt et al. (2017); Shen et al. (2017), a direct demonstration of the quantum-classical correspondence has not yet been made. Studies in this direction include Akila et al. (2017); Bianchi et al. (2018); Scaffidi and Altman ; Borgonovi et al. and Rammensee et al. .
Here, we investigate the OTOC for the Dicke model Dicke (1954). Comparing with one-body systems, the Dicke model is a step up toward an explicit quantum-classical correspondence for interacting many-body systems, since it contains atoms interacting with a quantized field. It was originally proposed to explain the collective phenomenon of superradiance: the field mediates interatomic interactions, which causes the atoms to act collectively Dicke (1954); Hepp and Lieb (1973); *Wang1973; *Carmichael1973. Superradiance has been experimentally studied with ultracold atoms in optical cavities Baumann et al. (2010, 2011); Ritsch et al. (2013); Baden et al. (2014); Klinder et al. (2015); Kollár et al. (2017). The Dicke Hamiltonian has also found applications beyond superradiance in various different fields. It has been employed, for instance, in studies of ground-state and excited-state quantum phase transitions Hepp and Lieb (1973); *Wang1973; *Carmichael1973; Castaños et al. (2005); Pérez-Fernández et al. (2011a); Brandes (2013); Bastarrachea-Magnani et al. (2014a); Larson and Irish (2017), entanglement creation Schneider and Milburn (2002); *Lambert2004; *Kloc2017, nonequilibrium dynamics Pérez-Fernández et al. (2011b); Altland and Haake (2012); Lerma-Hernández et al. ; Kloc et al. (2018), quantum chaos Lewenkopf et al. (1991); Emary and Brandes (2003a); *Emary2003; Bastarrachea-Magnani et al. (2014b); *Bastarrachea2015; *Bastarrachea2016PRE; Chávez-Carlos et al. (2016), and monodromy Babelon et al. (2009); Kloc et al. (2017b)
In the classical limit, the Dicke model presents regular and chaotic regions depending on the Hamiltonian parameters and excitation energies Chávez-Carlos et al. (2016). This allows us to benchmark the OTOC growth against the presence and absence of chaos. The results in the chaotic region display three different temporal behaviors: a sinusoidal evolution at short times, followed by an exponential growth, that holds up to the saturation of the dynamics. Our approach, based on the use of an efficient basis for the convergence of the eigenstates, enables the treatment of systems that are large enough to reveal the exponential part of the dynamics. We find that the exponential growth rate is in close agreement with the classical Lyapunov exponent.
Quantum and Classical Hamiltonian.– The Dicke model has two-level atoms of level spacing coupled with a single mode of a quantized radiation field of frequency . The Hamiltonian is given by
where ; and are the quadratures of the bosonic field and is the annihilation (creation) operator; the collective atomic pseudo-spin operators, , are the sums of the Pauli matrices for each atom ; is the atom-field interaction strength; and is the eigenvalue of the total spin operator . The critical point marks the transition from a normal phase () to a superradiant phase (). We set in the illustrations below and work with the symmetric atomic subspace (), where the ground state lies. The models has two degrees of freedom.
The classical Hamiltonian is built by employing Bloch coherent states and Glauber coherent states Ribeiro et al. (2006); Bakemeier et al. (2013); Chávez-Carlos et al. (2016). The first are given by where and is the ground state for the atoms, and the Glauber coherent states are where and in the photon vacuum. The canonical variables and are given in terms of the coherent state parameters and , respectively. Deriving the classical Hamiltonian is basically equivalent to replacing the operators with the canonical variables and as , , , . It reads
Since the classical limit is reached for , the effective Planck constant is .
We denote the energy per particle as , which is independent of . Since the number of bosons in the field is unlimited, the range of values of is only limited from below. The ground state energy is given by for and by for .
With the classical Hamiltonian, we solve the Hamilton equations of motion and obtain a map of the degree of chaoticity of the system as a function of the energy and the interaction strength , as shown in Fig. 1. The task of drawing the map is quite demanding. For each value of and , we consider a large sample of initial conditions distributed homogeneously in the energy shell. The largest Lyapunov exponent is evaluated for each initial condition. If , the initial condition is chaotic and for , the initial condition is regular. The percentage of chaos is defined as the ratio of the number of chaotic initial conditions over the total number of initial conditions in the sample. This percentage is shown in Fig. 1 with a color gradient: dark indicates that most initial conditions are regular and light indicates that most are chaotic. (Notice that one should look only at the results above the thick solid line that marks the ground state.) Regularity predominates for , while for , most regular trajectories have low energies and large energies are associated with chaos. This map guides our analysis of the OTOC below.
Method.– The OTOC quantifies the degree of non-commutativity in time between two Hermitian operators with small or null commutator at time . In terms of position and momentum, it is written as
where and are the eigenstates and eigenvalues of . In Ref. Hashimoto et al. (2017), is called microcanonical OTOC. In the semiclassical limit, substituting the commutator by the Poisson bracket, one gets for a classically chaotic system, , where is the classical Lyapunov exponent. This suggests the connection between the exponential growth rate of the OTOC and , and justifies referring to as the quantum Lyapunov exponent.
where the matrix elements
with , , and . Since the Dicke Hamiltonian is of the form and ,
which simplifies the calculations. The OTOC is obtained by evaluating numerically only the matrix elements of in the energy eigenbasis. For this, instead of employing the usual photon number (Fock) basis, we resort to an efficient basis that guarantees convergence of the eigenvalues and wave functions for a broad part of the spectrum (see Bastarrachea-Magnani and Hirsch (2014); *Bastarrachea2014PSb).
Quantum Lyapunov Exponent.– In this Letter, we concentrate our analysis on chaotic eigenstates. They are chosen along the vertical line in Fig. 1, where the coupling parameter is strong, . This line exhibits regular and chaotic regions. From the ground state to , the dynamics is regular. From to , regular and chaotic trajectories coexist. For larger energies, , chaos cover almost the whole energy shell. We select a group of sixty eigenstates in the chaotic energy region with . They are indicated with a circle in Fig. 1.
In Fig. 2 (a), we show that even for a single representative eigenstate, the behavior of the OTOC is clearly exponential from up to the saturation of the dynamics. The growth rate is obtained by fitting the curve with a straight line indicated with stars in the figure.
The exponential behavior is robust with respect to two different probes:
(i) It holds when we use the commutator for the operator at different times, , as also shown in Fig. 2 (a). The associated fit, indicated with circles, provides . Both exponential fits lead essentially to the same quantum Lyapunov exponents.
(ii) The exponential growth rates are very similar for the sixty different states selected in the chaotic region.
The log-log plot in Fig. 2 (b) makes evident the appearance of different behaviors at different time scales. For , the dynamics of [similarly for ] is controlled by the diagonal matrix elements in Eq. (5), , with few states contributing significantly to the sum, all with energy differences . The short-time evolution is therefore approximately described by the square of a cosine function [sine for ]. The two sinusoidal curves are shown with dotted lines in the inset of Fig. 2 (b).
At long times, the quantum dynamics saturates to the infinite-time average,
which is obtained from Eqs. (3) and (5) using that for . and are shown in Fig. 2 (b) with dotted lines. These averages are related with the square of the size of the available phase space Hashimoto et al. (2017). For the Dicke Hamiltonian, it scales with and with the number of bosons in the system, which grows with the excitation energy.
After the exponential growth, the OTOC fluctuates around its asymptotic value, as seen in Fig. 2 (b), with a standard deviation . We define the saturation time as the time when the OTOC reaches for the first time the value . The values of for and are marked in Fig. 2 (a) with a triangle and a square, respectively. The saturation time marks the point beyond which quantum effects are strong and the quantum-classical correspondence no longer holds, therefore the association between and the Ehrenfest time. The saturation of the dynamics for finite quantum systems is in contrast to what one finds for classical systems, where the spectrum is continuous. As increases and the system approaches the classical limit, grows and increases with it.
Quantum-classical correspondence.– To associate a classical Lyapunov exponent to each particular Hamiltonian eigenstate, we employ the Husimi functions, which are the square of the overlaps of the eigenstate with the coherent states used to build the classical description. They are defined as
The subtleties of the evaluation of the Husimi functions, in connection with the efficient coherent basis used to get the eigenstates Bastarrachea-Magnani and Hirsch (2014); *Bastarrachea2014PSb, are detailed in Ref. Bastarrachea-Magnani et al. (2016).
The Poincaré surface of section of the Husimi functions for three eigenstates with energies close to are shown in Fig. 3 (a,b,c), projected on the plane for . The bright structures indicate large overlaps. For comparison, we also show in Fig. 3 (d) the classical map of chaos for the same energy and plane. The color code represents the values of the classical Lyapunov exponent for each initial point in the phase space.
With the Husimi function, we compute the average Lyapunov exponent for the corresponding eigenstate. To this end, we assign weights , given by the discretized and normalized Husimi function, to each point in the phase space, where and is the number of points in the phase space over the plane with . The weighted average is
The purpose of writing the last term above is to emphasize that is the average of logarithms, while the maximum value of the Lyapunov exponent over the region where the Husimi function is not null is the logarithm of the average,
In Fig. 4, we compare the quantum Lyapunov exponent (circles), the average classical Lyapunov exponent (squares), and the maximum Lyapunov exponent (stars). Obviously, and both exponents are very stable for the sixty states. This is because the classical Lyapunov exponents in most of the Poincaré surface of section in Fig. 3 (d) have very similar values. The quantum exponent, on the other hand, fluctuates much more, due to the oscillations that modulate the exponential growth and finite size effects. Increasing the value of would reduce this uncertainty.
In general, is larger than , since the latter is the average of logarithms [see Eq. (8)], while the quantum Lyapunov exponent is obtained from the logarithm of the fit, as discussed in Rozenbaum et al. (2017). For this reason, the quantum Lyapunov exponent is closer to . It is worth to mention that the average value of for the energy region in Fig. 4 is approximately , while for both and , we find .
Discussion.– We showed that for the Dicke model in the chaotic region, the OTOC grows exponentially fast in time with a rate comparable to the average classical Lyapunov exponent and very close to the maximum classical Lyapunov exponent averaged by the Husimi function. These results confirm that the quantum-classical correspondence established by means of the OTOC is not exclusive to one-body systems, but is valid also for interacting systems with more than one degree of freedom. This work provides a proof-of-principle and should motivate similar studies in other interacting systems.
We stress that to clearly identify the quantum exponential growth and extract its rate, we need to have access to large system sizes. This was possible here, because we resorted to an efficient basis to construct the eigenstates.
The instrument of our analysis was the microcanonical OTOC [Eq. (3)] corresponding to the eigenstate expectation value of the commutator of two operators. Its use in stadium billiards Hashimoto et al. (2017) prevented the observation of the quantum exponential growth, which was only possible with the introduction of Gaussian states Rozenbaum et al. . In our case, however, the eigenstates were excellent probe states for revealing the OTOC exponential growth. This is a very important result for future studies of interacting systems, since the eigenstates are essential building blocks for thermal averages.
We conclude that it is possible to employ only quantum tools to characterize chaos in the phase space. Just as the Husimi distribution function can be associated with regular or chaotic sectors of the classical Poincaré surface of section, the exponential growth rate of the OTOC works as a quantum equivalent of the Lyapunov exponent.
We acknowledge financial support from Mexican CONACyT project CB2015-01/255702, DGAPA- UNAM project IN109417 and RedTC. MABM is a post-doctoral fellow of CONACyT. PS is supported by the Charles University Research Center UNCE/SCI/013 and is grateful to P. Cejnar for stimulating discussions. LFS is supported by the NSF grant No. DMR-1603418.
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