Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases

Real-time dynamics in a quantum many-body system are inherently complicated and hence difficult to predict. There are, however, a special set of systems where these dynamics are theoretically tractable: integrable models. Such models possess nontrivial conserved quantities beyond energy and momentum. These quantities are believed to control dynamics and thermalization in low-dimensional atomic gases as well as in quantum spin chains. But what happens when the special symmetries leading to the existence of the extra conserved quantities are broken? Is there any memory of the quantities if the breaking is weak? Here, in the presence of weak integrability breaking, we show that it is possible to construct residual quasiconserved quantities, thus providing a quantum analog to the KAM theorem and its attendant Nekhoreshev estimates. We demonstrate this construction explicitly in the context of quantum quenches in one-dimensional Bose gases and argue that these quasiconserved quantities can be probed experimentally.

Popular Summary

The Kolmogorov-Arnold-Moser (KAM) theorem is a famed theorem in classical mechanics governing the crossover from integrability to chaos in the dynamics of a classical system. According to the KAM theorem, chaos is not ubiquitous; special classical systems exist with a number of exotic conserved quantities (beyond energy and momentum) known as integrable systems. The motion of bodies in an integrable system appears to be nonchaotic. However, an open question is how generic integrable systems are. In nature, no system is ever truly integrable; there are always small corrections that spoil integrability. The KAM theorem states that there is a smooth crossover between integrable and chaotic dynamics; breaking the special symmetries of an integrable model does not immediately lead to chaotic behavior.

Here, we provide a direct quantum analog to the KAM theorem. Using a one-dimensional Bose gas in a one-body parabolic trap that is then released suddenly into a small, one-body cosine potential, we demonstrate a direct quantum variant of the KAM theorem. Absent the one-body potential, the gas is described by the Lieb-Liniger model, an integrable model. In the presence of the cosine potential, however, integrability is broken. We show that the conserved quantities of the integrable Lieb-Liniger model are not immediately destroyed in the presence of the integrability breaking cosine. Rather, the model’s conserved quantities are deformed and, at least on the low-energy Hilbert space, remain nearly conserved. This deformation amounts to taking linear combinations of the original charges.

We demonstrate that by increasing the number of charges in the linear combination, we can control the quality of the quasiconservation.

Figure 1

Quench protocol: We prepare the one-dimensional Bose gas in its ground state in a harmonic trap. At time $t=0$, we release the gas into a cosine potential and track the subsequent dynamics. The shaded green regions are illustrations of the equilibrium density profiles of the gas in the presence of the confining potentials.

Figure 2

The density profile of the gas at selected times post-quench as computed with the NRG. Here, this time dependence is computed after releasing an $N=L=14$, $c=7200$ gas prepared in a parabolic potential with $m{\omega }_0^2{L}^2/2E_F=10.36$ [shown with a green dashed line in the $t=0t_F$ frame, $t_F=1/E_F$, $E_F=k_F^2/(2m)$, and $k_F=\pi (N-1)/L$] into a cosine potential $V_{\cos }(x)/E_F=0.35(\cos [(4\pi /L)x]+1)$ (plotted with a dashed line in the $t=43t_F$ frame). In the $t=0$ frame, we show the density profile as computed analytically in the hard-core limit (see Appendix pp1-s2). Using the NRG, we can run the time evolution as far out as $t=85t_F$ before dephasing exceeds 1%. We see, however, by $t=43t_F$ the gas’s density profile has already come close to its long time average (black dashed line in the final panel).

Figure 3

(a) The post-quench time evolution of the Lieb-Liniger charges normalized by their mean value as described in the text, ${\widehat{Q}}_i/⟨{\widehat{Q}}_i{⟩}_{\mathrm{a}\mathrm{v}}$. Here, the time dependence is computed after releasing an $N=L=8$, $c=10$ gas prepared in a parabolic potential of strength $m{\omega }_0^2{L}^2/2E_F=3.24$ into a cosine potential $\cos [(4\pi /L)x]$. We show this behavior at late times (for details of how long we can run the simulation, see Appendix pp1-s1). (b) The post-quench time evolution of a sequence of effective charges with the mean subtracted off, $\mathcal{Q}(N_Q)={\sum }_{m=1}^{N_Q}{a}_{2m}{\widehat{Q}}_{2m}$, for $N_Q=2,4$, and 8. (c) Top panel: The standard deviation of the fluctuations of two sequences of effective charges $\mathcal{Q}$. We build the first sequence (in black) using linear combinations of the charges $\{{\widehat{Q}}_{2m}{\}}_{m=1}^{m=8}$, while the second sequence (in red) is formed with the next eight Lieb-Liniger charges, i.e., $\{{\widehat{Q}}_{2m}{\}}_{m=9}^{m=16}$. Bottom panel: We show the fluctuations of the two effective charges built following the quench of a $c=1$ gas prepared in a parabolic trap of strength $m{\omega }_0^2{L}^2/2E_F=0.13$ and released into the same cosine potential, $\cos [(4\pi /L)x]$.

Figure 4

We plot the fluctuations in time for $\mathcal{Q}$ as a function of $N_Q$ for $N=L=4,8$, and 16 for a quench in a $c=10$ gas from a parabolic potential of strength $m{\omega }_0^2{L}^2/2E_F=2.33$ to a cosine of amplitude $V_{\cos }(x)=0.26E_F\cos (2\pi x/L)$. We do so using the charges constructed at $c=\infty$, as discussed in Appendix pp2, as a partial demonstration that such charges work well at finite $c$.

Figure 5

(a) We plot the intensity of the off-diagonal matrix elements of ${\widehat{Q}}_2/⟨{\widehat{Q}}_2{⟩}_{\mathrm{a}\mathrm{v}}$, comparing it to (b) the off diagonal m.e.’s of $\mathcal{Q}(8)$ for the quench of the $c=1$ gas discussed in Fig. 3. (c) We plot the average size of the off-diagonal matrix elements of two sequences of effective charges $\mathcal{Q}(N_Q)$; in black is the sequence constructed from ${\widehat{Q}}_{2m}$, $m=1,\dots ,8$, while in red is the sequence constructed from ${\widehat{Q}}_{2m}$, $m=9,\dots ,16$. We show this for both the $c=1$ [same quench as in (a) and (b)] and the $c=10$ case [same quench as described in Figs. 3].

Figure 6

We demonstrate that the effective charges constructed analytically at $c=\infty$, as described in detail in Appendix pp2, have suppressed temporal fluctuations for quenches with finite $c=1,10$.

Figure 7

We show that the fluctuations in the effective charges $\mathcal{Q}$ constructed from a quench from a stronger to a weaker parabolic potential, like their parabola to cosine counterparts, die out rapidly with $N_Q$. We consider two quenches of this type, one with the gas at $c=7200$ and one with $c=10$. For the $c=7200$ case, we quench from a parabolic potential with strength, ${\omega }_{0,\text{init}}$ given by $m{\omega }_{0,\text{init}}^2{L}^2/2E_F=6.48$ into a parabolic potential with strength ${\omega }_{0,\text{fin}}$ given by $m{\omega }_{0,\text{fin}}^2{L}^2/2E_F=2.11$. And for the $c=10$ case, we quench from a parabola described by $m{\omega }_{0,\text{init}}^2{L}^2/2E_F=3.24$ into one given by $m{\omega }_{0,\text{fin}}^2{L}^2/2E_F=1.06$.

Figure 8

We plot the values of the diagonal matrix elements of $\mathcal{Q}(8)$ in the post-quench eigenbasis as derived for the $c=1$ quench discussed previously in Figs. 3.

Figure 9

The lower bounds on ${\chi }_k$ due to the effective charge $\mathcal{Q}$ for correlators involving the MDF and the density operators. Left-hand panel: We plot the lower bound on MDF correlations for two different quenches in a $c=10$ gas. In the first (the left sets of bars), we quench into a cosine potential $V_{\cos }\cos (2\pi x/L)$ of amplitude $V_{\cos }=0.26E_F$. In the second (the right set of bars), we quench into a flat potential, i.e., $V_{\cos }=0$, where the post-quench Hamiltonian is then integrable. We present ${\chi }_k$ for three different system sizes $N=L=4,8$, and 16 and three different values of $k$, $k_n=2\pi n/L$, $n=0,1$, and 2. The initial state of the quench is given by the ground state of a gas in a parabolic potential of strength $\omega =2.4/N$. Right-hand panel: We similarly plot the lower bound on density correlations. Here, we only consider the case of quenching into $V_{\cos }=2$, as ${\chi }_k$ for the density operator is identically zero in the absence of the breaking of translational invariance. We again compute the lower bound at three different system sizes and three different wave vectors $k_1,k_2$, and $k_3$. In both cases, we see no obvious dependence on system size. We believe that the fluctuations seen between different system sizes results from the particular construction of $\mathcal{Q}$ at any given system size. We construct $\mathcal{Q}$ to minimize time fluctuations of a particular initial condition rather than construct it to maximize its overlap with a particular observable as was done in Ref. [74].

Figure 10

A plot of the energy spectra for an $N=14$ gas with $c=7200$ in a cosine potential of amplitude $A/E_F=0.35$ (as in Fig. 2 of the main text). The analytic results are given in red, while in black are the corresponding numerics. On the r.h.s., we expand a range of energy with a dense number of states so as to better exhibit agreement between the numerics and the analytics. We can determine the first 365 states (up to energies of $E=65$) with accuracy of $10^{-3}$.

Figure 11

(a) The post-quench time evolution of the same normalized Lieb-Liniger charges shown in Fig. 2 of the main text but at times $t<100t_F$. (b) The post-quench time evolution of the sequence of effective charges, $\mathcal{Q}(N_Q)-a_0\mathrm{I}=\sum _{m=1}^{N_Q}{a}_{2m}{\widehat{Q}}_{2m}$, shown in Fig. 2 for $N_Q=2,4$, and 8 for the same range of time.

Figure 12

Here we show at $c=\infty$ how the matrix elements zeroed out for the commutator $C_1$ remain zero for the higher order commutators $C_{n>1}$. In the top block we graphically display how the matrix elements of $C_1$ are zeroed out for states with quantum numbers less than $N_{max}$. In the second block, we show that the matrix elements of the second and third order commutators $C_2$ and $C_3$ are also zero for this same set of states. It is only for the fourth and fifth order commutators, $C_4$ and $C_5$, that the set of states with zero matrix elements begins to shrink to states with quantum numbers less than $N_{max}-n_{cos}$ (as pictured in the third block). The set of states with zero matrix elements on $C_6$ and $C_7$ shrinks further to states with quantum numbers less than $N_{max}-2n_{cos}$. This shrinkage in the set of states by steps of $n_{cos}$ continues at higher order, one step for every two orders of commutators.

Figure 13

The magnitude of the off-diagonal matrix elements of the analytically constructed effective charges composed from (a) four Lieb-Liniger charges, ${\mathcal{Q}}_{N_{max}}(4)$, and (b) eight Lieb-Liniger charges, ${\mathcal{Q}}_{N_{max}}(8)$. (c) The size of the post-quench temporal fluctuations of the effective charges ${\mathcal{Q}}_{N_{max}}(N_Q)$ as a function of the number of charges in the linear combination. Here, the quench is performed by preparing an $N=8$, $c=7200$ gas in a parabolic potential of strength $m{\omega }_0^2{L}^2/2E_F=6.48$ released into a cosine potential of strength $A=1$. We show the size of the temporal fluctuations for two sequences of effective charges, the first (in black) constructed from Lieb-Liniger charges, $Q_{2m}$, $m=1,\dots ,8$ and the second (in red) constructed from $Q_{2m}$, $m=9,\dots ,16$. (d) The size of the off-diagonal matrix elements of these same two sequences of ${\mathcal{Q}}_{N_{max}}$ as a function of the number $N_Q$ of Lieb-Liniger charges in the linear combination.