Relaxation dynamics and dissipative phase transition in quantum oscillators with period tripling

Period tripling in driven quantum oscillators reveals unique features absent for linear and parametric drive, but generic for all higher-order resonances. Here we focus at zero temperature on the relaxation dynamics towards a stationary state starting initially from a domain around a classical fixed point in phase space. Beyond a certain threshold for the driving strength, the long-time dynamics is governed by a single time constant that sets the rate for switching between different states with broken time translation symmetry. By analyzing the lowest eigenvalues of the corresponding time evolution generator for the dissipative dynamics, we find that near the threshold the gap between these eigenvalues nearly closes. The closing becomes complete for a vanishing quantum parameter. We demonstrate that this behavior, reminiscent of a quantum phase transition, is associated with a transition from a stationary state which is localized in phase space to a delocalized one. We further show that switching between domains of classical fixed points happens via quantum activation, however, with rates that differ from those obtained by a standard semiclassical treatment. As period tripling has been explored with superconducting circuits mainly in the quasiclassical regime recently, our findings may trigger new activities towards the deep quantum realm.

Figure 1

(a) Classical bifurcation diagram in a rotating frame depicting steady state amplitudes (in units of $C=\sqrt{8{\omega }_0\delta \omega /3\alpha }$) versus driving strength [scaled as in (5)] for $\gamma /{\omega }_0=0.01, {\omega }_F/{\omega }_0=3.02$; the threshold beyond which multiple orbits coexist is $f_{c,\mathrm{cl}}\approx 0.49$. (b) and (c) Steady state orbits in the laboratory frame solving (1) for $f=0.76$. The period-1 orbit (b) is oscillating with the driving frequency ${\omega }_F$ (note the scale of the amplitude). Stable period-3 orbits (c) oscillating with ${\omega }_F/3\approx {\omega }_0$ occur with relative phase shifts $2\pi /3, 4\pi /3$; other parameters are $\alpha L^2/{\omega }_0^2=0.003, F_{0}L/{\omega }_0^2=0.05$ with arbitrary length scale $L$.

Figure 2

Classical quasienergy $g(Q,P)$ according to (4) for different driving strengths $f=0.1$ (a) and $f=0.5$ (b). The quasienergy has a $C_3$ symmetry in $QP$ space, where the local maximum at $Q_0=0,P_0=0$ corresponds to the period-1 orbit in Fig. 1 and the three wells located at $(Q_m,P_m), m=1,2,3$ are related to the period-3 orbits in Fig. 1.

Figure 3

Wigner distribution ${\rho }_W(Q,P,t)$ starting initially from a locally relaxed state in the well $Q_1,P_1$ of the quasienergy surface at different time steps and for $\kappa =1.5, \mathrm{\Lambda }=0.1688, f=0.9$: $t=33$ (a), $t=1636$ (b), $t=4975$ (c), and asymptotic state (d). The population dynamics of a closed area indicated by a white box in (a) is depicted in Fig. 4, such that the population $p_1\left(0\right)\approx 1$ at time $t=0$.

Figure 4

Decay of the population in the well around $Q_1,P_1$ [in the white box in Fig. 3] versus dimensionless times for $\kappa =1.5, \mathrm{\Lambda }=0.1688, f=0.9$. On long timescales it is governed by a single time constant $1/{\mathrm{\Gamma }}_L$. The black line is a fit function, where $p_1\left(\infty \right)=0.32$. The inset shows the transient behavior on very short timescales (blue); also shown is the decay of a state initially located around the origin at $Q=P=0$ (green). The initial distribution at $Q=P=0$ decays much faster than the initial population around the well $Q_1,P_1$.

Figure 5

Real (a) and imaginary (b) parts of the lowest lying eigenvalues of the Lindbladian in (10) versus the driving for $\mathrm{\Lambda }=0.1688$ and $\kappa =1.5$. Shown are eigenvalues corresponding to the class ${\rho }^{\left(3k\right)}$ (blue), namely, ${\lambda }_0$ ($⧫$) and ${\lambda }_3$ ($•$), and eigenvalues corresponding to the classes ${\rho }^{(3k+1)},{\rho }^{(3k+2)}$ (black), namely, $\mathrm{Re}\left\{{\lambda }_1\right\}=\mathrm{Re}\left\{{\lambda }_2\right\}$ ($\nabla$). For very weak driving $\mathrm{Re}\left\{{\lambda }_{1,2}\right\}$ describe relaxation of a harmonic oscillator near $Q=P=0$, while beyond a threshold $f_c$ (here at the maximum of ${\lambda }_3$) the eigenvalues become exponentially small [see inset of (a) with a log scale] and capture interwell transition processes. The imaginary part (b) of ${\lambda }_{1,2}$ also decreases towards ${\lambda }_0$.

Figure 6

(a) Transition matrix elements $|a_{\nu }\left(g\right)|$ at two different quasienergies $g=-1.2$ (diamonds), $g=-0.2$ (dots) for $f=0.66$ ($g_{\text{min}}=-3.55, g_{\text{saddle}}=0.22$). Shown are $|a_{\nu }|$ (blue) and $|a_{-\nu }|$ (red), where $|a_{\nu }|<|a_{-\nu }|(\nu >0)$. The matrix elements decrease exponentially with growing $\nu$ and increase as energy grows towards $g_{\text{saddle}}$. (b): Detailed balance ratio $r_{\mu \nu }$ according to (21) for $r_{42}$ (blue) and $r_{31}$ (red) at $f=0.5$. The ratio $r_{\mu \nu }$ differs from 1 in both cases so that the detailed balance condition is not fulfilled.

Figure 7

(a) Action $R\left(g\right)$ at $f=0.66$ ($g_{\min }=-3.55,g_{\mathrm{saddle}}=0.22$) and for different temperatures: $\overline{n}=0$ (black), $\overline{n}=0.1$ (blue), $\overline{n}=0.2$ (red). The action increases linearly with $g$. (b) Saddle point action $R\left(g_{\mathrm{saddle}}\right)$ determining the rate for quantum activation versus the driving strength at $\overline{n}=0$ (black), $\overline{n}=0.1$ (blue), $\overline{n}=0.2$ (red). $R\left(g_{\mathrm{saddle}}\right)$ increases almost linearly with increasing driving beyond $f_c$.

Figure 8

Semiclassical rate for interwell transitions via quantum activation ${\mathrm{\Gamma }}_{\mathrm{scl}}$ according to (23) with $D_0=1/T\left(g_{\min }\right)$ (purple and blue) and $D_0=\kappa$ (orange and turquoise) for $\mathrm{\Lambda }=0.1688$ and $\mathrm{\Lambda }=0.1125$ together with the numerically obtained switching rates ${\mathrm{\Gamma }}_L=\left|\mathrm{Re}\left\{{\lambda }_{1,2}\right\}\right|$ at $\mathrm{\Lambda }=0.1688$ (red crosses) and at $\mathrm{\Lambda }=0.1125$ (green crosses) versus the friction constant $\kappa$ at $f=0.8$.

Figure 9

Condition for quantum activation: The tunneling time $\left|\tau \right(g\left)\right|$ exceeds the slope of the action $R\left(g\right)$ for all energies $g$ at $f=0.66$ ($g_{\min }=-3.55,g_{\mathrm{saddle}}=0.22$) so that the condition is always fulfilled. Consequently, the relaxation dynamics towards a steady state is dominated by quantum activation.

Figure 10

(a)–(d) Steady state Wigner density ${\rho }_W^{\left(0\right)}(Q,P)$ of $||{\nu }_0\rangle \rangle$; (e)–(h) Wigner density ${\rho }_W^{(3,c)}(Q,P)$ of ${||{\nu }_3\rangle \rangle }_c$ for various driving strengths $f=0.27, f=0.47, f=0.6, f=0.67$ [from (a) to (d) and (e) to (h)] at $\overline{n}=0, \kappa =1.5$, and $\mathrm{\Lambda }=0.1688$, where $f_c\approx 0.63$ [cf. Fig. 5]. ${\rho }_W^{\left(0\right)}(Q,P)$ makes a transition from a localized state (ground state) around $Q=P=0$ for small driving to a delocalized distribution predominantly localized in the wells $Q_m,P_m,m=1,2,3$ corresponding to classical period-3 fixed points. ${\rho }_W^{(3,c)}(Q,P)$ exhibits the complementary behavior.

Figure 11

Fock state representation of the steady state density $\langle n\left|{\rho }^{(3,c)}\right|m\rangle$ corresponding to the eigenvector ${||{\nu }_3\rangle \rangle }_c$ orthogonal to the steady state for $f=0.27, f=0.47, f=0.6, f=0.67$ [from (a) to (d)] and $\kappa =1.5, \mathrm{\Lambda }=0.1688$. The density has a dominant contribution in the first excited Fock state for small driving which turns into a dominant contribution in the ground state for strong driving.

Figure 12

The overlap $|\langle \langle {\nu }_0||||{\nu }_3\rangle \rangle |$ for $\kappa =1.5, \mathrm{\Lambda }=0.1688$ (blue line) exhibits a maximum followed by a sharp drop to almost zero when $||{\nu }_3\rangle \rangle$ becomes least unstable ($|{\lambda }_3|$ has a minimum); it sharply increases again with growing driving. Most of this behavior results from the components of the eigenvectors $||{\nu }_0\rangle \rangle$ and $||{\nu }_3\rangle \rangle$ in the subspace spanned by the two lowest lying Fock states $\left\{\right|0\rangle ,|1\rangle \}$ ($|\langle \langle {\nu }_{0,r}||||{\nu }_{3,r}\rangle \rangle |$, black line).

Figure 13

Critical driving $f_c$ (blue line) located at the minimum of the eigenvalue $|{\lambda }_3|$ with changing $\mathrm{\Lambda }$ ($\kappa =1.5$ and $\overline{n}=0$). For decreasing $\mathrm{\Lambda }$ the systems turns into the semiclassical regime and approaches the classical threshold $f_{c,\text{cl}}\approx 0.49$ for $\mathrm{\Lambda }\to 0$. $\text{min}\left(\right|\mathrm{Re}\left({\lambda }_3\right)\left|\right)$ (green line) decreases as $\mathrm{\Lambda }$ goes to zero.

Figure 14

Phase diagram (detuning versus driving amplitude) of a dissipative quantum oscillator driven by three times its fundamental frequency. The black line separates the classical regions of single (left) and multiple (right) solutions. Quantum mechanically, the green ($\overline{n}=0$)/blue ($\overline{n}=0.1$) line separates the domain, where the steady state is localized around the origin of the $QP$ plane (left) from the domain, where it is localized with dominant contributions around classical period-3 fixed points (right). In comparison to the classical prediction for the bifurcation line in the absence of noise which separates the regime of period-1 orbits from the one of period-3 orbits, the quantum system exhibits ranges with a reentrant behavior when increasing the detuning for fixed driving. Parameters are $\gamma /{\omega }_0=0.01, \alpha q_0^2/{\omega }_0^2=0.0015$ with the new quantum parameter $q_0^2=\hslash /\left(2M{\omega }_0\right)=1/2$. The red point corresponds to the transition point in Fig. 1 ($f_{c,\mathrm{cl}}=0.49$).