Energy cat states induced by a parity-breaking excited-state quantum phase transition

We show that excited-state quantum phase transitions (ESQPTs) in a system in which the parity symmetry is broken can be used to engineer an energy cat state—a Schrödinger cat state involving a quantum superposition of both different positions and energies. By means of a generalization of the Rabi model, we show that adding a parity-breaking term annihilates the ground-state quantum phase transition between normal and superradiant phases, and induces the formation of three excited-state phases, all of them identified by means of an observable with two eigenvalues. In one of these phases, level crossings are observed in the thermodynamic limit. These allow us to separate a wave function into two parts: one, with lower energy, trapped within one region of the spectrum, and a second one, with higher energy, trapped within another. Finally, we show that a generalized microcanonical ensemble, including two different average energies, is required to properly describe equilibrium states in this situation. Our results illustrate yet another physical consequence of ESQPTs.

Figure 1

Classical potential $V\left(x\right)=V(x;b,c)$, Eq. (1), for several values of its parameters. (a–c) The symmetric case $c=0$ for (a) $b=1$, (b) $b=0$, and (c) $b=-3/2$. (d–f) The asymmetric case $c=4/5$ for (d) $b=0$, (e) $b=-1.29266$, and (f) $b=-3$. Black circles mark the critical values $x_c$ for which $V\left(x\right)$ is minimal (ground state). Black triangles mark critical points for which $V\left(x\right)$ attains other local (not global) minima. Black squares represent critical points for which $V\left(x\right)$ attains a local maximum or an inflection point. The triangle and squares can be associated with ESQPTs characterized by a jump discontinuity in the density of states and a logarithmic divergence in the density of states, respectively. Gray dashed lines mark the various energies $V\left(x\right)$ for which either kind of ESQPT takes place.

Figure 2

Classical energies corresponding to the critical points of the Hamiltonian Eq. (3) as a function of $g/g_{*}$ for (a) the usual Rabi model, $\alpha =0$, and (b) the deformed Rabi model with $\alpha =1/2$. Yellow (light gray) lines represent the ground-state energy, and orange (medium gray) and red (dark gray) lines show the energy at which ESQPTs take place.

Figure 3

Classical phase space of the semiclassical analog Eq. (3) for $\alpha =1/2$, $\omega =1$, ${\omega }_0=300$ and different values of the coupling strength $g$. Below $g_{*}(\alpha =1/2)\approx 1.7872$ the phase space is compact for all energies, while above $g_{*}$ there is a critical energy whose contour connects two previously disconnected regions.

Figure 4

Classical level density $\varrho \left(\epsilon \right)$ of Eq. (3) for a deformation strength $\alpha =1/2$ and several values of the coupling strength $g$. The finite jump nonanalyticity, ${\epsilon }_{c1}$, is signaled by dotted orange lines, and the logarithmic singularities at ${\epsilon }_{c2}$ by dashed red lines. These various critical energies are (b) ${\epsilon }_{c1}=-0.8620$; (c) ${\epsilon }_{c1}=-2.0426$, ${\epsilon }_{c2}=-0.9584$; and (d) ${\epsilon }_{c1}=-4.1596$, ${\epsilon }_{c2}=-0.9784$.

Figure 5

Indicators of ground-state quantum phase transitions. (a, b) Ground-state energy ${\epsilon }_{\text{GS}}$ and second derivative with respect to the coupling strength $g$. (c, d) Number of photons in the ground state ${\langle \widehat{N}\rangle }_{\text{GS}}$ and its derivative with respect to $g$. Red (light gray) curves correspond to the usual Rabi model ($\alpha =0$), while blue (dark gray) curves are for the deformed Rabi model ($\alpha =1/2$).

Figure 6

Diagonal expectation values of the operators (a) ${\widehat{J}}_x$ and (b) ${\widehat{a}}^{†}+\widehat{a}$ as a function of energy. Orange (dotted) and magenta (dashed) vertical lines mark the ESQPTs' critical energies ${\epsilon }_{c1}\approx -4.1596$ and ${\epsilon }_{c2}\approx -0.9785$, respectively. Model parameters are $\omega =1$ ${\omega }_0=100$, $\alpha =1/2$, and $g/g_{*}=2$. Green (light gray) triangles represent the diagonal expectation values corresponding to eigenstates in the region where the constant $\widehat{\mathcal{C}}$ does not exist; red (medium gray) points correspond to eigenstates for which $\langle \epsilon \left|\widehat{\mathcal{C}}\right|\epsilon \rangle \le -0.95$; and blue (dark gray) points to eigenstates for which $\langle \epsilon \left|\widehat{\mathcal{C}}\right|\epsilon \rangle \ge 0.95$. Number of photons is 2000.

Figure 7

(a) Energy flow diagram of the quantum model Eq. (2) as a function of the coupling strength, $\epsilon =\epsilon (g/g_{*})$. The lowest energy line represents the ground state, while upper lines show increasingly high excited states of the system. Model parameters are $\omega =1$, ${\omega }_0=20$, and $\alpha =1/2$. Black lines represent energy levels in the region where the constant $\widehat{\mathcal{C}}$ does not exist; red (light gray) lines show energy levels for which $\langle \epsilon \left|\widehat{\mathcal{C}}\right|\epsilon \rangle \le -0.95$; and blue (dark gray) lines the energy levels for which $\langle \epsilon \left|\widehat{\mathcal{C}}\right|\epsilon \rangle \ge 0.95$. The yellow thick line shows the energy of the quenched state ${\epsilon }_g=\langle \mathrm{\Psi }\left(g\right)|\widehat{\mathcal{H}}\left(g\right)|\mathrm{\Psi }\left(g\right)\rangle$. Green points show special cases $(g/g_{*},{\epsilon }_g)$ whose energy distribution is shown in Fig. 12. (b) Magnification of the flow diagram. Number of photons is 570.

Figure 8

Region of level flow diagram for $1.995\le g\le 2.01$. Model parameters are $\omega =1$, ${\omega }_0=300$, and $\alpha =1/2$. Numerical energy levels, corresponding to the eigenvalues of the quantum model Eq. (2), are shown with empty circles. Color lines represent the classical energy levels obtained via quantization, Eq. (6); yellow (light gray) lines correspond to levels with $\langle \widehat{\mathcal{C}}\rangle =-1$, and magenta (dark gray) lines to $\langle \widehat{\mathcal{C}}\rangle =+1$. Inset: magnification of a single level crossing. The number of photons goes up to 5000.

Figure 9

(a, b) Diagonal expectation values of $\widehat{\mathcal{C}}$ in the eigenstate $|{\epsilon }_p\left(g\right)\rangle$ [(blue (dark gray)] such that ${\epsilon }_p(g/g_{*}=1.6)\approx -0.8$ and in the eigenstate with energy closest and below the previous level, $|{\epsilon }_{p-1}\left(g\right)\rangle$ [magenta (light gray)], as the coupling parameter $g$ is varied. The system-size parameter leading to the thermodynamic limit is ${\omega }_0=55$ in (a) and ${\omega }_0=603$ in (b). (c) Finite-size scaling of the amplitude of the transient behavior region $\mathrm{\Delta }{g}_{\text{T}}$. We obtain the power-law behavior $\mathrm{\Delta }{g}_{\text{T}}\sim 1/{\omega }_0^z$ with $z\approx 1$. The number of photons varies in $n_{ph}\in [812,8977]$ depending on ${\omega }_0$.

Figure 10

(a, b) Expectation values of the interaction term of the Hamiltonian ${\widehat{\mathcal{H}}}_{\text{int}}$ in adjacent energy eigenstates of the full Hamiltonian Eq. (8) as a function of energy for (a) ${\omega }_0=55$ and ${\omega }_0=603$. (c, d) Product of the diagonal expectation values of $\widehat{\mathcal{C}}$ in adjacent energy eigenstates, $C_{nn}$ and $C_{n+1n+1}$, as a function of energy, for (c) ${\omega }_0=55$ and (d) ${\omega }_0=603$. Model parameters are $\omega =1$, $\alpha =1/2$, and $g/g_{*}=1.65$. Black dashed lines mark the energy of the first ESQPT, ${\epsilon }_{c1}=-2.5886$, while full black lines mark the second ESQPT, ${\epsilon }_{c2}=-0.9668$. Number of photons ranges in $n_{ph}\in [752,8135]$ depending on ${\omega }_0$.

Figure 11

(a, c) Estimation of the minimum distance between levels at an avoided crossing, $\mathrm{\Delta }E$, as a function of the control parameter width, $\mathrm{\Delta }g$ for $\alpha =1/2$. Results for different values of the thermodynamic limit parameter ${\omega }_0$ (see legend) are shown with different symbols. The crossings are all located around $\epsilon \approx -4$ in (a), while in (c) they are at $\epsilon \approx -2$. (b, d) Scaling of the saturation of the minimum distance at an avoided crossing as a function of ${\omega }_0$ (squares) for the crossings in (a) and (c), respectively. In (b) the black line represents the exponential decay $\mathrm{\Delta }{E}_{\times }\left({\omega }_0\right)\sim {10}^{-\delta {\omega }_0}$ with $\delta \approx 1.19$, while in (d) the exponent is $\delta \approx 0.34$.

Figure 12

Distribution of populated energy levels by the quenched state $|\mathrm{\Psi }\left(g\right)\rangle$ at a certain value of the coupling strength, Eq. (12). Model parameters are $\omega =1$, ${\omega }_0=100$, and $\alpha =1/2$. The value of the coupling strength $g/g_{*}$ is (a) 1.05, (b) 1.34, (c) 1.54, (d) 1.94, (e) 2.14, and (f) 2.25. Orange (dotted) and magenta (dashed) vertical lines mark the ESQPTs critical energies ${\epsilon }_{c1}$ and ${\epsilon }_{c2}$ for each case (cf. Fig. 2). Number of photons for the whole process is 1900.

Figure 13

Probability Eq. (13) of finding the quenched state $|\mathrm{\Psi }\left(g\right)\rangle$ at position $q_n$. Model parameters are $\omega =1$, ${\omega }_0=100$, and $\alpha =1/2$. The values of the coupling strength $g/g_{*}$ are the same as in Fig. 12: (a) 1.05, (b) 1.34, (c) 1.54, (d) 1.94, (e) 2.14, and (f) 2.25. Number of photons is 1900.

Figure 14

Thermalization of physically relevant observables. Full black points represent the exact long-time average Eq. (14); empty triangles represent the standard microcanonical ensemble Eq. (15); and empty circles represent the result for the modified microcanonical ensemble Eq. (16). Model parameters are $\omega =1$, ${\omega }_0=100$, and $\alpha =1/2$. The microcanonical ensemble has been obtained by considering a window of 30 energy levels to each side of the mean energy ($N=61$). In the modified microcanonical ensemble each window has size 15 ($N_{+}=N_{-}=31$); $p_{+}=0.3984$ and $p_{-}=0.6016$. Number of photons is 1900.