Critical quantum geometric tensors of parametrically-driven nonlinear resonators

The system under investigation is a parametrically-driven Kerr resonator, described by the following Hamiltonian ($\hbar=1$ is set)

where $a$ and $a^{\dagger}$ represent the photon annihilation and creation operators, respectively, with the canonical commutation relation $[a,a^{\dagger}]=1$. The Kerr nonlinearity is denoted by $K$, and $P$ is the amplitude of the two-photon drive with detuning $\delta$. Notably, this Hamiltonian exhibits a discrete $\mathbb{Z}_{2}$ symmetry, being invariant under the parity transformation $U=e^{i\pi a^{\dagger}a}$. To analyze how the QPT is influenced by a finite value of $\delta/K$, we perform a finite-size scaling analysis, based on the effective “system size” defined as $L=\delta/K$, with $L$ governing the proximity to the thermodynamic limit. To investigate the geometric properties of the model, we can parametrize $P=\delta\varepsilon\exp(-i\phi)/2$, and study QPTs in terms of two-photon drive amplitude $\varepsilon$ and phase $\phi$. Then the Hamiltonian can be rewritten as

Before delving into QPTs, it is pertinent to discuss the existing phases in the model. As a first step, we can analytically obtain quantum phases under different limits when $K\rightarrow 0$. On the one hand, for $\varepsilon<1$, the first term can be neglected, and a squeezing transformation $S(r_{\rm n})=e^{(r_{\rm n}^{*}a^{2}-r_{\rm n}a^{\dagger 2})/2}$ is applied, with $r_{\rm n}=\frac{1}{4}\ln\frac{1-\varepsilon}{1+\varepsilon}e^{-i\phi}$. This transforms the Hamiltonian to the normal phase Hamiltonian:

where $\omega_{\rm n}^{e}=\delta\sqrt{1-\varepsilon^{2}}$ and $\omega_{\rm n}^{g}=\delta(\sqrt{1-\varepsilon^{2}}-1)/2$ are the excitation and ground-state energies in the normal phase, respectively. Therefore, the system ground state is given by:

It is worth noting that this wavefunction is a squeezed vacuum state, which preserves the $\mathbb{Z}_{2}$ symmetry. The squeezing parameter is dependent on $\varepsilon$ and $\phi$.

The energy gap vanishes and the squeezing parameter tends to infinity at the critical point $\varepsilon_{c}=1$. To reveal the ground state in the regime $\varepsilon>1$, a displacement transformation $D(\pm\alpha)$ is performed, with $\alpha=\sqrt{L(\varepsilon-1)/2}\exp(-i\phi/2)$. Discarding higher-order terms due to the large $L$, the original Hamiltonian becomes:

where $\delta=\delta(2\varepsilon-1)$ and $\Omega=\delta/2$. Then we perform the squeezing transformation $S(r_{\rm s})=e^{(r_{\rm s}^{*}a^{2}-r_{\rm s}a^{\dagger 2})/2}$ and set $r_{\rm s}=\frac{1}{4}\ln\frac{\varepsilon-1}{\varepsilon}e^{-i\phi}$, transforming the Hamiltonian to the superradiant phase Hamiltonian:

where $\omega_{\rm s}^{e}=2\delta\sqrt{\varepsilon(\varepsilon-1)}$ and $\omega_{\rm s}^{g}=\delta(\sqrt{\varepsilon(\varepsilon-1)}-\varepsilon+1/2)-L\delta^{2}(\varepsilon-1)^{2}/4$. Consequently, the system has two degenerate ground states, given by

both breaking the $\mathbb{Z}_{2}$ parity symmetry.

The above results indicate that the system undergoes a transition from the normal phase to the superradiant phase at the critical point $\varepsilon_{c}=1$. In general, when $K$ takes a finite value, obtaining the ground state analytically becomes challenging. However, we can numerically diagonalize the Hamiltonian in the representation of the Fock basis and obtain the corresponding eigenstates by solving the secular equation. This method requires the photon number cut-off much larger than the average photon number of the system so that all the information on the quantum state can be included. In this work, unless explicitly stated otherwise, we set the cut-off to $N_{\text{cut}}=800$.

To explore the global phase diagram of the model, we numerically calculate the average photon density $\rho=\langle a^{\dagger}a\rangle/L$ as a function of the two-photon drive amplitude $\varepsilon$ and phase $\phi$. This average photon density serves as the order parameter in the theory of QPT. As depicted in Fig. 1(a), for a specific $\phi$, when $\varepsilon\textless 1$, the average photon density remains at 0, indicating the stability of the normal phase even with finite $K$. Conversely, when $\varepsilon\textgreater 1$, the average photon density abruptly increases [see Fig. 1(b)], signifying that, photons spontaneously condense, breaking the $\mathbb{Z}_{2}$ parity symmetry and exhibiting the superradiant phase. Moreover, it is observed that the normal-to-superradiant phase transition point is unaffected by the two-photon driving phase $\phi$, which will be discussed in the context of the QGT below.

Having established the global phase diagram, the subsequent crucial task is to investigate the QPT between different phases and determine the universality class to which it belongs. Various methods exist for detecting quantum critical points and determining their universality class[47, 3]. In the following sections, we introduce the basic concept of the critical QGT and employ it to examine QPT within the phase diagram.

The next question is what is the critical behavior of the parametrically-driven nonlinear resonator with finite Kerr nonlinearity and to which universality class it belongs. To this end, we perform a numerical diagonalization of the Hamiltonian in the Fock basis and subsequently evaluate the components of the QGT using Eq. (9). The numerical results depend on the dimensions of the Hilbert space, and convergence is observed as the photon number cut-off increases. We extract the convergent results as a function of the driving parameter $\varepsilon$ for different $L$. Subsequently, we conduct fits to obtain the corresponding scaling dimension $\Delta_{jk}$ and correlation length exponent $\nu$.

An interesting finding is that the first diagonal component of the QGT, denoted as $g_{\varepsilon\varepsilon}$, is equivalent to the fidelity susceptibility [56, 57, 41, 42, 53, 58, 59, 58, 60, 57] (see Appendix A). The corresponding scaling dimension is related to the correlation length exponent as $\Delta_{\varepsilon\varepsilon}=2/\nu$. Illustrated in Fig. 2(a), the rescaled quantum metric tensor $g_{\varepsilon\varepsilon}/L$ exhibits a peak that becomes sharper with increasing size at the pseudo-critical point $\varepsilon_{c}(L)$. This implies that $L$ plays a role analogous to the system size in an ordinary phase transition. In the inset of Fig. 2(a), we perform finite-size scaling of the pseudo-critical point $\varepsilon_{c}(L)$ as a function of inverse effective system sizes $1/L$. The quantum critical point $\varepsilon_{c}^{*}$ is estimated by polynomial fitting: $\varepsilon_{c}(L)=\varepsilon_{c}^{*}+aL^{-b}$. Consequently, at the critical point $\varepsilon_{c}$, the corresponding scaling dimension $\Delta_{\varepsilon\varepsilon}$ is obtained by fitting the function in Eq.(10), and the correlation length exponent $\nu$ is calculated. As shown in Fig. LABEL:fig3(a), we find that the scaling dimension $\Delta_{\varepsilon\varepsilon}\approx 1.325$ and correlation length exponent $\nu$ converges to $1.510$ as the thermodynamic limit is reached, closely matching the result $1.5$ for quantum Rabi models in the infinite qubit-field frequency ratio limit [16].

To verify the universal scaling behavior of the QGT, according to Eq.(11), the tensor can be scaled by $L^{-\Delta_{jk}}g_{\varepsilon\varepsilon}$ as a function of $L^{1/\nu}|\varepsilon-\varepsilon_{c}^{*}|$ in the vicinity of the quantum critical point $\varepsilon_{c}^{*}$. Upon inserting the obtained critical point $\varepsilon_{c}^{*}$ and correlation length exponent $\nu$ into Eq.(11), all metric tensors for different $L$ collapse into a single one [Fig. 2(b)], indicating that $g_{\varepsilon\varepsilon}$ is a universal function of $L^{1/\nu}|\varepsilon-\varepsilon_{c}^{*}|$, and the estimated critical point and critical exponent $\nu\approx 1.510$ are accurate.

Furthermore, we observe that the second diagonal component of the QGT $g_{\phi\phi}$ also diverges at the critical point $\varepsilon_{c}^{*}$ due to gaplessness in the limit $L\rightarrow\infty$. As shown in Fig. LABEL:fig3(b), the values of $g_{\phi\phi}$ at the critical point diverge as $L$ increases, with a scaling dimension $\Delta_{\phi\phi}\approx 0.643$.

Finally, we delve into the imaginary part of the off-diagonal component of the QGT, denoted as the Berry curvature $\mathcal{F}_{\varepsilon\phi}$. As illustrated in Fig. 4(a), the corresponding scaling dimension is $\Delta_{\varepsilon\phi}\approx 1.0$. We obtatin the correlation length exponent and critical point by fitting the metric tensor into Eq. (11). Similarity, all Berry curvatures for different $L$ collapse into a single one in Fig. 4(b), which also suggests $\mathcal{F}_{\varepsilon\phi}$ is a universal function of $L^{1/\nu}|\varepsilon-\varepsilon_{c}^{*}|$.

To summarize, our numerical findings collectively affirm that the critical point of the parametrically-driven Kerr resonator aligns with the same universality class as the Rabi model [16].

As an additional note, combined with the previous scaling analysis, we have determined the scaling dimensions $\Delta_{jk}$ of the geometric tensor $Q_{jk}$. Referring to Eq.(6) in Ref. [51], we can derive the scaling dimensions for $\varepsilon$ and $\phi$ perturbations, respectively. The results reveal that $\Delta_{\varepsilon}=0.3375$ and $\Delta_{\phi}=0.6785$. Additionally, the scaling dimension of the Berry curvature is $2-\Delta_{\varepsilon}-\Delta_{\phi}=0.984$, closely matching the result obtained from the finite-size scaling $\Delta_{\varepsilon\phi}\approx 1.0$. Crucially, as $\Delta_{\varepsilon}\textless\Delta_{\phi}$, this implies that in the renormalization group sense, $\varepsilon$ perturbation is more relevant than $\phi$ perturbation. In other words, the driving parameter for the QPT is the $\varepsilon$ perturbation, consistent with the findings illustrated in our phase diagram.

When $K=0$, the model reduces to a two-photon driven bosonic mode [35]. A natural question is whether the inclusion of finite Kerr nonlinearity alters the universality class of the phase transitions. For this purpose, we specifically examine the case without Kerr nonlinearity. Diverging from the previous situation, the photon number cut-off $N_{{\rm{cut}}}$ serves as the effective system size to quantify the approach to the thermodynamic limit. At this time, the finite-size scaling relation of the first diagonal component of the QGT at the quantum critical point $\varepsilon^{*}_{c}=1$ can be written as $g_{\varepsilon\varepsilon}(\varepsilon^{*}_{\rm c})\propto N_{{\rm{cut}}}^{\gamma_{1}}$, where $\gamma_{1}$ signifies the scaling dimension of $g_{\varepsilon\varepsilon}$ concerning the photon number cut-off. The numerical result is depicted in Fig. 5(a), revealing a fitted scaling dimension of approximately $\gamma_{1}\approx 3.996$. Similarly, the finite-size scaling relation of the Berry curvature is given by $\mathcal{F}_{\varepsilon\phi}(\varepsilon^{*}_{\rm c})\propto N_{cut}^{\gamma_{2}}$, where $\gamma_{2}$ represents the scaling dimension of $\mathcal{F}_{\varepsilon\phi}$ with respect to the photon number cut-off. As illustrated in Fig. 5(b), the fitted scaling dimension is approximately $\gamma_{2}\approx 2.997$. For easier comparison, we write $g_{\varepsilon\varepsilon}$ and $\mathcal{F}_{\varepsilon\phi}$ as scaling relations for the average number of photons $\bar{N}$ for both $K\neq$ and $K=0$. Intuitively, when the photon number cutoff $N_{{\rm{cut}}}$ is increased, the average photon number of the system $\bar{N}=\langle a^{\dagger}a\rangle$ becomes larger, that is, $\bar{N}\propto N_{{\rm{cut}}}^{\alpha}$, as shown in the Fig. 5(c), where we find $\alpha\approx 0.998$. Then the finite-size relation in terms of average photon number $\bar{N}$ can be expressed as ($K=0$)

On the other hand, in the previous section IV.1, the scaling relation of $g_{\varepsilon\varepsilon}$ and $\mathcal{F}_{\varepsilon\phi}$ concerning effective system size $L=\delta/K$ are expressed as $g_{\varepsilon\varepsilon}\propto L^{\Delta_{\varepsilon\varepsilon}}$ and $\mathcal{F}_{\varepsilon\phi}\propto L^{\Delta_{\varepsilon\phi}}$, respectively. Moreover, the scaling relation between the average number of photons and the effective system size is $\bar{N}(\varepsilon^{*}_{\rm c})\propto L^{\Delta_{\bar{N}}}$. As shown in Fig. 5(d), $\Delta_{\bar{N}}\approx 0.33$. Combining the aforementioned scaling formulas, the finite-size scaling of $g_{\varepsilon\varepsilon}$ concerning the average photon number is determined as ($K\neq 0$)

Here $\beta_{1}^{\prime}$ and $\beta_{2}^{\prime}$ indicate the scaling dimension of $g_{\varepsilon\varepsilon}$ and $\mathcal{F}_{\varepsilon\phi}$ concerning the average photon number, respectively. With all the numerical results obtained, we ascertain the scaling dimension of the quantum metric tensor concerning the average number of photons $\beta_{1}^{\prime}=\Delta_{\varepsilon\varepsilon}/\Delta_{\bar{N}}\approx 4.04$, which is approximately equal to $\beta_{1}=\alpha\gamma_{1}\approx 3.99$. Similarly, the Berry curvature with respect to the average photon number yields $\beta_{2}^{\prime}=\Delta_{\varepsilon\phi}/\Delta_{\bar{N}}\approx 3.03$, roughly equal to $\beta_{2}=\alpha\gamma_{2}\approx 2.99$. These results indicate that finite Kerr nonlinearity does not alter quantum critical behavior and always belongs to the same universality class as the quantum Rabi model.