GHZ-like states in the Qubit-Qudit Rabi Model

The quantum Rabi model (QRM) describes the interaction between a two-level system and a single quantized harmonic oscillator mode. It is one of the most celebrated models in atomic physics for light-matter interaction [1]. In quantum technology, Rabi-like models are widely employed to describe the effective physics emerging in a variety of different contexts ranging from spintronics [2, 3] to trapped ions [4], and from circuit quantum electrodynamics (cQED) [5] to atom-superconducting qubit hybrid schemes [6]. Despite its simple form, the Rabi model was solved exactly only recently [7, 8]. The ground state of the quantum Rabi model consists of a non-classical highly entangled state of two-level system and bosonic mode [7, 9]. In cQED, different regimes of interaction between the two-level system and the bosonic field can be explored. In particular, weak and strong coupling regimes are routinely exploited for read-out and coherent state transfer [10]. Recent research has demonstrated the possibility of reaching the ultrastrong and deep strong coupling regimes, too [11, 12].

Here, we formulate and study a Rabi-type minimal model describing qubit-qudit interaction mediated by a single mode quantum bosonic field. This type of models has emerged recently in several studies of specific systems where atoms, solid state devices (such as superconducting or quantum dot qubits) are assembled together to form hybrid quantum networks [6, 13, 14, 15, 16, 17, 18, 19, 20]. In this context, entangling quantum systems of heterogeneous nature is sought intensively  [21, 22, 23], as such hybrid entangled states could become useful in converting quantum information between different encodings  [24].

We shall see that the physics of our model is particularly interesting in the ultrastrong coupling regime [25, 18, 26, 27, 28]. In particular, the ground state of the system turns out to be defining a Greenberger-Horne-Zeilinger (GHZ) entangled state. GHZ states present great significance among all types of multipartite entanglement [29]. These states exhibit maximal correlations between three or more quantum systems. GHZ states have been considered a key resource in fundamental physics since the early stages of quantum information. They have also been proven useful in various quantum technologies, including quantum error-correcting codes [30] and quantum metrology beyond the Heisenberg limit [31].

We point out that the generation of GHZ hybrid entanglement defines a challenging problem in quantum technology. In cQED, for example, GHZ hybrid entanglement has been achieved by state-dependent phase shift operations which involve complicated control and feedback sequences [32, 23]. In this context, exploration of the ultrastrong coupling regime has been demonstrated beneficial for GHZ state preparation [33, 34]. Indeed, our scheme guarantees a straightforward preparation of hybrid GHZ states, as such state could appear in the ground state of the QRM at large coupling strength.

The article is organized as follows: in Sec. II, we introduce a generalization of the quantum Rabi model to describe the qubit-qudit interaction through the resonator bosonic field. We demonstrate, through an analytical approach based on adiabatic approximation and perturbative expansion that hybrid GHZ states constitute the ground state solutions in the ultrastrong coupling limit. In Sec. III, we study the bipartite entanglement between the qubit and qudit mediated by the common resonator, quantified by negativity [35]. We show that the presence of the GHZ state induces an exponential suppression of the negativity for large values of the coupling strengths. Dynamical features are investigated in Sec. IV. In particular, we show the dynamics after quenching the coupling strength, and propose adiabatic state preparation schemes for the hybrid GHZ states. A short discussion of the main results of the manuscript is presented in Sec. V.

We investigate the physical system schematically pictured in Fig. 1. The scheme features a qubit (two-level quantum system) and a qudit ($d$-level quantum system) individually coupled to a common quantum resonator described by bosonic degrees of freedom. The Hamiltonian reads (here, and in the rest of this manuscript, we work in units $\hbar=1$)

Here, $\hat{\sigma}_{x,y,z}$ are the Pauli matrices for the qubit with transition frequency $\Omega_{1}$, $\hat{J}_{d}^{z,\pm}$ are the spin $(d-1)/2$ operators with level spacing $\Omega_{2}$, and $\hat{a}(\hat{a}^{\dagger})$ is the annihilation (creation) operator for the bosonic field with frequency $\omega$. The coupling strengths $g_{1,2}$ in Eq. (1) quantify the vacuum-Rabi splittings. Employing a jargon that is widely used in the literature, we will denote the qubit and the qudit as “artificial atoms”. For $d=2$, our model is equivalent (up to a sign convention) to the two-qubit quantum Rabi model [36, 37, 38, 39]. In contrast to the single qubit Rabi model, this generalized model is not integrable for general parameter values [37, 38].

The eigenvalues and eigenstates of $\hat{H}$ can be readily obtained numerically. Here, we devise analytical approximation schemes both in the weak-coupling and in the ultrastrong coupling regimes.

The weak coupling limit ($g_{1},g_{2}\ll\omega$), in the presence of strong qubit/qudit-resonator detuning ($\Omega_{1},\Omega_{2}\ll\omega$), can be treated by means of a Schrieffer-Wolff transformation [5, 40]. In particular, we apply the following unitary transformation to the Hamiltonian Eq. (1):

where we choose

In the weak coupling limit considered here $\epsilon_{i},\xi_{i}\ll 1$. In particular, at the lowest order in the expansion, the effective Hamiltonian $\hat{H}_{\rm eff}=\hat{V}\hat{H}\hat{V}^{\dagger}$ reads:

with renormalized frequencies ¹¹1Here our notations is slightly abusive, since $\tilde{\omega}$ contains the operators $\hat{\sigma}_{z},\hat{J}_{d}^{z}$. However, since we focus on the $N=0$ subspace of the resonator degree of freedom, this notation simplifies the subsequent discussion.:

and effective coupling $g_{\rm eff}=[g_{1}(\epsilon_{2}+\xi_{2})+g_{2}(\epsilon_{1}+\xi_{1})]/2$. In Eq. (5), $\hat{H}_{0}=\omega\hat{a}^{\dagger}\hat{a}-\frac{\Omega_{1}}{2}\hat{\sigma}_{z}+\Omega_{2}\hat{J}_{d}^{z}$ is the uncoupled Hamiltonian and $[...,...]$ denotes the commutator.

Within our approximation, the energy spectrum consists of different manifolds characterized by a fixed value of resonator photon number operator $\hat{N}=\hat{a}^{\dagger}\hat{a}$ (the interactions between different manifolds can be neglected due to the large resonator frequency compared to other energy scales). For the qubit ($d=2$), we have $(\hat{J}_{2}^{\pm})^{2}=0$, $(\hat{J}_{2}^{z})^{2}=1/4$, and the spectrum of the Hamiltonian in Eq. (5) can be found by diagonalizing a $4\times 4$ matrix consisting of two $2\times 2$ blocks. In the lowest manifold ($N=0$), the four eigenenergies are given by

In the general qudit case ($d\geq 3$), the eigenenergies can be obtained by computing the roots of the product of two degree $d$ characteristic polynomials of submatrices of dimension $2\times d$. Simplified expression can be obtained by neglecting the $(\hat{J}_{d}^{\pm})^{2}$ terms in Eq. (5), and performing an approximation similar to the standard rotating wave approximation $\hat{\sigma}_{x}(\hat{J}_{d}^{+}+\hat{J}_{d}^{-})\sim\hat{\sigma}_{+}\hat{J}_{d}^{-}+\hat{\sigma}_{-}\hat{J}_{d}^{+}$, valid for $\Omega_{1}\sim\Omega_{2}$. The approximate results in the $N=0$ subspace for the qutrit and the ququart are reported in Appendix A.

In the ultrastrong coupling regime, the numerical results to be presented below are corroborated by an analytical approach combining the adiabatic approximation in the displaced oscillator basis [42] and degenerate perturbation theory. More precisely, we first obtain the exact spectrum of the reduced Hamiltonian

neglecting the free Hamiltonian of the qubit and qudit in the limit where $\Omega_{1},\Omega_{2}\ll\omega$, and $g_{1},g_{2}\lesssim\omega$. Then, these terms are restored within a perturbative expansion. The eigenstates of $\tilde{H}$ are product states $\ket{\sigma\ m\ N_{\sigma,m}}=\ket{\sigma}\otimes\ket{m}\otimes\ket{N_{\sigma,m}}$. Here, $\sigma=\uparrow,\downarrow$ are the eigenstates of $\hat{\sigma}_{x}$ with eigenvalues $\pm 1$, $\ket{m=0,1,\dots,d-1}$ are the eigenstates of the qudit operator $(\hat{J}_{d}^{+}+\hat{J}_{d}^{-})$ with eigenvalues $\lambda_{m}=-(d-1)+2m$, and $\ket{N_{\sigma,m}}$ are displaced Fock states [42, 36, 37] (see Appendix B). The system yields a two-fold degenerate ground state, obtained from a displacement of the vacuum state in the resonator $\{\ket{\uparrow,+,0_{\uparrow,+}},\ket{\downarrow,-,0_{\downarrow,-}}\}$, with energy $E_{0}=-[g_{1}+(d-1)g_{2}]^{2}/\omega$, where $\ket{+}(\ket{-})$ is the eigenstate of the operator $(\hat{J}_{d}^{+}+\hat{J}_{d}^{-})$ with the largest(lowest) eigenvalue, i.e., $d-1(-d+1)$. The corrections to the spectrum of $\tilde{H}$ are then evaluated through perturbation theory in $\hat{H}^{\prime}=-\frac{1}{2}\Omega_{1}\hat{\sigma}_{z}+\Omega_{2}\hat{J}_{d}^{z}$. The lowest (second) order corrections to the energy are obtained as (see Appendix B and Appendix C for a detailed derivation):

where $\delta_{i,j}$ is the Kronecker delta ²²2We have verified that for $d=2$, when both equations (9) and (11) are applicable (that is, when taking $g_{i},\,\Omega_{i}\to 0$) they agree at next to leading order.. Notably, the two-fold degeneration of ground state is only resolved at $d$-order perturbation theory in $\hat{H}^{\prime}$, with a correction proportional to $\Omega_{1}\Omega_{2}^{d-1}$ (see Appendix C). The corresponding eigenstates read

where $\mathcal{C}$ is the normalization factor, and the functions $C_{\sigma,m}(g_{1},g_{2})\propto e^{-g_{1}^{2}/\omega^{2}},\,e^{-g_{2}^{2}/\omega^{2}}$ are exponentially suppressed with the coupling strengths. As discussed above, the analytical expression in Eqs. (11)-(12) are expected to hold in the regime where the free Hamiltonian terms of the atoms are treated as perturbations to the interacting system, i.e. $\Omega_{1},\Omega_{2}\ll\omega,g_{1},g_{2}$.

In Fig. 2, we display the low-energy spectrum of the Hamiltonian in Eq. (1) as a function of the coupling strength (with $g_{1}=g_{2}$) for $d=2$ (panels a,d), $d=3$ (panels b,e), and $d=4$ (panels c,f). For visualization purposes, we plot the energy as a function of $dg_{1}$ (with $d$ number of levels in the qudit), since the ground state energy for $g_{1}=g_{2}$ scales as $E_{0}/\omega\sim-(dg_{1}/\omega)^{2}$ for large values of $g_{1}$. The analytical expressions obtained in the low-coupling limit (dotted-dashed) and through perturbation theory in the ultrastrong coupling regime (dashed) are compared with the solutions obtained through numerical diagonalization (solid). In the low coupling regime, the analytical expression of Eq. (9) gives a very accurate description of the spectrum for $dg_{1}\lesssim 0.4\Omega$ (see Fig. 2a), and Fig. 2d)). For the general qudit case, the expressions obtained through RWA approximation reproduce the numerical results in a less satisfactory way. Still, they correctly reproduce the spectrum for $dg_{1}\lesssim 0.3\Omega$.

For $\Omega_{1}=0.15\omega,\Omega_{2}=0.1\omega$ (top panels), an excellent agreement between the strong coupling regime approximations and numerical solutions arises for $dg_{1},dg_{2}\gtrsim 0.3\omega$. For higher $\Omega_{1}$ and $\Omega_{2}$ values ($\Omega_{1}=0.55\omega,\Omega_{2}=0.5\omega$, bottom panels), the adiabatic approximation breaks down below $dg_{1},dg_{2}\sim 0.75\omega$ (Fig. 2b).

With increasing coupling strength $g_{1},g_{2}$, the higher order correction terms in Eq. (12) are suppressed exponentially, and the states $|\Psi_{\pm}\rangle$ approach the GHZ-type states. Note that the states $|0_{\uparrow,+}\rangle=D^{\dagger}(\frac{g_{1}+(d-1)g_{2}}{\omega})|0\rangle$ and $|0_{\downarrow,-}\rangle=D^{\dagger}(-\frac{g_{1}+(d-1)g_{2}}{\omega})|0\rangle$ are coherent states with opposite displacement in the phase space, which are asymptotically orthogonal in the limit $g_{1},g_{2}\rightarrow\infty$. As a result, the ground state under such large coupling assumption can be approximated as:

The validity of this approximation is investigated in Fig. 3, where the fidelity between the GHZ state and the ground state of the Hamiltonian obtained through numerical diagonalization is plotted as a function of the coupling strength $g_{1}=g_{2}$. In this manuscript, we define the fidelity between two pure states $\ket{\phi},\ket{\psi}$ as $\mathcal{F}=|\braket{\phi}{\psi}|$. In agreement with our perturbative calculation, the fidelity of the ground state with the GHZ state approaches the unit value in the limit $g_{1}\gg\Omega_{1},\Omega_{2}$.

In our scheme, it is interesting to investigate the bipartite entanglement between the qubit and qudit. We choose to compute negativity [35] as the measure of entanglement. For a bipartite pure state $|\varphi\rangle_{AB}$ in a $d\otimes d^{\prime}(d\leq d^{\prime})$ quantum system, the negativity is defined as

where $|\varphi\rangle_{AB}\langle\varphi|^{T_{B}}$ is the partial transpose of $|\varphi\rangle_{AB}\langle\varphi|$ and $\|\cdot\|_{1}$ is the trace norm. To extract bipartite pair-wise entanglement in a tripartite system, we use the reduced density matrix of $|\varphi\rangle_{ABC}$ on subsystem $A\otimes B$ by tracing over subsystem $C$: $\rho_{AB}=\textbf{tr}_{C}|\varphi\rangle_{ABC}\langle\varphi|$.

Figure 4 displays the density plot of the negativity as a function of the coupling strengths $g_{1},g_{2}$ in the qubit (Fig. 4a), qutrit (Fig. 4b), ququart (Fig. 4c) cases, for $\Omega_{1}=\Omega_{2}=0.1\omega$. Note that the negativity is clearly symmetric under the exchange $g_{1}\leftrightarrow g_{2}$ in the qubit case (since $\Omega_{1}=\Omega_{2})$, and it becomes gradually more asymmetric by increasing the number of levels. The bipartite entanglement between the two artificial atoms has a nontrivial response to the coupling strengths between subsystems. In particular the negativity is maximum for intermediate values of the couplings, and it is strongly suppressed at large couplings. The position of the maximum depends on the number of levels in the qudit and reads: $g_{1}=g_{2}\simeq 0.24\omega$ (qubit), $g_{1}\simeq 0.21\omega,\,g_{2}\simeq 0.17\omega$ (qutrit), $g_{1}\simeq 0.21\omega,\,g_{2}\simeq 0.14\omega$ (ququart).

For a better visualization, in Fig. 4d we consider cuts of Figs. 4a-c at a fixed value of $g_{2}=0.2\omega$. The negativity first rises to a maximum with increased coupling strength $g_{1}$, before decaying to zero exponentially (as we will discuss below). This phenomenon is reported in Ref. 44 where an approximate expression is derived to explain the curve at weaker coupling. Here, we obtain an analytical expression for the decaying curve. In addition, our approach demonstrates that the entanglement suppression is a consequence of the structure of the entanglement encoded in the ground state [see Eq. (13)]: the tripartite GHZ state at large coupling limit($g_{1},g_{2}\rightarrow\infty$), hinders the bipartite entanglement obtained after tracing out one of the subsystems, that asymptotically vanishes. This property of the GHZ states results in a counter-intuitive implication: the strong coupling can destroy entanglement between the two quantum systems connected by the resonator.

Now we show that the approximate expression of the ground state, i.e., the GHZ state of Eq. (13), leads to an accurate prediction for the entanglement at large coupling; we can easily calculate the negativity for those states. Indeed, the corresponding reduced density matrix $\rho^{\prime}$ with resonator degree-of-freedom traced out is a ($2d\times 2d$) matrix and has only four non-zero matrix elements:

with $K=\exp\{-2[g_{1}+(d-1)g_{2}]^{2}/\omega^{2}\}$. Therefore the analytical expression for the negativity of the ground state in Eq. (13) is

These approximate expressions are displayed (dashed) in Fig. 4. Note that the approximation describes very accurately the exponential decay of the negativity at large coupling.

We close the section by noting that the negativity measure is not a sufficient test of entanglement for systems with dimensions beyond $2\times 3$. Under such circumstances, a state with zero negativity could possibly be a PPT or “bound entangled” state, which is argued to be metrologically useful [45, 46, 47].

In this section, we discuss the dynamical evolution of the coupled system. We consider two complementary cases: the quench dynamics starting from the non interacting state and the adiabatic preparation of the GHZ state.

We formulated and studied a quantum Rabi type model describing the interaction between a two-level and a multi-level system mediated by a single mode bosonic field (in quantum technology such bosonic field is realized by a resonator). In the weak and in the strong coupling limits, we devised two different analytical schemes. In the weak-coupling limit, the effective Hamiltonian is obtained through a suitable Schrieffer-Wolff transformation. Assuming the qudit degenerating in a two-level system, the spectrum of the effective Hamiltonian has been obtained exactly. In the strong coupling limit, the ground state of the system is provided by a tripartite-entangled state of the GHZ type. A known feature of the GHZ states is that they do not allow bipartite entanglement between the three partners. As a result, qubit and qudit cannot be highly entangled, and the correlation between them drops exponentially with increasing coupling in the ultrastrong coupling regime. Such analysis is corroborated by the study of the negativity providing the sufficient condition for the qubit-qudit entanglement.

We analyzed the system dynamics both under quenching and adiabatic switching of the couplings between the atoms and the resonator. The non-adiabatic nature of the quantum quench leads to the generations of photons in the resonator, with a magnified effect by increasing the number of levels in the qudit. By this procedure, the GHZ states can be adiabatically prepared with high fidelity. Both the analysis of the spectrum and the dynamics indicate that the interaction is effectively magnified by the number of the levels of the qudit. The study of the dynamics gives preliminary information on the qubit-qudit coherent state transfer. Our work provides relevant information for applications in quantum technology, particularly for hybrid quantum networks design [24, 22, 21, 50]. The proposed scheme for the GHZ preparation may be implemented in cQED platforms, where both ultrastrong and deep-strong coupling regimes have been studied theoretically [51, 52, 53] and implemented experimentally [11, 54, 55, 12, 56]. Very recently, the ultrastrong coupling regime has been reached in hybrid semiconducting-superconducting technology [57], too.

The generalization of the Rabi model to the qudit system can be applied to discuss superconducting circuits based on transmon qubits. Indeed, the multi-level nature of the energy spectrum cannot be ignored in these elements due to the weak anharmonicity. These circuits have been extensively investigated, even in combination with semiconducting qubits [58]. On the theoretical side, our scheme can be investigated in a more general setting. A certainly interesting direction to go would be studying the system dynamics in presence of dissipation [59].