Cavity-induced switching between Bell-state textures in a quantum dot

The optical control of structural, vibrational, and transport properties in nanostructure systems has led to new research areas at the intersection of quantum optics and condensed matter [1, 2]. It has been shown that ultra-strong coupling (USC) between light and matter is crucial for exploring quantum states of matter [3, 2], quantum phase transitions, manipulating crystal structures and symmetry-breaking [4], obtaining tunable light-matter entangled states [5], including Floquet-Bloch states [6]. Quantum electrodynamic microcavities in solid-state systems are widely utilized to entangle photon modes and condensed matter states, leading to the creation of emergent phases with enhanced non-local correlations and the development of complex quantum materials [1, 2]. This interplay between cavities and atoms enables the control of spin states using resonant optical fields [7]. Additionally, the introduction of a magnetic field or quantized polarized light can trigger chiral and topological effects [8], which have promising applications for magnetic data storage technologies [9].

Nanostructures such as semiconductor quantum dots (QDs) represent promising solid-state platforms in a wide range of contexts, including the fundamental study of strongly correlated many-body systems [10] and applications as ideal solid-state sources of highly coherent light [11, 12]. QDs integrated into cavity systems, have attracted a lot of attention as candidate quantum emitters. Extensive research has also been conducted on the effects of magnetic fields on nanostructures, with a specific focus on spin-dependent interactions [13, 14]. Zeeman–Spin-Orbit-Coupling (SOC) competition, which is influenced by magnetic fields, is a key factor in controlling electronic behavior [15, 16, 17]. Additionally, topological insulator states in quantum matter have emerged as a result of SOC effects in few-particle systems [18].

Recent theoretical calculations have shown that the optical and transport properties of QDs can be modified by external magnetic fields [19, 20], and the SOC’s strengths can be tuned by gate electric fields and in-plane magnetic fields [21, 22]. In particular, the manipulation of single and two-spin electron qubits can benefit from the generation and quantification of individual electron correlations [23, 24]. The potential transformation of semiconductor structures into bits of quantum information processing, depends on the ability to trap electrons in QDs and regulate their quantum states [25]. Studies in optical spectroscopy have shed light on related processes [26, 27] leading to the presence of stable topological properties such as spin vortices [16, 17]. However, a microscopic knowledge of the spatially inhomogeneous QD electronic spin state, which is essential for regulating their mesoscopic behavior, is still lacking.

Experiments have demonstrated the successful coupling of charge and spin degrees of freedom in semiconductor QDs with visible and microwave photon cavities [26, 27]. In particular, the polarization (linear or circular) of confined light can be controlled in an efficient way by means of semiconductor-superconductor cavities [27, 28, 29], demonstrating anticorrelation electron effects. The main focus is to understand electron spin textures in semiconductor nanostructures [30], as well as achieving high-sensitivity spin sensing using tip-based technologies such as scanning tunneling microscopy (STM) [31, 32]. In particular, double-tip STM [33] and spin-polarized STM (SP-STM) are powerful techniques that can explore the correlation between electrical and magnetic properties with atomic resolution, and provide measurements to probe non-local spatial spin electron correlations [34]. As a result, the non-local spin-photon optical properties in an hybrid QD-cavity system effectively open the way to studying quantum gates for quantum information.

In this paper, we analyze the interplay between the electron SOC, the magnetic field, and the cavity coupling in a QD system comprising two interacting electrons. Though obviously simplified in terms of real material complications, our purpose is precisely to focus in on the interplay of these terms – and just these terms – in the system’s Hamiltonian and hence its quantum properties. In the presence of dominant Dresselhaus SOC and mixed Dresselhaus-Rashba SOC, our main interest is on the quantum correlations between matter and light states. We show that SOC causes the development of new polariton modes as a result of the decoupling between the cavity and the two-electron center of mass (CM) mode. By adjusting the magnetic field and the cavity coupling, we tune the spin states into resonance, and demonstrate the entanglement of the relative modes with the cavity. We also demonstrate how tuning the cavity coupling controls the distribution of the Bell spatial states and the spin correlation for the two electrons.

We show that: (1) highly spin-topological properties can be tailored by the microwave cavity field; (2) the violation of the Kohn theorem (KT) [35] is shown to be more pronounced in the USC regime and can be adjusted by the microwave cavity field; and (3) besides their fundamental interest, our results demonstrate that measurements of the ground state (GS) transitions enable tuning of correlated spins and the generation of spatially inhomogeneous Bell states.

The paper is organized as follows. Firstly, in Section II the basic theoretical background is provided for incorporating hybrid SOC-cavity effects in an interacting two-electron QD, focusing on the central quantities of interest: the charge density and the spin correlations. Subsequently, in Section III we present the results for two cases: (A) Dresselhaus SOC alone and (B) mixed Dresselhaus-Rashba SOC. Finally, in Section IV we discuss the numerical resolved spin Bell state distribution in a QD. Our main conclusion are summarized in Section V and the Appendices contain particular details of our calculations.

The hybrid QD-cavity system is schematically shown in Fig. 1a. Two electrons are confined in a parabolic two-dimensional QD embedded in a single-mode cavity, in the presence of a perpendicular magnetic field $B\hat{z}$. The Hamiltonian of the total system (with $c=\hbar=1$):

where $\sigma_{k}^{(j)}$ with $k=\{x,y,z\}$ refers to a Pauli matrix for particle $j$.

The harmonic QD is characterised by energy $\omega_{o}$ and length $\ell_{o}=[1/(m^{*}\omega_{o})]^{1/2}$. The electron-electron interaction, usually taken as the Coulomb repulsion, is here incorporated through an effective potential $V_{e-e}(|\vec{r}|)=\frac{\alpha}{r^{2}}$ (where $\vec{r}=\vec{r}_{1}-\vec{r}_{2}$ is the two-electron relative coordinate and $\alpha$ is an adjustable parameter to reproduce the Coulomb results), whose higher-order power mimics the presence of mirror charges in the semiconductor QD. This effective dipole-like interacting two-electron model has attracted significant interest in the literature since, despite its simplicity, it yields nontrivial results which are comparable with the bare Coulomb interaction results. Without including SOC and cavity effects, an analytical solution in the presence of an external magnetic field is obtained [13], which depends on Landau levels $n_{cm}$ ($n_{r}$) and angular momentum quantum number $m_{cm}$ ($m_{r}$) of CM (relative) space and yields analytic matrix elements, improving the efficiency of numerical results and allowing the use of numerical exact diagonalization. Moreover, this model has been extended to explore new physical results in multi-electron QDs [36, 37], Quantum Hall systems [38, 39] as well as nuclear-electron quantum logic gates [40].

The external magnetic and cavity fields are included through the vector potentials which determine the kinetic momentum $\vec{\Pi}_{j}=\vec{p}_{j}+e\vec{A}^{(c)}_{j}+e\vec{A}^{(q)}$. In what follows, we choose the symmetric gauge vector potential $\vec{A}^{(c)}_{j}=\frac{B}{2}(-y_{j},x_{j},0)$. The single-mode cavity radiation which couples to matter through the electric dipolar approximation, is represented by $\vec{A}^{(q)}=A(a^{\dagger}\vec{\epsilon}^{*}+a\vec{\epsilon})$ and is independent from the particle label $j$, and $a(a^{\dagger})$ is the annihilation (creation) operator of a single-mode photon with energy $\omega_{c}$. We consider a right circularly polarized light (CPL) field as specified by $\vec{\epsilon}=\frac{1}{\sqrt{2}}(1,i,0)$. In addition, the magnetic field causes Zeeman splitting which is given by $\Delta_{Z}=g_{L}\mu_{B}B$, where $g_{L}$ is the Landé factor and $S_{z}=\frac{1}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})$.

Only the CM degrees of freedom are connected to the cavity due to dipolar interactions (see Appendix A). However, nonzero SOC violates this idea since it introduces a link between relative modes and light. Resonance between photon modes and the spin Zeeman splitting is required ($\omega_{c}=|\Delta_{z}|$), which makes this violation stronger. In the presence of electron-electron interactions, the relative energy levels of the non-interacting case ($g=\lambda_{D}=\lambda_{R}=0$) exhibit crossings due to the conservation of total angular momentum $L_{z}$. Small energy differences between relative states thus resonate with spin and radiation (indirectly). Additionally, cavity-mediated SOC introduces interactions between the resonant spaces and motivates our investigation of the relatively unexplored optical properties, where the allowable transitions are determined by the SOC type and the type of CPL.

The non-interacting crossings, depicted by the dashed curves in Figs. 1b and 1c, experience splittings due to cavity-mediated SOC among states sharing the same conserved quantity (see Appendix A.1). Specifically, when considering only the Dresselhaus SOC, the first terms in Eq. 2 result in the conserved quantity $J_{D}=N+L_{z}-S_{z}$, where $N$ represents the photon number operator. Eq. 2a gives rise to the two splittings (shown by more blue colors in Fig. 1b). The Fock light states $\ket{0}$ and $\ket{1}$ have weights in the ground and excited states that are involved in the anticrossings, respectively. Moreover, interactions with CM states produce asymmetries in the splittings with respect to the non-interacting crossings (dashed curves). When states intersect along the blue curves at $\omega_{b}=1.78$ in Fig. 1b, the cavity (through Eq. 2c) introduces an anticrossing depicted in more red colors and shifts the existing splittings. Consequently, relative-spin-photon polaritons with $J_{D}=-1$ result from Dresselhaus SOC anticrossings in resonance. Larger state transitions are allowed by considering both SOCs, leading to splitting processes between states with different $J_{D}$ values.

In order to assess spin textures, we resort to the calculation of the spatial variation of total spin projections at two diametrically opposite positions around the QD origin. This means that the CM position is fixed at $\vec{R}=\frac{1}{2}(\vec{r}_{1}+\vec{r}_{2})=0$, while $\vec{r}$ is varied. This way of detection allows us to observe changes in relative coordinates due to the cavity. Specifically, the relative density and spin field of the ground state $\ket{GS}$ are defined respectively as:

where the traces run over the spin and photon spaces and total spin in the $k$ direction: $S_{k}=\frac{1}{2}(\sigma^{(1)}_{k}+\sigma^{(2)}_{k})$. The photon density $\rho_{ph}=\sum_{i,j}C_{ij}|n_{i}\rangle\langle n_{j}|$ is obtained by tracing over matter states. Bell state densities consider the spin states: singlet $\ket{\phi_{-}}=\ket{A}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$, the symmetric state of the triplet $\ket{\phi_{+}}=\ket{S}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})$, and the linear combinations of triplet states $\ket{\psi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{\uparrow\uparrow}\pm\ket{\downarrow\downarrow})$.

As a concrete example, we consider a two-electron GaAs QD, with relatively small Zeeman interaction but strong SOC, immersed in a microwave cavity. The values of the QD parameters are as follows: $m^{*}=0.067m_{e}$ is the electron effective mass, $g_{L}=-0.44$ is the Landé g-factor [41] and $\ell_{o}=50$ nm ($\omega_{o}=0.45$meV) is the confinement length. Cyclotron frequency $\omega_{b}=1.9$ is chosen and SOC strength is changed from zero to $\lambda=0.3$ [41]. Cavity coupling is tuned up to $g=0.3$, which corresponds to the USC regime with $g/\Omega\sim 1$. $\Omega=\omega_{c}+2\frac{g^{2}}{\omega_{o}}$ is the effective radiation frequency associated with the diamagnetic term $|\vec{A}^{(q)}|^{2}$. SOC effects from Dresselhaus interaction ($\lambda_{D}\neq 0$), which is characterized by a conserved quantity, and the Dresselhaus-Rashba combination ($\lambda_{D}=\lambda_{R}\neq 0$) are discussed separately using a numerically exact diagonalization.

Energy level convergence was ensured through a two-step process. Firstly, a basis of uncoupled oscillators was formulated for the Hamiltonian, taking into account the dipolar interaction with the CM coordinates through a Bogoliubov transformation, which is explained in Appendix A.2, and the maximum quantum numbers were specified. Secondly, the Hamiltonian matrix was reorganized into blocks based on conserved quantities, which provided us with both computational and analytical advantages. In fact, the ground state’s optoelectronic properties exhibit distinct trends due to the presence of symmetries.

The transitions in the QD’s total spin field described in the previous sections provide evidence of the localization of spin Bell states. The possibility of observing spin-entangled states is controlled by tuning SOC strengths, due to resonance between spin and relative states. However, the presence of the cavity is crucial for shaping these spin states since it facilitates new transitions through jumps between Fock states. These transitions exhibit a staircase-like pattern of correlations (see Appendix B) which depends on $g$ and leads to modifications in the densities of the Bell states.

The cavity successfully maintains the trend of $\langle S_{z}\rangle<0$ due to the Dresselhaus SOC, enabling transitions at lower values of $\lambda_{D}$ (see Fig. 2a). Fig. 4a illustrates how the singlet state begins to transition into a state with a higher weight of $\ket{\downarrow\downarrow}$. Then, in Fig. 4b, the densities of the $\ket{\psi_{-}}$ ($\ket{\phi_{+}}$) exhibit an asymmetric emergence, with radial localization at $r=2$ and maximum values at angles of $0^{\circ}$ ($90^{\circ}$) caused by the mixture with relative modes. The cavity-mediated SOC also introduces contributions from other triplet spin states, resulting in uncorrelated spins and an increased probability density of finding the state $\ket{\phi_{+}}$ in Fig. 4c. While it possesses a clear azimuthal symmetry, local spin transformations [24] could enhance its strength and facilitate its detection.

Due to the lack of a conserved quantity in the presence of both SOCs, the densities of the Bell states show azimuthal asymmetry while spin correlations are smoothed and tend to zero as the couplings increase. Transitions to the $\ket{\downarrow\downarrow}$ and $\ket{\uparrow\uparrow}$ (caused by Rashba SOC) are facilitated by the cavity at lower $\lambda$ despite the SOCs competition. As seen in Fig. 5b, the singlet state is concentrated at coordinates $\pm(1.75,-1.75)$, and with increased $g$-coupling, the probability to get $\ket{\psi_{\pm}}$ states improves considerably. In Fig. 5b, the non-homogeneous localization of these states is concentrated in distinct peaks $\pm(1,-1.6)$ for $\ket{\psi_{-}}$ and $\pm(1.6,-1)$ for $\ket{\psi_{+}}$. The different localization within the QD is attributed to the exclusive combination of each $\ket{\psi_{\pm}}$ states with light and relative states, demonstrating spatial spin correlations in microcavities. In Fig. 3c, the probability of finding the $\ket{\phi_{+}}$ state increases, exhibiting an azimuthally symmetric density concentrated at $r=1.5$. The presence of triplet states at large $g$ is explained by the direct coupling of light with these states in Eq. 2c, and it is consistent with the trend towards uncorrelated spins. Additionally, densities demonstrate a loss of asymmetry effect, as illustrated also in Appendix C, revealing an increasing of confinement as a result of strong couplings with the cavity and mixture with a greater number of light states.

The interaction of a two-electron QD with cavity coupling, magnetic fields, and spin-orbit coupling has been investigated in order to understand how these factors contribute to the formation of polariton states and complex spin-spatial correlation inhomogeneities. The cavity-mediated spin-orbit-coupling enables interactions between the center-of-mass and relative modes of the cavity, resulting in mixed light-matter states. These states provide measurable evidence of Kohn’s Theorem violation, where the Coulomb hole is modified by the cavity and electron-electron interactions alter the light states. Furthermore, the control of spin correlations can be attributed to the mediation in the radiation-matter coupling.

Dresselhaus spin-orbit-coupling, on one hand, gives rise to staircase-like spin correlations due to its connection with transitions involving the conserved quantity $J_{D}$. On the other hand, the inclusion of Rashba spin-orbit-coupling leads to the smoothing of correlations and a tendency towards uncorrelated spins, caused by the inclusion of virtual spin-radiation transitions and subsequent interactions with matter. The change in correlations is linked to the generation of Bell states with inhomogeneous spatial distributions. In this way, quantum dots in microwave cavities can potentially be used as an experimental platform for full solid-state Bell tests [42], where the detection of light states reflects the spin properties. Additionally, the flexible control and measurement of spin states contribute to the development of protocols for quantum information processing with few-particle systems, the generation of entangled systems, and applications in spintronics [15].

S.S.B-R, F.J.R. and L.Q. are thankful for financial support from Facultad de Ciencias- UniAndes Projects No. INV2021-128-2292 and INV-2023-162-2833, and Universidad de los Andes High Performance Computing (HPC) facility. N.F.J. is supported by U.S. Air Force Office of Scientific Research awards FA9550-20-1-0382 and FA9550-20-1-0383.