Topologically protected Bell-cat states in a simple spin model

A cat state is a coherent superposition of two or more quantum states that exhibit significant differences on a macroscopic scale. Initially introduced in a thought experiment by Erwin Schrödinger [1] to underscore the striking distinctions between quantum and classical physics, cat states became a useful concept in philosophical discussions about the interpretations of quantum mechanics. More recently, progress in controlling quantum systems at the level of single atoms and photons has led to the successful realization of cat states in experiments. These include the superposition of a single laser cooled ion in two spatially separated locations [2], and the superposition of two or more coherent states of photons localized in different regions of phase space in a light pulse [3], in microwave fields in cavities [4], and in circuit-QED systems [5]. Additionally, cat states have been created using the spin degrees of freedom of trapped ions [6], and also by placing the vibrational state of a trapped ion in a coherent superposition dependent on its internal spin [8, 9, 10]. Such entanglement between different types of degrees of freedom has also been achieved between the polarization of spatial mode degrees of freedom with photons [11].

This progress in our ability to generate and control quantum states has technological implications because cat states formed from a pair of coherent states have the potential to be used as qubits [12]. When sufficiently separated, each coherent state has exponentially small overlap, so they can form the basis states $|0\rangle$ and $|1\rangle$ of the qubit. Coherent states have significantly larger Hilbert spaces than traditional qubits which allows for the possibility of encoding other information within these superpositions, and in some cases, facilitating more effective error correction protocols.

A widely implemented technique for creating cat states involves the use of an ancillary qubit. The first step is to initialize the qubit in an equal superposition of its two states, which for an atomic qubit would be achieved by applying a $\pi/2$ laser pulse, and the second step is to entangle the qubit with a coherent state of a larger quantum system, thereby generating a cat state in the latter system. This can be accomplished by making the phase of the coherent state dependent on the state of the qubit. For example, a force applied to a mechanical oscillator that depends on the state of a qubit can generate a macroscopic superposition of two coherent states of phonons where each coherent state has a different phase [13], or the state of an atom resonantly coupled to an optical cavity can be used to apply a phase flip to incoming coherent pulses of light [14]. In numerous instances, the interaction between the ancillary qubit and the coherent state results in maximal entanglement between the two, giving rise to states known as Bell-cat (BC) states, see [15] and references therein (these states are also known by other names such as squeezed wavepacket entangled states [9]). BC states represent more complex counterparts to regular Bell states, which in their original setting are maximally entangled states only of pairs of qubits. The complexity arises from the fact that coherent states possess more degrees of freedom than a single qubit, and this has lead to BC states being employed in research exploring information extraction and entanglement generation beyond the scope of qubit Bell states [15]. The continuous nature of the relative phase between coherent states means BC states might find a role in the continuous variable form of quantum information processing [16].

In this paper, we show that a BC state can be created via adiabatic driving in a simple version of the Mermin central spin (CS) model [17, 18, 19, 20]. The CS model consists of a central two-state particle playing the role of the ancillary qubit surrounded by $N$ identical non-self-interacting two-state particles. The driving protocol makes use of two important features of the system: $(1)$ There is a topological phase transition where the topologically nontrivial phase hosts a pair of zero energy bound states in the Fock space of the identical spins; $(2)$ The inhomogeneity of the Fock space of the identical spins allows one to vary the locations of the bound states by tuning the strength of an the applied magnetic field.

Highly controllable versions of the CS model can potentially be realized experimentally using the physical systems already mentioned. For example, the two polarization states of photons means they can naturally play the role of the non-self-interacting two-state particles. Alternatively, the two topologically distinguished clockwise and anticlockwise orbital angular momentum states of photons in a structured Gaussian light beam can also act in this capacity [21, 22]. Coupling photons to a real atom [4], or a circuit-QED setup in the form of a superconducting transmon qubit in a microwave waveguide cavity resonator [5, 15], would allow the realization of the CS model, as would the spin-motion entanglement that can be utilized in trapped ion systems [9].

Yet another way to realize the CS model is using ultracold atoms, for example by immersing an atomic quantum dot in a two-state Bose-Einstein condensate (BEC), the latter also being known as a bosonic Josephson junction. The two states of the BEC atoms can either be two internal spin states [23] or two external modes (double well potential) [24, 25], and the atomic quantum dot can be a single impurity atom [26, 27, 28, 29, 30, 31] or ion [32, 33, 34] that also has either two internal or two external states. Feshbach resonances can be used to independently control the boson-boson interactions and impurity-boson interactions. Recent experimental realizations of the controlled coupling between individual localized Cs impurity atoms immersed in a Rb BEC in have been reported in references [35, 36, 37]. We also note that bosonic Josephson junctions can also be realized using structured light in an aberrated optical cavity [38].

The system we will be investigating is a version of the CS model which describes $N$ identical two-state particles interacting with a distinguishable central two-state particle. Due to each particle having access to two states, we will refer to them as spin-1/2 particles. The $N$ identical spins do not interact with each other, but interact equally with the CS leading to the Hamiltonian

where the CS operators are represented explicitly as Pauli matrices and the large spin is represented as the sum of the identical spins’ Pauli matrices $\hat{S}_{\pm}=\sum_{i}^{N}\hat{\sigma}_{\pm}^{i}$. The parameters $v$ and $w$ represent the CS spin-flip energy and the energy of the spin-exchange between the CS and identical spins, respectively. We will work in the eigenbasis of the $\hat{S}_{z}$ and $\hat{\sigma}_{z}$ operators, $\{|n,m\rangle\}$, where $n=(N_{\uparrow}-N_{\downarrow})/2$ is half the difference between the number of up and down identical spins and $m=\uparrow,\downarrow$ are the two states of the CS. This basis defines the Fock states of the system and physically speaking it gives the magnetization along the $z$ axis.

The usefulness of this basis comes from the fact that the CS model bears a close resemblance to the Su-Schrieffer-Heeger (SSH) model [7] which describes the electronic properties of a 1D polymer with alternating single and double bonds. Under this comparison the Fock state label $n$ is analogous to the unit cell label and $m$ is analogous to the intra-unit cell label. In addition, like the SSH model, the CS model undergoes a topological phase transition from a trivial phase when $v>w$ to a nontrivial phase when $v<w$ which is marked by the presence of a pair of topologically protected bound states with energy $E=0$. However, unlike the SSH model whose bound states only appear at the edges of the polymer, the CS model’s bound states can be centered at any value of $n$. Their protection relies on the chiral symmetry which the Hamiltonian in Eq. (1) obeys and is defined in terms of the chiral operator $\hat{\sigma}_{z}$ and the relation $\hat{\sigma}_{z}\hat{H}\hat{\sigma}_{z}=-\hat{H}$. The chiral symmetry means that $\hat{\sigma}_{z}$ transforms states with energy $E$ to states with energy $-E$ and since the bound states have $E=0$, they are eignenstates of $\hat{\sigma}_{z}$. This also means that each bound state exists entirely in either the CS $\uparrow$ or $\downarrow$ subspace and that any perturbations respecting the chiral symmetry will not destroy them.

In condensed matter systems, topologically protected bound states are well localized in the material. For now we will assume that the same is true for the protected bound states in the Fock space of the CS model and provide evidence for it later. The assumption of localization allows us to take a mean field approximation of the identical spins’ operators by taking their coherent spin expectation values

This approximation labels the states of the identical spins as points on the surface of a spin-$N/2$ Bloch sphere where $\theta$ and $\phi$ are the polar and azimuthal angles, respectively. The mean field approximation of Eq. (1) is

where $\bm{\hat{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ and the vector $\bm{d}(\theta,\phi)=\left(d_{x}(\theta,\phi),d_{y}(\theta,\phi),d_{z}(\theta,\phi)\right)$ has components

Equation (3) resembles a Bloch Hamiltonian of a two-band model describing a periodic lattice under the tight-binding approximation, however, there is one major difference. In a typical Bloch Hamiltonian the vector $\bm{d}$ is parameterized by the quasi-momentum which is $2\pi$-periodic, so it maps points from a circle (1D) or a torus (2D) to, in general, a 3D vector, whereas $\bm{d}$ in Eq. (3) maps points from the mean-field Bloch sphere to a 2D vector (because $d_{z}=0$). The azimuthal angle $\phi$ is indeed $2\pi$-periodic, so it plays the role of a quasi-momentum in the CS model, but the polar angle $\theta$ is not $2\pi$-periodic, so it cannot be considered as another quasi-momentum. Instead, it gives the ‘position’ in Fock space from the relation $n=\frac{N}{2}\cos\theta$.

In 1D lattice models which undergo a topological phase transition, the transition is quantified in terms of the winding number which answers the question, how many times does $\bm{d}$ wind around its origin as the quasi-momentum is cycled through its range once. Due to the fact that $\bm{d}$ maps one periodic surface to another, the winding number takes integer values and is nonzero in the topologically nontrivial phase. In the SSH model, the components of $\bm{d}$ can be written in a similar form to Eq. (4), but with $\theta$ fixed to $\theta=\pi/2$ and $\phi\to k$ where $k$ is the quasimomentum of the lattice:

Therefore, as $k$ varies by $2\pi$, $\bm{d}$ traces out a circle of radius $w$ in the $d_{x}$, $d_{y}$ plane centered at $(v,0)$. When $v<w$, the circle encloses the origin and the winding number is unity which signifies that the system is in the topologically nontrivial phase. The components of $\bm{d}$ for the CS model are more general than those of the SSH model, however, we can ask a similar question: how many times does $\bm{d}$ wind around the origin of the $d_{x}$, $d_{y}$ plane for any fixed value of $\theta$ as $\phi$ varies by $2\pi$. We answer this question with an explicit calculation of the winding number

where $\bm{\hat{d}}=\bm{d}/|\bm{d}|$. The topologically trivial phase is marked by $W=0$ which occurs when $v>w$ and the topologically nontrivial phase is marked by $W=1$ which occurs when $v<w$. However, due to the $\theta$ dependence of the winding number, only part of the Bloch sphere of the identical spins has nontrivial topology in the latter case. The nontrivial part lies in the range $\theta_{1}<\theta<\theta_{2}$ where $\theta_{1}=\arcsin\left(v/w\right)$ and $\theta_{2}=\pi-\arcsin\left(v/w\right)$ and outside of this range there is trivial topology. The boundary angles mark the polar coordinates on the Bloch sphere of topologically protected bound states [7] and we emphasize that the appearance of a single bound state at each boundary provides support for the bulk-boundary correspondence. The correspondence asserts that a bulk topological invariant, such as the winding number, determines the number of bound states at a particular boundary separating two regions of different topology. In the CS model, the presence of two boundaries is a consequence of the existence of two ‘edges’ in the Fock space of identical spins located at $n_{\mathrm{edge}}=\pm N/2$.

Figure 1 gives a geometric interpretation of the topological properties of the system. The vector extending from the sphere’s center to its surface is $\frac{2}{N}\langle\bm{\hat{S}}\rangle$ (scaled by $w$) and represents the mean field state of the identical spins while the vector in the $d_{x}$, $d_{y}$ plane is $\bm{d}(\theta,\phi)$ whose direction denotes the eigenstate of the mean-field Hamiltonian in Eq. (3). For a fixed value of $\theta$, as $\phi$ changes by $2\pi$, the vector on the sphere traces out a circle on its surface and the vector $\bm{d}(\theta,\phi)$ traces the same path projected directly down onto the $d_{x}$, $d_{y}$ plane. The $d_{x}$ coordinate of the center of the sphere is $v$, whereas the sphere’s radius is $w$. When $v>w$, the projected path fails to enclose the origin of the plane, causing $\bm{d}(\theta,\phi)$ to oscillate without fully winding as $\phi$ changes by $2\pi$. When $v<w$, there exist certain values of $\theta$ for which the trajectory projected onto the plane encircles the origin, causing $\bm{d}(\theta,\phi)$ to wrap around the origin once. The range of $\theta$ exhibiting nonzero winding is $\theta_{1}<\theta<\theta_{2}$, with the boundaries determined by the points where the projected trajectory intersects the origin of the plane. In Figure 1, the red circles represent the trajectories for $\theta=\theta_{1}$ and show that when $\phi=\pi$, $|\bm{d}(\theta_{1},\pi)|=0$ leading to the intersection of the trajectory with the origin and the closure of the energy gap, $E_{\mathrm{gap}}=2|\bm{d}|$, at that point. Although not shown, due to the spherical symmetry, the same thing happens at $(\theta,\phi)=(\theta_{2},\pi)$. Consequently, these points correspond to the locations of the protected bound states.

Figure 1 also gives a clear picture of how modifications to Eq. (1) affect the topological properties of the system. For instance, if the term $u\hat{\sigma}_{y}$ is introduced, then the center of the sphere will have its coordinates shifted to $(v,u)$ in the plane. The condition for the existence of a nontrivial topological region is that the radius of the sphere be larger than the distance from the origin to its center projected onto the $d_{x}$, $d_{y}$ plane which means the topological phase transition will occur at $w=\sqrt{v^{2}+u^{2}}$. Having nonzero $u$ also changes the direction from the origin to the sphere, so the gap will no longer close at $\phi=\pi$, but at the more general angle $\phi=\pi+\arctan(u/v)$. Note that such a change preserves the chiral symmetry of the Hamiltonian and in turn preserves the existence of protected bound states. Modifications that introduce terms proportional to $\hat{\sigma}_{z}$ break the chiral symmetry by lifting the projected trajectories out of the $d_{x}$, $d_{y}$ plane, so they are no longer forced to intersect the origin. In that case, the energy gap no longer closes and the protected bound states no longer exist.

We have demonstrated that in a CS model obeying chiral symmetry, a topological phase transition takes place. In the topologically nontrivial phase, a pair of protected bound states emerges within the Fock space of the identical spins surrounding the CS. Each member of this pair is coupled to a different state of the CS. Notably, these bound states exhibit a unique characteristic — their positions can be adjusted by tuning parameters in the system, distinguishing them from typical bound states found in many condensed matter systems. This distinction stems from the spherical geometry of the Hilbert space of identical spins, which differs from the circular and toroidal geometry found in the quasi-momentum of 1D and 2D periodic lattices, respectively.

The mobility of the bound states is pivotal in the protocol employed for generating BC states. Our approach involves having a large overlap between an initial spin coherent state and a target eigenstate of the Hamiltonian which will become the protected bound states, then adiabatically driving the system to induce the splitting of the bound states into a macroscopic superposition within Fock space. We achieve the large overlap by initializing the system in the topologically trivial phase where the bound states have the same form as the spin coherent state. To ensure an equal overlap with the two bound states, we initialize the CS in an equal superposition of its two states. The splitting is achieved by adiabatically driving the system through the transition point into the topologically nontrivial phase where the bound states become mobile. The driving results in the CS spin-up part of the state moving in one direction in Fock space while the spin-down part moves in the other direction. When the system is driven deep enough into the nontrivial phase, the two parts of the state have exponentially small overlap and a BC state is formed.

Although the version of the CS model we analyzed is quite simple, it can be expanded to be more general because the main criterion needed to generate a BC state is the presence of mobile topologically protected bound states. This requirement means that the Hamiltonian must obey chiral symmetry which leaves room for additional terms of the form $\hat{\sigma}_{x}f(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ and $\hat{\sigma}_{y}g(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ where $f$ and $g$ denote general functions of the identical spins’ operators. The zero energy eigenstates (there may be many pairs of such states depending on the functions) will also be eigenstates of the chiral symmetry operator, $\hat{\sigma}_{z}$, and will serve as the target BC state(s) in similar driving protocols to the one considered in this paper. In experiments, it may be impossible to neglect chiral symmetry-breaking terms. Two such terms that may appear in the CS model are $v_{z}\hat{\sigma}_{z}$ and $w_{z}\hat{\sigma}_{z}\hat{S}_{z}$. Separately they destroy the bound states, however, together it may be possible to tune the parameters of the system such that their contributions cancel for a given bound state, $v_{z}\hat{\sigma}_{z}+w_{z}\hat{\sigma}_{z}\langle\hat{S}_{z}\rangle_{B}=0$. Although further analysis is necessary to explore this possibility, it could potentially offer a workaround of certain persistent symmetry-breaking terms.