Fermion parity switches imprinted in the photonic field of cavity embedded Kitaev chains

TL;DR

This work addresses detecting finite-size topological features of a Kitaev chain embedded in a single-mode cavity. It uses exact diagonalization of an electron-photon Hamiltonian with a quantum Peierls substitution to compute the many-body spectrum, revealing ground-state parity switches at $μ_{ps}^1$ and photonic signatures such as peaks in photon number and dips/peaks in quadratures at those points. Ground-state properties are only weakly affected by the cavity, while excited states show strong dependence on the cavity frequency and anticrossings due to light-matter hybridization; squeezing of the photonic field accompanies these effects, and mean-field decoupling captures some trends but misses odd-photon probabilities. Overall, the study suggests a feasible optical route to read out finite-size topological features in cavity QED systems, linking Majorana-like physics to measurable photonic observables.

Abstract

We study a finite-length Kitaev chain coupled to a single mode photonic cavity. The topological phase of the finite-length Kitaev chain is characterized by the presence of fermion parity switching points that correspond to the degeneracy between even and odd parity ground states. Using exact diagonalization, we compute the many-body energy spectrum of the electron-photon Hamiltonian and we find that the ground state in the topological phase of the Kitaev chain is only weakly affected by the cavity coupling. This is in contrast with the excited states showing strong dependence on the cavity frequency. We find that the photon number and the photonic field quadratures peak at values of the chemical potential corresponding to parity switching points revealing a property of the finite-length Kitaev chain in the topological phase. This later finding suggests that quantum optics experiments could be used to detect topological features of the Kitaev chain embedded into a photonic cavity. Moreover, calculations of photonic quadratures reveal squeezed states that are both captured by the exact diagonalization technique and mean field decoupling. However, the mean field approach fails to correctly capture the photonic probability in the odd photonic states.

Figures (14)

• Figure 1: Kitaev chain embedded in a single mode photonic cavity with frequency $\omega_c$ and photonic vector potential $\mathbf{A}=(g/e)(a^{\dagger} +a)\mathbf{e}_x$, where $\mathbf{e}_x$ points along the chain and $g$ is the light-matter coupling constant. The Kitaev chain is defined by the hopping $t$, chemical potential $\mu$, superconducting gap $\Delta$ and the number of sites $N$.
• Figure 2: (a) Many-body energy spectrum of the Kitaev chain coupled to a single mode cavity as a function of cavity frequency $\omega_c$. The blue solid (red dashed) lines correspond to the even (odd) parity states obtained with ED of Eq. \ref{['eq:Hc']}. For comparison, the many-body spectrum of the isolated Kitaev chain ($g=0$) is also shown in gray dot-dashed lines. The green dotted vertical line corresponds to the gap $\Delta_{\text{g}}/t = 0.44$ of the isolated Kitaev chain. (b) Amplitude of the Majorana operators $\gamma_j^A = c^{\dagger}_j + c_j$ and $\gamma_j^B = i(c^{\dagger}_j - c_j)$ between the even $|\Psi^e_0\rangle$ and odd parity $|\Psi^o_0\rangle$ ground states. The black solid, red dashed and blue dotted lines correspond to the amplitude $|\langle \Psi_0^e|\gamma_j^A|\Psi_0^0\rangle|^2$ for the isolated Kitaev chain ($g=0$), and cavity frequencies $\omega_c/t = 3$ and $\omega_c/t = 0.15$. The pink solid, green dashed and gray dotted lines correspond to the amplitude $|\langle \Psi_0^e|\gamma_j^B|\Psi_0^0\rangle|^2$ for the isolated Kitaev chain ($g=0$), and cavity frequencies $\omega_c/t = 3$ and $\omega_c/t = 0.15$. The localization of the amplitude of the Majorana operators at the edges of the chain shows small changes as a function of frequency. The parameters of the system are: $\mu/t = 0.75$$\Delta/t = 0.2$, $g=0.25$, $N=11$, and $N_c = 20$.
• Figure 3: (a) Many-body energy spectrum, ground state energy subtracted, of the cavity-embedded Kitaev chain as a function of the chemical potential $\mu$. States with even (odd) parity are plotted with straight (dashed) lines. For $\mu/t < \mu_{ps}^1$, where $\mu_{ps}^1 = 2\sqrt{t^2-\Delta^2}\cos\left(\pi/(N+1)\right)$, the parity of the ground state alternates between even and odd, and a doubly degenerate ground state takes place at the parity switching points. The cavity frequency considered here is $\omega_c/t = 0.5$ and no considerable changes are found in the behavior of the ground state within the interval $0.1 < \omega_c/t < 10$. (b) Photon number at the ground state as a function of chemical potential for frequencies indicated in the figure. The photon number shows peaks at the parity switching points. Dotted vertical lines are introduced as a guide to the eye to indicate where the parity switches. The parameters of the system are the same as in Fig. \ref{['fig:ManyBodySpectrum']}.
• Figure 4: Photonic quadratures (a) $\langle X^2\rangle$ and (b) $\langle P^2\rangle$, with $X=(a+a^{\dag})/\sqrt{2}$ and $P=-i(a-a^{\dag})/\sqrt{2}$, computed at the ground state as a function of chemical potential $\mu/t$ for different cavity frequencies $\omega_c/t$, indicated directly in the plot. While the values of $\langle P^2\rangle$ decrease with increasing $\omega_c$ independently of $\mu$, $\langle X^2\rangle$ shows a richer behavior with respect to $\mu$ and $\omega_c$. Importantly, the quadratures also show particular features at the parity switching points $\mu_{ps}$, dips in $\langle X^2\rangle$ and peaks in $\langle P^2\rangle$. For $\mu/t \gtrsim 2$ corresponding to the ground state with a well defined parity, the values of the quadratures approach $1/2$ as in the uncoupled case. Dotted vertical lines are introduced as a guide to the eye to indicate where the parity switches. The parameters of the system are the same as in Fig. \ref{['fig:ManyBodySpectrum']}.
• Figure 5: Probability $P_n(\Psi)=|\langle n| \Psi \rangle|^2$ to find the light-matter state $|\Psi\rangle$ in the photon state $|n\rangle$ at frequencies $\omega_c/t= 0.35$ (a) and $\omega_c/t = 2.0$ (b). The light-matter states considered are: the even and odd parity ground state ($|\Psi_0^e\rangle$ and $|\Psi_0^o\rangle$) and the excited states $|\Psi_1^e\rangle$ and $|\Psi_1^o\rangle$. Notice that the probabilities $P_{n=0}(\Psi_0^e)$ and $P_{n=0}(\Psi_0^o)$ are approximately one and weakly dependent on frequency. On the other hand, the probabilities $P_{n=0}(\Psi_1^{e(o)})$, $P_{n=1}(\Psi_1^{e(o)})$, and $P_{n=2}(\Psi_1^{e(o)})$ change considerably with frequency. The photon probabilities computed with MFD are also shown in the figure with empty diamonds (squares) for the even (odd) parity ground state. The parameters of the system are the same as in Fig. \ref{['fig:ManyBodySpectrum']}.