Probing quantum nonlinearity of cavity-QED systems with quantum light

Giant optical circular birefringence (GCB) induced by a single quantum-dot-confined spin in an optical microcavity finds wide applications in quantum and optical technologies, such as quantum gates, quantum transistors, quantum repeaters, quantum routers, etc. If the system is probed with a classical light, such as the laser light where the photons are uncorrelated with each other, then the single-photon GCB is detected. In this work we develop a general approach to investigate the many-body dynamics of $n$ photons bound to one quantum emitter and apply this method to calculate the $n$-photon GCB probed with quantum light in Fock states. With suppressed atomic saturation, quantum light loads photons into dressed states more efficiently than classical light such that the whole Jaynes-Cummings energy ladder and the quantum nonlinearity related to the multiphoton transitions can be observed. The $n$-photon GCB lying at the cavity mode resonance allows the spin-cavity quantum gates and quantum transistors to extend from single-photon to $n$-photon operations and generate entangled photonic Fock states, i.e., the NOON states which are useful for quantum metrology and lithography.

Figure 1

Structure of the spin-cavity unit. (a) A negatively charged QD is embedded in a double-sided pillar microcavity with two symmetric distributed Bragg reflectors as the end mirrors. The pillar cross section is made circular with diameter bigger than the wavelength so that the pillar cavity supports the propagation of longitudinal circularly polarized light. (b) The spin selection rules for optical transitions in a charged QD: A photon in the $|L\rangle$ state couples to the transition $|\uparrow \rangle \leftrightarrow |\uparrow \downarrow \Uparrow \rangle$ only, whereas a photon in the $|R\rangle$ state couples to the transition $|\downarrow \rangle \leftrightarrow |\downarrow \uparrow \Downarrow \rangle$ only due to the conservation of angular momentum and the Pauli exclusion principle.

Figure 2

Calculated reflection and transmission spectra probed with classical light. GCB is manifested as the different reflection/transmission coefficients between the hot and cold cavity. (a) Reflection/transmission spectra of a hot cavity at different input light powers. (b) Reflection/transmission spectra of a cold cavity. Note that these spectra are independent of the input light power. (c) Saturation spectra of a quantum dot strongly coupled to the cavity at different input light powers. (d) Resonance fluorescence spectra of a quantum dot strongly coupled to the cavity driven at the cavity resonance frequency at different input light powers. The dotted lines are a guide to the eye.

Figure 3

Calculated reflection and transmission spectra probed with quantum light in Fock states. GCB is manifested as the different reflection/transmission coefficients between the hot and cold cavity. (a) Reflection and transmission spectra of a hot cavity at different photon number $n$. (b) Reflection and transmission spectra of a cold cavity. These spectra are $n$ independent. (c) The reflectance and transmittance at $\omega ={\omega }_c$ and (d) the gate fidelity versus the photon number $n$. (e) The reflectance and transmittance at $\omega ={\omega }_c$ and (f) the gate fidelity versus the QD-cavity coupling strength $g$ normalized by the cavity loss rate.

Figure 4

Two applications of the $n$-photon GCB probed with quantum light in Fock states. (a) A deterministic photon-Fock-state-spin entangling gate, with which $n$ photons in Fock state get entangled with the spin in a deterministic way. (b) A spin-cavity quantum transistor. First, the spin is initialized to $|+\rangle =\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$, and the gate photon is prepared in an arbitrary state $|{\psi }^{\mathrm{ph}}\rangle =\alpha |R\rangle +\beta |L\rangle$; second, the gate photon state is rotated to $\alpha |1,R\rangle +r_h^{\left(n\right)}\left({\omega }_c\right)\beta |1,L\rangle$; third, the gate photon state is transferred to the spin by measuring the photon in the $\left\{\right|H\rangle ,|V\rangle \}$ basis in the transmission and reflection ports; fourth, the spin state is transferred to $n$ photons in $|L\rangle$ (or $|R\rangle$) by measuring the spin in the $\left\{\right|+\rangle ,|-\rangle \}$ basis. As a result, an arbitrary quantum state enclosed on the gate photon is transferred (or “amplified”) to $n$ photons. PBS-1 and PBS-2 (polarizing beam splitters), D1–D4 (single photon detectors).