Hybrid quantum-state joining and splitting assisted by quantum dots in one-side optical microcavities

Quantum state joining has been recently experimentally demonstrated [C. Vitelli et al., Nat. Photon. 7, 521 (2013)] which can transfer two input photonic qubits into a photonic ququart. Here, we revisit these processes from a hybrid point of view. By exploring the giant optical circular birefringence induced by quantum-dot spins in one-sided optical microcavities, we introduce some deterministic joining schemes including two quantum-dot spin joining, hybrid photon and quantum-dot spin joining, and two-photon joining. The input quantum information is represented by one photon with polarization and spatial mode degrees of freedom (DOFs). These schemes are also adapted to the inverse processes called quantum state splitting because all the joining procedures are unitary and do not require projection and feed-forward steps. The fused photon is convenient for realizing elementary logic gates such as the controlled-not (cnot) gate, swap gate, and Toffoli gate. These hybrid fusion and splitting schemes provide flexible synthesis of the quantum-dot spin and photon in quantum applications. The transmission superiority of photons and storage superiority of the quantum-dot spin may be combined for quantum network communication or quantum computations.

Figure 1

(a) A charged QD inside an one-sided micropillar microcavity interacting with circularly polarized photons. ${\widehat{a}}_{\mathrm{in}}$ and ${\widehat{a}}_{\mathrm{out}}$ are the input and output field operators of the waveguide, respectively. (b) Optical selection rules due to the Pauli exclusion principle.

Figure 2

(a) Quantum spin fusion in a postselection mechanism. ${\mathrm{cPS}}_i$ represent the circularly polarizing beam splitters that transmit the right circularly polarizing photon $|R\rangle$ and reflect the left circularly polarizing $|L\rangle$, respectively. $a_i$ denote the spatial modes of the auxiliary photon $A$. $e_i$ denote quantum-dot spins with initial states $|{\psi }_i\rangle$. $H_i$ denote the half-wave plates to perform the Hadamard transformation $|R\rangle \to \frac{1}{\sqrt{2}}\left(\right|R\rangle +|L\rangle )$ and $|L\rangle \to \frac{1}{\sqrt{2}}\left(\right|R\rangle -|L\rangle )$. $X_i$ denote the wave plates to perform the polarization flip transformation $|R\rangle \langle L|+|L\rangle \langle R|$. ${\mathrm{cBS}}_i$ represent the 50:50 beam splitters to perform the Hadamard operation $|a_1\rangle \to \frac{1}{\sqrt{2}}\left(\right|a_1\rangle +|a_2\rangle )$ and $|a_2\rangle \to \frac{1}{\sqrt{2}}\left(\right|a_1\rangle -|a_2\rangle )$ on the spatial mode DOF of the photon. (b) Quantum spin disentangling. $W_i$ denote the Hadamard transformation $|\uparrow \rangle \to |+\rangle =\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle +|\downarrow \rangle )$ and $|\downarrow \rangle \to |-\rangle =\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle -|\downarrow \rangle )$ performed on the spins. If there are two input lines of one cavity, the photon represented with red lines passes through the cavity first, and then the photon represented with black lines passes through the cavity.

Figure 3

(a) Hybrid joining scheme based on the measurement feedback. $e^{\prime }$ denotes an auxiliary quantum-dot spin in the state $|+\rangle$ located in the cavity ${\mathrm{Cy}}_1$. ${\mathrm{cPS}}_i,H_i,X_i,{\mathrm{cBS}}_i$, and $W_i$ are the same as those defined in Fig. 2. (b) Schematic disentangling. If there are two input lines of one cavity, the photon represented with red lines passes through the cavity first, and then the photon represented with black lines passes through the cavity. If there are three input lines of one cavity, the photon represented with red lines passes through the cavity first, and then the photon represented with purple lines passes through the cavity while the photon represented with black lines passes through the cavity last.

Figure 4

(a) Photon fusion based on the measurement feedback. $a_i$ denote the spatial modes of an auxiliary photon $A$ in the state $|R\rangle |a_1\rangle$. $e_i$ denote the auxiliary quantum-dot spins in the state $|+\rangle$. ${\mathrm{cPS}}_i,H_i,X_i,{\mathrm{cBS}}_i$, and $W_i$ are the same as those defined in Fig. 2. (b) Schematic disentangling circuit. If there are two input lines of one cavity, the photon represented with the red lines passes through the cavity first, and then the photon represented with the black lines passes through the cavity. If there are three input lines of one cavity, the photon represented with the red lines passes through the cavity first, and then the photon represented with the purple lines passes through the cavity while the photon represented with the black lines passes through the cavity last.

Figure 5

Elementary logic gates on one photon state with two DOFs. (a) cnot gate. The polarization qubit is the controlling qubit while the spatial qubit is the target qubit. (b) cnot gate. The polarization qubit is the target qubit while the spatial qubit is the controlling qubit. (c) swap gate. ${\mathrm{cPS}}_i,H_i,X_i$, and $W_i$ are the same as those defined in Fig. 2. (d) Toffoli gate. The photon with two DOFs is the controlling qubit while the electron spin is the target qubit. (e) Toffoli gate. The photon with two DOFs is controlling qubit while the photon with only a polarization DOF is the target qubit. ${\mathrm{cPBS}}_i$ represents the polarizing beam splitter in the circular basis, which transmits the photon in polarization $|L\rangle$ and reflects the photon in polarization $|R\rangle$, respectively.

Figure 6

(a) The average fidelity $F_{SS}$ of spin joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. (b) The average fidelity $F_{PS}$ of photon-spin joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. (c) The average fidelity $F_{PP}$ of photon joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. The coupling strength is defined by $\varsigma =0.1{\eta }_s$.

Figure 7

(a) The efficiency $P_{SS}$ of spin joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. (b) The efficiency $P_{PS}$ of photon-spin joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. (c) The efficiency $P_{PP}$ of photon joining versus ${\eta }_s/\eta$ and $g/(\eta +{\eta }_s)$. The coupling strength is defined by $\varsigma =0.1{\eta }_s$.