Nondestructive Rydberg parity meter and its applications

Parity meters, utilized to distinguish bipartite quantum states with even or odd parity, have many applications in quantum information processing. We propose a potentially practical scheme to construct the nondestructive Rydberg parity meter (NRPM) in ground-state subspace using Rydberg atoms. Based on the proposed NRPM, the parity of the qubit atoms could be known in a deterministic and nondestructive way, implying that the output state of the NRPM can be further used in the rest of the process for quantum information tasks. In addition, we study the applications of the proposed NRPM in entangled state fusion, entangled state analysis, and quantum teleportation, as well as quantum logic gate in a decoherence-free subspace regarding collective-dephasing errors. We analyze the performance of the NRPM under the influence of decoherence and consider more general situations for the feasibility of the scheme.

Figure 1

(a) Schematic of the NRPM, where three Rydberg atoms are located equidistantly as a triangle. Rydberg atoms 1 and 2 are encoded as qubits, while Rydberg atom $a$ works as the auxiliary to acquire the parity information of the qubit atoms 1 and 2. (b) Level structure of two qubit atoms under the laser drives, where the ground state $|0\rangle$ ($|1\rangle$) is coupled to the Rydberg state $|R\rangle$ ($|r\rangle$) with the Rabi frequency ${\mathrm{\Omega }}_R$ (${\mathrm{\Omega }}_r$) and the blue detuning ${\mathrm{\Delta }}_R$ (${\mathrm{\Delta }}_r$). $V_{mn}$ denotes the vdW-type RRI between Rydberg states $|m\rangle$ and $|n\rangle$. (c) Level structure of the auxiliary atom $a$ under the laser drive, with initial state $|+\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. $|1\rangle$ ($|0\rangle$) coupled to $|r\rangle$ ($|R\rangle$) with Rabi frequency ${\mathrm{\Omega }}_a$. $|0\rangle$ and $|1\rangle$ are ground states. We set ${\mathrm{\Delta }}_R={\mathrm{\Delta }}_r=\mathrm{\Delta }$ and ${\mathrm{\Omega }}_R={\mathrm{\Omega }}_r=\mathrm{\Omega }$ for simplicity, and $\mathrm{\Delta }\gg \mathrm{\Omega }$ should be satisfied.

Figure 2

Excitations in steps (i) and (iii) for the proposed NRPM. The system dynamics of even parity is governed by Rydberg antiblockade mechanism, while that of odd parity is blocked due to the large detuning. The Stark-shift-induced phase could be discarded due to the symmetry of the different components for the parity state. $|T_1\rangle =\left(\right|0R\rangle +|R0\rangle )/\sqrt{2}$, $|T_2\rangle =\left(\right|0r\rangle +|R1\rangle )/\sqrt{2}$, $|T_3\rangle =\left(\right|r0\rangle +|1R\rangle )/\sqrt{2}$, $|T_4\rangle =\left(\right|0r\rangle +|r0\rangle )/\sqrt{2}$.

Figure 3

Schematic of the entangled state fusion via the proposed NRPM. The connected chain denotes entanglement. Left panel: Atoms 1 and 2, 3 and 4 are initially in the nonmaximal entangled state with the same form $\alpha |00\rangle +\beta |11\rangle$. Auxiliary atom a is initially in state $|+\rangle$. Right panel: Four-body maximal entangled state induced by the NRPM due to the odd-parity measurement result.

Figure 4

Diagram of quantum teleportation based on the proposed NRPM, where Alice owns Rydberg atoms 1 and 2, and Bob owns Rydberg atom 3. The state to be teleported is prepared in atom 1. Atoms 2 and 3 are initially entangled with each other. H and CC indicate a single-qubit Hadamard gate and classical communication, respectively. P denotes the identity matrix or one of the Pauli matrices based on the results of NRPMs.

Figure 5

(a) Diagram of NRPM in collective-dephasing DFS, where L1 and L2 denote the logical qubits. (b) Diagram of controlled-not in DFS, where $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$, $A_{\mathrm{in}}$ and $A_{\mathrm{out}}$, and $T_{\mathrm{in}}$ and $T_{\mathrm{out}}$ represent the input and output states of the controlled, auxiliary, and target qubits, respectively. $\tilde{H}$ denotes the Hadamard gate operation in DFS.

Figure 6

(a)–(d): Fidelities of even- and odd-parity cases versus the fluctuations of characteristic parameters, in which we set $\alpha =\beta =\gamma =\delta =0.5$. (e) Fidelity of odd-parity case versus $\beta$, where we set $\gamma =5-\beta$, $\alpha =\delta =0.5$. (f) Fidelity of even-parity case versus $\alpha$, where we set $\delta =5-\alpha$, $\beta =\gamma =0.5$. The rest parameters are chosen as $\mathrm{\Omega }/2\pi =10$ MHz, ${\mathrm{\Omega }}_a=\mathrm{\Omega },\mathrm{\Delta }=10\mathrm{\Omega },V=2\mathrm{\Delta }$, $V^{\prime }=0$, ${\gamma }_R={\gamma }_r=2$ kHz.

Figure 7

Schematic of the auxiliary atom's excitation in step (ii) conditional on the qubit atoms in $|RR\rangle$ or $|rr\rangle$. $V_{jj,k}$ ($\{j,k\}\in \{r,R\}$) denotes the RRI strength between the qubit atoms' states $|jj\rangle$ and the auxiliary atom's state $|k\rangle$. Two horizontal circles denote the case $V^{\prime }\ll {\mathrm{\Omega }}_a$ in Sec. 2. The upper two circles indicate the failure of the scheme due to the unwanted blockade induced by $V^{\prime }\gg {\mathrm{\Omega }}_a$, which yields the output state of the auxiliary atom as ${|+\rangle }_a$. However, as we have shown in step (ii) of Sec. 2, the output state of auxiliary atom should be ${|-\rangle }_a$ conditional on the systematic qubit in $|RR\rangle$ or $|rr\rangle$, and thus the scheme would not work. The bottom two circles show how to address the issue for the case $V^{\prime }\gg {\mathrm{\Omega }}_a$ via turning off the driving lasers acting on $|0\rangle \to |R\rangle$ of the auxiliary atom. For the former case in the horizontal circles, the measurement result ${|+\rangle }_a$ (${|-\rangle }_a$) corresponds to the odd (even) parity. But for the cases in bottom circles, the measurement result ${|+\rangle }_a$ (${|-\rangle }_a$) corresponds to the even (odd) parity.

Figure 8

Energy level scheme and laser driving process of a single qubit atom. (a) Two-photon process exemplified by ${}^{87}\mathrm{Rb}$. ${\mathrm{\Omega }}_{0p}$ and ${\mathrm{\Omega }}_{pR}$ denote the coupling strengths of $|0\rangle \to |p\rangle$ and $|p\rangle \to |R\rangle$, respectively. ${\mathrm{\Delta }}_{R0}$ is the effective detuning for the two-photon process $|0\rangle \to |p\rangle \to |R\rangle$, during which the detuning ${\mathrm{\Delta }}_{p0}$ is applied to the intermediate state $|p\rangle$. ${\mathrm{\Omega }}_{1p}$ and ${\mathrm{\Omega }}_{pr}$ denote the coupling strengths of $|1\rangle \to |p\rangle$ and $|p\rangle \to |r\rangle$, respectively. ${\mathrm{\Delta }}_{r1}$ is the effective detuning for the two-photon process $|1\rangle \to |p\rangle \to |r\rangle$, during which the detuning ${\mathrm{\Delta }}_{p1}$ is applied to $|p\rangle$. (b) Single-photon process to implement the scheme, exemplified by ${}^{133}\mathrm{Cs}$.

Figure 9

(a) Energy-level scheme and laser driving process for atoms 1, 2 and $a$ to construct ${\widehat{U}}_3$ in a single step without atomic individual addressing. (b) Fidelity of ${\widehat{U}}_3$ with respect to the evolution time. The system is initially in the state $|\psi \rangle =\left(\right|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle +|101\rangle +|110\rangle +|111\rangle )/2\sqrt{2}$ and would evolve finally to ${\widehat{U}}_3|\psi \rangle$ in an ideal case. The parameters are set as ${\mathrm{\Omega }}_R=1.5{\mathrm{\Omega }}_r$, ${\gamma }_R={\gamma }_r=0.25\times {10}^{-4}{\mathrm{\Omega }}_r$, ${\mathrm{\Delta }}_r=12{\mathrm{\Omega }}_r$, ${\mathrm{\Delta }}_R=\sqrt{{\mathrm{\Omega }}_R^3/{\mathrm{\Omega }}_r^3}{\mathrm{\Delta }}_r$, $V_{RR}={\mathrm{\Delta }}_R-{\mathrm{\Omega }}_R^2/\left(4{\mathrm{\Delta }}_R\right)$, $V_{rr}={\mathrm{\Delta }}_r-{\mathrm{\Omega }}_r^2/\left(4{\mathrm{\Delta }}_r\right)$, and $V_{Rr}=V_{rR}=0.5V_{rr}$.