Assessment of a quantum phase-gate operation based on nonlinear optics

We analyze in detail the proposal for a two-qubit gate for travelling single-photon qubits recently presented by Ottaviani et al. [Phys. Rev. A 73, 010301(R) (2006)]. The scheme is based on an ensemble of five-level atoms coupled to two quantum and two classical light fields. The two quantum fields undergo cross-phase modulation induced by electromagnetically induced transparency. The performance of this two-qubit quantum phase gate for travelling single-photon qubits is thoroughly examined in the steady-state and transient regimes, by means of a full quantum treatment of the system dynamics. In the steady-state regime, we find a general trade-off between the size of the conditional phase shift and the fidelity of the gate operation. However, this trade-off can be bypassed in the transient regime, where a satisfactory gate operation is found to be possible, significantly reducing the gate operation time.

Figure 1

Energy levels of the $M$ scheme. ${\Omega }_j$ are the Rabi frequencies of classical fields, while $g_{p,t}$ denote couplings of the quantized probe and trigger fields to their respective transitions. ${\delta }_j$ are the detuning of the fields from resonance.

Figure 2

(Color online) A schematic plot of the assumed single-photon pulse propagation through the gas cell of length $L$ and diameter $d$. The pulse length is assumed to coincide with the cell length $L$, and the pulse waist $w$ is assumed to be of order of cell diameter $d$.

Figure 3

Conditional phase shift (left) and the fidelity of Eq. (12) (right) as a function of the interaction time for $N_a={10}^6$, ${\delta }_1={\delta }_3=7.5\gamma$, ${ϵ}_{12}={ϵ}_{34}=0.05\gamma$, $g_p=g_t=0.0011\gamma$, ${\Omega }_1={\Omega }_4=1.875\gamma$, and ${\gamma }_{kk}={\gamma }_{ph}={10}^{-3}\gamma$, $\forall k$. We have taken equal decay rates, ${\gamma }_{21}={\gamma }_{23}={\gamma }_{25}={\gamma }_{41}={\gamma }_{43}={\gamma }_{45}=\gamma ∕3$, with $\gamma =2\pi \times 6\mathrm{MHz}$. Left-hand side, solid line represents the phase shift, as calculated from the full master equation, while the dashed line gives the perturbative prediction of Eq. (16). Right-hand side, solid line is the unconditional gate fidelity $\mathcal{F}\left(t\right)$, while the dotted-dashed line is the conditional one, ${\mathcal{F}}^c\left(t\right)$.

Figure 4

Conditional phase shift (left) and the fidelity (right) as a function of the interaction time for $N_a={10}^8$, ${\delta }_1={\delta }_3=9.5\gamma$, ${ϵ}_{12}={ϵ}_{34}=0.2\gamma$, $g_p=g_t=0.018\gamma$, ${\Omega }_1={\Omega }_4=19\gamma$, and ${\gamma }_{kk}={\gamma }_{ph}={10}^{-3}\gamma$, $\forall k$. We have taken equal decay rates, ${\gamma }_{21}={\gamma }_{23}={\gamma }_{25}={\gamma }_{41}={\gamma }_{43}={\gamma }_{45}=\gamma ∕3$, with $\gamma =2\pi \times 6\mathrm{MHz}$. Left-handed side, solid line represents the phase shift, as calculated from a density matrix, while the dashed line gives the eigenvalue solution. Right-hand side, solid line is the unconditional gate fidelity $\mathcal{F}\left(t\right)$, while the dotted-dashed line is the conditional one, ${\mathcal{F}}^c\left(t\right)$.

Figure 5

Conditional phase shift (left) and the fidelity (right) of the QPG operation for $N_a={10}^6$, ${\delta }_1={\delta }_3=15\gamma$, ${ϵ}_{12}={ϵ}_{34}=0.01\gamma$, $g_p=g_t=0.0022\gamma$, ${\Omega }_1={\Omega }_4=4\gamma$, and ${\gamma }_{kk}={\gamma }_{ph}={10}^{-3}\gamma$, $\forall k$. We have taken equal decay rates, ${\gamma }_{21}={\gamma }_{23}={\gamma }_{25}={\gamma }_{41}={\gamma }_{43}={\gamma }_{45}=\gamma ∕3$, with $\gamma =2\pi \times 6\mathrm{MHz}$. Right-hand side, the unconditional fidelity (solid) and conditional fidelity (dashed) are shown. See text for details.

Figure 6

(Color online) Conditional phase shift (left) and the fidelity (right) of a QPG operation for $N_a={10}^8$, ${\delta }_2={\delta }_3=6\gamma$, ${ϵ}_{12}={ϵ}_{34}=0.05\gamma$, $g_p=g_t=0.0009\gamma$, and ${\Omega }_1={\Omega }_4=\gamma$. We have taken equal decay rates, ${\gamma }_{21}={\gamma }_{23}={\gamma }_{25}={\gamma }_{41}={\gamma }_{43}={\gamma }_{45}=\gamma ∕3$. For ${}^{87}\mathrm{Rb}$, $\gamma =2\pi \times 6\mathrm{MHz}$, giving the interaction time (i.e., pulse length) of $\simeq 25\mathrm{ns}$ for $\mid \varphi \mid \simeq \pi$. Right-hand side, solid line denotes the deterministic fidelity, while dotted-dashed line denotes the conditional fidelity.

Figure 7

Energy levels of the ladder scheme. $g_{p,t}$ denote couplings of the quantized probe and trigger fields to their respective transitions. ${\delta }_{p,t}$ are detunings of the probe and trigger fields from resonance.

Figure 8

Average fidelity (top figure) and conditional phase shift (bottom figure) as a function of time for the three-level ladder scheme of Fig. 7 and for $N_a={10}^8$, ${\delta }_p=10\gamma$, ${\delta }_t=0$, and $g_p=g_t=0.0022\gamma$. The spontaneous emission rate is ${\gamma }_{21}={\gamma }_{32}=\gamma =2\pi \times 6\mathrm{MHz}$.

Figure 9

(Color online) Scheme of the proposed experiment for a measurement of the nonlinear phase shift in a QPG. A Michelson-type interferometer with a two-photons input state $\mid 1_R⟩\mid 1_B⟩$, probe and trigger, respectively, allows to measure the nonlinear phase induced by the XPM on the logical basis of the qubits, which coincide with the two lowest Fock states. Two intense classical fields ${\alpha }_t$ (tuner with Rabi frequency ${\Omega }_1$) and ${\alpha }_c$ (coupler, with Rabi frequency ${\Omega }_4$), are necessary to the five-level XPM process. $L$ is a lens for the mode matching in the EIT medium. BS a 50/50 beam splitter. DL a delay line. ${\mathrm{FP}}_{1,2}$ Fabry-Perot cavities. $D_{R,B1,B2}$ are avalanche-photodiodes APDs. Left-hand inset: Scheme using a Sagnac interferometer for avoiding the optical path difference. $M$ is a mirror. Right-hand inset: Frequency spectrum (arb. units) for a $2\mathrm{mm}$ FP cavity length and finesse equal to ${10}^3$. The spectra of the probe and trigger photon are also shown in the plots as lines.

Figure 10

(Color online) Truth table of the QPG for a logical basis of the qubits determined by the two lowest Fock states. On the right-hand side, diagrams corresponding to the probability amplitudes after the action of the BS and the FPs on the two-photons input state are shown.

Figure 11

(Color online) Truth table of the QPG for a logical basis of the qubits determined by orthogonal circular polarization basis. On the right-hand side the diagrams corresponding to the probability amplitudes after the action of the BS and the FPs on the two-photons input state.

Figure 12

(Color online) Scheme of the proposed experiment for a complete characterization of the QPG. Two photons in the ${\sigma }^{+}$ polarization state (logical state $\mid 0_R⟩\otimes \mid 0_B⟩$) are transformed by a QWP, corresponding to two Hadamard single-qubit gates, and then to the XPM medium (QPG). A Fabry-Perot cavity with the same parameter as before transmits the trigger photon and reflect the probe to two tomographic measurement systems $\left(T_{R,B}\right)$ and detected by APDs.