---
abstract: |
    The transfer technique of quantum states from light to collective atomic excitations in a double $\Lambda$ type system is extended to matter waves in this paper, as a novel scheme towards making a continuous atom laser. The intensity of the output matter waves is found to be determined by the initial relative phase of the two independent coherent probe lights, which may indicate an interesting method for the measurement of initial relative phase of two independent light sources.\
    PACS numbers: 03.75.-b, 42.50.Gy, 03.65.Ta, 03.67.-a
author:
- 'Xiong-Jun Liu$^{a,b}$[^1], Hui Jing$^{c}$, Xiao-Ting Zhou$^d$ and Mo-Lin Ge$^{a,b}$'
title: Technique of quantum state transfer for a double $Lambda$ atomic beam
---

Recently, the novel mechanism of Electromagnetically Induced Transparency (EIT) @1 and its many important applications have attracted much attention in both experimental and theoretical aspects @2 [@3; @4], especially after Fleischhauer and Lukin @5 proposed their famous dark-states polaritons (DSPs) theory and thereby the developments of quantum memory technique, i.e., transferring the quantum states of photon wave-packet to collective Raman excitations in a loss-free and reversible manner. All these works are based on a field theory reformulation of the adiabatic approximation @6. In a very recent paper, by extending the quantum state transfer technique to propagating matter-waves, Fleischhauer and Gong formulated a wonderful scheme to generate continuous beams of three-level $\Lambda$ type atoms in non-classical or entangled states @7. This new method may provide an experimentally accessible way for the preparation of a continuous coherent atomic beam or an atom laser that has drawn much attention in current literatures, since the first pulsed atom laser was created in 1997 based on the remarkable realization of Bose-Einstein condensation in dilute atomic clouds @8 [@9]. On the other hand, the the controlled light storing in the medium composed of double $\Lambda$ type four-level atoms was also investigated by several authors quite recently @10 [@11]. As a natural extension, in this paper, we proceed to study the possible extending of the elegant Fleischhauer-Gong scheme @7 in the double $\Lambda$ atomic configuration, in particular the transfer technique of quantum states from a pair of probe pulses to matter waves. Besides, the intensity of the output matter waves is found to be determined by the initial relative phase of the two independent coherent probe lights, which may indicate an interesting way to measure the initial relative phase of two independent light sources.

We consider the quasi 1-dimensional system shown in Fig.1. A beam of double $\Lambda$ type atoms interacts with two coherent probe and two control Stokes fields and the former fields are taken to be much weaker than the later. Atoms in different internal states are described by four bosonic fields $\Psi_{\mu}(z,t)
(\mu=1,2,3,4)$. The Stokes fields can be described by the Rabi-frequencies $\Omega_j(z,t)=\Omega_{0j}(z)e^{-i\omega_{s_j}(t-z/c_j)}$ with $\Omega_{0j}~(j=1,2)$ being taken as real and $c_j$ denoting the phase velocities projected onto the $z$ axis, and the two coherent probe fields are characterized by the dimensionless positive frequency components $E^{(+)}_j(z,t)=\varepsilon_j(z,t)e^{-i\omega_{pj}(t-z/c)}$. Assuming $\omega_{p_1}-\omega_{s_1}=\omega_{p_2}-\omega_{s_2}$, we may introduce the slowly-varying amplitudes, and a decomposition into velocity classes $\Psi_1=\sum_l\Phi^l_1e^{i(k_lz-\omega_lt)}$, $\Psi_2=\sum_l\Phi^l_2e^{i[(k_l+k_{p_1})z-(\omega_l+\omega_{p_1})t]}$, $\Psi_3=\sum_l\Phi^l_3e^{i[(k_l+k_{p_j}-k_{s_j})z-(\omega_l+\omega_{p_j}
-\omega_{s_j})t]}$, and $\Psi_4=\sum_l\Phi^l_4e^{i[(k_l+k_{p_2})z-(\omega_l+\omega_{p_2})t]}$, where $\hbar\omega_l=\hbar^2k^2_l/2m$ is the corresponding kinetic energy in the $l$th velocity class, $k_{p_j} $ and $k_{s_j}$ $(j=1,2)$ are respectively the vector projections of the two probe and Stokes fields to the $z$ axis. The atoms have a narrow velocity distribution around $v_0=\hbar{k_0}/2m$ with $k_0\gg|k_{p_j}-k_{s_j}|$, and all fields are assumed to be in resonance for the central velocity class. Hence the equations of motion for the atomic fields are given by $$\begin{aligned}
\label{eqn:1}
(\frac{\partial}{\partial{t}}+\frac{\hbar{k_l}}{2m}\frac{\partial}
{\partial{z}})\Phi^l_1
=-ig_1\varepsilon^*_1\Phi^l_2-ig_2\varepsilon^*_2\Phi^l_4\end{aligned}$$ $$\begin{aligned}
\label{eqn:2}
(\frac{\partial}{\partial{t}}+\frac{\hbar{k_l}}{2m}\frac{\partial}
{\partial{z}})\Phi^l_3
=-i\Omega_{01}\Phi^l_2-i\Omega_{02}\Phi^l_4-i\delta_l\Phi^l_3\end{aligned}$$ $$\begin{aligned}
\label{eqn:3}
(\frac{\partial}{\partial{t}}+\frac{\hbar(k_l+k_{p_1})}{2m}
\frac{\partial}{\partial{z}})\Phi^l_2
=-(\gamma_2+i\Delta_l)\Phi^l_2-i\Omega_{01}\Phi^l_3-ig_1\varepsilon_1\Phi^l_1+F^l_2\end{aligned}$$ $$\begin{aligned}
\label{eqn:4}
(\frac{\partial}{\partial{t}}+\frac{\hbar(k_l+k_{p_2})}{2m}
\frac{\partial}{\partial{z}})\Phi^l_4
=-(\gamma_4+i\Delta_l)\Phi^l_4-i\Omega_{02}\Phi^l_3-ig_2\varepsilon_2\Phi^l_1+F^l_4\end{aligned}$$ where $g_j$ is the atom-field coupling constant between the states ${|1\!\!>}$ and ${|2\!\!>}$ (for $j=1$) or ${|1\!\!>}$ and ${|4\!\!>}$ (for $j=2$), $\gamma_{2,4}$ denotes the loss rate out of the excited state ${|2\!\!>}$ or ${|4\!\!>}$, $\Delta_l\approx\hbar{k_l}k_{p1}/m+(\omega_{21}-\omega_{p1})\approx\hbar{k_l}k_{p2}/m+(\omega_{41}-\omega_{p2})$ and $\delta_l\approx\hbar{k_l}(k_{pj}-k_{sj})/m+(\omega_{31}-\omega_{pj}-\omega_{sj})$ are the single and two-photon detunings and $F^l_{2,4}$ is the corresponding Langevin noise operator that will be omitted in the following derivation for simplicity. The propagation equations of the two probe fields read $$\begin{aligned}
\label{eqn:5}
(\frac{\partial}{\partial{t}}+c\frac{\partial}{\partial{z}})\varepsilon_1(z,t)
=-ig_1\sum_l{{\Phi^{\dag}_1}^l\Phi^l_2}\end{aligned}$$ $$\begin{aligned}
\label{eqn:6}
(\frac{\partial}{\partial{t}}+c\frac{\partial}{\partial{z}})\varepsilon_2(z,t)
=-ig_2\sum_l{{\Phi^{\dag}_1}^l\Phi^l_4}\end{aligned}$$

Consider a stationary input of atoms in state ${|1\!\!>}$, i.e. $\Psi_1(0,t)=\sqrt{n}$, where $n$ is the constant total density of atoms. In the limit of the two weak probe quantum fields and weak atomic excitation one finds:@7 $\Phi^l_1(z,t)\approx\Phi^l_1(0,t-2mz/\hbar{k_l})=\sqrt{n}\xi_le^{-i\varphi_l(z,t)}$, where $\sum_l{\xi_l}=1$ and $\varphi_l\equiv(k_lz-\omega_lt)$. From the formula (\[eqn:3\]) and (\[eqn:4\]) one can find the condition of the adiabatic evolution is fulfilled only in the special case: $\varepsilon_2(z,t)=\tan\vartheta(z)\varepsilon_1(z,t)$, where $\vartheta$ is defined according to $\tan\vartheta(z)=\frac{g_1\Omega_{02}(z)}{g_2\Omega_{01}(z)}$. In general, however, the two input probe fields do not satisfy the former special relation at the entrance region of the atoms. The process is then nonadiabatic from the very beginning. Raczyński et al.@10 show that, due to the self-adjusting of the two probe fields, the adiabatic condition will become fulfilled after the nonadiabatic process, i.e., $\varepsilon_j(0,t-\tau(\delta))\rightarrow\varepsilon'_j(\delta,t)
$, where $\varepsilon'_2(z,t)=\tan\vartheta\varepsilon'_1(z,t)$ and $\tau(\delta)$ is the time shift from the entrance region to an adjacent position $\delta$, $0<\delta<L$ with $L$ being the interaction length in $z$ direction. To obtain the motion equation of probe fields, in the region $\delta<z<L$, we definite a new probe field: $
\varepsilon_{12}(z,t)=\cos\vartheta(z)\varepsilon'_1(z,t)+\sin\vartheta(z)\varepsilon'_2(z,t)
$, therefore we can obtain $$\begin{aligned}
\label{eqn:9}
(\frac{\partial}{\partial{t}}+c\frac{\partial}{\partial{z}})
\varepsilon_{12}(z,t)=-i\frac{g_1g_2}{\Omega}\sum_l{(\Omega_{01}\Phi^l_2(z,t)
+\Omega_{02}\Phi^l_4(z,t))\Phi^{\dag}_1}^l(z,t)\end{aligned}$$ $$\begin{aligned}
\label{eqn:10}
\Phi^l_3(z,t)&=&-\frac{g_1g_2}{\Omega}\varepsilon_{12}(z,t)\sqrt{n}
\xi_le^{-i\varphi_l(z,t)}+\frac{i}{\Omega_{01}+\Omega_{02}}[(\frac{\partial}{\partial{t}}
+\frac{\hbar(k_l+k_{p_1})}{2m}\frac{\partial}{\partial{z}}+\gamma_2+i\Delta_l)\Phi^l_2(z,t)\nonumber\\
&&+(\frac{\partial}{\partial{t}}+\frac{\hbar(k_l+k_{p_2})}{2m}
\frac{\partial}{\partial{z}}+\gamma_4+i\Delta_l)\Phi^l_4(z,t)]\end{aligned}$$ where $\Omega=\sqrt{g^2_1\Omega^2_{02}+g^2_2\Omega^2_{01}}$. Here we only consider the perfect two photon resonance case for all atoms by omitting all the loss rates out of two excited states (the validity of this approximation will be discussed later), hence $\gamma_2=\gamma_4\equiv0$ and $\delta_l\equiv0$. In the adiabatic approximation, we obtain $\Phi^l_3(z,t)=-\frac{g_1g_2}{\Omega}\varepsilon_{12}(z,t)
\sqrt{n}\xi_le^{-i\varphi_l(z,t)}$ and $\Omega_1\Phi^l_2(z,t)+\Omega_2\Phi^l_4(z,t)=-ig_1g_2\sqrt{n}\xi_le^{-i\varphi_l(z,t)}
(\frac{\partial}{\partial{t}}+\frac{\hbar{k_l}}{2m}
\frac{\partial}{\partial{z}})\frac{\varepsilon_{12}(z,t)}{\Omega(z)}$. Substituting the latter results into the equation of the motion for new probe field $\varepsilon_{12}(z,t)$ yields $$\begin{aligned}
\label{eqn:13}
[(1+\frac{g^2_1g^2_2n}{\Omega^2(z)})\frac{\partial}{\partial{t}}
+c(1+\frac{g^2_1g^2_2n}{\Omega^2(z)}\frac{v_0}{c})
\frac{\partial}{\partial{z}}]\varepsilon_{12}(z,t)
=\frac{g^2_1g^2_2n}{\Omega^2(z)}v_0(\frac{\partial}{\partial{z}}
\ln\Omega(z))\varepsilon_{12}(z,t)\end{aligned}$$ with $v_0\equiv\sum_l{\xi_l}v_l$. The r.h.s. of this equation describes a reduction (enhancement) due to stimulated Raman adiabatic passage in two spatially varying Stokes fields for $v_0\neq0$, and the space-dependent Stokes fields in laboratory frame is equivalent to the time-dependent fields in the rest frame of the atoms. Introducing the mixing angle $\theta(z)$ according to $\tan^2\theta(z)\equiv\frac{g^2_1g^2_2n}{\Omega^2}\frac{v_0}{c}$, one can easily find the solution: $\varepsilon_{12}(z,t)=\varepsilon_{12}(\delta,t-\tau(z,\delta))
\frac{\cos\theta(z)}{\cos\theta(\delta)}$, where $\tau(z,\delta)=\tau(z)-\tau(\delta)=\int^z_{\delta}{dz'V^{-1}_g(z')}$ with the group velocity $V_g=c(1+\frac{g^2_1g^2_2n}{\Omega^2}\frac{v_0}{c})/(1
+\frac{g^2_1g^2_2n}{\Omega^2})$ that approaches $v_0$ for $\Omega(z)\rightarrow0$. The initial value of $\varepsilon_{12}$ can be calculated as@10: $\varepsilon_{12}(\delta,t)=(\cos\vartheta(0)\varepsilon_1(0,t-\tau(\delta))
+\sin\vartheta(0)\varepsilon_2(0,t-\tau(\delta)))\cos\theta(\delta)/\cos\theta(0)$. Assuming $\theta(0)=0$ and $\theta(L)=\pi/2$ at the input and output points respectively, the bosonic field $\Phi_3(z,t)$ is then obtained by $$\begin{aligned}
\label{eqn:14}
\Phi_3(z,t)=-\sqrt{\frac{c}{v_0}}(\cos\vartheta_0\varepsilon_1(0,t-\tau(L))
+\sin\vartheta_0\varepsilon_2(0,t-\tau(L)))\end{aligned}$$ where $\tau(L)=\tau(\delta)+\int^L_{\delta}{dz'V^{-1}_g(z')}$ and $\vartheta_0=\vartheta(0)$. The factor $\sqrt{c/v_0}$ accounts for the fact that the input light pulses propagate with velocity $c$ and the output matter field propagates with $v_0$. Furthermore $$\begin{aligned}
\label{eqn:15}
v_0|\Psi_3|^2_{out}=c(\cos^2\vartheta_0|\varepsilon_1|^2_{in}
+\sin^2\vartheta_0|\varepsilon_2|^2_{in}+2Re({\cos\vartheta_0\sin\vartheta_0\varepsilon^{*}_1\varepsilon_2})_{in})\end{aligned}$$ Noting the complex amplitudes $\varepsilon_j=|\varepsilon_j|e^{i\phi_j}~(j=1,2)$, where $\phi_j$ corresponds to the initial phase of the probe lights, the last term of the r.h.s of above formula then recasts into: $2\cos\vartheta_0\sin\vartheta_0|\varepsilon_1|_{in}|\varepsilon_2|_{in}\cos\phi$, where $\phi=\phi_1-\phi_2$ is the initial relative phase, and it gives a crucial influence on the intensity of output $\Phi_3$-atomic beam. As a result of the non-adiabatic evolution, from Eq.(\[eqn:15\]) one can easily find the output flux of $\Phi_3$-atoms is less than (or equal) the total input flux of photons, i.e. $v_0|\Psi_3|^2_{out}\leq
c(|\varepsilon_1|^2_{in}+|\varepsilon_2|^2_{in})$. Particularly, (a) if $g_2\Omega_{01}(0)\gg{g_1}\Omega_{02}(0)$, we have $\cos\vartheta(0)\simeq1$ and $v_0|\Psi_3|^2_{out}=c|\varepsilon_1|^2_{in}$, which indicates the output flux of $\Phi_3$-atoms equals to the input flux of the photons of the probe field $\varepsilon_1$, while all photons of $\varepsilon_2$ is damped in the non-adiabatic process from the very beginning; (b) If $g_2\Omega_{01}(0)\ll{g_1}\Omega_{02}(0)$, we have $\cos\vartheta(0)\simeq0$ and $v_0|\Psi_3|^2_{out}=c|\varepsilon_2|^2_{in}$, which indicates the output flux of $\Phi_3$-atoms equals to the input flux of the photons of the probe field $\varepsilon_2$,while $\varepsilon_1$ is damped in the non-adiabatic process. These results indicate that the quantum states of the continuous output matter waves can easily be steered by the two control stokes fields.

The formula (\[eqn:15\]) is the main result in this paper. For this we may present a possible application for the measurement of the initial relative phase $\phi$ of two independent coherent lights. The schematic measurement setup is shown in Fig.2. M is a semitransparent mirror splitter, through which the two input probe pulses $E^0_1(z,t)$ and $E^0_2(z',t)$ are split into four pulses with identical intensities, i.e. $E_1(z,t)$, $E_2(z,t)$, $E'_1(z't)$ and $E'_2(z',t)$ with their amplitudes $|\varepsilon_1|=|\varepsilon_2|=|\varepsilon'_1|=|\varepsilon'_2|=\varepsilon_0$. Among these splitters, $E'_1(z',t)$ and $E'_2(z',t)$ enter the (-) channel, while $E_1(z,t)$ and $E_2(z,t)$ enter the (+) one. G is a plate glass with its thickness $d=\pi/|(n-1)(k_{p1}-k_{p2})|$, where $n$ is the refractive index of the two probe pulses in the glass. The relative phase between $E_1(z,t)$ and $E_2(z,t)$ increases an additional value $\pi$ after passing through the glass G. Assuming that $g_1\Omega_2(0)={g_2}\Omega_1(0)$, the intensities of the outputs of $\Phi_3$-atom flux from channel (+) and (-) are then given by $$\begin{aligned}
\label{eqn:17}
I^+_3=I_0\sin^2\frac{\phi}{2},&& I^-_3=I_0\cos^2\frac{\phi}{2}\end{aligned}$$ where $I_3=|\Psi_3|^2_{out}$ and $I_0=\frac{v_0}{c}\varepsilon_0^2$. The recording of $I_3^{\pm}$ allows one to determine the absolute value of the relative phase $\phi$ of two independent coherent probe fields. Note that $\phi$ is an unpredictable random variable, which takes a different value for any new realization. We consider the case of $k_{\pm}$ detected $\Phi_3$-atoms in the ($\pm$) channel for a fixed number of measurements $k=k_{+}+k_-$. Each count occurs with probabilities $\sin^2\frac{\phi}{2}$ and $\cos^2\frac{\phi}{2}$ in the (+) and (-) channels. Note $P(k_+,k_-,\phi)$ as the probability for the result ($k_+,k_-$), when $k\gg1$, one can easily find $P(k_+,k_-,\phi)$ is maximal for $\phi=2\tan^{-1}\sqrt{k_+/k_-}$, and the shot noise on the signal in the two channels ($\pm$) can be neglected @12. The phase $\phi$ can be then well determined by the ratio $k_+/k_-$ detected on the counters.

Now we discuss in detail on the approximations invoked in the derivation of the above results. First, from the relation between $\Phi_3$ and $\varepsilon_{12}(z,t)$ one finds $g^2_1g^2_2\langle\varepsilon^{\dag}_{12}\varepsilon_{12}\rangle/\Omega^2
=\langle\Psi^{\dag}_3\Psi_3\rangle/n$. Then, as long as the condition of weak atomic excitation fulfills, in other words, when the input flux of atoms is much larger than the input flux of pump photons, the adiabatic approximation used in above discussion is valid. Another approximation is that we have assume the perfect two photon resonance. For the case of a non-vanishing but constant value of $\delta_l=\delta$ and equal loss rates $\gamma_2=\gamma_4=\gamma$, the accompanying contribution to $\Omega_{01}\Phi^l_2+\Omega_{02}\Phi^l_4$ in the lowest order is $$\begin{aligned}
\label{eqn:19}
\Omega_{01}\Phi^l_2+\Omega_{02}\Phi^l_4\rightarrow\Omega_{01}\Phi^l_2
+\Omega_{02}\Phi^l_4+\frac{(\Omega^2_{01}+\Omega^2_{02})\delta}{\Omega^2_{01}
+\Omega^2_{02}-\delta\Delta+i\gamma\delta}\frac{g_1g_2\sqrt{n}}{\Omega}
\varepsilon_{12}\xi_le^{-i(k_lz-\omega_lt)}\end{aligned}$$ Substituting above result to the formula (\[eqn:9\]), one can easily see that the additional imaginary (real) terms of the formula(\[eqn:19\]) bring a loss (a phase shift) of $\varepsilon_{12}$, i.e. $\varepsilon_{12}\rightarrow{e^{-\alpha_1+i\alpha_2}\varepsilon_{12}}$ with $\alpha_i(i=1,2)$ real, and then the output $\Phi_3$-atom flux intensity: $I^{\pm}_3\rightarrow{e^{-2\alpha_1}I^{\pm}_3}$. For $\Delta=0$, the parameter $\alpha_{1,2}$ can be calculated by $\alpha_1=\int^L_0{dz\sin^2\theta(z)\delta^2\gamma(\Omega^2_{01}+\Omega^2_{02})/(v_0((\Omega^2_{01}+\Omega^2_{02})^4+\delta^2\gamma^2))}$ and $\alpha_2=\int^L_0{dz\sin^2\theta(z)(\Omega^2_{01}+\Omega^2_{02})^2\delta^2/(v_0((\Omega^2_{01}+\Omega^2_{02})^4+\delta^2\gamma^2))}$. Since the detected value of $k_+/k_-$ depends on the ratio between $I^+_3$ and $I^-_3$ instead of on $I^+_3$ or $I^-_3$ only, the above derivation for the measurement of relative phase is also valid. On the other hand, to obtain a high efficiency in the transfer technique of quantum state from probe fields to matter waves, the value of $\alpha_1$ should be small enough. Generally, we may assume that $g_1\geq{g_2}$, one can then obtain $\alpha_1\leq{\eta\int^1_0{d\xi\frac{\cos^2\beta(\xi)\sigma^2}{\cot^4\beta(\xi)+\sigma^2}}}$, where $\beta$ is defined via $tan^2\beta=\frac{g^2_1n}{\Omega^2_{01}+\Omega^2_{02}}\frac{v_0}{c}$, $\xi=z/L$, $\sigma\equiv\delta\gamma/g^2_1n\frac{v_0}{c}$ and $\eta\equiv{g^2_1nL/\gamma{c}}$ is the opacity of the medium in the absence of EIT. The following discussions is then similar as that in ref.@7. Assuming $\sigma\ll1$ one can give an upper limit to the integral as $\int^1_0{d\xi\frac{\cos^2\beta(\xi)\sigma^2}{\cot^4\beta(\xi)+\sigma^2}}\leq|\sigma|/2$, consequently $\alpha_1\leq\eta|\sigma|/2$. Thus the case $|\delta|L/v_0\ll1$ meets the request of a small value of $\alpha_1$. A residual Doppler shift of the $1-3$ transition can result in a two-photon detuning through $\delta_j=\Delta
v(\vec{k}_{pj}-\vec{k}_{sj})\cdot\vec{e}_z(j=1,2)$, where $\Delta{v}$ denotes the difference of the velocity in $z$ direction with respect to the resonant velocity class. Then the condition for a small $\alpha_1$ can also reads $\Delta{v}/v_0\ll
min \{1/(k_{pj}-k_{sj})L \}$.

In conclusion we have extended the transfer technique of quantum states from a pair of probe fields to collective atomic excitation to matter-waves in a double $\Lambda$ type system. We observe that the quantum states of the continuous output matter-wave beam can be steered by the two control stokes fields, and their intensity can be determined by the initial relative phase of the lights. For this an interesting potential application in measuring the initial relative phase of two independent coherent sources is also briefly discussed. Taking into account of present difficulties in making a continuous atom laser in laboratory based on either the output of a trapped Bose condensate with finite atoms number or the moving magneto-optic trap and evaporative cooling technique @13, the problem studies in this paper raises an interest since it confirms the elegant Fleischhauer-Gong scheme @7 in this more general configuration also as a novel way in next generation of atom-laser experiments.

We thank Xu Jingjun, Zheng-Xin Liu, and Guimei Jiang for valuable discussions. This work is supported by NSF of China under grants No.10275036 and No.10304020. One of the authors (H. J.) is also supported by CAS K.-C. Wong education Fund (HongKong), China Postdoctoral Science Foundation and Postdoctoral Research Plan in Shanghai.

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![(a)Beam of double $\Lambda$ type atoms coupled to two control fields and two coherent probe fields. (b)To minimize effect of Doppler-broadening, geometry is chosen such that $(\vec{k}_{pi}-\vec{k}_{si})\cdot\vec{e}_z\approx0$ $(i=1,2)$.<span data-label=""></span>](fig1.eps){width="0.65\columnwidth"}

![The setup schematic for measurement. Here the geometry relation between the probe and control fields is the same as that in fig.1(b), for simplicity which is not shown in this figure. The output $\Phi_3$-atoms are detected on $D_{+}$ and $D_{-}$, from which the relative phase between $\varepsilon_1$ and $\varepsilon_2$ can be measured. <span data-label=""></span>](fig2.eps){width="0.6\columnwidth"}

[^1]: Electronic address: dan91@example.com

{'source': 'arXiv_out/quant-ph0401014_extract_paper1_liu'}