1.Key Laboratory of Materials Design and Quantum Simulation, School of Science, Changchun University, Changchun 130022, China 2.Department of Basic Courses, Changchun University of Science and Technology, Changchun 130000, China 3.College of Information Science and Technology, Qingdao University of Science and Technology, Qingdao 266046, China
Fund Project:Project supported by the National Nature Science Foundation of China (Grant Nos. 11204019, 11874004) and the “Spring Sunshine” Plan Foundation of Ministry of Education of China (Grant No. Z2017030).
Received Date:31 October 2018
Accepted Date:14 January 2019
Available Online:01 April 2019
Published Online:20 April 2019
Abstract:Transmission properties of a weak probe field traveling through a sample of interacting cold 87Rb atoms driven into the three-level ladder configuration, which is a typical Rydberg electromagnetically induced transparency (EIT) system, are investigated. Rydberg atoms are considered to be a perfect platform in the research fields of quantum optics and quantum information processing due to some exaggerated properties of Rydberg atoms with high principal quantum number, especially, the dipole-dipole interaction between Rydberg atoms leads to the so-called dipole blockade effect accommodating at most one Rydberg excitation within a mesoscopic volume. The dipole blockade effect may be mapped onto the spectrum of EIT, and the EIT exhibits the cooperative optical nonlinearity which is usually characterized by two indicators, i.e., the probe intensity and the photonic correlation. The cooperative optical nonlinearity is also found here in the phase of transmission spectrum, and the phase can be regarded as the third indicator of nonlinearity in Rydberg EIT. However, there are tremendous differences between the phase and probe transmission (photonic correlation) though they both originate from the conditional polarization. Specifically, the phase is not sensitive to neither the incident probe intensity nor the initial photonic correlation at the resonant probe frequency under the condition of the Autler-Townes (AT) splitting where two other indicators exhibit significant cooperative nonlinearity. The nonlinearity in phase spectrum occurs only in the regime between the resonant probe frequency and the AT splitting and especially is remarkable at the frequency where the probe field is classical. Finally, influence of the principal quantum number and the atomic density on the transmitted phase are examined. In the nonlinear regime, the absolute value of the phase becomes smaller and smaller as the principal quantum number and the atomic density are raised. This indicates that the nonlinearity is strengthened by increasing them. The probe phase provides an attractive supplement to study in depth the cooperative optical nonlinearity in Rydberg EIT and offers us the considerable flexibility to manipulate the propagation and evolution of a quantum light field. Keywords:electromagnetically induced transparency/ dipole-dipole interaction/ Rydberg atom/ superatom
4.数值结果讨论与分析考虑实验验证的可行性, 这里采用实际的实验参数来进行数值计算, 然后进行理论分析与讨论. 在超冷87Rb原子中, 选取$5{S_{1/2}}\left| {F = 2,{m_F} = 2} \right\rangle $、$5{P_{3/2}}\left| {F = 3,{m_F} = 3} \right\rangle $和$70{S_{1/2}}$分别对应于图1(b)中的基态$\left| g \right\rangle $、激发态$\left| e \right\rangle $和里德伯态$\left| r \right\rangle $. 弛豫速率${\gamma _e} = 3.0\;{\rm{MHz}}$和${\gamma _r} = 0.02\;{\rm{MHz}}$. 原子密度为$\rho (z) = $$ 1.5 \times {10^7}\;{\rm{m}}{{\rm{m}}^{ - 3}}$, 样品长度为$L = 1.5\;{\rm{mm}}$, vdW系数${C_6}/(2{\text{π}}) = 8.8 \times {10^{11}}\;{{\rm{s}}^{ - 1}} \cdot {\text{μ}}{{\rm{m}}^6}$, 其他具体参数见图下文字说明. 首先扫描探测场来观察透射光谱. 图2(a)和(b)显示, 当入射探测场拉比频率${\varOmega _{\rm{p}}}(0)$很弱时, 共振频率处透射率${I_{\rm{p}}}\left( L \right)/{I_{\rm{p}}}\left( 0 \right) \approx 1$, 而Autler-Townes (AT)劈裂处, 即${\varDelta _{\rm{p}}} = \pm {\varOmega _{\rm{c}}}$有${I_{\rm{p}}}\left( L \right)/{I_{\rm{p}}}\left( 0 \right) \approx 0$, 并且透射光一直为经典光$g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( L \right) = 1$, 表现为典型的线性EIT. 随着入射探测场拉比频率增大到${\varOmega _{\rm{p}}}(0)/(2{\text{π}}) = 0.3\;{\rm{MHz}}$, 出现明显的合作光学非线性效应, 具体表现为: 共振频率处有明显的吸收现象, 透明窗口由透明转为部分透明, 对应的二阶关联函数也从经典光$g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( L \right) = 1$变为反聚束光$g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( L \right) < 1$. 特别地, AT劈裂处则变为聚束光$g_{\rm{p}}^{\left( 2 \right)}\left( L \right) > 1$. 进一步增大入射场强, 非线性效应更加明显, 说明在未饱和之前, 合作光学非线性效应是强烈依赖于入射探测场强度的. 从图2(c)可以看出, 与探测场透射率和二阶关联函数相比, 探测场相位也表现出明显依赖于入射场强的非线性特性, 但是又有明显差异, 体现出独特的非线性效应. 具体来说, 在共振频率和AT劈裂处, 前者相位${\phi _{\rm{p}}}\left( L \right) = 0$保持不变而后者不敏感. 介于二者之间的频率区域才显示出非线性效应, 特别突出的是在${\varDelta _{\rm{p}}} \approx 1.4\;{\rm{MHz}}$(${\varDelta _{\rm{p}}} \approx - 1.4\;{\rm{MHz}}$)出现极大值(极小值)并且表现出明显的入射场强敏感性: 随着入射场强的增加, 极大值变小而极小值变大. 需要强调的是, ${\varDelta _{\rm{p}}} \approx \pm $$1.4\;{\rm{MHz}} $对应于经典光$g_{\rm{p}}^{\left( 2 \right)} = 1$. 与其他两种非线性标识一样, 相位的非线性特征也来源于条件极化率. 当探测场极弱的时候, 根本不存在里德伯激发, 表现为三能级透明结构的相位, 而当探测场足够强就转变为二能级吸收型原子相位特征. 图 2 (a)探测场透射率${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b)二阶关联函数$g_{\rm{p}}^{(2)}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c)探测场相位${\phi _{\rm{p}}}(L)/{\text{π}}$作为探测失谐${\varDelta _{\rm{p}}}/(2{\text{π}})$的函数. 黑色实线, 蓝色折线以及红色点线分别对应入射探测场拉比频率${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) = 0.01$, 0.3 MHz和0.6 MHz的情况. 单光子失谐${\varDelta _{\rm{c}}} = 0$, 控制场拉比频率${\varOmega _{\rm{c}}}/(2{\text{π}}) = 2.5\;{\rm{MHz}}$, 其他参数见正文描述 Figure2. (a) The transmitted probe intensity ${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b) the second-order correlation function $g_{\rm{p}}^{(2)}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c) probe phase ${\phi _{\rm{p}}}(L)/{\text{π}}$ as a function of the probe detuning ${\varDelta _{\rm{p}}}/(2{\text{π}})$. The black solid, blue dashed and red dotted curves are corresponding to incident probe Rabi frequencies ${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) = 0.01$, $0.3\;{\rm{MHz}}$ and $0.6\;{\rm{MHz}}$, respectively. The single-photon detuning ${\varDelta _{\rm{c}}} = 0$ and the Rabi frequency of control field ${\varOmega _{\rm{c}}}/(2{\text{π}}) = 2.5\;{\rm{MHz}}$. Other parameters are described in the text.
图3给出透射光谱的相空间结构, 用来重点考察探测场相位与其他两类非线性标识对频率和入射场强依赖的一致性. 为了尽量保证弱探测场的前提条件, 这里探测场拉比频率满足${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) \in $$ \left[ {0.01,0.75} \right]\;{\rm{ MHz}}$. 很明显, 相位分布关于共振点所在的对称轴呈现完美的反对称特征. 除了共振频率${\varDelta _{\rm{p}}}/(2{\text{π}}) = 0$, AT劈裂(${\varDelta _{\rm{p}}} = \pm {\varOmega _{\rm{c}}}$)直至大失谐($\left| {{\varDelta _{\rm{p}}}} \right|/$$(2{\text{π}}) > > 3\;{\rm{MHz}} $)以外, 都可以看出相位明显依赖于探测场强度, 特别是在${\varDelta _{\rm{p}}} \approx \pm 1.4\;{\rm{MHz}}$频率处. 此时, 相位的极值始终对应于经典光$g_{\rm{p}}^{\left( 2 \right)} = 1$, 增大探测场强度也不会改变. 原因在于条件极化率中不存在失谐对探测场强度的依赖关系, 这个从(5)—(8)式也很容易看出. 图 3 (a)探测场透射率${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b)二阶关联函数$g_{\rm{p}}^{(2)}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c)探测场相位${\phi _{\rm{p}}}(L)/{\text{π}}$作为探测失谐${\varDelta _{\rm{p}}}/(2{\text{π}})$和入射探测场拉比频率${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}})$的函数. 其他参数同图2 Figure3. (a) The transmitted probe intensity ${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b) the second-order correlation function $g_{\rm{p}}^{(2)}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c) the probe phase ${\phi _{\rm{p}}}(L)/{\text{π}}$ as a function of the probe detuning ${\varDelta _{\rm{p}}}/(2{\text{π}})$ and theRabi frequency of the incident probe field ${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}})$. Other parameters are the same as in Fig. 2.
文献[30]表明, 合作光学非线性除了表现在对探测场强度的依赖以外, 还具有入射场量子属性(光子关联)的敏感性. 图4选取较强的入射场强, 在易于产生非线性光学效应的前提下给出透射区光谱对频率和入射光子关联依赖的相空间结构. 与图3类似, 相位也呈现反对称空间结构, 敏感区域发生在共振频率和AT劈裂之间. 在${\varDelta _{\rm{p}}} \approx $$ \pm 1.4\;{\rm{MHz}}$频率处出现极值, 当初始关联函数改变较小的时候, 相位变化不明显, 但是整体上仍然可以看出对入射光子关联的依赖性. 图 4 (a)探测场透射率${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b)二阶关联函数$g_{\rm{p}}^{(2)}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c)探测场相位${\phi _{\rm{p}}}(L)/{\text{π}}$作为探测失谐${\varDelta _{\rm{p}}}/(2{\text{π}})$和初始二阶关联函数$g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( 0 \right)$的函数. 入射探测场拉比频率${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) = 0.3\;{\rm{MHz}}$, 其他参数同图2 Figure4. (a) The transmitted probe intensity ${I_{\rm{p}}}(L)/{I_{\rm{p}}}(0)$, (b) the second-order correlation function $g_{\rm{p}}^{{\rm{(2)}}}(L)/g_{\rm{p}}^{{\rm{(2)}}}(0)$, (c) probe phase $ {\phi _{\rm{p}}}(L)/{\text{π}} $ as a function of the probe detuning ${\varDelta _{\rm{p}}}/(2{\text{π}})$ and the initial second-order correlation function $g_{\rm{p}}^{\left( 2 \right)}\left( 0 \right)$. The Rabi frequency of the incident probe field ${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) = 0.3\;{\rm{MHz}}$ and other parameters are the same as in Fig. 2.
进一步把频率固定在${\varDelta _{\rm{p}}} \approx \pm 1.4\;{\rm{MHz}}$处, 考察相位对入射探测场强度和初始光子关联的敏感性. 从前面的研究可知, 这个频率对应的是相位的极值, 当发生非线性效应时候, 相位的绝对值会变小, 图5也能显示出这个特点. 另外, 由图5还可以推测出: 相位的非线性效应会在探测场强度和初始光子关联较大的情况下达到饱和. 图 5 探测场相位${\phi _{\rm{p}}}\left( L \right)/{\text{π}}$作为入射探测场拉比频率${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}})$和初始二阶关联函数$g_{\rm{p}}^{\left( 2 \right)}\left( 0 \right)$的函数 (a)探测失谐${\varDelta _{\rm{p}}} = 1.4\;{\rm{MHz}}$; (b)探测失谐${\varDelta _{\rm{p}}} = $$ - 1.4\;{\rm{MHz}}$. 其他参数同图2 Figure5. Probe phase ${\phi _{\rm{p}}}\left( L \right)/{\text{π}}$ as a function of the Rabi frequency of the incident probe field ${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}})$ and initial second-order correlation function $g_{\rm{p}}^{\left( 2 \right)}\left( 0 \right)$: (a) Probe detuning ${\varDelta _{\rm{p}}} = 1.4\;{\rm{MHz}}$; (b) probe detuning ${\varDelta _{\rm{p}}} = $$- 1.4\;{\rm{MHz}}$. Other parameters are the same as in Fig. 2.
最后检查主量子数和原子样品密度对相位的影响. 由文献[34]可知, vdW系数${C_6} \approx {n^{11}}\left( {{c_0} + } \right.$$ \left. {{c_1}n + {c_2}{n^2}} \right)$, $n$为主量子数, ${c_0} = 11.97$, ${c_1} =$$ - 0.8486$, ${c_2} = 0.003385$. 从图6可以看出共振频率处相位一直保持为零, 不受主量子数和密度变化影响. 而${\varDelta _{\rm{p}}} \approx \pm 1.4\;{\rm{MHz}}$频率处的相位, 随着主量子数和原子密度的增加而被不断被压缩, 非线性效应明显. 原因就在于每个超级原子内包含的原子数增多了, 前者是原子密度不变阻塞半径增大引起的, 而后者阻塞半径不变, 仅是原子密度增大的结果. 图 6 探测场相位${\phi _{\rm{p}}}\left( L \right)/{\text{π}}$作为(a)主量子数$n$和(b)原子密度$\rho $的函数. 入射探测场拉比频率${\varOmega _{\rm{p}}}\left( 0 \right)/$$(2{\text{π}}) = 0.3\;{\rm{MHz}}$, 初始二阶关联函数$g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( 0 \right) = 1$. 其他参数同图2 Figure6. Probe phase ${\phi _{\rm{p}}}\left( L \right)/{\text{π}}$ as a function of (a)the principal quantum number $n$ and (b) the atomic density $\rho $. The incident probe intensity ${\varOmega _{\rm{p}}}\left( 0 \right)/(2{\text{π}}) = 0.3\;{\rm{MHz}}$ and the initial second-order correlation function $g_{\rm{p}}^{\left( {\rm{2}} \right)}\left( 0 \right) = 1$. Other parameters are the same as in Fig. 2.
图 1 (a)量子探测场
图 2 (a)探测场透射率
图 3 (a)探测场透射率
图 4 (a)探测场透射率
图 5 探测场相位
图 6 探测场相位
