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压缩态在量子通信[1,2]、量子计算[3,4]和精密测量[5,6]等领域中具有重要应用. 已被用于提高引力波探测器的灵敏度和测量精度等方面[7,8]. 现今实验上已经可以通过光学参量振荡器[9-13]、四波混频[14,15]以及原子与光力耦合等方法[16,17]制备压缩光场, 最大压缩度可达15 dB[9]. 作为一种非经典光场, 通常压缩光的光子统计呈明显的聚束效应, 但通过干涉或减光子操作, 还可表现出光子反聚束效应[18,19]. 更重要的是, 在压缩光中引入平移操作, 构建制备压缩相干态, 在一定的平移和压缩条件下, 可产生明显反聚束效应的压缩光场[20,21]. 利用反聚束效应的光场可以实现超越衍射极限的超分辨成像, 结合更高阶反聚束测量还可提高成像精度[22,23]; 而且具有光子反聚束效应的压缩相干光展现出了较好的粒子特性, 利用此性质可抑制多光子发射的概率, 制备高效单光子源, 从而为长距离量子密钥分发提供优质的光源[24,25]. 因此, 在单光子水平上研究压缩相干态的反聚束效应及量子统计特性已成为研究的热点.
光子的反聚束效应可由Glauber[26,27]引入的二阶光子关联函数g(2)(τ)来表征, 用以区分判别光场的非经典特性, 即在延迟时间τ = 0时的二阶光子关联g(2)(0)可作为区分不同光场的量度: g(2)(0) > 1表明光场不相干或者光子聚束效应[28-31]; g(2)(0) = 1表明光场处于相干态; 而g(2)(0) < 1表明光场具有反聚束效应, 是具有亚泊松分布的非经典态[32]. 实验上通常采用Hanbury Brown-Twiss (HBT)方案测量光场的二阶光子关联[18,19]. 但是, 二阶光子关联g(2)只能反应光子统计分布的方差, 为了获得更多光子统计分布和光场非经典性的信息, 需要测量更高阶的光子关联[33,34]. 其中三阶和四阶光子关联可反映光子数分布的偏度和峭度, 且能够揭示非高斯散射等新的效应. 于是基于多个单光子计数器扩展的HBT方案, 已开展研究了不同光场的高阶光子关联[33-36], 如压缩真空态的超聚束效应、热态的聚束效应及Fock态的反聚束效应, 并应用于鬼成像[37-39]、单光子探测器的表征[40]、激子动力学的研究[41]以及NV色心荧光发射 [42]等领域. 同时, 目前对压缩相干态光子统计特性的研究中, 主要针对二阶光子关联随着平移振幅和压缩参数的变化[20,21], 及利用二阶光子关联研究压缩态与不同光场的干涉现象[43,44]和光子阻挡效应[18], 然而并未深入研究压缩相位对压缩相干态反聚束效应的影响, 且缺少在考虑背景噪声和系统效率条件下对压缩相干态更高阶光子关联的分析.
本文基于4个单光子计数器扩展的双HBT测量方案, 研究分析了相位可变压缩相干态的高阶光子关联及其反聚束效应. 研究了平移和压缩操作不同顺序下压缩相干态的高阶光子反聚束效应. 通过调控压缩相位, 研究了压缩相干态反聚束效应对应的相位区间. 同时考虑系统效率和背景噪声的情况下, 研究了相位可变压缩相干态的入射平移、压缩相位、探测平均光子数对高阶光子反聚束效应的影响, 验证该方法的实验可行性, 为该量子光源在精密测量与保密通信领域的应用奠定基础.
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利用传统的HBT测量方法只能提取光场的二阶光子关联g(2), 但二阶光子关联只能揭示光场的部分光子统计特性, 因此需要测量更高阶的光子关联, 为了同时获得光场的二阶g(2)、三阶g(3)和四阶g(4)光子关联, 对传统HBT方案进行扩展并结合多路单光子探测技术, 构建如图1所示的双HBT测量模型. 实验上可以利用一束相干光和一束经过倍频、光学参量放大制备的压缩光产生相位可变的压缩相干态. 相干光注入强度决定了平移α的大小, 产生压缩光的压缩强度与压缩参数r有关, 而相干光和压缩光的相对相位决定了压缩相位θ. 相位可变压缩相干态|ξ(θ), α
$\rangle $ = S(ξ)D(α)|0$\rangle $ 的光子数分布可表示为[45]
图 1 测量相位可变压缩相干态高阶光子关联的双HBT原理示意图
Figure 1. Schematic diagram of double HBT model for measuring high-order photon correlation of phase-variable squeezed state.
$\begin{split} {P_{{\text{scs}}}}(n) = \;&\frac{1}{{n!u}}{\left| {\frac{v}{{2u}}} \right|^n}{\left| {{H_n} \Big(\frac{\alpha }{{\sqrt {2uv} }}\Big)} \right|^2}\\ & \times\exp\left[ { - {{\left| \alpha \right|}^2} + \frac{v}{{2u}}{\alpha ^2} + \frac{{{v^ * }}}{{2u}}{\alpha ^{*2}}} \right] . \end{split}$ 其中, u = coshr, v = eiθsinhr. 对于|α', ξ(θ)
$\rangle $ = D(α')S(ξ)|0$\rangle $ , 则有α = uα'+vα'*, α'也是位移. 在一定的压缩参数r、位移α和压缩相位θ下, 压缩相干态的光强I和平均光子数成正比, 为I = |μ|2 + sinh2r, 其中μ = α(coshr – eiθsinhr). 于是可利用压缩相干态的光子数分布和平均光子数来得到入射态的高阶光子关联.在具有光子数分布(1)的相位可变压缩相干态进入双HBT探测系统之前, 经过一个具有透射率为η的分束器, 用以模拟系统总的探测效率. 同时, 在入射态中引入背景噪声|β
$\rangle $ , 其平均光子数为γ. 考虑了总的探测效率η和背景噪声γ后相位可变压缩相干态的光子数分布为$ \begin{split} \;& {P}_{\text{mix}}=\displaystyle \sum _{m=0}^{L}\frac{{\gamma }^{L-m}{\eta }^{m}}{m!\left(L-m\right)!} \\ & \times \displaystyle \sum _{n=m}^{\infty } \frac{{2}^{-n}{\left(1 - \eta \right)}^{n-m}{\mathrm{coth}}^{-n}{\left|{H}_{n} \left( {\text{e}}^{-\text{i}\theta /2}\alpha \sqrt{\mathrm{csch}2r}\right) \right|}^{2}}{\left(n-m\right)!}\\ & \times {\rm{e}}^{-\gamma }r{\text{e}}^{{\alpha }^{2}\left(\mathrm{cos}\theta \mathrm{tanh}r\text{-1}\right)}\mathrm{sech}r\text{, }\\[-10pt] \end{split} $ 其中,
${H_n}\left( x \right) = {\left( { - 1} \right)^n}{{\text{e}}^{{x^2}}}\dfrac{{{{\text{d}}^n}}}{{{\text{d}}{x^n}}}{{\text{e}}^{ - {x^2}}}$ 是n阶厄米多项式.随后入射态进入由4个单光子探测器构成的双HBT装置, 完成对光场高阶光子关联的测量. 假定探测器D1, D2, D3和D4的性能相同, 每个响应时间内只能分辨一个光子数, 因此可以得到被测光子数的概率
$ {P_{{\text{click}} \otimes i}} $ 和系统检测的总平均光子数$ \left\langle n \right\rangle = \displaystyle\sum\nolimits_{i = 0}^\infty {i{P_{{\text{click}} \otimes i}}} $ , 其中i表示响应探测器的数量. 可通过联合概率分布方法[46]从(2)式得到(3)式, 由定义${g^{\left( m \right)}} = \dfrac{{\left\langle {{n_1}{n_2} \cdots {n_m}} \right\rangle }}{{\left\langle {{n_1}} \right\rangle \left\langle {{n_2}} \right\rangle \cdots \left\langle {{n_m}} \right\rangle }}$ 可得到相位可变压缩相干态的二阶、三阶和四阶光子关联:$ \begin{split} {g^{\left( 2 \right)}} =\;& \frac{{\left\langle {{n_1}{n_2}} \right\rangle }}{{\left\langle {{n_1}} \right\rangle \left\langle {{n_2}} \right\rangle }} = \frac{{8{P_{{\text{clic}}{{\text{k}}^{ \otimes 2}}}} + 24{P_{{\text{clic}}{{\text{k}}^{ \otimes 3}}}} + 48{P_{{\text{clic}}{{\text{k}}^{ \otimes 4}}}}}}{{3{{\left\langle n \right\rangle }^2}}} \\ =\;& \frac{8}{3}\frac{{{P_{{\text{clic}}{{\text{k}}^{ \otimes 2}}}}}}{{{{\left\langle n \right\rangle }^2}}} + \frac{{\left\langle n \right\rangle }}{2}{g^{\left( 3 \right)}} + \frac{{{{\left\langle n \right\rangle }^2}}}{{16}}{g^{\left( 4 \right)}}, \end{split} \tag{3a}$ $ \begin{split} {g^{\left( 3 \right)}} =\;& \frac{{\left\langle {{n_1}{n_2}{n_3}} \right\rangle }}{{\left\langle {{n_1}} \right\rangle \left\langle {{n_2}} \right\rangle \left\langle {{n_3}} \right\rangle }} = \frac{{16{P_{{\text{clic}}{{\text{k}}^{ \otimes 3}}}}}}{{{{\left\langle n \right\rangle }^3}}} \\=\;& \frac{2}{{\left\langle n \right\rangle }}\left( {{g^{\left( 2 \right)}} - \frac{{{{\left\langle n \right\rangle }^2}}}{{16}}{g^{\left( 4 \right)}} - \frac{8}{3}\frac{{{P_{{\text{clic}}{{\text{k}}^{ \otimes 2}}}}}}{{{{\left\langle n \right\rangle }^2}}}} \right), \end{split} \tag{3b}$ $\begin{split} {g^{\left( 4 \right)}}=\;& \frac{{\left\langle {{n_1}{n_2}{n_3}{n_4}} \right\rangle }}{{\left\langle {{n_1}} \right\rangle \left\langle {{n_2}} \right\rangle \left\langle {{n_3}} \right\rangle \left\langle {{n_4}} \right\rangle }} = \frac{{256{P_{{\text{clic}}{{\text{k}}^{ \otimes 4}}}}}}{{{{\left\langle n \right\rangle }^4}}} \\=\;& \frac{{16}}{{{{\left\langle n \right\rangle }^2}}}\left( {{g^{\left( 2 \right)}} - \frac{{\left\langle n \right\rangle }}{2}{g^{\left( 3 \right)}} - \frac{8}{3}\frac{{{P_{{\text{clic}}{{\text{k}}^{ \otimes 2}}}}}}{{{{\left\langle n \right\rangle }^2}}}} \right). \end{split} \tag{3c}$ 由(3)式可知, 基于双HBT测量模型得到二阶g(2)、三阶g(3)、四阶g(4)光子关联存在相互关系.
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在考虑压缩相位θ的条件下, 制备压缩相干态的过程中, 压缩和位移操作顺序至关重要. 故压缩相干态可表示为|ξ(θ), α
$\rangle $ 和|α, ξ(θ)$\rangle $ , 其中D(α) = eαa†-α*a为位移算符, S(ξ) = exp(ξ*a2 – ξa†2)为压缩算符, ξ(θ) = reiθ是与压缩相位θ相关的压缩因子, 压缩参数为r = |ξ|, a和a†分别是湮灭和产生算符. 平移α为注入的相干振幅, 根据实际实验条件α为实数. 压缩相干态|ξ(θ), α$\rangle $ 表示先对真空态进行位移操作, 再进行压缩; 而压缩相干态|α, ξ(θ)$\rangle $ 则是相反的过程, 即先对真空态进行压缩操作, 然后位移. 针对上述两种情况的压缩相干态, 以下研究对应的高阶光子关联.当入射的相位可变压缩相干态为|ξ(θ), α
$\rangle $ 时, 可得理想情况下光场的高阶光子关联为$ {g^{\left( 2 \right)}} = 1 - \frac{1}{{{A^2}}}\left[ {B - \left( {2{{\left| \Omega \right|}^2} + \cosh 2r} \right){{\sinh }^2}r} \right], \tag{4a}$ $\begin{split} {g^{\left( 3 \right)}} =\;& 1 - \frac{1}{{{A^3}}}\left\{ 3B\left( {{{\left| \Omega \right|}^2} + 3{{\sinh }^2}r} \right)\right. \\ &- \left( {2 + 7\cosh 2r} \right){{\sinh }^4}r \\& \left. { - 3{{\left| \Omega \right|}^2}{{\sinh }^2}r\left[ {2{{\left| \Omega \right|}^2} + \left( {4\cosh 2r - 1} \right)} \right]} \right\}, \\[-5pt] \end{split} \tag{4b}$ $\begin{split} {g^{\left( 4 \right)}} =\;& 1 - \frac{1}{{{A^4}}}\Big\{ 6B\left[ {{{\left| \Omega \right|}^4} - 8{{\left| \Omega \right|}^2}{{\sinh }^2}r - 3C{{\sinh }^2}r} \right] \\& - 4D{{\left| \Omega \right|}^2}{{\sinh }^4}r - 6{\left| \Omega \right|^4}{\sinh ^2}r\left( {2C + 3\cosh 2r} \right)\\& -12{\left| \Omega \right|^6}{\sinh ^2}r{\sinh ^4}r \\& -3{B^2}-\left( {3 - 7\cosh 2r + 13\cosh 4r} \right) \Big\}, \end{split} \tag{4c}$ 其中, Ω = α(coshr – eiθsinhr), A = |Ω|2 + sinh2r, B = (Ω*2eiθ + Ω2e–iθ)coshrsinhr, C = cosh(2r) + 3sinh2r, D = 13 cosh2r + 23cosh(2r).
类似地, 当入射的相位可变压缩相干态为|α, ξ(θ)
$\rangle $ 时, 可得理想情况下光场的高阶光子关联为$ {g^{\left( 2 \right)}} = 1 - \frac{{F - 2{{\left| \alpha \right|}^2}{{\sinh }^2}r - \cosh 2r{{\sinh }^2}r}}{{{E^2}}} \text{, } \tag{5a}$ $ \begin{split} {g^{\left( 3 \right)}} =\;& 1 - \frac{1}{{{E^3}}}\left[ 3F\left( {E + 2{{\sinh }^2}r} \right)\right. \\& - \left( {2 + 7\cosh 2r} \right){{\sinh }^4}r \\& \left. { - 3{{\left| \alpha \right|}^2}\left( {2{{\left| \alpha \right|}^2} + 4\cosh 2r - 1} \right){{\sinh }^2}r} \right], \end{split} \tag{5b}$ $\begin{split} {g^{\left( 4 \right)}} =\;& 1 - \frac{1}{{{E^4}}}\left\{ 6F\left[ {{{\left| \alpha \right|}^4}} - 8{{\left| \alpha \right|}^2}{{\sinh }^2}r { - 3C{{\sinh }^2}r} \right] \right.\\& - 3{F^2}- 6{\left| \alpha \right|^4}{\sinh ^2}r\left( {2C + 3\cosh 2r} \right) \\& - 4D{\left| \alpha \right|^2}{\sinh ^4}r + \left( 3 - 7\cosh 2r \right.\\& \left.\left.+ 13\cosh 4r \right){{\sinh }^4}r - 12{{\left| \alpha \right|}^6}{{\sinh }^2}r \right\}. \end{split} \tag{5c}$ 其中E = |α|2 + sinh2r, F = (Ω*2eiθ + Ω2e–iθ). 上述结果表明, 理想情况下两种方式制备的相位可变压缩相干态的高阶光子关联与压缩参数r、平移振幅α以及压缩相位θ密切相关.
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当压缩相位θ = 0, 相应的压缩相干态为振幅压缩时, |ξ(θ), α
$\rangle $ 和|α, ξ(θ)$\rangle $ 两种方式得到的压缩相干态高阶光子反聚束(g(n) < 1, n = 2, 3, 4)结果如图2所示. 图2(a1)—(c1)是压缩相干态|ξ(θ), α$\rangle $ 的二阶、三阶和四阶光子关联随压缩参数r和平移振幅α的变化结果, 图2(a2)—(c2)为|α, ξ(θ)$\rangle $ 入射下光场的高阶光子关联g(n)随r和α变化的反聚束结果. 图2中的白色实线表示g(n) = 0.5, 黑色实线表示g(n) = 1, 有色变化区域表示g(n) < 1的反聚束区域. 结果表明, 压缩相位θ = 0时, 两种压缩相干态入射条件下的高阶光子反聚束结果略有不同但变化规律相近. |α, ξ(θ)$\rangle $ 的高阶光子关联的反聚束区域比|ξ(θ), α$\rangle $ 的反聚束区域更大, 同时两种压缩相干态的高阶光子关联具有相似的变化趋势: 当压缩参数r为较小的定值时, 随着平移振幅α的增大, 在反聚束区域(g(n) < 1)高阶光子关联g(n)先减小到最低反聚束值, 再增大并逐渐趋于1; 当平移振幅α为较小的定值时, 随着压缩参数r的增大, 在反聚束区域g(n)从1减小到最小反聚束值, 进而再增大. 同时结果还表明, 随着阶数n的增大, g(n) < 0.5的强反聚束区域明显增大, 即测量更高阶光子关联更易获得明显的反聚束效应. 此外, 当|ξ(θ), α$\rangle $ 的平移振幅为α, 而|α, ξ(θ)$\rangle $ 的平移振幅为Ω = α(coshr – eiθsinhr)时, 两种相位可变压缩态|ξ(θ), α$\rangle $ 和|α, ξ(θ)$\rangle $ 入射条件下还可得到相同的高阶光子关联. 因上述两种压缩相干态入射条件下得到的高阶光子关联结果规律相近, 且两光子关联之间又存在转换关系, 故以下只分析相位可变压缩相干态|ξ(θ), α$\rangle $ 入射下的结果.
图 2 压缩相位为θ = 0时, 压缩相干态 (a1)—(c1) |ξ(θ), α
$\rangle $ 和(a2)—(c2) |α, ξ(θ)$\rangle $ 的高阶光子关联随平移振幅α和压缩参数r变化的反聚束结果Figure 2. High-order photon antibunching of squeezed coherent states (a1)–(c1) |ξ(θ), α
$\rangle $ and (a2)–(c2) |α, ξ(θ)$\rangle $ versus displacement amplitude α and squeezing parameter r for squeezing phase θ = 0.理想情况下, 当平移振幅为α = 0.1时, 相位可变压缩相干态的g(n)随着压缩参数r和压缩相位θ的变化结果如图3所示. 图3(a)—(c)的结果表明: 理想情况下, 高阶光子关联g(n)随着压缩相位θ呈现2π的周期性变化, 图中的白色实线表示g(n) = 0.5, 黑色实线为g(n) = 1, 有色变化区域为g(n) < 1的反聚束区域. 随着阶数n的增大, 达到同样强反聚束效应(g(n) < 0.5)所需的压缩参数变小, 即强的光子反聚束效应的获得, 可在较弱压缩度条件下测量更高阶光子关联, 高阶光子反聚束效应测量对压缩度的要求降低. 当压缩参数r固定为较小的值时, 随着压缩相位θ的变化, 最小的g(n)值总是在相位θ = 0时, 并且以2π周期变化. 当相位θ为[0, π/2]或其他反聚束区域一个固定值时, 随着压缩参数r的增大, 高阶光子关联g(n)从1减小到最低反聚束值, 然后再逐渐增大, 直至反聚束效应消失.
图 3 当平移振幅α = 0.1时, 相位可变压缩相干态高阶光子关联g(n)随压缩参数r和压缩相位θ变化的反聚束结果
Figure 3. Photon antibunching results of high-order correlation g(n) of phase-variable squeezed coherent state versus squeezing parameter r and squeezing phase θ when displacement amplitude α = 0.1.
当压缩参数r固定为较小值0.01时, 理想情况下压缩相干态的g(n)随着平移振幅α和压缩相位θ的变化结果如图4所示. 结果表明: g(n)随着压缩相位θ的变化呈现2π的周期性, 图中的白色实线为g(n) = 0.5, 黑色实线为g(n) = 1, 有色区域表示g(n) < 1的反聚束区域, 无色区域表示g(n) > 1的聚束区域. 随着阶数n的增大, 更高阶光子关联g(n) < 0.5 (n = 3, 4)的强反聚束区域明显大于g(2) < 0.5的区域, 即更高阶光子关联更易测得强的反聚束效应. 当平移振幅α固定为较小值, 压缩相位在3π/2
$\leqslant $ θ$\leqslant $ 5π/2时, 随着θ的增大, 高阶光子反聚束效应先增强后减弱, 直至g(n)趋于1.当相位取3π/2$\leqslant $ θ$\leqslant $ 5π/2时, 随着平移振幅α的增大, g(n)从聚束区域降至1, 进入反聚束区域, 降至最低反聚束值, 而后又增大逐渐趋于1.
图 4 当压缩参数r = 0.01时, 相位可变压缩相干态的高阶光子关联g(n)随平移振幅α和压缩相位θ变化的反聚束结果
Figure 4. Photon antibunching results of g(n) of phase-variable squeezed coherent state versus displacement amplitude α and squeezing phase θ when squeezing parameter r = 0.01.
综合以上结果, 可通过控制压缩相干态的压缩相位θ, 在特定的平移压缩α-r参数区间获得显著的光子反聚束效应; 同时, 通过测量高阶光子关联g(n), 也可获取压缩态的相位信息. 此外在较低压缩度及较小平移条件下, 测量更高阶光子关联, 可获得同样明显反聚束效应的结果, 即更高阶光子反聚束效应的测得对压缩度和平移的强度要求更低.
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理想情况下, 在压缩相位θ = 0, 压缩参数r = 10–4时, 可得压缩相干态高阶光子反聚束的最小值分别为g(2) = 4.006 × 10–4, g(3) = 1.3594 × 10–4和g(4) = 6.6352 × 10–5, 此时相应的平移振幅分别为α = 0.01, α = 0.0173和α = 0.0233. 在实际实验过程中, 由于背景噪声γ和探测效率η的影响, 会导致测量的反聚束值偏离理想值. 于是, 基于双HBT测量模型, 理论分析了在压缩相位θ = 0时, 背景噪声γ和探测效率η对压缩相干态|ξ, α
$\rangle $ 高阶光子反聚束最小值g(2), g(3)和g(4)的影响, 从而获得实验中噪声的范围以及系统所需的探测效率, 分析结果如图5所示. 结果表明, 探测效率η越高, g(n)趋于1的速度越慢, 反聚束效应减弱的速度越慢, 对噪声的鲁棒性越好; 背景噪声γ < 10–5时, 压缩相干态处于g(n) < 0.1的强反聚束效应区域. 当γ < 10–3时, 仍可获得g(n) (n = 2, 3, 4) < 1的光子反聚束效应; 当γ < 10–8时, 探测效率η几乎不影响反聚束效应值; 而当γ > 10–3时, 高阶光子关联g(n)趋于1并饱和, 此时提高探测效率也无法测得较显著的光子反聚束效应. 这主要是由于随着背景噪声的增大, 由弱压缩、弱位移主导的压缩相干态逐渐被噪声湮没, 变为由泊松分布背景噪声主导的光场.
图 5 探测效率分别为η = 0.1, 0.5和1时, 压缩态的(a)二阶、(b)三阶和(c)四阶光子关联最小值随背景噪声γ变化的结果
Figure 5. Minimum second-order (a), third-order (b), and fourth-order (c) photon correlations of squeezed coherent state |ξ, α
$\rangle $ versus background noise γ when detection efficiencies η = 0.1, 0.5, and 1.此外, 随着光子关联阶数n的增大, 在达到同等反聚束效应值及相同探测效率条件下, 更高阶光子关联g(n) (n = 3, 4)比二阶光子关联g(2)所对应的背景噪声大, 即测量更高阶光子关联反聚束效应所容忍的背景噪声更大, 对噪声的鲁棒性更好. 同时在相同背景噪声条件下, 获得同样的反聚束效应值, 更高阶光子关联比二阶光子关联对探测效率的要求更低. 当探测效率低至η = 0.1, 背景噪声γ = 10–6时, 仍可获得较强的光子反聚束效应: g(2) = 0.1740, g(3) = 0.0432, g(4) = 0.0149, 此时四阶光子关联g(4)的比二阶光子关联g(2)低了1个数量级. 因此, 更高阶光子关联的测量对探测效率的要求更低、对背景噪声的鲁棒性更强, 在实验上更易测得高阶光子反聚束效应.
为了研究实验可行条件下, 系统测量的平均光子数
$ \left\langle n \right\rangle = \displaystyle\sum\nolimits_{i = 0}^\infty {i{P_{{\text{click}} \otimes i}}} $ 与高阶光子关联的关系, 分析了背景噪声为γ = 10–6, 探测效率为η = 0.5时的高阶光子关联, 结果如图6所示. 结果表明, 随着被测平均光子数$\langle $ n$\rangle $ 的增大, 相位可变压缩相干态的高阶光子关联g(n)的反聚束值先减小到最小值, 后逐渐增大并趋于1. 在压缩相干态的被测平均光子数$\langle $ n$\rangle $ 较小时可以获得较好的反聚束效应, 更高阶光子关联的反聚束最小值对应较强的被测平均光子数, 但高阶光子关联的强反聚束效应测量只需较弱的待测光子数. 利用压缩参数r, 可得光场的压缩度S = 10 lge–2r. 当压缩度非常小时(S$\leqslant $ 10–8dB), g(n)的反聚束效应较弱, 这是由于背景噪声较大时, 被测光子数分布由泊松分布主导, 因此g(n)在1附近. 同时, 随着压缩度S的增强, 反聚束效应区间向被测平均光子数$\langle $ n$\rangle $ 增大的方向移动, 相应$\langle $ n$\rangle $ 区间内反聚束效应呈现先增强后减弱的变化.
图 6 背景噪声γ = 10–6, 探测效率η = 0.5, 压缩度分别为 (a) r = 10–8 (8.6859 × 10–8 dB); (b) r = 10–4 (8.6859 × 10–4 dB); (c) r = 10–1 (8.6859 × 10–1 dB)时, 压缩相干态的g(n)随平均光子数
$\langle $ n$\rangle $ 的变化Figure 6. g(n) of squeezed coherent state versus measured mean photon number
$\langle $ n$\rangle $ when background noise is γ = 10–6, detection efficiency is η = 0.5, and squeezing degrees are (a) r = 10–8 (8.6859 × 10–8 dB); (b) r = 10–4 (8.6859 × 10–4 dB); (c) r = 10–1 (8.6859 × 10–1 dB) respectively.图7为反聚束区域内, 压缩相干态高阶光子关联g(n)随压缩度S的变化结果. 不同于图2和图3中理想情况下高阶光子反聚束随压缩参数r的变化, 图7(a)—(c)是在实际可行探测效率η = 0.5和背景噪声γ = 10–6条件下, 平移振幅分别为α = 0.01, α = 0.1和α = 1时压缩相干态|ξ, α
$\rangle $ 的高阶光子关联的变化结果. 结果表明: 随着压缩度S的增大, 入射光场从相干光逐渐转变为压缩相干态, 压缩参量r和平移振幅α共同作用, g(n)先从1减小到最小值, 然后逐渐增大; 此外更高阶光子关联相较于二阶光子关联的反聚束最小值更小, 反聚束最小值对应的压缩度S也较小. 同时, 当平移振幅较小时, 如图7(a) α = 0.01, 相位可变压缩相干态中的相干平移成分所占比重较小, 受泊松分布背景噪声的影响, 其反聚束效应相较更强平移α = 0.1[图7(b)]时较弱; 当平移继续增大至α = 1(图7(c))时, 相干光成分变大, 此时反聚束效应又变弱. 随着平移α的增大, 反聚束最小值对应的压缩度S变大, 从α = 0.01时g(4)最小值对应压缩度S的10–4量级, 增至α = 0.1和α = 1时g(4)最小值分别对应压缩度S的10–2和100量级. 值得注意的是, 更高阶光子反聚束极小值对应的压缩度S更小, 对压缩度的强度要求不高, 在实验上更易制备, 可利用双HBT装置测得较低压缩度相位可变压缩态的反聚束效应.
图 7 背景噪声γ = 10–6, 探测效率η = 0.5, 位移分别为 (a) α = 0.01, (b) α = 0.1和(c) α = 1时, 相位可变压缩相干态高阶光子关联g(n)随压缩度S的变化
Figure 7. g(n) of phase-variable squeezed coherent state versus squeezing degree S when background noise is γ = 10–6, detection efficiency is η = 0.5, and displacement amplitudes are (a) α = 0.01, (b) α = 0.1, and (c) α = 1, respectively.
最后分析背景噪声γ和探测效率η对不同压缩相位下压缩相干态高阶光子关联g(n)的影响, 结果如图8和图9所示. 当压缩相位取θ = 0, 此时为各阶光子关联最小值对应的结果, 即在压缩参数为r = 10–4, 位移振幅分别为α = 0.01, α = 0.0173和α = 0.0233的条件下. 图8和图9中白色实线表示g(n) = 0.5, 黑色实线表示g(n) = 1, 有色变化区域表示g(n) < 1的反聚束区域. 高阶光子关联g(n)随压缩相位θ呈2π周期性变化, 由图中(–π/2, π/2)的区间内结果可知, 高阶光子关联g(n)随着背景噪声γ的增大而逐渐趋于1, 反聚束效应逐渐减弱, 光子统计分布逐渐从亚泊松分布趋于泊松分布, 最终变为由泊松分布噪声主导的光场; 同时图8结果还显示g(n) < 0.5 (n = 3, 4)的反聚束效应区域明显大于g(2) < 0.5的区域, 表明更高阶光子关联对背景噪声的鲁棒性更好. 图9结果也表明在不同探测效率η下, 更高阶光子关联的显著反聚束效应区域大于二阶光子关联相应的反聚束区域, 并且利用更高阶光子关联表征光场反聚束效应时, 对探测效率η的要求降低. 可以发现, 利用双HBT模型测量相位可变压缩相干态的更高阶光子反聚束效应, 可降低背景噪声γ和探测效率η的影响, 测量结果更趋于理想值, 更高阶光子关联相较于二阶光子关联更适于揭示光场的反聚束效应.
图 8 探测效率η = 0.5, 压缩参数r = 10–4, 平移振幅分别为 (a) α = 0.01, (b) α = 0.0173和(c) α = 0.0233时, 相位可变压缩相干态的高阶光子关联随压缩相位θ和背景噪声γ变化的分布图
Figure 8. Maps of high-order photon correlations g(n) of phase-variable squeezed coherent state versus squeezing phase θ and background noise γ when detection efficiency is η = 0.5, squeezing parameter is r = 10–4 and displacement amplitudes are (a) α = 0.01, (b) α = 0.0173, and (c) α = 0.0233 respectively.
图 9 背景噪声γ = 10–6, 压缩参数r = 10–4, 平移振幅分别为 (a) α = 0.01, (b) α = 0.0173和(c) α = 0.0233时, 压缩相干态的高阶光子关联随压缩相位θ和探测效率η变化的分布图
Figure 9. Maps of g(n) of the squeezed coherent state versus squeezing phase θ and detection efficiency η when background noise is γ = 10–6, squeezing parameter is r = 10–4 and displacement amplitudes are (a) α = 0.01, (b) α = 0.0173, and (c) α = 0.0233, respectively.
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基于4个单光子探测器的双HBT方案, 研究分析了理想情况下相位可变压缩相干态的高阶光子反聚束随压缩参数r、平移α和压缩相位θ的变化. 光场反聚束效应随压缩相位θ呈2π周期性变化, 当压缩相位处于[0, π/2]时, 在较大的平移压缩α-r参数变化区间均可获得压缩相干态的反聚束效应, 理想情况下高阶反聚束效应最小值趋于0, 低至10–5量级. 同时, 考虑实际实验条件下, 背景噪声和系统效率对压缩相干态高阶光子关联的影响, 在较低探测效率和较高背景噪声时, 仍可获得明显的高阶光子反聚束效应, 并且证明了更高阶光子关联相较于二阶光子关联对背景噪声和系统效率具有更强的鲁棒性. 另外, 还分析验证了不同被测平均光子数
$\langle $ n$\rangle $ 和压缩度情况下, 压缩相干态高阶光子关联随相位θ变化的结果, 表明在较低待测平均光子数和较弱压缩度条件下, 可获得更强的压缩相干态反聚束效应, 而且更高阶光子反聚束效应对压缩度的要求更低. 综上本文研究结果表明, 通过对压缩相干态压缩相位的控制, 同时综合考虑背景噪声、系统效率、平移振幅、压缩度等因素影响, 可制备具有显著反聚束效应的相位可变压缩相干态, 并可利用扩展的HBT方案测量更高阶光子关联实现对反聚束效应的更高效提取, 突破传统测量极限并实现光场表征, 为高质量量子光源制备及其在精密测量等领域的应用提供支持.
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基于扩展的Hanbury Brown-Twiss方案研究相位可变压缩相干态的高阶光子关联及反聚束效应. 通过调控压缩参数r、平移α和压缩相位θ, 压缩相干态的高阶光子关联呈明显的反聚束效应. 在压缩相位θ∈[0,π/2]范围内, 较大α-r参数区间都可获得光场的高阶反聚束效应, 理想情况下最小的反聚束值为g(4) = 6.6352 × 10–5. 研究了背景噪声γ和系统探测效率η对高阶光子反聚束的影响, 在较低探测效率η = 0.1, 背景噪声γ = 10–6时, 仍可获得明显的高阶反聚束效应g(4) = 0.0149, 验证了更高阶光子关联的反聚束效应对实验环境具有较强的鲁棒性. 此外, 研究了相位可变压缩相干态的反聚束效应随探测平均光子数
$\langle$ n$\rangle $ 和压缩度S的变化, 在探测平均光子数远小于1、压缩参数10–4以下时, 仍可得到g(n)$\ll $ 0.5的显著的光子反聚束效应. 结果表明利用对压缩相位θ的调控可制备具有明显反聚束效应的压缩相干态, 在量子精密测量及保密通信领域有着潜在的重要应用.Squeezed state has important applications in quantum communication, quantum computing, and precision measurement. It has been used to improve the sensitivity and measurement accuracy of gravitational wave detectors. Currently, squeezed state can be prepared by optical parametric oscillators, four-wave mixing, and atom–optomechanical coupling. As a typical non-classical light, the photon statistics of squeezed state usually shows obvious bunching effect, but it can also present photon antibunching effect through interference or photon subtraction operation. More importantly, squeezed coherent state is prepared by performing displacement operation on the squeezed state. In the case of certain displacement and squeezing operations, squeezed coherent state with obvious antibunching effect can be produced. The squeezed coherent state with photon antibunching effect can be employed to achieve super-resolution imaging beyond the diffraction limit, and the state exhibits good particle features which can suppress the multiphoton emission. Then it has become a focus for studying the antibunching effect and quantum statistical properties of squeezed coherent state at a single-photon level. The photon antibunching effect can be characterized by the second-order photon correlation g(2)(τ), which is introduced by Glauber to determine the non-classical properties of the light field. Namely, the second-order photon correlation g(2) can be used as a metric to distinguish different lights. Hanbury Brown-Twiss (HBT) scheme is used to measure the second-order photon correlation experimentally. However, the second-order photon correlation g(2) can reflect only the variance of the photon-number statistical distribution. In order to obtain more information about the photon statistical distribution and non-classical features, it is necessary to measure higher-order photon correlations. Then the higher-order photon correlations for different light fields are studied by extending the traditional HBT scheme and combining with multiplex single-photon detection technology. This method can be applied to ghost imaging, characterization of single-photon detectors, research of exciton dynamics, and analysis of NV center fluorescence emission. However, the research on photon statistics of the squeezed state focuses mainly on the second-order photon correlation and the effect of displacement amplitude on the statistical properties. The effect of squeezed phase on photon antibunching and higher-order photon correlation of squeezed coherent states, with background noise and detection efficiency taken into consideration, have not been investigated. In this paper, we study high-order photon correlations and antibunching effect of phase-variable squeezed coherent state based on an extended HBT scheme. The photon statistics of the squeezed coherent state manifests prominent antibunching effect by adjusting the squeezing parameter r, displacement amplitude α and squeezing phase θ. The antibunching effect of the state can be obtained in a wide range of α-r parameter space when squeezing phase θ∈[0,π/2]. In an ideal case, the minimum antibunching values of the squeezed coherent state are g(2) = 4.006 × 10–4, g(3) = 1.3594 × 10–4 and g(4) = 6.6352 × 10–5. When the detection efficiency η = 0.1 and background noise γ = 10–6, the strong antibunching effect can still be observed, specifically, g(2) = 0.1740, g(3) = 0.0432, g(4) = 0.0149. The results indicate that the antibunching effect of higher-order photon correlation has strong robustness against the experimental environment. In addition, the antibunching effect of the phase-variable squeezed coherent state is studied as a function of the measured mean photon number <n> and the squeezing degree S. When the measured mean photon number is much less than 1 and the squeezing parameter is less than 10–4, a prominent photon anti-bunching effect of g(n) $\ll $ 0.5 can still be obtained. The results show that the control of the squeezing phase θ can be used to prepare the squeezed coherent state with obvious antibunching effect, which has potentially important applications in quantum metrology and secure communication.-
Keywords:
- high-order photon correlation /
- squeezed coherent state /
- squeezing phase /
- antibunching effect
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图 1 测量相位可变压缩相干态高阶光子关联的双HBT原理示意图
Fig. 1. Schematic diagram of double HBT model for measuring high-order photon correlation of phase-variable squeezed state.
图 2 压缩相位为θ = 0时, 压缩相干态 (a1)—(c1) |ξ(θ), α
$\rangle $ 和(a2)—(c2) |α, ξ(θ)$\rangle $ 的高阶光子关联随平移振幅α和压缩参数r变化的反聚束结果Fig. 2. High-order photon antibunching of squeezed coherent states (a1)–(c1) |ξ(θ), α
$\rangle $ and (a2)–(c2) |α, ξ(θ)$\rangle $ versus displacement amplitude α and squeezing parameter r for squeezing phase θ = 0.
图 3 当平移振幅α = 0.1时, 相位可变压缩相干态高阶光子关联g(n)随压缩参数r和压缩相位θ变化的反聚束结果
Fig. 3. Photon antibunching results of high-order correlation g(n) of phase-variable squeezed coherent state versus squeezing parameter r and squeezing phase θ when displacement amplitude α = 0.1.
图 4 当压缩参数r = 0.01时, 相位可变压缩相干态的高阶光子关联g(n)随平移振幅α和压缩相位θ变化的反聚束结果
Fig. 4. Photon antibunching results of g(n) of phase-variable squeezed coherent state versus displacement amplitude α and squeezing phase θ when squeezing parameter r = 0.01.
图 5 探测效率分别为η = 0.1, 0.5和1时, 压缩态的(a)二阶、(b)三阶和(c)四阶光子关联最小值随背景噪声γ变化的结果
Fig. 5. Minimum second-order (a), third-order (b), and fourth-order (c) photon correlations of squeezed coherent state |ξ, α
$\rangle $ versus background noise γ when detection efficiencies η = 0.1, 0.5, and 1.
图 6 背景噪声γ = 10–6, 探测效率η = 0.5, 压缩度分别为 (a) r = 10–8 (8.6859 × 10–8 dB); (b) r = 10–4 (8.6859 × 10–4 dB); (c) r = 10–1 (8.6859 × 10–1 dB)时, 压缩相干态的g(n)随平均光子数
$\langle $ n$\rangle $ 的变化Fig. 6. g(n) of squeezed coherent state versus measured mean photon number
$\langle $ n$\rangle $ when background noise is γ = 10–6, detection efficiency is η = 0.5, and squeezing degrees are (a) r = 10–8 (8.6859 × 10–8 dB); (b) r = 10–4 (8.6859 × 10–4 dB); (c) r = 10–1 (8.6859 × 10–1 dB) respectively.
图 7 背景噪声γ = 10–6, 探测效率η = 0.5, 位移分别为 (a) α = 0.01, (b) α = 0.1和(c) α = 1时, 相位可变压缩相干态高阶光子关联g(n)随压缩度S的变化
Fig. 7. g(n) of phase-variable squeezed coherent state versus squeezing degree S when background noise is γ = 10–6, detection efficiency is η = 0.5, and displacement amplitudes are (a) α = 0.01, (b) α = 0.1, and (c) α = 1, respectively.
图 8 探测效率η = 0.5, 压缩参数r = 10–4, 平移振幅分别为 (a) α = 0.01, (b) α = 0.0173和(c) α = 0.0233时, 相位可变压缩相干态的高阶光子关联随压缩相位θ和背景噪声γ变化的分布图
Fig. 8. Maps of high-order photon correlations g(n) of phase-variable squeezed coherent state versus squeezing phase θ and background noise γ when detection efficiency is η = 0.5, squeezing parameter is r = 10–4 and displacement amplitudes are (a) α = 0.01, (b) α = 0.0173, and (c) α = 0.0233 respectively.
图 9 背景噪声γ = 10–6, 压缩参数r = 10–4, 平移振幅分别为 (a) α = 0.01, (b) α = 0.0173和(c) α = 0.0233时, 压缩相干态的高阶光子关联随压缩相位θ和探测效率η变化的分布图
Fig. 9. Maps of g(n) of the squeezed coherent state versus squeezing phase θ and detection efficiency η when background noise is γ = 10–6, squeezing parameter is r = 10–4 and displacement amplitudes are (a) α = 0.01, (b) α = 0.0173, and (c) α = 0.0233, respectively.
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