在 TikZ/PGF 中绘制沿 z 轴方向的平面布洛赫球

Question

我想绘制一个 Bloch 球体，该球体在某个 z 值处有平面（可以指定）。例如像这样：

我不需要阴影相似，只需要平面在 3D Blosh 球体中清晰即可。我目前有以下输出：

它是由以下 MWE 生成的：

\documentclass[tikz,11pt]{standalone} 
\usepackage{pgfplots}  
\usetikzlibrary{arrows.meta,arrows,angles} 
\usepackage{subfig}

\begin{document}

\begin{tikzpicture}[line cap=round, line join=round, >=Triangle]
\clip(-2.1,-2.1) rectangle (2.38,2.58);

\draw[ball color=gray!20!white, fill opacity=0.6] (0,0) circle (2cm);   
\draw [rotate around={0.:(0.,0.)},dash pattern=on 3pt off 3pt] (0,0) ellipse (2cm and 0.9cm);   \draw (0,0)-- (0.70,1.07);


\draw[-to,line width=0.7pt]   (0,0) -- +(0,2);   \draw[-to,line width=0.7pt]   (0,0) -- +(-0.83,-0.81);   \draw[-to,line width=0.7pt]  (0,0) -- +(2,0);    


\draw [-Latex, <-, >=stealth', shift={(0,0)}, black, fill opacity=1] (56.7:0.4) arc (56.7:90.:0.4);   
\begin{scope}[rotate around x=10, y=10, xshift=1, yshift=-4.6]   
\draw [-Latex, ->, >=stealth', shift={(0,0)}, black, fill opacity=1] (-135.7:0.4) arc (-135.7:-33.2:0.4);   
\end{scope}

\draw [dotted] (0.7,1)-- (0.7,-0.46);   
\draw [dotted] (0,0)-- (0.7,-0.46); 
\draw [fill] (0,0) circle (1.5pt);   
\draw [fill] (0.7,1.07) circle (0.5pt); 
\end{tikzpicture}

\end{document}

Answer 1

这将在某些纬度处绘制平面，如果需要，可以将其更改为接受 z 值作为输入。视角 theta 可以随意更改。

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{65}{00}
\begin{tikzpicture}[declare function={R=2;}]
 \begin{scope}[tdplot_main_coords,samples=101]
  \fill[blue,opacity=0.8] plot[domain=0:360,variable=\t,smooth cycle] 
      (xyz spherical cs:radius=R,latitude=-10,longitude=\t);
  \fill[red,opacity=0.8] plot[domain=0:360,variable=\t,smooth cycle] 
      (xyz spherical cs:radius=R,latitude=30,longitude=\t);
 \end{scope}
 \shade[ball color=red,opacity=0.4] (0,0) circle[radius=R];
\end{tikzpicture}
\end{document}

还可以指定z。

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{75}{00}
\begin{tikzpicture}[declare function={R=2;angle(\z)=asin(\z/R);}]
 \begin{scope}[tdplot_main_coords,samples=101]
 \path[draw,fill=red] plot[domain=90:270,variable=\t,smooth] 
      (xyz spherical cs:radius=R,latitude={angle(0.5)},longitude=\t)
      plot[domain=270:90,variable=\t,smooth] 
      (xyz spherical cs:radius=R,latitude={angle(0.5)+3*cos(\t)},longitude=\t);   
  \path[draw,opacity=0.8,inner color=red!40,outer color=red!80] plot[domain=0:360,variable=\t,smooth cycle] 
      (xyz spherical cs:radius=R,latitude={angle(0.5)},longitude=\t);
 \end{scope}
 \shade[ball color=red,opacity=0.4] (0,0) circle[radius=R];
\end{tikzpicture}
\end{document}

或者

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{75}{110}
\begin{tikzpicture}[
    declare function={R=4;angle(\z)=asin(\z/R);myz=0.8;
        myphi=50;mytheta=60;rr=4;},
 >=stealth]
 \begin{scope}[tdplot_main_coords,samples=101]
  \path[draw,fill=red] plot[domain=90-\tdplotmainphi:270-\tdplotmainphi,variable=\t,smooth] 
       (xyz spherical cs:radius=R,latitude={angle(myz)},longitude=\t)
       plot[domain=270-\tdplotmainphi:90-\tdplotmainphi,variable=\t,smooth] 
       (xyz spherical cs:radius=R,latitude={angle(myz)+3*cos(\t+\tdplotmainphi)},longitude=\t);   
   \path[draw,opacity=0.8,inner color=red!40,outer color=red!80] 
     plot[domain=0:360,variable=\t,smooth cycle] 
       (xyz spherical cs:radius=R,latitude={angle(myz)},longitude=\t);
  \draw[->] (0,0,myz) -- ({R*cos(asin(myz/R))},0,myz);
  \draw[->] (0,0,myz) -- (0,{R*cos(asin(myz/R))},myz);
  \draw[->] (0,0,myz) -- (0,0,R);
  \draw (0,0,myz) -- (xyz spherical cs:radius=rr,
    latitude=mytheta,longitude=myphi) coordinate(P) node[circle,fill,inner
    sep=1pt]{};
  \draw[dotted] (P) -- 
    ({rr*sin(myphi)*cos(mytheta)},{rr*cos(myphi)*cos(mytheta)},{myz})
    -- (0,0,myz);
  \draw[->] plot[domain=90:{atan((rr*sin(mytheta)-myz)/(rr*cos(mytheta)))},
    variable=\t,smooth] ([yshift=myz*1cm]xyz spherical cs:radius=1,latitude={\t},longitude=myphi);
  \draw[->] plot[domain=90:myphi,
    variable=\t,smooth] ({0.6*sin(\t)},{0.6*cos(\t)},myz);
 \end{scope}
 \shade[ball color=red,opacity=0.4] (0,0) circle[radius=R];
\end{tikzpicture}
\end{document}

或者

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{75}{110}
\begin{tikzpicture}[
    declare function={R=4;angle(\z)=asin(\z/R);myz=0.8;
        myphi=50;mytheta=60;rr=4;a=myz/(rr*sin(mytheta));},
 >=stealth]
 \begin{scope}[tdplot_main_coords,samples=101]
  \draw[->] (0,0,0) -- ({R*cos(asin(myz/R))},0,0);
  \draw[->] (0,0,0) -- (0,{R*cos(asin(myz/R))},0);
  \draw[-] (0,0,0) -- (0,0,myz);
  \draw (0,0,0) -- (xyz spherical cs:radius=rr*a,
    latitude=mytheta,longitude=myphi) coordinate(P');
  \path[draw,fill=red] plot[domain=90-\tdplotmainphi:270-\tdplotmainphi,variable=\t,smooth] 
       (xyz spherical cs:radius=R,latitude={angle(myz)},longitude=\t)
       plot[domain=270-\tdplotmainphi:90-\tdplotmainphi,variable=\t,smooth] 
       (xyz spherical cs:radius=R,latitude={angle(myz)+3*cos(\t+\tdplotmainphi)},longitude=\t);   
  \path[draw,opacity=0.8,inner color=red!40,outer color=red!80] 
     plot[domain=0:360,variable=\t,smooth cycle] 
       (xyz spherical cs:radius=R,latitude={angle(myz)},longitude=\t);
  \draw[->] (0,0,myz) -- (0,0,R);
  \draw (P') -- (xyz spherical cs:radius=rr,
    latitude=mytheta,longitude=myphi) coordinate(P) node[circle,fill,inner
    sep=1pt]{};
 \end{scope}
 \shade[ball color=red,opacity=0.4] (0,0) circle[radius=R];
\end{tikzpicture}
\end{document}

不幸的是，我不知道 x 轴和 y 轴应该位于哪个平面。

请注意，还有blochsphere专门用于 Bloch 球的包。