How to insert equal height brackets on both sides while wrapping a new line？

Question

I'd like to input a long equation, so I had to cut it into two lines. But, there's a couple of {} at the start of the first line of the equation and the end of the second line of the equation, and the 'equal height brackets' doesn't support \\ in it. I've used the \qty{} command of the package physics, or the \ab\{\} command of the package physics2, just like the code below:

\documentclass[9pt,a4paper]{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{extarrows}
\usepackage{fixdif,physics2}
\def\e{\mathrm{e}}
\usephysicsmodule{ab,ab.legacy,braket,nabla.legacy}
\def\Re{\mathrm{Re}}
\begin{document}
\begin{equation}
    \begin{aligned}
        J_L(t)&=\frac{2e}{\hbar}\Re\ab\{\sum_{k,\alpha\in L}{V_{k\alpha,n(t)}G_{n,k\alpha}^<(t,t)}\}\\
        &\xlongequal{\sum_{n}\sum_m=\sum_{n,m}}\frac{2e}{\hbar}\Re\ab\{\sum_{\substack{k,\alpha\in L\\n,m}}V_{k\alpha,n}(t)\int_{-\infty}^t\d t_1V_{k\alpha,m}^*\ab(t_1)\times\ab[G_{nm}^r\ab(t,t_1)\times\ab(\i f\ab(\epsilon_{k\alpha}^0)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\\
        &\ \ +G_{nm}^<\ab(t,t_1)\times\ab(\i\theta\ab(-t_1+t)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\vphantom{\int_t^{t_1}}]\}\\
      \end{aligned}
\end{equation}
\end{document}

they all returned me errors:

Extra }, or forgotten \right.
<template> }
Missing } inserted.
<inserted text> 

Then, I used the command \left\{ at the at the start of the first line of the equation and \right\} at the end of the second line of the equation, however it also returned me the errors above.

Then, I used the command \left\{ at the start of the first line and \right. at the end of the first line, used the command \left. at the start of the second line and \right\} at the end of the second line, the errors disappeared, however, the height of the right bracket } doesn't equal to the height of the left bracket {.

The code is just like below:

\documentclass[9pt,a4paper]{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{extarrows}
\usepackage{fixdif,physics2}
\def\e{\mathrm{e}}
\usephysicsmodule{ab,ab.legacy,braket,nabla.legacy}
\def\Re{\mathrm{Re}}
\begin{document}
\begin{equation}
    \begin{aligned}
        J_L(t)&=\frac{2e}{\hbar}\Re\ab\{\sum_{k,\alpha\in L}{V_{k\alpha,n(t)}G_{n,k\alpha}^<(t,t)}\}\\
        &\xlongequal{\sum_{n}\sum_m=\sum_{n,m}}\frac{2e}{\hbar}\Re\left\{\sum_{\substack{k,\alpha\in L\\n,m}}V_{k\alpha,n}(t)\int_{-\infty}^t\d t_1V_{k\alpha,m}^*\ab(t_1)\times\left[G_{nm}^r\ab(t,t_1)\times\ab(\i f\ab(\epsilon_{k\alpha}^0)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\right.\right.\\
        &\ \ \left.\left.+G_{nm}^<\ab(t,t_1)\times\ab(\i\theta\ab(-t_1+t)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\vphantom{\int_t^{t_1}}\right]\right\}\\
      \end{aligned}
\end{equation}
\end{document}

Finally, I found that I can add a 'virtual' height of the first line, is just add the command \vphantom{\sum_{\substack{k,\alpha\in L\\n,m}}} before the \right\} at the end of the second line, just like the code below:

\documentclass[9pt,a4paper]{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{extarrows}
\usepackage{fixdif,physics2}
\def\e{\mathrm{e}}
\usephysicsmodule{ab,ab.legacy,braket,nabla.legacy}
\def\Re{\mathrm{Re}}
\begin{document}
\begin{equation}
    \begin{aligned}
        J_L(t)&=\frac{2e}{\hbar}\Re\ab\{\sum_{k,\alpha\in L}{V_{k\alpha,n(t)}G_{n,k\alpha}^<(t,t)}\}\\
        &\xlongequal{\sum_{n}\sum_m=\sum_{n,m}}\frac{2e}{\hbar}\Re\left\{\sum_{\substack{k,\alpha\in L\\n,m}}V_{k\alpha,n}(t)\int_{-\infty}^t\d t_1V_{k\alpha,m}^*\ab(t_1)\times\left[G_{nm}^r\ab(t,t_1)\times\ab(\i f\ab(\epsilon_{k\alpha}^0)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\right.\right.\\
        &\ \ \left.\left.+G_{nm}^<\ab(t,t_1)\times\ab(\i\theta\ab(-t_1+t)\exp\ab[-\i\int_{t}^{t_1}{\d t_2\epsilon_{k\alpha}(t_2)}])\vphantom{\int_t^{t_1}}\right]\vphantom{\sum_{\substack{k,\alpha\in L\\n,m}}}\right\}\\
      \end{aligned}
\end{equation}
\end{document}

The question is 'solved', this is the effect that I want:

However, this is too much trouble... I'd like to know if there's a simple command just like the command \qty of package physics or \ab of package physics2 can achieve the effect I want?

Answer 1

Rather than define and use various \vphantom contstructs, do learn how to use \Bigg[lr] and \bigg[lr]. Incidentally, your formula needs to be spread across four rows, not just three, in order for it not to protrude into the margins. Oh, and don't use \def unless you are willing to take the risk of clobbering pre-existing macros; the macro \Re is a case in point.

\documentclass[%9pt, % "9pt" is not a recognized option
               a4paper]{article}
\usepackage{geometry}
\usepackage{mathtools} % for "\smashoperator" macro
\usepackage{extarrows,fixdif}
\let\Re\relax % first, undefine the existing '\Re' macro
\DeclareMathOperator\Re{\mathrm{Re}} % next, redefine '\Re'
%\def\e{\mathrm{e}} % not needed 
\providecommand\I{\mathrm{i}} % imag. unit. Use only in math mode
\begin{document}
\begin{equation}
\begin{aligned}[b]
J_L(t)
  &=\frac{2e}{\hbar} \Re\Biggl{ ,
    \smashoperator[r]{\sum_{k,\alpha\in L}}
    V_{k\alpha,n(t)}G_{n,k\alpha}^<(t,t)\Biggr}\
  &\xlongequal{\sum_{n}\sum_m=\sum_{n,m}}
    \frac{2e}{\hbar} \Re\Biggl{ ,
    \smashoperator[r]{\sum_{\substack{k,\alpha\in L\n,m}}}
    V_{k\alpha,n}(t)
    \int_{-\infty}^t \d t_1 V_{k\alpha,m}^* (t_1)\
  &\quad
    \times\Biggl[G_{nm}^r(t,t_1)\times
    \biggl(\I f(\epsilon_{k\alpha}^0)
    \exp\biggl[-\I\int_{t}^{t_1} \d t_2\epsilon_{k\alpha}(t_2) 
    \biggr]\biggr) \
  &\qquad
    +G_{nm}^< (t,t_1)\times\biggl(\I\theta(-t_1+t)
    \exp\biggl[-\I\int_{t}^{t_1} \d t_2\epsilon_{k\alpha}(t_2)
    \biggr]\biggr)
    \Biggr]\Biggr}
\end{aligned}
\end{equation}
\end{document}