The Annihilation and Creation commands return annihilation and creation operators, respectively, which act on the quantum numbers of state vectors, Kets or Bras, of a space of quantum states. The first argument, A, is the label of the basis of quantum states on which the operator returned acts. The second and subsequent arguments are expected to be positive integers indicating the position of the quantum numbers on which the operator returned acts; if not given, it is assumed that there is only one quantum number.

Note: Annihilation and creation operators can be created to operate only on Bras and Kets belonging to discrete spaces of states, which is the default case when you define quantum operators - for details on that, see Setup and its keyword quantumdiscretebasis and quantumcontinuousbasis.

By default, the annihilation and creation operators returned are displayed as $a^{-}$ and $a^{+}$, respectively (in the underlying computational implementation, $a^{-}$ and $a^{+}$ are local variables that you can invoke in an input line using the global variables `a-` and `a+`). Thus, the default returned operator tells neither the basis of states nor the position of the quantum numbers onto which it acts. You can override this default by using the optional argument notation = explicit, as well as label = ... where the right-hand side is a name to by used instead of $a$. See Options below.

The annihilation and creation operators satisfy the commutation relation, ${\left[a^{-}\,a^{+}\right]}_{-}=1$, where ${\left[a^{-}\,a^{+}\right]}_{-}$ is the Commutator, unless the label of the basis of states on which the operators act is itself an anticommutative variable, in which case the operators satisfy the anticommutation relation, ${\left[a^{-}\,a^{+}\right]}_{+}=1$, where ${\left[a^{-}\,a^{+}\right]}_{+}$ is the AntiCommutator. When these operators are constructed by calls to Annihilation and Creation, these (anti)commutation relations are automatically set, and so are taken into account by the Commutator and AntiCommutator commands.

When setting different Hilbert spaces using Setup, for example three spaces with only one operator acting on each of them via $\mathrm{Setup}\left(\mathrm{hilbertspaces}=\left\{\left\{A\right\}\,\left\{B\right\}\,\left\{C\right\}\right\}\right)$, all the (anti)commutators between any two of $A$, $B$ and $C$ are automatically set equal to 0. Likewise, using Annihilation and Creation to produce annihilation and creation operators corresponding to $A$, $B$ or $C$, automatically set equal to 0 all the commutators (or anticommutators) between annihilation and creation operators corresponding to any two of them. In connection, when computing with different Hilbert spaces it is particularly useful the option label = ... to distinguish between annihilation and creation operators acting on different spaces. For example, using, respectively label = a, label = b and label = c for each of the operators $A$, $B$ and $C$.

The %Annihilation and %Creation commands are the inert forms of Annihilation and Creation. That is, they represent the same mathematical operations while returning the operators unevaluated. To evaluate the inert operations, use the value command.

notation

The default value is notation = implicit. You can override this value by using the optional argument notation = explicit. This option is useful, for example, to distinguish among various annihilation (or creation) operators within a single worksheet, and to make $a^{-}$ and $a^{+}$ (entering the returned indexed annihilation and creation operators) be exports of Physics, not local variables. Alternatively, for the same purpose, you can use the alias command. See the Examples for a demonstration.

label

The default value is label = a, so that the annihilation and creation operators returned are displayed as $a^{-}$ and $a^{+}$, respectively. You can override this value by using the optional argument label = ... where the right-hand side is any Maple name or string to replace the default value $a$. This option is useful, for example, to distinguish among various annihilation (or creation) operators without having to use the notation = explicit that can make the output have too much information.

phaseconvention

Given a basis of states with label A, and a single quantum number, n, the annihilation operator satisfies the equation $a^{-}\left|A_0\right.⟩\=0$ . So the state vector $\left|A_0\right.⟩$ is defined up to a constant factor. Assuming $\left|A_0\right.⟩$ is normalized to 1, this factor is of the form ${\ⅇ}^{i\mathrm{\theta }}$ with $\mathrm{\theta }$ real. By requiring, additionally, that $\left|A_1\right.⟩\=c_1{a}^{+}\left|A_0\right.⟩$, with $c_1$ real and positive, and taking into account the (anti)commutation relation satisfied by $a^{-}$ and $a^{+}$, the factor $c_1$ is equal to 1.

In the bosonic case, considering now that $\left|A_n\right.⟩\=c_n{a}^{+}\left|A_{n-1}\right.⟩$, by using the same assumptions, the following phase convention arises:

You can override this phase convention with a procedure pc having one argument, by using the optional argument phaseconvention = pc.

In the fermionic case, recalling Pauli's exclusion principle, each quantum number can only assume the values 0 or 1, so:

In the case of many quantum numbers, for example with $a^{-}$ and $a^{+}$ acting on the $n^{\mathrm{th}}$ quantum number $q_n$, let $M=\sum _{m=1}^{n-1}{q}_m$ represent the total number of particles occupying states represented by the previous $q_m$ quantum numbers. Then the phaseconvention is

The default value for this option in the Annihilation command is phaseconvention = sqrt; the default in the Creation command is phaseconvention = n -> sqrt(n + 1).

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{am}≔\mathrm{Annihilation}\left(\mathrm{\phi }\,1\right)$

$\mathrm{am}≔a^{-}$

$\mathrm{ap}≔\mathrm{Creation}\left(\mathrm{\phi }\,1\right)$

$\mathrm{ap}≔a^{+}$

$\mathrm{am}·\mathrm{Ket}\left(\mathrm{\phi }\,n\right)$

$\sqrt{n}\left|{\mathrm{\phi }}_{n-1}\right.⟩$

$\mathrm{Bra}\left(\mathrm{\phi }\,n\right)·\mathrm{am}$

$\sqrt{n+1}\left.⟨{\mathrm{\phi }}_{n+1}\right|$

$\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\phi }\,n\right)$

$\sqrt{n+1}\left|{\mathrm{\phi }}_{n+1}\right.⟩$

$\mathrm{Bra}\left(\mathrm{\phi }\,n\right)·\mathrm{ap}$

$\sqrt{n}\left.⟨{\mathrm{\phi }}_{n-1}\right|$

$\mathrm{Commutator}\left(\mathrm{am}\,\mathrm{ap}\right)$

$1$

$\mathrm{c\_am\_ap}≔\mathrm{\%Commutator}\left(\mathrm{am}\,\mathrm{ap}\right)$

$\mathrm{c\_am\_ap}≔{\left[a^{-}\,a^{+}\right]}_{-}$

$\mathrm{value}\left(\mathrm{c\_am\_ap}\right)$

$1$

$\mathrm{am2}≔\mathrm{Annihilation}\left(\mathrm{\psi }\,1\,\mathrm{notation}=\mathrm{explicit}\right)$

$\mathrm{am2}≔{a^{-}}_{{\mathrm{\psi }}_1}$

$\mathrm{ap2}≔\mathrm{Creation}\left(\mathrm{\psi }\,2\,\mathrm{notation}=\mathrm{explicit}\right)$

$\mathrm{ap2}≔{a^{+}}_{{\mathrm{\psi }}_2}$

$\mathrm{am2}·\mathrm{Ket}\left(\mathrm{\psi }\,n\,m\right)$

$\sqrt{n}\left|{\mathrm{\psi }}_{n-1\,m}\right.⟩$

$\mathrm{ap2}·\mathrm{Ket}\left(\mathrm{\psi }\,n\,m\right)$

$\sqrt{m+1}\left|{\mathrm{\psi }}_{n\,m+1}\right.⟩$

$\mathrm{alias}\left(\mathrm{Am1}=\mathrm{Annihilation}\left(\mathrm{\psi }\,1\right)\right)\:$

$\mathrm{alias}\left(\mathrm{Am2}=\mathrm{Annihilation}\left(\mathrm{\psi }\,2\right)\right)\:$

$\mathrm{Am1}$

$\mathrm{Am1}$

$\mathrm{Am1}·\mathrm{Ket}\left(\mathrm{\psi }\,n\,m\right)$

$\sqrt{n}\left|{\mathrm{\psi }}_{n-1\,m}\right.⟩$

$\mathrm{Am2}$

$\mathrm{Am2}$

$\mathrm{Am2}·\mathrm{Ket}\left(\mathrm{\psi }\,n\,m\right)$

$\sqrt{m}\left|{\mathrm{\psi }}_{n\,m-1}\right.⟩$

$\mathrm{Physics}:-\mathrm{Setup}\left(\mathrm{anticommutativeprefix}=\mathrm{\theta }\right)$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\right]$

$\mathrm{am}≔\mathrm{Annihilation}\left(\mathrm{\theta }\,3\,\mathrm{notation}=\mathrm{explicit}\right)$

$\mathrm{am}≔{a^{-}}_{{\mathrm{\theta }}_3}$

$\mathrm{ap}≔\mathrm{Creation}\left(\mathrm{\theta }\,3\,\mathrm{notation}=\mathrm{explicit}\right)$

$\mathrm{ap}≔{a^{+}}_{{\mathrm{\theta }}_3}$

$\mathrm{a\_am\_ap}≔\mathrm{\%AntiCommutator}\left(\mathrm{am}\,\mathrm{ap}\right)$

$\mathrm{a\_am\_ap}≔{\left[{a^{-}}_{{\mathrm{\theta }}_3}\,{a^{+}}_{{\mathrm{\theta }}_3}\right]}_{+}$

$\mathrm{value}\left(\mathrm{a\_am\_ap}\right)$

$1$

$\mathrm{am\_K}≔\mathrm{`\%.`}\left(\mathrm{am}\,\mathrm{Ket}\left(\mathrm{\θ}\,1\,0\,1\right)\right)$

$\mathrm{am\_K}≔{a^{-}}_{{\mathrm{\theta }}_3}·\left|{\mathrm{\theta }}_{1\,0\,1}\right.⟩$

$\mathrm{value}\left(\mathrm{am\_K}\right)$

$-\left|{\mathrm{\theta }}_{1\,0\,0}\right.⟩$

$\mathrm{am\_K}≔\mathrm{`\%.`}\left(\mathrm{am}\,\mathrm{Ket}\left(\mathrm{\θ}\,1\,1\,1\right)\right)$

$\mathrm{am\_K}≔{a^{-}}_{{\mathrm{\theta }}_3}·\left|{\mathrm{\theta }}_{1\,1\,1}\right.⟩$

$\mathrm{value}\left(\mathrm{am\_K}\right)$

$\left|{\mathrm{\theta }}_{1\,1\,0}\right.⟩$

$\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\theta }\,1\,0\,1\right)$

$0$

$\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\theta }\,1\,0\,0\right)$

$-\left|{\mathrm{\theta }}_{1\,0\,1}\right.⟩$

$\mathrm{Setup}\left(\mathrm{quantumoperators}=\left\{A\,B\,C\right\}\,\mathrm{hilbertspaces}=\left\{\left\{A\right\}\,\left\{B\right\}\,\left\{C\right\}\right\}\right)$

$\left[\mathrm{disjointedspaces}=\left\{\left\{A\right\}\,\left\{B\right\}\,\left\{C\right\}\right\}\,\mathrm{quantumoperators}=\left\{A\,B\,C\right\}\right]$

$a,\mathrm{ap}≔\left(\mathrm{Annihilation}\,\mathrm{Creation}\right)\left(A\right)$

$a,\mathrm{ap}≔a^{-},a^{+}$

$b,\mathrm{bp}≔\left(\mathrm{Annihilation}\,\mathrm{Creation}\right)\left(B\,\mathrm{label}=b\right)$

$b,\mathrm{bp}≔b^{-},b^{+}$

$c,\mathrm{cp}≔\left(\mathrm{Annihilation}\,\mathrm{Creation}\right)\left(C\,\mathrm{label}=c\right)$

$c,\mathrm{cp}≔c^{-},c^{+}$

$a-\mathrm{`a-`}$

$0$

$\mathrm{bp}-\mathrm{`b+`}$

$0$

$\left(\mathrm{\%Commutator}=\mathrm{Commutator}\right)\left(a\,b\right)$

${\left[a^{-}\,b^{-}\right]}_{-}=0$

$\left(\mathrm{\%Commutator}=\mathrm{Commutator}\right)\left(b\,c\right)$

${\left[b^{-}\,c^{-}\right]}_{-}=0$

$\left(\mathrm{\%Commutator}=\mathrm{Commutator}\right)\left(\mathrm{bp}\,\mathrm{cp}\right)$

${\left[b^{+}\,c^{+}\right]}_{-}=0$

$H≔\mathrm{\ω\_\_a}\mathrm{ap}a+\mathrm{\ω\_\_b}\mathrm{bp}b+\mathrm{U\_\_b}\mathrm{bp}\mathrm{bp}bb+\mathrm{\ω\_\_c}\mathrm{cp}c+\mathrm{g\_\_ab}\left(a\mathrm{bp}+\mathrm{ap}b\right)+\mathrm{g\_\_bc}\left(b\mathrm{cp}+\mathrm{bp}c\right)+\mathrm{g\_\_ac}\left(a\mathrm{cp}+\mathrm{ap}c\right)$

$H≔\mathrm{\ω\_\_a}{a}^{+}{a}^{-}+\mathrm{\ω\_\_b}{b}^{+}{b}^{-}+\mathrm{U\_\_b}{b^{+}}^2{b^{-}}^2+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}+\mathrm{g\_\_ab}\left(a^{-}\otimes b^{+}+a^{+}\otimes b^{-}\right)+\mathrm{g\_\_bc}\left(b^{-}\otimes c^{+}+b^{+}\otimes c^{-}\right)+\mathrm{g\_\_ac}\left(a^{-}\otimes c^{+}+a^{+}\otimes c^{-}\right)$

$\mathrm{g\_\_ac}≔0\;$$\mathrm{g\_\_bc}≔0$

$\mathrm{g\_\_ac}≔0$

$\mathrm{g\_\_bc}≔0$

$S≔\mathrm{\lambda }\left(a\mathrm{bp}-\mathrm{ap}b\right)$

$S≔\mathrm{\lambda }\left(a^{-}\otimes b^{+}-a^{+}\otimes b^{-}\right)$

$\mathrm{\lambda }≔\frac{\mathrm{g\_\_ab}\cdot 1}{-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}}$

$\mathrm{\lambda }≔\frac{\mathrm{g\_\_ab}}{-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}}$

$\mathrm{\%H}+\mathrm{\%Commutator}\left(\mathrm{\%S}\,\mathrm{\%H}\right)+\frac{1}{2}\mathrm{\%Commutator}\left(\mathrm{\%S}\,\mathrm{\%Commutator}\left(\mathrm{\%S}\,\mathrm{\%H}\right)\right)$

$H+{\left[S\,H\right]}_{-}+\frac{{\left[S\,{\left[S\,H\right]}_{-}\right]}_{-}}{2}$

$\mathrm{value}\left(\right)$

$\mathrm{\ω\_\_a}{a}^{+}{a}^{-}+\mathrm{\ω\_\_b}{b}^{+}{b}^{-}+\mathrm{U\_\_b}{b^{+}}^2{b^{-}}^2+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}+\mathrm{g\_\_ab}\left(a^{-}\otimes b^{+}+a^{+}\otimes b^{-}\right)+\frac{\mathrm{g\_\_ab}\left(\mathrm{\ω\_\_a}{a}^{-}\otimes b^{+}-\mathrm{\ω\_\_b}{a}^{-}\otimes b^{+}-2\mathrm{U\_\_b}{a}^{-}\otimes {b^{+}}^2{b}^{-}+\mathrm{g\_\_ab}\left(-a^{-}{a}^{+}+b^{-}{b}^{+}\right)+\mathrm{\ω\_\_a}{a}^{+}\otimes b^{-}-\mathrm{\ω\_\_b}{a}^{+}\otimes b^{-}-2\mathrm{U\_\_b}{a}^{+}\otimes b^{+}{b^{-}}^2-\mathrm{g\_\_ab}\left(a^{+}{a}^{-}-b^{+}{b}^{-}\right)\right)}{-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}}+\frac{{\mathrm{g\_\_ab}}^2\left(2\mathrm{U\_\_b}{a^{-}}^2\otimes {b^{+}}^2-4\mathrm{g\_\_ab}{a}^{-}\otimes b^{+}+\mathrm{\ω\_\_a}\left(-a^{-}{a}^{+}+b^{-}{b}^{+}\right)-\mathrm{\ω\_\_b}\left(-a^{-}{a}^{+}+b^{-}{b}^{+}\right)-2\mathrm{U\_\_b}\left(-2a^{-}{a}^{+}\otimes b^{+}{b}^{-}+b^{+}{b^{-}}^2{b}^{+}\right)-\mathrm{\ω\_\_a}\left(a^{+}{a}^{-}-b^{+}{b}^{-}\right)+\mathrm{\ω\_\_b}\left(a^{+}{a}^{-}-b^{+}{b}^{-}\right)+2\mathrm{U\_\_b}\left(2a^{+}{a}^{-}\otimes b^{+}{b}^{-}-{b^{+}}^2{b^{-}}^2\right)-4\mathrm{g\_\_ab}{a}^{+}\otimes b^{-}+2\mathrm{U\_\_b}{a^{+}}^2\otimes {b^{-}}^2\right)}{2{\left(-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}\right)}^2}$

$\mathrm{Gtaylor}\left(\,\mathrm{g\_\_ab}\,3\right)$

$\frac{-2\mathrm{U\_\_b}\left(\mathrm{g\_\_ab}+\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)\left(\mathrm{g\_\_ab}-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}\right){b^{+}}^2{b^{-}}^2+2{a^{+}}^2\otimes {b^{-}}^2\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2+2{a^{-}}^2\otimes {b^{+}}^2\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2+4\mathrm{U\_\_b}\mathrm{g\_\_ab}\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)a^{-}\otimes {b^{+}}^2{b}^{-}+4\mathrm{U\_\_b}\mathrm{g\_\_ab}\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)a^{+}\otimes b^{+}{b^{-}}^2-2b^{+}{b^{-}}^2{b}^{+}\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2+4a^{+}{a}^{-}\otimes b^{+}{b}^{-}\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2+4a^{-}{a}^{+}\otimes b^{+}{b}^{-}\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2+\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)\left(\left({\mathrm{g\_\_ab}}^2+2\mathrm{\ω\_\_a}\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)\right)a^{+}{a}^{-}+\left(-{\mathrm{g\_\_ab}}^2+2\mathrm{\ω\_\_b}\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)\right)b^{+}{b}^{-}+2\mathrm{\ω\_\_c}\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)c^{+}{c}^{-}+{\mathrm{g\_\_ab}}^2\left(a^{-}{a}^{+}-b^{-}{b}^{+}\right)\right)}{2{\left(\mathrm{\ω\_\_a}-\mathrm{\ω\_\_b}\right)}^2}$

$\mathrm{collect}\left(\mathrm{Simplify}\left(\right)\,\left[\mathrm{ap}a\,{\mathrm{ap}}^2{a}^2\,\mathrm{bp}b\,{\mathrm{bp}}^2{b}^2\,\mathrm{g\_\_ab}\right]\,\mathrm{simplify}\right)$

$\left(\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_a}\right)a^{+}{a}^{-}+\left(\frac{\left(\mathrm{U\_\_b}-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}\right){\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{\ω\_\_b}\right)b^{+}{b}^{-}+\left(-\frac{\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{U\_\_b}\right){b^{+}}^2{b^{-}}^2-\frac{\mathrm{U\_\_b}\left(b^{+}{b}^{-}+{b^{+}}^2{b^{-}}^2-{a^{-}}^2\otimes {b^{+}}^2-4a^{+}{a}^{-}\otimes b^{+}{b}^{-}-{a^{+}}^2\otimes {b^{-}}^2\right){\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\frac{2\mathrm{U\_\_b}\left(a^{-}\otimes {b^{+}}^2{b}^{-}+a^{+}\otimes b^{+}{b^{-}}^2\right)\mathrm{g\_\_ab}}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}$

$\mathrm{subs}\left({\mathrm{bp}}^{2}ab=0\,\mathrm{ap}\mathrm{bp}{b}^2=0\,{\mathrm{ap}}^2{b}^2=0\,{\mathrm{bp}}^2{a}^2=0\,\right)$

$\left(\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_a}\right)a^{+}{a}^{-}+\left(\frac{\left(\mathrm{U\_\_b}-\mathrm{\ω\_\_a}+\mathrm{\ω\_\_b}\right){\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{\ω\_\_b}\right)b^{+}{b}^{-}+\left(-\frac{\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{U\_\_b}\right){b^{+}}^2{b^{-}}^2-\frac{\mathrm{U\_\_b}\left(b^{+}{b}^{-}+{b^{+}}^2{b^{-}}^2-4a^{+}{a}^{-}\otimes b^{+}{b}^{-}\right){\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}$

$\mathrm{H\_\_approx}≔\mathrm{collect}\left(\,\left[\mathrm{ap}a\,{\mathrm{ap}}^2{a}^2\,\mathrm{bp}b\,{\mathrm{bp}}^2{b}^2\,\mathrm{g\_\_ab}\right]\,\mathrm{simplify}\right)$

$\mathrm{H\_\_approx}≔\left(\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_a}\right)a^{+}{a}^{-}+\left(-\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_b}\right)b^{+}{b}^{-}+\left(-\frac{2\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{U\_\_b}\right){b^{+}}^2{b^{-}}^2+\frac{4a^{+}{a}^{-}\otimes b^{+}{b}^{-}\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}$

$\mathrm{Setup}\left(\mathrm{tensorproductnotation}=\mathrm{false}\right)$

$\left[\mathrm{tensorproductnotation}=\mathrm{false}\right]$

$\mathrm{H\_\_approx}$

$\left(\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_a}\right)a^{+}{a}^{-}+\left(-\frac{{\mathrm{g\_\_ab}}^2}{-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}}+\mathrm{\ω\_\_b}\right)b^{+}{b}^{-}+\left(-\frac{2\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{U\_\_b}\right){b^{+}}^2{b^{-}}^2+\frac{4a^{+}{a}^{-}{b}^{+}{b}^{-}\mathrm{U\_\_b}{\mathrm{g\_\_ab}}^2}{{\left(-\mathrm{\ω\_\_b}+\mathrm{\ω\_\_a}\right)}^2}+\mathrm{\ω\_\_c}{c}^{+}{c}^{-}$

Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics. Chapter II. Paris, France: Hermann, 1977.