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eemg
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I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ However, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a derivation of these matrices or justsimply state the result for the 3x3 case.

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ However, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a derivation of these matrices or just state the result for the 3x3 case.

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ However, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a derivation of these matrices or simply state the result for the 3x3 case.

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Qmechanic ♦
Qmechanic ♦
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I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$$$$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}$$$$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ howeverHowever, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a reference to the derivation of these matrices or just state the result for the 3x3 case.

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}$$ however, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a reference to the derivation of these matrices or just state the result for the 3x3 case.

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ However, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a derivation of these matrices or just state the result for the 3x3 case.

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ZeroTheHero
ZeroTheHero
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Matrix representation of the SU$\mathfrak{su}(1,1)$ $K$ operators

I am trying to find the matrix representation of the $SU(1,1)$$\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}$$ however, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a reference to the derivation of these matrices or just state the result for the 3x3 case.

Matrix representation of the SU(1,1) $K$ operators

I am trying to find the matrix representation of the $SU(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}$$ however, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a reference to the derivation of these matrices or just state the result for the 3x3 case.

Matrix representation of the $\mathfrak{su}(1,1)$ $K$ operators

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}$$ however, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a reference to the derivation of these matrices or just state the result for the 3x3 case.

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eemg
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