Spherical vector components and transition-frequency convention

PDF p. 13: Eq. (S3.16) and preceding sentence

Let $H={\mathbb{C}}^d$, $d\ge 1$, be a finite operator space with its specified orthonormal basis, and let $X_x,X_y,X_z:H\to H$ be linear operators. In the displayed phase convention their spherical components are $X_{+1}=-(X_x+i{X}_y)\mathrm{/}\sqrt{2}$, $X_{-1}=(X_x-i{X}_y)\mathrm{/}\sqrt{2}$, and $X_0=X_z$. Other conventions may multiply these components by convention-dependent unit phases. The name specifies this change of labels and combinations; by itself it does not assert that an arbitrary triple satisfies spherical-tensor commutation relations.

Let $H_0$ be a linear operator diagonal in the specified basis, with real eigenvalues $E_n$ and basis eigenvectors $\mathrm{\mid }n⟩$. For the directed transition from $\mathrm{\mid }n⟩$ to $\mathrm{\mid }m⟩$, define ${\omega }_{mn}=E_n-E_m$. Thus ${\omega }_{mn}>0$ exactly when the source energy $E_n$ exceeds the destination energy $E_m$.

Finite operator and ket conventions

Calibrated diagonal lifts and transition phase differences

The finite operator conventions make each displayed complex linear combination a well-defined operator. In the ordered input labels $(x,y,z)$ and output labels $(+1,-1,0)$, the coefficient matrix is $U=\left(\begin{array}{ccc}-1\mathrm{/}\sqrt{2}& -i\mathrm{/}\sqrt{2}& 0\\ 1\mathrm{/}\sqrt{2}& -i\mathrm{/}\sqrt{2}& 0\\ 0& 0& 1\end{array}\right)$. Here $U$ acts on the component-label space ${\mathbb{C}}^3$ with its standard inner product, independently of the dimension $d$ of $H$.

Direct multiplication gives $U{U}^{†}=U^{†}U=I_3$, so $U^{†}$ is an inverse. For every coordinate column $v\in {\mathbb{C}}^3$, the induced norm satisfies $\mathrm{\parallel }Uv{\mathrm{\parallel }}^2=(Uv{)}^{†}(Uv)=v^{†}{U}^{†}Uv=v^{†}v=\mathrm{\parallel }v{\mathrm{\parallel }}^2$. Thus $U$ is an invertible linear isometry, hence a unitary operator by Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces. This verifies the needed implication directly, without assuming a matrix characterization of unitarity.

For phases $p_{+1},p_{-1},p_0$ with $\mathrm{\mid }{p}_q\mathrm{\mid }=1$, let $P=\mathrm{diag}(p_{+1},p_{-1},p_0)$. Then $P^{†}P=P{P}^{†}=I_3$, so $U^{†}{P}^{†}$ is an inverse of $PU$, and $\mathrm{\parallel }PUv{\mathrm{\parallel }}^2={\sum }_q\mathrm{\mid }{p}_q(Uv{)}_q{\mathrm{\mid }}^2={\sum }_q\mathrm{\mid }(Uv{)}_q{\mathrm{\mid }}^2=\mathrm{\parallel }Uv{\mathrm{\parallel }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}^2$. Therefore multiplying output components by unit phases also gives a unitary coordinate change under the same definition.

The directed energy difference agrees with the source-minus-destination convention in Calibrated diagonal lifts and transition phase differences, and its sign statement follows by subtraction of real numbers.