Coherent interaction of multistate quantum systems possessing the Majorana and Morris-Shore dynamic symmetries with pulse trains

The Majorana decomposition has been presented in several papers, see eg Refs. [16, 30, 31, 19]. It stems from the rotation group theory for the angular momentum. The Hamiltonian has the tridiagonal form [31]

$$\mathbf{H}_{M}\!=\!\left[\!\begin{array}[]{cccccc}H_{11}&H_{12}&0&\cdots&0&0\\ H_{21}&H_{22}&H_{23}&\cdots&0&0\\ 0&H_{32}&H_{33}&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&H_{M-1,M-1}&H_{M-1,M}\\ 0&0&0&\cdots&H_{M,M-1}&H_{MM}\end{array}\!\right]\!,$$

(8)

where the nonzero matrix elements read

	$\displaystyle H_{kk}(t)$	$\displaystyle=\Big{(}k-\frac{M+1}{2}\Big{)}\Delta(t),$		(9a)
		$\displaystyle(k=1,2,...,M),$
	$\displaystyle H_{k+1,k}(t)$	$\displaystyle=H^{*}_{k,k+1}(t)=\frac{1}{2}\sqrt{k(M-k)}\,\Omega(t),$		(9b)
		$\displaystyle(k=1,2,...,M-1),$

In terms of the angular momentum quantum numbers $j$ and $m$, we have the relations $M=2j+1$ and $k=j+1-m$.

The matrix elements of the propagator $\mathbf{U}_{M}$ are [33, 34, 31]

	$\displaystyle U_{kl}$	$\displaystyle=\sum_{r}\frac{\sqrt{(k-1)!(l-1)!(M-k)!(M-l)!}}{(l-1-r)!(M-k-r)!(r-l+k)!r!}$
		$\displaystyle\times a^{M-k-r}(a^{*})^{l-1-r}b^{r}(-b^{*})^{r-l+k},$		(10)

where $r$ runs from $r_{\text{min}}$ to $r_{\text{max}}$, with

	$\displaystyle r_{\text{min}}$	$\displaystyle=\text{min}[0,k+l+1-M],$		(11a)
	$\displaystyle r_{\text{max}}$	$\displaystyle=\text{max}[k-1,l-1].$		(11b)

For $M=2$ states, the matrix is the one shown already in Eq. (5). The propagator for $M=3$ and $M=4$ states reads

	$\displaystyle\mathbf{U}_{3}$	$\displaystyle=\left[\begin{array}[]{ccc}a^{2}&ab\sqrt{2}&b^{2}\\ -ab^{*}\sqrt{2}&|a|^{2}-|b|^{2}&ba^{*}\sqrt{2}\\ b^{*2}&-a^{*}b^{*}\sqrt{2}&a^{*2}\\ \end{array}\right],$		(12d)
	$\displaystyle\mathbf{U}_{4}\!$	$\displaystyle=\!\!\left[\!\!\begin{array}[]{cccc}a^{3}&a^{2}b\sqrt{3}&ab^{2}\sqrt{3}&b^{3}\\ -a^{2}b^{*}\sqrt{3}&\!a(|a|^{2}-2|b|^{2})&\!b(2|a|^{2}-|b|^{2})&\!b^{2}a^{*}\sqrt{3}\\ ab^{*2}\sqrt{3}&\!b^{*}(|b|^{2}-2|a|^{2})&\!a^{*}(|a|^{2}-2|b|^{2})&\!ba^{*2}\sqrt{3}\\ -b^{*3}&a^{*}b^{*2}\sqrt{3}&-a^{*2}b^{*}\sqrt{3}&a^{*3}\\ \end{array}\!\!\right]\!\!.$		(12i)

We shall derive the Majorana propagator of an $M$-state system after $N$ repetitions of the interaction described by the Hamiltonian $\mathbf{H}_{M}$ of Eq. (8) in two alternative manners. First we shall use the analogy of $M$-state to two-state dynamics and then we shall derive it by diagonalization of the propagator $\mathbf{U}_{M}$.

The propagator $\mathbf{U}_{3}$ shown in (12d) was constructed by the equation for the matrix elements $U_{kl}$ of Eq. (III.1). The Hamiltonian is defined by Eq. (8) and it is

$$H_{3}(t)=\left[\begin{array}[]{ccc}-\Delta(t)&\frac{\Omega(t)}{\sqrt{2}}&0\\ \frac{\Omega^{*}(t)}{\sqrt{2}}&0&\frac{\Omega(t)}{\sqrt{2}}\\ 0&\frac{\Omega^{*}(t)}{\sqrt{2}}&\Delta(t)\end{array}\right],$$

(31)

The system is shown schematically in Fig. 1. The multi-pass propagator $\mathbf{U}_{3}^{N}$ can be obtained from the single one $\mathbf{U}_{3}$ by the substitution $a\to a_{N}$ and $b\to b_{N}$ according to Eq. (7),

$$\mathbf{U}_{3}^{N}=\left[\begin{array}[]{ccc}a_{N}^{2}&\sqrt{2}a_{N}b_{N}&b_{N}^{2}\\ -\sqrt{2}a_{N}b_{N}^{*}&|a_{N}|^{2}-|b_{N}|^{2}&\sqrt{2}b_{N}a_{N}^{*}\\ b_{N}^{*2}&-\sqrt{2}a_{N}^{*}b_{N}^{*}&a_{N}^{*2}\end{array}\right].$$

(32)

Therefore, already for $M=3$ states we have quadratic powers in the Cayley-Klein parameters and this amplification is further boosted by the application of $N$ interactions.

For example, if the Cayley-Klein parameter $a$ is real then $a^{2}=\cos^{2}\theta$ is the probability for no transition and $p_{2}=1-a^{2}=\sin^{2}\theta$ is the transition probability for a single-pass in the two-state system. The $N$-pass transition probability reads

$$p_{2}^{(N)}=p_{2}\frac{\sin^{2}N\theta}{\sin^{2}\theta}=\sin^{2}N\theta.$$

(33)

Correspondingly, the transition probability in the three-state Majorana system from state 1 to state 3 is $p_{3}=\sin^{4}\theta$, and the $N$-pass transition probability reads

$$p_{3}^{(N)}=p_{3}\frac{\sin^{4}N\theta}{\sin^{4}\theta}=\sin^{4}N\theta.$$

(34)

A real $a$ occurs for resonant excitation ($\Delta=0$) and also when the Rabi frequency is a symmetric function of time while $\Delta$ is anti-symmetric [29].

The results in this example can be used for efficient tomography of qutrit gates as well as for quantum sensing with qutrits. Similar relations can be derived for larger number of states $M$.

The Morris-Shore transformation (MST) [4, 12, 32, 5, 14, 38, 13] is a powerful tool for reducing the dynamics of a certain class of multistate systems to the dynamics of one or more two-state systems. The MS system is shown schematically in Fig. 2. It consists of two sets of states: a ground set with $L$ states and an exited set with $M$ states. There are couplings, quanttified by the Rabi frequencies $\Omega_{lm}f(t)$, only between states from different levels ($l=1,2,...,L$; $m=1,2,...,M$), and $f(t)$ is a common time dependence of all couplings. All couplings have the same detunings $\Delta(t)$. The Hamiltonian of the MS system can be written as

$$\mathbf{H}(t)=\frac{1}{2}\left[\begin{array}[]{cc}\mathbf{O_{L}}&f(t)\mathbf{\Omega}\\ f(t)\mathbf{\Omega}^{\dagger}&\Delta(t)\mathbf{1_{M}}\\ \end{array}\right],\quad\\$$

(35)

where the constant matrix $\mathbf{\Omega}$ is $L\times M$ dimensional,

$$\mathbf{\Omega}=\left[\begin{array}[]{cccc}\Omega_{11}&\Omega_{12}&\cdots&\Omega_{1M}\\ \Omega_{21}&\Omega_{22}&\cdots&\Omega_{2M}\\ \cdots&\cdots&\ddots&\vdots\\ \Omega_{L1}&\Omega_{L2}&\cdots&\Omega_{LM}\\ \end{array}\right],$$

(36)

and $\Omega_{lm}$ ($l=1,2,...,L;\quad m=1,2,...,M$) are arbitrary complex constants . The MS transformation reduces the multi-state dynamics, which have a Hilbert space dimension of $L+M$ [Fig. 2 (top)] to a set of $M$ independent two-state systems with additional $d=L-M$ decoupled (dark) states [Fig. 2 (bottom)]. In the MS basis, the Hamiltonian has the block matrix form

$$\widetilde{\mathbf{H}}(t)=\mathbf{S}\mathbf{H}(t)\mathbf{S}^{\dagger}=\frac{1}{2}\left[\begin{array}[]{cc}\mathbf{O_{L}}&f(t)\widetilde{\mathbf{\Omega}}\\ f(t)\widetilde{\mathbf{\Omega}}^{\dagger}&\Delta(t)\mathbf{1_{M}}\\ \end{array}\right],$$

(37)

where $\mathbf{S}$ is a constant unitary matrix, defined by two square unitary matrices $\mathbf{S}_{L}$ and $\mathbf{S}_{M}$ with dimensions of $L$ and $M$, respectively,

$$\mathbf{S}=\left[\begin{array}[]{cc}\mathbf{S}_{L}&\mathbf{O}\\ \mathbf{O}&\mathbf{S}_{M}\\ \end{array}\right],\quad\mathbf{S}\mathbf{S}^{\dagger}=\mathbf{S}^{\dagger}\mathbf{S}=\mathbf{1_{(L+M)}}.$$

(38)

Then, by using Eqs. (37) and (38), the transformed coupling matrix $\widetilde{\mathbf{\Omega}}$ can be expressed as

$$\widetilde{\mathbf{\Omega}}=\mathbf{S}_{L}\mathbf{\Omega}\mathbf{S}_{M}^{\dagger}$$

(39)

The matrices $\mathbf{S}_{L}$ and $\mathbf{S}_{M}$ are defined by the condition that they diagonalize $\mathbf{\Omega}^{\dagger}\mathbf{\Omega}$ and $\mathbf{\Omega}\mathbf{\Omega}^{\dagger}$, i.e.

	$\displaystyle\mathbf{S}_{L}\mathbf{\Omega}\mathbf{\Omega}^{\dagger}\mathbf{S}_{L}^{\dagger}$	$\displaystyle=\mathbf{1_{L}},$		(40a)
	$\displaystyle\mathbf{S}_{M}\mathbf{\Omega}^{\dagger}\mathbf{\Omega}\mathbf{S}_{M}^{\dagger}$	$\displaystyle=\mathbf{1_{M}},$		(40b)

and are found by solving these equations. The $M$-dimensional square matrix $\mathbf{\Omega}^{\dagger}\mathbf{\Omega}$ has $M$ generally nonzero eigenvalues $\lambda^{2}_{m}$ $(m=1,2,...,M)$. The $L$-dimensional square matrix $\mathbf{\Omega}\mathbf{\Omega}^{\dagger}$ has the same $M$ eigenvalues as $\mathbf{\Omega}^{\dagger}\mathbf{\Omega}$ and additional $d=L-M$ zero eigenvalues, corresponding to the dark states.

The transformed Hamiltonian  (37) acquires the form

$$\widetilde{\mathbf{H}}(t)=\left[\begin{array}[]{cc}\mathbf{O}_{d\times d}&\mathbf{O}_{d\times 2M}\\ \mathbf{O}_{2M\times d}&\widetilde{\mathbf{H}}_{c}(t)\end{array}\right],$$

(41)

where $\widetilde{\mathbf{H}}_{c}(t)$ is formed by 4 square $M\times M$ matrices,

$$\widetilde{\mathbf{H}}_{c}(t)=\left[\begin{array}[]{cc}\mathbf{O}&{\bm{\Lambda}}f(t)\\ {\bm{\Lambda}}f(t)&\Delta(t)\mathbf{1}\\ \end{array}\right],$$

(42)

where

$${\bm{\Lambda}}=\left[\begin{array}[]{cccc}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{M}\end{array}\right].$$

(43)

With an appropriate reordering of the states (i.e., the rows and the columns), described by a matrix ${\bf R}$, $\widetilde{\mathbf{H}}_{c}(t)$ can be cast into the block matrix form ${\bf R}^{-1}\widetilde{\mathbf{H}}_{c}(t){\bf R}=\widetilde{\mathbf{H}}_{b}(t)$, with

$$\widetilde{\mathbf{H}}_{b}(t)=\left[\begin{array}[]{cccc}\widetilde{\mathbf{H}}_{1}(t)&\bm{0}&\cdots&\bm{0}\\ \bm{0}&\widetilde{\mathbf{H}}_{2}(t)&\cdots&\bm{0}\\ \vdots&\vdots&\ddots&\vdots\\ \bm{0}&\bm{0}&\cdots&\widetilde{\mathbf{H}}_{M}(t)\end{array}\right],$$

(44)

with

$$\widetilde{\mathbf{H}}_{m}(t)=\left[\begin{array}[]{cc}0&\lambda_{m}f(t)\\ \lambda_{m}f(t)&\Delta(t)\end{array}\right].$$

(45)

being the Hamiltonian describing the $m$-th independent two-state system in the MS basis.

Each of these $M$ independent two-state Hamiltonians generates $M$ independent two-state propagators with $M$ different pairs of CK parameters $a_{k}$ and $b_{k}$,

$$\widetilde{\mathbf{U}}_{m}=\left[\begin{array}[]{cc}a_{m}&b_{m}\\ -b^{*}_{m}e^{-i\delta}&a^{*}_{m}e^{-i\delta}\end{array}\right],$$

(46)

where $|a_{m}|^{2}+|b_{m}|^{2}=1$ and

$$\delta=\int_{0}^{T}\Delta(t)\,dt$$

(47)

is a common accumulated phase for all $M$ independent systems. The propagator in the reordered MS basis has the same block matrix structure as $\widetilde{\mathbf{H}}_{b}(t)$,

$$\widetilde{\mathbf{U}}_{b}=\left[\begin{array}[]{cccc}\widetilde{\mathbf{U}}_{1}&\bm{0}&\cdots&\bm{0}\\ \bm{0}&\widetilde{\mathbf{U}}_{2}&\cdots&\bm{0}\\ \vdots&\vdots&\ddots&\vdots\\ \bm{0}&\bm{0}&\cdots&\widetilde{\mathbf{U}}_{M}\end{array}\right].$$

(48)

After the reordering operation ${\bf R}\widetilde{\mathbf{U}}_{b}{\bf R}^{-1}=\widetilde{\mathbf{U}}_{c}$ we find the propagator of the full system in MS basis in a block matrix form of four $M\times M$ square matrices,

$$\widetilde{\mathbf{U}}=\left[\begin{array}[]{cc}\mathbf{1}_{d\times d}&\mathbf{O}_{d\times 2M}\\ \mathbf{O}_{2M\times d}&\widetilde{\mathbf{U}}_{c}\end{array}\right],$$

(49)

where

$$\widetilde{\mathbf{U}}_{c}=\left[\begin{array}[]{cc}\mathbf{A}&\mathbf{B}\\ -\mathbf{B}^{*}e^{-i\delta}&\mathbf{A}^{*}e^{-i\delta}\end{array}\right],$$

(50)

with

$${\mathbf{A}}=\left[\begin{array}[]{cccc}a_{1}&0&\cdots&0\\ 0&a_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&a_{M}\end{array}\right],\quad{\mathbf{B}}=\left[\begin{array}[]{cccc}b_{1}&0&\cdots&0\\ 0&b_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&b_{M}\end{array}\right].$$

(51)

The original propagator obeys the same transformation as the original Hamiltonian,

$$\mathbf{U}=\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}\mathbf{S}.$$

(52)

With the expression (52) for the single propagator ${\mathbf{U}}$ at hand, we can easily find the $N$-pass propagator by taking the $N$-th power of ${\mathbf{U}}$,

$$\mathbf{U}^{N}=\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}\mathbf{S}\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}\mathbf{S}\cdots\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}\mathbf{S}=\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}^{N}\mathbf{S},$$

(53)

where $\widetilde{\mathbf{U}}^{N}$ is the transformed MS propagator. As the MS system is decomposed into $M$ independent two-state systems, then all of them have $M$ independent evolutions, governed by the respective propagators $\widetilde{\mathbf{U}}_{k}$ of Eq. (46). Indeed, we have from Eq. (49)

$$\widetilde{\mathbf{U}}^{N}=\left[\begin{array}[]{cc}\mathbf{1}_{d\times d}&\mathbf{O}_{d\times 2M}\\ \mathbf{O}_{2M\times d}&\widetilde{\mathbf{U}}_{c}^{N}\end{array}\right].$$

(54)

We have $\widetilde{\mathbf{U}}_{c}^{N}={\bf R}\widetilde{\mathbf{U}}_{b}^{N}{\bf R}^{-1}$, and $\widetilde{\mathbf{U}}_{b}^{N}$ is readily derived,

$$\widetilde{\mathbf{U}}_{b}^{N}=\left[\begin{array}[]{cccc}\widetilde{\mathbf{U}}_{1}^{N}&\bm{0}&\cdots&\bm{0}\\ \bm{0}&\widetilde{\mathbf{U}}_{2}^{N}&\cdots&\bm{0}\\ \vdots&\vdots&\ddots&\vdots\\ \bm{0}&\bm{0}&\cdots&\widetilde{\mathbf{U}}_{M}^{N}\end{array}\right].$$

(55)

We can find all $\widetilde{\mathbf{U}}_{m}^{N}$ using Eqs. (5) and (6), and then construct the propagator of the full system.

In order to find $\widetilde{\mathbf{U}}_{m}^{N}$, we first note that it does not possess the SU(2) symmetry as its determinant is $e^{-i\delta}$, and hence we cannot use Eqs. (5) and (6) directly because they require SU(2) symmetry. Therefore, we represent $\widetilde{\mathbf{U}}_{m}$ as

$$\widetilde{\mathbf{U}}_{m}=e^{-i\delta/2}\left[\begin{array}[]{cc}a_{m}e^{i\delta/2}&b_{m}e^{i\delta/2}\\ -b_{m}^{*}e^{-i\delta/2}&a_{m}^{*}e^{-i\delta/2}\end{array}\right],$$

(56)

where the matrix on the right-hand side is now SU(2) symmetric, which allows us to apply relations (5) and (6). We have

$$\widetilde{\mathbf{U}}_{m}^{N}=e^{-iN\delta/2}\left[\begin{array}[]{cc}a_{m}e^{i\delta/2}&b_{m}e^{i\delta/2}\\ -b_{m}^{*}e^{-i\delta/2}&a_{m}^{*}e^{-i\delta/2}\end{array}\right]^{N}.$$

(57)

Then, by introducing the notation

$$a_{m}^{\prime}=a_{m}e^{i\delta/2},\quad b_{m}^{\prime}=b_{m}e^{i\delta/2},$$

(58)

and using Eqs. (5) and (6), we find

$$\widetilde{\mathbf{U}}_{m}^{N}=\left[\begin{array}[]{cc}a_{mN}^{\prime}&b_{mN}^{\prime}\\ -b^{\prime*}_{mN}e^{-iN\delta}&a^{\prime*}_{mN}e^{-iN\delta}\end{array}\right],$$

(59)

where

	$\displaystyle a_{mN}^{\prime}$	$\displaystyle=\left[\cos(N\theta_{m}^{\prime})+i\,\textrm{Im}(a_{m}^{\prime})\dfrac{\sin(N\theta_{m}^{\prime})}{\sin(\theta_{m}^{\prime})}\right]e^{-iN\delta/2},$		(60a)
	$\displaystyle b_{mN}^{\prime}$	$\displaystyle=b_{k}^{\prime}\dfrac{\sin(N\theta_{m}^{\prime})}{\sin(\theta_{m}^{\prime})}e^{-iN\delta/2},$		(60b)
	$\displaystyle\theta_{m}^{\prime}$	$\displaystyle=\arccos(\textrm{Re}\,a_{m}^{\prime}).$		(60c)

The relations (60), together with  (58) give the connection between the single $\widetilde{\mathbf{U}}_{m}$ and the repeated $\widetilde{\mathbf{U}}_{m}^{N}$ propagators. Thereby we find the multi-pass propagator of the $M$ MS two-state systems [cf. Eq. (54)],

$$\widetilde{\mathbf{U}}_{c}^{N}=\left[\begin{array}[]{cc}\mathbf{A}_{N}&\mathbf{B}_{N}\\ -\mathbf{B}_{N}^{*}e^{-iN\delta}&\mathbf{A}_{N}^{*}e^{-iN\delta}\end{array}\right],$$

(61)

with

	$\displaystyle\mathbf{A}_{N}$	$\displaystyle=\left[\begin{array}[]{cccc}a_{1N}^{\prime}&0&\cdots&0\\ 0&a_{2N}^{\prime}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&a_{MN}^{\prime}\end{array}\right],$		(62e)
	$\displaystyle\mathbf{B}_{N}$	$\displaystyle=\left[\begin{array}[]{cccc}b_{1N}^{\prime}&0&\cdots&0\\ 0&b_{2N}^{\prime}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&b_{MN}^{\prime}\end{array}\right].$		(62j)

Next, we find $\widetilde{\mathbf{U}}^{N}$ from Eqs. (54) and (61). Finally, we find the original $N$-pass propagator $\mathbf{U}^{N}$ by the transformation  (53), ie $\mathbf{U}^{N}=\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}^{N}\mathbf{S}$.

Because $\mathbf{S}$ is a constant matrix and it appears in both $\mathbf{U}$ of Eq. (52) and $\mathbf{U}^{N}$ of Eq. (53) in the same manner, it follows that $\mathbf{U}^{N}$ can be obtained from $\mathbf{U}$ by the substitutions $a_{m}\to a_{mN}^{\prime}$ and $b_{m}\to b_{mN}^{\prime}$ and $e^{i\delta}\to e^{iN\delta}$, according to the connections (60).

As it has been noticed in  [5], in the interaction representation for a single propagator, one can get rid of the phase $\delta$ by removing the phase factor $e^{-i\delta}$. This means that for single propagators, the phase $\delta$ can be considered as unimportant. Now we see that for the multi-pass MS propagator we could also remove the phase factor $e^{-iN\delta}$ by switching to the interaction representation, so that only $a^{\prime}_{mN}$ and $b^{\prime}_{mN}$ will remain. However, these CK parameters depend on $\delta$ according to (60). Which means, that for repeated MS propagators the phase $\delta$ becomes already important.