Figure 1

Three-state chainwise connected quantum systems. (a) System with the SU(2) dynamic symmetry. (b) System with the MS symmetry (${\Omega }_1$ and ${\Omega }_2$ have the same time dependance).Reuse & Permissions

Figure 2

Transition probability $P_{1\to N}$ for three- and five-state systems with SU(2) dynamic symmetry, described by the Hamiltonian of Eq. (12a), vs the pulse area $A$ for a single pulse and for composite sequences of 3, 5, and 15 pulses (denoted on the curves), with phases from Eq. (9). Solid curves, $N=3$ states; dashed curves, $N=5$ states.Reuse & Permissions

Figure 3

Transition probability $P_{1\to 3}$ for a three-state system with SU(2) dynamic symmetry, described by the Hamiltonian of Eq. (15), vs the pulse area $A$ for a single pulse and for NB composite sequences of 5, 9, and 17 pulses (denoted on the curves) (top frame), and PB composite sequences of 7 and 15 pulses (bottom frame) with phases (approximately) [9]: N5, $(0,1.161,0.580,1.161,0)\pi$; N9, $(0,1.129,0.822,0.108,1.386,0.108,0.822,1.129,0)\pi$; N17, $(0,1.604,0.553,1.091,0.888,0.620,1.535,0.149,$$1.569,0.149,1.535,0.620,0.888,1.091,0.553,1.604,0)\pi$; P7, $(0,0.704,1.186,1.834,1.186,0.704,0)\pi$; P15, $(0,0.473,0.421,1.624,1.050,1.081,1.469,0.259,$$1.469,1.081,1.050,1.624,0.421,0.473,0)\pi$.Reuse & Permissions

Figure 4

Transition probability $P_{1\to 3}$ for a three-state system with SU(2) symmetry versus the detuning for a single hyperbolic-secant pulse and for a sequence of five $3\pi /5$ pulses with phases (approximately) $(0,0.747,0.424,0.747,0)\pi$ and nine $4\pi /9$ pulses with phases (approximately) $(0,1.308,1.153,1.251,0.562,1.251,1.153,1.308,0)\pi$.Reuse & Permissions

Figure 5

Contour plots of the transition probability $P_{1\to 3}$ for a three-state system with SU(2) symmetry versus pulse area and detuning for a single pulse (top) and composite pulse of five pulses with phases $(0,5/6,1/3,5/6,0)\pi$.Reuse & Permissions

Figure 6

Morris-Shore transformation of a $W$ system ($N_a=3$, $N_b=2$) and a $V$ system ($N_a=2$, $N_b=1$), which are independent from each other. The top part of the figure shows the states in the original basis of the $W$ and $V$ systems, while the bottom part displays the states in the corresponding MS basis. In this example, we have used simplified notation for the MS states: $|{\tilde{\psi }}_1^d\rangle =|d_{W,1}\rangle$, $|{\tilde{\psi }}_k^a\rangle =|e_{W,k}\rangle$ and $|{\tilde{\psi }}_k^b\rangle =|b_{W,k}\rangle$ ($k=1,2$) for the $W$ system, and $|{\tilde{\psi }}_1^d\rangle =|d_{V,1}\rangle$, $|{\tilde{\psi }}_1^a\rangle =|e_{V,1}\rangle$ and $|{\tilde{\psi }}_1^b\rangle =|b_{V,1}\rangle$ for the $V$ system. The $W$ system is transformed into two independent two-state systems with couplings ${\lambda }_1$ and ${\lambda }_2$ and a decoupled state. The $V$ system itself is transformed into one independent two-state system with coupling $\lambda$ and a decoupled state.Reuse & Permissions

Figure 7

Total population in the excited sublevels of a degenerate two-level system formed of the sublevels of the transition $J_g=1\leftrightarrow J_e=2$ following the action of a BB resonant composite pulse sequence vs pulse area $A=2\left|\Omega \right|{\int }_{t_i}^{t_f}f\left(t\right)dt$, where $\Omega =\sqrt{{\Omega }_{+}^2+{\Omega }_{-}^2}$ [Eq. (36)]. Comparison is made between the inversion induced by a single pulse and BB sequences of $n=3$, 5, and 9 pulses with phases from Eq. (9). Additionally, a composite pulse, labeled 5b, with phases: $(0,0.269,0.771,0.269,0)\pi$ is shown to demonstrate a composite sequence of five $3\pi /7$ pulses. The system is prepared initially in an equal coherent superposition of the ground sublevels, $|{\psi }_i\rangle =\left(\right|-1\rangle +|0\rangle +|1\rangle )/\sqrt{3}$. The ellipticity is $\varepsilon =0$ (linear polarization) and the rotation angle is $\beta =0$. The dotted curves show the total population in sublevels $m_e=-1$ and 1 (the upper states in the V system, the maximum population in which is $\frac{1}{3}$), whereas the dashed curves show the total population in sublevels $m_e=-2$, 0, 2 (the upper states in the W system, the maximum population in which is $\frac{2}{3}$). The solid curves show the total population in all five excited sublevels.Reuse & Permissions