Two-level systems with periodic $N$-step driving fields: Exact dynamics and quantum state manipulations

In this work, we derive exact solutions of a dynamical equation, which can represent all two-level Hermitian systems driven by periodic $N$-step driving fields. For different physical parameters, this dynamical equation displays various phenomena for periodic $N$-step driven systems. The time-dependent transition probability can be expressed by a general formula that consists of cosine functions with discrete frequencies, and, remarkably, this formula is suitable for arbitrary parameter regimes. Moreover, only a few cosine functions (i.e., one to three main frequencies) are sufficient to describe the actual dynamics of the periodic $N$-step driven system. Furthermore, we find that a beating in the transition probability emerges when two (or three) main frequencies are similar. Some applications are also demonstrated in quantum state manipulations by periodic $N$-step driving fields.

Figure 1

(a) Schematic diagram of distinct partitions of the same two-step sequence with period $T$. The emergence of the first jump is quantified by the parameter $\lambda$. In the top panel, the evolution operator is divided into two factors (shaded with different colors), $\mathbb{U}(t,t_3)\mathbb{U}(t_3,t_0)$, while it is divided into three factors, $\mathbb{U}(t,t_2)\mathbb{U}(t_2,t_1)\mathbb{U}(t_1,t_0)$, in the bottom panel. (b) Population evolution of level $|2\rangle$ for different $\lambda$, where the initial state is $\left(\right|1\rangle +|2\rangle )/\sqrt{2}$. The black-dotted curves represent the predictions given by the effective Hamiltonian $H_{\mathrm{eff}}\left(\lambda \right)$. Each dotted curve almost coincides with the envelope of its solid counterpart, indicating that $H_{\mathrm{eff}}\left(\lambda \right)$ contains relevant information caused by different starting times of the driving fields. The parameters chosen here are $\{{\epsilon }_n/{\epsilon }_1\}=\{1,2\}, \{{\mathrm{\Delta }}_n/{\epsilon }_1\}=\{50,40\}$, and $E_1{\tau }_1=E_2{\tau }_2=\pi /2$.

Figure 2

(a) Schematic representation of various parameter regimes. The effective Hamiltonian in Eq. (7) is valid in region I for an arbitrary ${\tau }_1$. The beat phenomenon can emerge in regions II and III, and is more obvious on the light blue axis or the origin of coordinates (${\mathrm{\Delta }}_1=0$ or ${\mathrm{\Delta }}_2=0$). (b) ${\varepsilon }^m\left(t_s\right)$ vs ${\mathrm{\Delta }}_n$, where $\left\{{\theta }_n\right\}=\{0,\pi /3\}, {\epsilon }_2=2{\epsilon }_1, {\tau }_1=0.2/{\epsilon }_1$, and ${\tau }_2$ is chosen to be ${\mathrm{\Delta }}_{\mathrm{eff}}=0$. The yellow solid curves, red dotted curves, and blue solid curves correspond to ${\varepsilon }^m\left(t_s\right)=0.05,0.08,0.1$, respectively. The results verify the classification in Fig. 2. (c, d) Time-dependent transition probabilities $P_{12}\left(t\right)$ and $P_{12}^m\left(t\right)$ with different detunings in different regions, where the red solid (blue dashed) curves are plotted by $P_{12}\left(t\right) \left[P_{12}^m\left(t\right)\right]$. The appreciable overlap between these two curves implies that the actual dynamics at arbitrary time can be well described by $P_{12}^m\left(t\right)$. Note that $P_{12}^m\left(t\right)$ contains three main frequencies in (e).

Figure 3

Time-dependent transition probabilities $P_{12}\left(t\right)$ and $P_{12}^m\left(t\right)$, which show distinct beat phenomena in different regions of Fig. 2. The red solid (blue dashed) curves are plotted by $P_{12}\left(t\right) \left[P_{12}^m\left(t\right)\right]$, and the appreciable overlap between these two curves shows the validity of Eq. (13). Note that $P_{12}^m\left(t\right)$ contains three main frequencies in (e) and (f).

Figure 4

Time-dependent transition probability $P_{12}\left(t\right)$ in the case of an unknown phase in the two-step driven system, where other parameters are ${\theta }_1=0, {\epsilon }_1={\epsilon }_2, \{{\mathrm{\Delta }}_n/{\epsilon }_1\}=\{0,40\}$, and ${\vartheta }_n=1/2$. The red solid curves represent the actual dynamics and the green (black) dashed envelope curves are described by the functions $P_b\left(t\right)=1/2(1\pm cos{\omega }_{b}t)$. According to the envelope waveform of $P_{12}\left(t\right)$, we can acquire the beat frequency ${\omega }_b$ by the green (black) dashed envelope curves. Different ${\omega }_b$ reflect different phases, given by Eq. (16).

Figure 5

Time-dependent transition probability $P_{12}\left(t\right)$ and the corresponding waveform of physical quantities: (a) detuning modulation, (b) coupling strength modulation, and (c) phase modulation. In (a), (b), and (c), by merely modulating one parameter with a square-wave sequence, we achieve the population inversion in the TLS, respectively. (d) Population inversion by merely modulating $\left\{{\tau }_n\right\}$ (upper panel, ${\epsilon }_2={\epsilon }_1/2$) or ${\epsilon }_2$ (lower panel, $\left\{{\tau }_n\right\}=\{0.1,0.2\}/{\epsilon }_1$). Panel (d) shows that the transition time is adjustable by choosing different quantities in the resonant pulses.

Figure 6

(a) Example of coherent destruction of tunneling for periodic $N$-step driven (PNSD) systems. The parameters used here are $\left\{{\theta }_n\right\}=\{0,\pi \}, \left\{{\vartheta }_n\right\}=\{0.05,0.05\}, \{{\mathrm{\Delta }}_n/{\epsilon }_1\}=\{0,0\}$, and ${\epsilon }_1={\epsilon }_2$. (b–d) Examples of the complete (incomplete) population transition in the PNSD system. The parameters are found in Table 2. The red solid curves represent the actual dynamics and the blue dashed curves correspond to the predictions given by $P_{12}^m\left(t\right)$. The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing $P_{12}^m\left(t\right)$ with a few main frequencies.

Figure 7

(a) Examples of periodic evolution in the PNSD system. The parameters used here are found in Table 2. The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing $P_{12}^m\left(t\right)$ with a few main frequencies.

Figure 8

Examples of the step evolution in the PNSD system. The parameters used here are found in Table 2. The appreciable overlap between the red solid and the blue dashed curves verifies that the actual dynamics of the PNSD system can be well described by only employing $P_{12}^m\left(t\right)$ with a few main frequencies.

Figure 9

(a) ${\varepsilon }^{\mathrm{m}}\left(t_s\right)$ vs ${\mathrm{\Delta }}_6$ with different $n_1$ in a periodic $N$-step driven system, where $N=8$ and we properly select a set of physical quantities (in units of ${\epsilon }_1$): ${\epsilon }_2=1.8, {\epsilon }_3=1.5, {\epsilon }_4=1.1, {\epsilon }_5=1.2, {\epsilon }_6=1.6, {\epsilon }_7=1.9, {\epsilon }_8=1.3, {\mathrm{\Delta }}_1=0, {\mathrm{\Delta }}_2=67, {\mathrm{\Delta }}_3=60, {\mathrm{\Delta }}_4=46, {\mathrm{\Delta }}_5=83, {\mathrm{\Delta }}_6=36, {\mathrm{\Delta }}_7=40$, and ${\mathrm{\Delta }}_8=91$. In the legend, each $k$ indicates that the $k\mathrm{th}$ Hamiltonian is in the resonant regime. (b, c) Time-dependent transition probabilities $P_{12}\left(t\right)$ with (b) $n_1=1$ and (c) $n_1=2$ in a periodic $N$-step driven system. The red solid curves represent the actual dynamics and the blue dashed curves correspond to the predictions given by Eq. (D1). Those results show that when $N$ is even, the actual dynamics can be well described by the beating formula (D1) with the frequencies (D4) and (D5).

Figure 10

(a) ${\varepsilon }^{\mathrm{m}}\left(t_s\right)$ vs ${\mathrm{\Delta }}_6$ with different $n_1$ in a periodic $N$-step driven system, where $N=7$ and we properly select a set of physical quantities (in units of ${\epsilon }_1$): ${\epsilon }_2=1.8, {\epsilon }_3=1.5, {\epsilon }_4=1.1, {\epsilon }_5=1.2, {\epsilon }_6=1.6, {\epsilon }_7=1.9, {\epsilon }_8=1.3, E_1{\tau }_1=\pi , {\mathrm{\Delta }}_1=0, {\mathrm{\Delta }}_2=55, {\mathrm{\Delta }}_3=62, {\mathrm{\Delta }}_4=47, {\mathrm{\Delta }}_5=88, {\mathrm{\Delta }}_6=35$, and ${\mathrm{\Delta }}_7=65$. In the legend, each $k$ indicates that the $k\mathrm{th}$ Hamiltonian is in the resonant regime. (b, c) Time-dependent transition probabilities $P_{12}\left(t\right)$ with (b) $n_1=2$ and (c) $n_1=3$ in the periodic $N$-step driven system. The red solid curves represent the actual dynamics and the blue dashed curves correspond to the predictions given by Eq. (D1). Those results show that when $N$ is odd, the actual dynamics can be well described by the beating formula (D1) with the frequencies (D8) and (D9).