Optical control of the complex phase of a quantum ground-state amplitude

We discuss how coherent driving of a two-level quantum system can be used to induce a complex phase on the ground state and we discuss its geometric and dynamic contributions. While the global phase of a wave function has no physical significance, the coherent dynamics in a two-level subspace provides relative phases and is an essential building block for more advanced dynamics in larger systems. In this regard, we note that one must be careful with intuitive accounts of the phase dynamics as it depends on the interaction picture applied. To mitigate ambiguities in practical analyses, we suggest to complement the Bloch sphere picture with the path taken by the ground-state amplitude in the complex plane, and we show how the two-level pure-state dynamics can serve as a starting point for the study of the dynamics explored in three-level lambda systems, four-level tripod systems, and open quantum systems.

Figure 1

Illustrations of the Bloch vector trajectory for four operations, (a) resonant drive, (b) off-resonant drive, (c) far off-resonant drive, and (d) capture and release of the energy eigenstate, are shown in the interaction picture described by Eq. (1). All operations induce a phase $\theta =\pi /2$ on $|g\rangle$ [the trajectory in panel (c) is encircled five times]. Panel (e) depicts the same operations in a complementary $c_g$ plot, which shows the real part and the imaginary part of the complex amplitude $c_g$. The ground-state population $|c_g{|}^2$ is read from the figure as the square of the distance from the center of the disk (1 at the disk's edge, $1/2$ at the dashed black circle, and 0 in the center). The color gradient indicates the amplitude of the excited-state coefficient $|c_e|$. The phase $\theta$ is read as the angle between the initial-state (black dot) and the final-state (gray dot) amplitudes.

Figure 2

Depiction of the capture and release operation described in Sec. 2a4 in the frame that evolves with a fixed frequency equal to the atomic frequency, i.e., Eq. (1) applies with $\mathrm{\Delta }=0$ and the complex phase $\varphi$ varies with time.

Figure 3

The (a) integrated excited-state population, (b) detuning, and (c) total duration of the four operations depicted in Fig. 1 are shown as a function of the phase $\theta$ added to $|g\rangle$. All operations use a constant Rabi frequency of $\mathrm{\Omega }=1.0\mathrm{MHz}$, which determines all gate parameters for the resonant and off-resonant operations, while the far off-resonant operation performs $N=20$ loops. The capture and release gate uses the same initial detuning as the far off-resonant operation and thus reaches the same maximum excited-state population [note that in panel (b) we depict only the detuning used in the locked part of this gate operation]. The excited-state population is obtained by numerically solving the Schrödinger equation, but the approximately correct analytical expressions presented in Secs. 2a3 and 2a4 in the limit of large detunings provide very similar results.

Figure 4

(a) Lambda three-level system with time-dependent complex Rabi frequencies ${\mathrm{\Omega }}_{ge}\left(t\right)$ and ${\mathrm{\Omega }}_{ae}\left(t\right)$. (b) Tripod four-level system with qubit levels $|0\rangle$ and $|1\rangle$, as well as a third ground state $|a\rangle$ and an excited state $|e\rangle$.

Figure 5

(a–c) Time-dependent shaping function $s\left(t\right)$, (d–f) Rabi frequencies, and (g–i) populations, for three different examples of possible lambda dynamics (columns) that return to state $|g\rangle$ with a phase $\theta =\pi /2$ upon completion. In all three examples, the dynamics in the two-level $\{|g\rangle ,|p\left(t\right)\rangle \}$ subspace is described by the off-resonant operation discussed in Sec. 2a2 and shown as the dotted blue lines in Figs. 1 and 1, except state $|e\rangle$ should be replaced by $|p\left(t\right)\rangle$. The Rabi frequencies are obtained from Eq. (C9), where $\mathrm{\Omega }\left(t\right)$ changes as $|{\mathrm{\Omega }}_{ge}\left(t\right)|$ shown in panel (d) and $s\left(t\right)$ as shown in panels (a)–(c), and both are created using two smoothstep functions. Furthermore, $\alpha \equiv \frac{\mathrm{\Delta }\left(t\right)}{\mathrm{\Omega }\left(t\right)}=\text{sign}\left(\theta \right)tan[arcsin(1-\left|\theta \right|/\pi \left)\right]$ is determined from the choice of performing the off-resonant operation in Sec. 2a2. The populations shown in panels (g)–(i) are obtained by numerically solving the Schrödinger equation, but the analytical expressions of Eqs. (7) and (C7) give the same results.

Figure 6

The errors of the four operations discussed in Sec. 2a when $\theta =\pi /2$ are shown as a function of the excited state lifetime $T_1$ based on (a) Lindblad master equation simulations, Eqs. (10) and Eq. (11), and (b) analytical estimations using Eq. (12). The operations use the same parameters as in Fig. 3. Panel (c) shows the density matrix element $2{\rho }_{gf}\left(t\right)$ in the complex plane for the four operations when $T_1=1\mu \mathrm{s}$ and the initial state is $\frac{1}{\sqrt{2}}(|g\rangle +|f\rangle )$. This quantity plays the same role and keeps track of the phase evolution of state $|g\rangle$ in the same way as the complex amplitude $c_g\left(t\right)$ in the pure-state case. Note that the far off-resonant and capture and release operations use higher detunings compared to those used in Fig. 1.

Figure 7

(a) Shows the trajectories for three different initial states, indicated by the black dots, when $\mathrm{\Delta }\left(t\right)=0$. The phase of $-i{e}^{-i\varphi }{c}_e\left(0\right)$ is set so that the initial velocity, black arrows, points toward the left, and the speed is given by Eq. (A3). The trajectories are identical to how a long-swinging pendulum would swing in the limit of small amplitudes. (b) Shows three examples with different ratios of $\mathrm{\Delta }\left(t\right)/\mathrm{\Omega }\left(t\right)$. The detuning acts as a steering force, turning the pendulum toward the right (left) if the detuning is positive (negative). Larger detunings results in a stronger steering force.