Dressed-state analysis of two-color excitation schemes

To coherently control a few-level quantum emitter, typically pulses with an energy resonant to the transition energy are applied, making use of the Rabi mechanism, while a single off-resonant pulse does not result in a population inversion. Surprisingly, a two-color excitation with a combination of two off-resonant pulses is able to invert the system. In this paper, we provide an in-depth analysis of two-color excitation schemes within the dressed-state picture. We show that such schemes can be understood as a driving of the transition between the dressed states. In the two-level system this allows us to derive analytic expressions for the pulse parameters yielding a population inversion. We relate our findings to the Swing-UP of quantum EmitteR population (SUPER) mechanism, where the two pulses are red detuned. Our interpretation also holds in the case where one pulse is red detuned while the other pulse is blue detuned, as in dichromatic excitation schemes. We extend our considerations of the SUPER mechanism to the three-level system, where a strong mixing between the bare states takes place. The insights gained from these results help in finding accessible regimes for further experimental realizations of two-color excitation schemes.

Figure 1

Two-level system with ground state $|g\rangle$ and excited state $|x\rangle$, showing laser energy and detuning. In the dressed-state basis, the laser leads to an energy splitting with the Rabi energy.

Figure 2

Time evolution of several quantities: (a) Energies of the dressed states for a rectangular pulse with smoothened edges with $\hslash {\mathrm{\Delta }}_1=-5\mathrm{meV},\tau =40\mathrm{ps},\kappa =1{\mathrm{ps}}^{-1},\hslash {\mathrm{\Omega }}_0=4\mathrm{meV}$. (b) Energy difference of upper and lower dressed state excluding (violet) and including (green) the influence of the second pulse. The dashed black line indicates the detuning of the second pulse, which in the rotating frame is $|\hslash {\omega }_{\mathrm{\Delta }}|$. (c) Dressed-state occupation under the second pulse with ${\sigma }_2=4\mathrm{ps},{\alpha }_2=9.13\pi ,\hslash {\mathrm{\Delta }}_2=-11.46\mathrm{meV}$, where the second pulse is indicated by the dashed line. (d) Occupation of the excited state in the bare-state picture as well as contributions of the dressed states.

Figure 3

Same as Fig. 2 but for a positive detuning of $\hslash {\mathrm{\Delta }}_1=+5\mathrm{meV}$, which yields as parameters for the second pulse ${\sigma }_2=4\mathrm{ps},{\alpha }_2=1.12\pi ,\hslash {\mathrm{\Delta }}_2=-1.41\mathrm{meV}$.

Figure 4

(a) Dressed-state energies $E_{\pm }$ induced by a single off-resonant Gaussian pulse, ${\sigma }_1=2.4\mathrm{ps},\hslash {\mathrm{\Delta }}_1=-8\mathrm{meV},{\alpha }_1=22.65\pi$, as indicated by the dashed line. (b) Energy difference of the dressed states, (violet) without and (green) including the interaction Hamiltonian ${\tilde{H}}_{{\mathrm{\Omega }}_2}$ (second pulse parameters: ${\sigma }_2=3.04\mathrm{ps},\hslash {\mathrm{\Delta }}_2=-19.163\mathrm{meV},{\alpha }_2=19.29\pi )$. The dashed black line depicts the energy of the second laser. (c) Dressed-state and excited-state occupation. The dashed blue line shows the shape of the second pulse.

Figure 5

Three-level system with ground state $|g\rangle$, exciton state $|x\rangle$, and biexciton state $|xx\rangle$. The detuning of the first laser is tuned $3\mathrm{meV}$ below the two-photon resonance (TPR), which is located at $-2\mathrm{meV}$.

Figure 6

(a) Dressed-state energies $E_{1,2,3}$ for the biexciton system interacting with a single off-resonant pulse, ${\sigma }_1=3\mathrm{ps},\hslash {\mathrm{\Delta }}_1=-5\mathrm{meV},{\alpha }_1=27\pi$. The color indicates how the dressed states mix from the bare states. (b) Coupling coefficients given in Eq. (27), corresponding to the pulse in (a) as well as the laser envelope (arb. units). The time interval as well as the strength of the coupling is different for all three possible transitions, with the $|{\psi }_2\rangle \leftrightarrow |{\psi }_3\rangle$ transition showing by far the strongest coupling.

Figure 7

(a) Energy differences of the dressed states including the interaction Hamiltonian ${\tilde{H}}_{{\mathrm{\Omega }}_2}$ containing a pulse with ${\sigma }_2=3\mathrm{ps},\hslash {\mathrm{\Delta }}_2=-11.83\mathrm{meV},{\alpha }_2=22.8\pi$. The black dashed line depicts the energy of the second laser in this frame of reference. (b) Dressed-state occupations. The dashed blue line indicates the second pulse. (c) Occupation of exciton and biexciton.

Figure 8

Same as Fig. 7 but addressing the biexciton. For this, the detuning and pulse area of the second pulse are changed to $\hslash {\mathrm{\Delta }}_2=-10.64\mathrm{meV},{\alpha }_2=17.49\pi$.