Shortcuts to Adiabaticity for Open Quantum Systems and a Mixed-State Inverse Engineering Scheme

We propose a fast mixed-state control scheme to transfer the quantum state along designable trajectories in Hilbert space, which is robust to multiple decoherence noises. Starting with the dynamical invariants of open quantum systems, we present the shortcuts to adiabaticity (STA) of open quantum systems at first, then apply the STA to speed up the adiabatic steady process. Our scheme drives open systems from a initial steady state to a target steady state by a controlled Liouvillian that possesses the same form as the reference (original) one that is accessible in present-day experiments. An experimental design with currently available parameters for the nitrogen-vacancy (N-$V$) center in diamond is suggested and discussed.

Figure 1

The level structure of the N-$V$ center. (a) A schematic illustration of the level structure of the N-$V$ center. The optical zero photon line (ZPL) at 637 nm is related to the transition from ${}^{3}E$ to ${}^3{A}_2$ and the ZPL at 3371 nm corresponds to the transition from ${}^{1}E$ to ${}^3{A}_2$. (b) State transfer in a N-$V$ center $\mathrm{\Lambda }$ system by the protocol with initial-to-final state coupling. (c) State transfer in a N-$V$ center $\mathrm{\Lambda }$ system by the protocol without initial-to-final state coupling.

Figure 2

(a) The control fields and (b) the populations on $|m_s=-1⟩$ (blue solid line), $|A_2⟩$ (green dash line), and $|+1⟩$ (red dot-dash line) as a function of time for pulse length $\tau =16.8\mathrm{ns}$. (c) The final population on states except $|m_s=+1⟩$ as a function of the pulse length $\tau$ for $\mathrm{\Omega }=2\pi \times 122\mathrm{MHz}$, ${\mathrm{\Gamma }}_{-1}=2\pi \times 4.3\mathrm{MHz}$, ${\mathrm{\Gamma }}_{+1}=2\pi \times 8.4\mathrm{MHz}$, ${\mathrm{\Gamma }}_d=2\pi \times 8.8\mathrm{MHz}$, and $N=1.9\times {10}^{-33}$.

Figure 3

(a) The control fields, (b) the mean excitation numbers, and (c) the populations on states $|m_s=0⟩$ (blue lines), $|{}^{1}E⟩$ (green lines), and $|m_s=+1⟩$ (red lines) as a function of time for $\mathrm{\Omega }=18.1\mathrm{KHz}$, $\mathrm{\Gamma }=2\pi \times 1.5\mathrm{MHz}$, ${\mathrm{\Gamma }}_d=2\pi \times 8.8\mathrm{MHz}$, and $N=6.55\times {10}^{-7}$. The pulse length is $\tau =20\mu \mathrm{s}$.

Figure 4

The final population not on $|m_s=+1⟩$ (red lines) as a function of (a) the pulse length $\tau$ for $\mathrm{\Omega }=154\mathrm{KHz}$ and (c) the control fields $\mathrm{\Omega }$ for $\tau =5\mu \mathrm{s}$. The main excitation number $N_{+1}^i$ as a function of time with (b) different pulse lengths for $\mathrm{\Omega }=154\mathrm{KHz}$ and (d) different control fields for $\tau =5\mu \mathrm{s}$. The other parameters in the master equation are chosen as $\mathrm{\Gamma }=2\pi \times 1.5\mathrm{MHz}$, ${\mathrm{\Gamma }}_d=2\pi \times 8.8\mathrm{MHz}$, and $N=6.55\times {10}^{-7}$.

Figure 5

(a) The control field ${\mathrm{\Omega }}_p$, (b) the population on $|{}^{1}E⟩$, (c) the ratio of the control field strengths between the mixed-state and the pure-state inverse engineering schemes, and (d) the population out of $|+1⟩$ versus the dimensionless time $t/\tau$ for the mixed-state trajectory (red dash lines) and the pure state trajectory with ${\gamma }_0=0.1$ (green dot-dash lines) and ${\gamma }_0=0.01$ (blue solid lines). The decay rate is chosen as $\mathrm{\Gamma }=10/\tau$.