Harmonic dual dressing of spin-1/2 systems

Controlled modifications of the magnetic response of a two-level system are produced in dressed systems by one high-frequency, strong, and nonresonant electromagnetic field. This quantum control is greatly enhanced and enriched by a harmonic, commensurable, and orthogonally oriented dual dressing, as discussed here. The secondary field enables a fine tuning of the qubit response, with control parameter amplitude, harmonic content, spatial orientation, and phase relation. Our analysis, mainly based on a perturbative approach with respect to the driving strength, includes also nonperturbative numerical solutions. The Zeeman response becomes anisotropic in a triaxial geometry and includes a nonlinear quadratic contribution. The long-time dynamics is described by an anisotropic effective magnetic field representing the handle for the system full engineering. Through the low-order harmonic mixing, the bichromatic driving generates a synthetic static field modifying the system dynamics. The spin temporal evolution includes a micromotion at harmonics of the driving frequency whose role in the spin detection is examined. Our dressing increases the two-level energy splitting, improving the spin detection sensitivity. In the weak-field direction it compensates the static fields applied in different geometries. The results presented here lay a foundation for additional applications to be harnessed in quantum simulations.

Figure 1

Schematic of an atomic spin system dressed by the $B_x$ and $B_y$ oscillating fields in the presence of static field ${\mathit{B}}_0$ arbitrarily oriented. Initially the spins are optically pumped into an eigenstate.

Figure 2

First Brillouin zone ${\lambda }_{+},{\lambda }_{-}$ eigenvalues vs ${\omega }_{0z}$. The $|{\mathit{h}}_s|$ modulus is the energy separation at zero field. The $\left|\mathit{g}\right|$ amplitude is the absolute value for the derivative of that separation. Parameters ${\omega }_{0x}={\omega }_{0y}=0, {\mathrm{\Omega }}_x=5.11,{\mathrm{\Omega }}_y=3, p=1, \mathrm{\Phi }=\pi /2$. For this case, at ${\omega }_0\simeq 0, {\mathrm{\Omega }}_L={\lambda }_{+}-{\lambda }_{-}$ is larger than ${\omega }_0$.

Figure 3

Cosine driving synthetic and effective fields for ${\omega }_{0z}=0.1$ and ${\omega }_{0x}={\omega }_{0y}=0$. In (a) and (b) synthetic fields derived from the perturbation treatment vs ${\mathrm{\Omega }}_x$ at ${\mathrm{\Omega }}_y=0.1$. In (a) black line for $h_{s,z}^{\left(1\right)}$ at $p=1$ and $\mathrm{\Phi }=\pi /2$; red line $h_{s,y}^{\left(1\right)}$ at $p=2$ and $\mathrm{\Phi }=\pi /6$. In (b) second-order $h_{s,x}^{\left(2\right)}$ vs ${\mathrm{\Omega }}_x$ at ${\mathrm{\Omega }}_y=0.2$ and $(p=1,2)$. In (c) numerical results for the $h_z$ effective field in a $({\mathrm{\Omega }}_x,{\mathrm{\Omega }}_y)$ two-dimensional (2D) plot at $p=1$ and $\mathrm{\Phi }=\pi /2$.

Figure 4

Cosine driving results for the $\mathsf{g}$ components. In (a) ${\mathsf{g}}_{xx}^{\left(1\right)}={\mathsf{g}}_{yy}^{\left(1\right)}$ components vs ${\mathrm{\Omega }}_x$, given by the $J_0$ Bessel function and independent of ${\mathrm{\Omega }}_y$. In (b) ${\mathsf{g}}_{ij}^{\left(2\right)}$ tensor components vs ${\mathrm{\Omega }}_x$ at ${\mathrm{\Omega }}_y=0.2, p=1$, and $\mathrm{\Phi }=\pi /3$. In (c) perturbation-derived lines separating the $({\mathrm{\Omega }}_x,{\mathrm{\Omega }}_y)$ regions of real principal values, below the parabole, and of real or complex principal values, above the parabole. Parameters $p=1$ and different $\mathrm{\Phi }$ values.

Figure 5

Results for the $q_0$ and $q_s$ functions determining the ${\mathsf{f}}^{\left(2\right)}$ tensor components vs ${\mathrm{\Omega }}_x$. They are independent of the ${\mathrm{\Omega }}_y$ parameter.

Figure 6

The ${\mathrm{\Omega }}_L$ Larmor frequency (left axis) and the $P_{S_x}$ peak of the ${\mathrm{\Omega }}_L$ oscillating Faraday signal peak (right axis) vs the applied ${\omega }_{0x}$ magnetic field measured in the Cs dual-dressing experiment [21]. Parameters: ${\omega }_{0z}=0.1993\left(2\right), {\mathrm{\Omega }}_x=1.833\left(5\right), p=1, {\mathrm{\Omega }}_y=0.0118\left(1\right)$, and $\mathrm{\Phi }=\pi /2$.

Figure 7

In (a) numerical results for $a{\mathrm{\Omega }}_L$ in the 2D $({\mathrm{\Omega }}_x,{\mathrm{\Omega }}_y)$ plane at ${\omega }_{0z}=0.001, {\omega }_{0x}={\omega }_{0y}=0, p=3$, and $\mathrm{\Phi }=\pi /2$. The $\simeq 1000$ maximum value is reached at $({\mathrm{\Omega }}_x\simeq \pm 3.9,{\mathrm{\Omega }}_y\simeq \pm 2.4)$ and $({\mathrm{\Omega }}_x\simeq \pm 7.5,{\mathrm{\Omega }}_y\simeq \pm 3.6)$ values. In (b) from the perturbation treatment, $a{\mathrm{\Omega }}_L$ vs $\mathrm{\Phi }$ derived at $p=3, {\omega }_{0z}=0.001, {\mathrm{\Omega }}_x=3.9,{\mathrm{\Omega }}_y=1$, with maximum $\approx 43$.

Figure 8

Numerical results for $P{v}_l$, the largest principal value of the $\mathsf{g}$ tensor, vs ${\mathrm{\Omega }}_y$, for ${\mathrm{\Omega }}_x=3.5, p=1$, and $\mathrm{\Phi }=\pi /2$. Real $\mathrm{Re}(P{v}_l$) and complex $\mathrm{Im}(P{v}_l$) parts are plotted. The ${\mathrm{\Omega }}_y$ parameter is varied in the transition region from one real and two complex principal values to three real values. Notice that the largest absolute $\mathrm{Re}(P{v}_l$) value is negative.

Figure 9

On the left scale, compensation results for the $h_y,h_z$ effective transverse magnetic fields vs the strong ${\mathrm{\Omega }}_x$ dressing field in a ${\omega }_{0x}=0.1, {\omega }_{0y}={\omega }_{0z}=1\times {10}^{-4}$ configuration, with weak dressing fields ${\mathrm{\Omega }}_{z5}=0.6\times {10}^{-5}, {\mathrm{\Omega }}_{y6}=2.4\times {10}^{-5}$. The plot of the normalized $h_y/{\omega }_{0y},h_z/{\omega }_{0z}$ ratios evidences the magic compensations. On the right scale the $\partial h_y/\partial {\mathrm{\Omega }}_x,\partial h_z/\partial {\mathrm{\Omega }}_x$ derivatives vs ${\mathrm{\Omega }}_x$ in the second-order magic compensation search. Even if the null derivatives are not exactly coincident for the magic compensation, the sensitivity to the strong dressing fluctuations is reduced by three orders of magnitude. This appears in the inset plot for the normalized effective fields in the ${\mathrm{\Omega }}_d$ (2.87,2.92) range.