Coherent multiple-period states of periodically modulated qubits

We consider multiple-period states in systems of periodically modulated qubits. In such states the discrete time-translation symmetry imposed by the modulation is broken. We explicitly show how multiple-period states emerge in the simplest quantum system, a single qubit subjected to a pulsed resonant modulation and/or a pulsed modulation of the transition frequency. We also show that a qubit chain with the qubit coupling modulated at twice the qubit frequency has symmetry that allows mapping it on the Kitaev chain and thus provides an example of a topologically nontrivial Floquet system. An explicit solution for a two-qubit system illustrates the effect of resonant period doubling for coupled qubits, whereas in a long chain period doubling is topologically protected.

Figure 1

The quasienergy ${\varepsilon }_1$ of a qubit in the rotating frame. The qubit is driven at its eigenfrequency, ${\omega }_F={\omega }_0$. The driving amplitude is pulsed with period $T\gg 2\pi /{\omega }_F$, the dimensionless area of a pulse is $F_1$. The blue, magenta, green, and red curves (the curves from bottom to top, for $F_1=2\pi$) refer to the Rabi frequency $\mathrm{\Omega }=2T^{-1},T^{-1},0.5T^{-1}$, and $0.05T^{-1}$.

Figure 2

The rotating-frame energies of a two-qubit system with the coupling periodically modulated at frequency ${\omega }_F$ close to twice the qubit transition frequency ${\omega }_0$, Eq. (14). The dashed and solid curves refer to the states 1, 2 and 3, 4, respectively. The frequency detuning is $\frac{1}{2}{\omega }_F-{\omega }_0\equiv \mu =0.6J$. The green circles indicate where the rotating-frame quasienergies in the laboratory frame differ by ${\omega }_F/2$. There is no anticrossing between the corresponding quasienergy levels.

Figure 3

The rotating-frame energies of the first (a) and second (b) lowest excited states of the qubit chain with a resonantly modulated coupling. The energies are given by the eigenvalues of the Hamiltonian (15) for a 16-site chain of fermions. The chemical potential of the fermions $\mu$ is given by the frequency detuning $({\omega }_F-2{\omega }_0)/2$; see Eq. (11). The blue, magenta, and green curves (the lowest to the highest curves) correspond to the scaled modulation amplitude $F/J=0.3,1.2$, and 3.