Theory of Multi-photon Processes for Applications in Quantum Control

Abstract

We present a general theoretical framework for evaluating multi-photon processes in periodically driven quantum systems, which have been identified as a versatile tool for engineering and controlling nontrivial interactions in various quantum technology platforms. To achieve the accuracy required for such applications, the resulting effective coupling rates, as well as any drive-induced frequency shifts, must be determined with very high precision. Here, we employ degenerate Floquet perturbation theory together with a diagrammatic representation of multi-photon processes to develop a systematic and automatable approach for evaluating the effective dynamics of driven quantum systems to arbitrary orders in the drive strength. As a specific example, we demonstrate the effectiveness of this framework by applying it to the study of multi-photon Rabi oscillations in a superconducting fluxonium qubit, finding excellent agreement between our theoretical predictions and exact numerical simulations, even for large driving amplitude.

Figures (9)

• Figure 1: A multi-photon driven quantum system. (a) Sketch of the circuit of a driven fluxonium qubit, which is used in our analysis as a specific example of a multi-photon driven quantum system. (b) As discussed in more detail in Sec. \ref{['sec:fluxonium']} below, the fluxonium qubit is described by a double-well potential $V_{\rm pot} (\varphi)$ for the quantized phase variable $\varphi$, which determines the eigenstates $|k\rangle$ and eigenenergies $E_k$ of this system. Driving the circuit by an external AC current $I(t)\sim \cos(\omega_d t)$ induces additional periodic modulations of the potential, which can be used to implement multi-photon transitions between two or several of the bare eigenstates whenever the frequency difference matches a multiple of the drive frequency $\omega_d$. (c) Identification of a set of quasi-resonant states as discussed in Sec. \ref{['subsec:QuasiResonantSubspace']}. The black lines represent the energies $E_k$, which are decomposed according to Eq. \ref{['eq:energy_decomposition']}. The green dashed lines indicate integer multiples of the drive frequency $\omega_d$, added to the energy of the reference state $|0\rangle$. In the example configuration depicted in this plot, the state $\ket{0}$ is coupled to state $\ket{2}$ via a $n_2=4$ photon process and with a residual detuning of $|\epsilon_2|\ll\omega_d$. The same level configuration then maps to an exact degeneracy between the eigenstates $\mathinner{|{0,0}\rangle\space\rangle}$ and $\mathinner{|{2,4}\rangle\space\rangle}$ of the extended Hamiltonian $\mathcal{H}_0$ in the Sambe representation used in Eq. \ref{['eq:FloquetStateOrder0']}.
• Figure 2: Degenerate Floquet perturbation theory. This figure summarizes the general theoretical framework used for the description of a multi-photon-driven quantum system. The starting point is the time-dependent Hamiltonian, for which we identify the relevant set of quasi-resonant states $|k\rangle \in D$ (indicated in orange and blue) with energies $\tilde{E}_k=\tilde{E}_0+n_k \omega_d$. This Hamiltonian is then mapped onto a time-independent Hamiltonian $\mathcal{H}$ in the extended Sambe space, where the states $|k\rangle \in D$ are mapped onto degenerate unperturbed states $\mathinner{|{k,n_k}\rangle\space\rangle}$ located in different photon-number sectors (in orange and blue). Finally, degenerate perturbation theory is used to find a transformation $W^\dag$ such that the dynamics of this degenerate subspace is decoupled from the other states and described by an effective Hamiltonian $\mathcal{H}_\textrm{eff}$. Once the evolution in this transformed subspace has been obtained, it can be mapped back onto the corresponding time-evolved states in the full Sambe space and, successively, onto the time-evolved state $|\psi(t)\rangle$ in the physical Hilbert space.
• Figure 3: Diagrammatic representation. (a) The general diagram representing a second-order process with initial and final states $\ket{k}$ and $\ket{l}$ (black dots), and involving the absorption of $n_l-n_k$ photons in two steps. The first step is an absorption of $p$ photons with a virtual transition to the state $\ket{a}$ (red dot), which is proportional to the matrix element $V_{p,ak}$. The second step is the transition from the virtual state $\ket{a}$ to the final state $\ket{l}$ with an absorption of the remaining $n_l-n_k-p$ photons and with matrix element $V_{n_l-n_k-p,la}$. The whole process is suppressed by an energy denominator (red arrow) that is the difference $\tilde{E}_k + p\omega_d-\tilde{E}_a$ between the virtual state energy and the energy of the system after the first step (blue dashed line). (b) Diagrams representing the second-order couplings given in Eq. \ref{['eq:Rabi_order2_XZ']} for the two-photon $XZ$-model. Both contributions describe the successive absorption of two individual photons, $p=1$, but with either $\ket{a}=\ket{0}$ or $\ket{a}=\ket{1}$ as the virtual intermediate state. (c) Diagrams representing the second-order Stark shift given in Eq. \ref{['eq:Stark_order2_XZ']} for the two-photon $XZ$-model. In total four processes contribute to the shift $\delta_0^{(2)}$, in which photons are virtually absorbed ($p=1$) or emitted ($p=-1$). Since the longitudinal coupling does not change the initial state, the two processes with $|a\rangle=|0\rangle$ have energy denominators with exactly opposite sign and cancel each other.
• Figure 4: Sub-harmonic Rabi model with monochromatic drive. (a,b) Diagrammatic representation of the lowest order coupling rate for $n_1=3$ and $n_1=5$, following the same notation as explained in Fig. \ref{['fig:diagr_order2_general']}. (c) Diagrammatic representation of one type of process for the fifth-order coupling rate at $n_1=3$. The encircled red dot is a resonant step that would naively induce a diverging process and must be assigned a nontrivial multiplicity coefficient.
• Figure 5: Dynamics of the $n_1=3$ sub-harmonic Rabi model. The green solid lines show the full numerical solutions for the population of state $\ket{1}$ for (a) $\Omega_x/\omega_{01}=0.05$ and (b,c) $\Omega_x/\omega_{01}=0.25$. The vertical dotted lines in red, blue and yellow indicate the half Rabi period $t_\pi=\pi/\Omega_R^{[r_{H}]}$ predicted at orders $r_{H}=3,5,7$, respectively. All other lines show the predicted state evolution obtained from Eq. \ref{['eq:evol_amplitude_allOrders']} from the computation of the effective Hamiltonian and the operator $W$ at the orders indicated in the legend.