Nature of the Lamb shift in weakly anharmonic atoms: From normal-mode splitting to quantum fluctuations

When a two-level system (TLS) is coupled to an electromagnetic resonator, its transition frequency changes in response to the quantum vacuum fluctuations of the electromagnetic field, a phenomenon known as the Lamb shift. Remarkably, by replacing the TLS by a harmonic oscillator, normal-mode splitting leads to a quantitatively similar shift, without taking quantum fluctuations into account. In a weakly anharmonic system, lying in between the harmonic oscillator and a TLS, the origins of such shifts can be unclear. An example of this is the dispersive shift of a transmon qubit in circuit quantum electrodynamics (QED). Although often referred to as a Lamb shift, the dispersive shift observed in spectroscopy in circuit QED could contain a significant contribution from normal-mode splitting that is not driven by quantum fluctuations, raising the question: how much of this shift is quantum in origin? Here we treat normal-mode splitting separately from shifts induced by quantum vacuum fluctuations in the Hamiltonian of a weakly anharmonic system, providing a framework for understanding the extent to which observed frequency shifts can be attributed to quantum fluctuations.

Figure 1

The origin of different energy shifts in a weakly anharmonic atom. (a) Replacing the linear inductance of an $LC$ oscillator with a Josephson junction (JJ) results in a weakly anharmonic artificial atom. To first order, the energy level $n$ is shifted proportionally to $\langle n|{\widehat{\varphi }}^4|n\rangle$, where $|n\rangle$ is a Fock state of the harmonic system. (b) Two coupled harmonic oscillators undergo normal-mode splitting, resulting in a frequency shift ${\delta }_{\text{NM}}$. The flux traversing one of the inductances $\varphi$ is then composed of the flux from both normal-mode oscillations $\varphi ={\varphi }_a+{\varphi }_r$. Replacing an inductor with a JJ leads to the same shift as in the isolated atom ${\chi }_a\langle {\widehat{\varphi }}_a^4\rangle$, but also to a shift due to quantum fluctuations of the coupled oscillator ${\chi }_{ar}\langle {\widehat{\varphi }}_a^2\rangle \langle {\widehat{\varphi }}_r^2\rangle$.

Figure 2

Fraction of the atomic energy shift due to quantum vacuum fluctuations. (a) Dressed frequency of the ground-to-first-excited state transitions of the harmonic oscillator (black) and atom (blue) as a function of detuning $\mathrm{\Delta }={\omega }_r-{\omega }_a$. Bare frequencies ($g=0$) are shown as dashed lines, We fixed $\lambda /{\omega }_a=0.01$ and $g/{\omega }_a=0.02$. (b) Total frequency shift $\delta {\omega }_a$ of the atom, decomposed into its two main components: normal-mode splitting ${\delta }_{\text{NM}}$ and a shift resulting from vacuum fluctuations ${\chi }_{ar}$. Coupling also changes the anharmonicity ${\chi }_a$, this results in a small shift absorbed here in ${\delta }_{\text{NM}}$. (c) The vacuum-fluctuations-induced shift ${\chi }_{ar}$ as a fraction of the total frequency shift of the atom $\delta {\omega }_a$ for increasing anharmonicity $\lambda$ and fixed detuning $\mathrm{\Delta }={\omega }_a/4$. For a TLS, all of the energy shift arises from quantum fluctuations, ${\chi }_{ar}/\delta {\omega }_a=1$. In all panels, the dotted lines are computed from Eqs. (9), full lines correspond to a numerical diagonalization of Eq. (4). In (c), ${\chi }_{ar}$ is computed from numerics as half the shift resulting from adding a photon in the oscillator.

Figure 3

Vacuum-fluctuations-induced shift ${\chi }_{ar}$ beyond the RWA fixing $\lambda /{\omega }_a=0.01$ and $g/{\omega }_a=0.02$. Numerical calculation (full red line), are compared to the analytical expression of Eq. (9) (dashed blue line) and Eq. (6) (dashed green). Resonances invalidating our approximations are denoted by red bars.