Observation of the Phononic Lamb Shift with a Synthetic Vacuum

In contrast to classical empty space, the quantum vacuum fundamentally alters the properties of embedded particles. This paradigm shift allows one to explain the discovery of the celebrated Lamb shift in the spectrum of the hydrogen atom. Here, we engineer a synthetic vacuum, building on the unique properties of ultracold atomic gas mixtures, offering the ability to switch between empty space and quantum vacuum. Using high-precision spectroscopy, we observe the phononic Lamb shift, an intriguing many-body effect originally conjectured in the context of solid-state physics. We find good agreement with theoretical predictions based on the Fröhlich model. Our observations establish this experimental platform as a new tool for precision benchmarking of open theoretical challenges, especially in the regime of strong coupling between the particles and the quantum vacuum.

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According to quantum field theory, a vacuum is not empty but is instead filled with ephemeral particles that pop into life and die an immediate death. Therefore, even an electron moving through “empty space” is surrounded by a constantly changing cloud of short-lived particles; it is impossible to observe the bare electron itself. This situation has several observable consequences, such as a change in the mass of the electron. Because the vacuum cannot be switched off, it is impossible to observe the mass of the bare electron directly. However, an observable effect of the quantum vacuum persists for an electron bound to a proton in the hydrogen atom, which leads to a tiny shift in the electronic energy levels (i.e., the Lamb shift). Here, we synthesize a model system for such a quantum vacuum, which affords us the ability to switch on and off the effect that it has on the immersed particles.

We base our system on the high flexibility and control of properties of ultracold atomic gas mixtures. In particular, we emulate the quantum vacuum by a large cloud of particles (roughly 1,000,000 sodium atoms) that are all condensed into the same quantum state. Particles of a second species (a few thousand lithium atoms) are then immersed into this cloud and tightly confined in an optical trap such that their energy spectra become quantized. The entire experimental setup is held at 0.000000350 K. Using high-precision spectroscopy and switching the quantum vacuum on and off, we directly observe the appearance of the phononic Lamb shift, an analog to the tiny energy shift known from the hydrogen atom, in the energy spectrum of the immersed particles. We find that our observations are consistent with theoretical predictions.

We expect that our findings will pave the way for future studies of quantum vacuum fluctuations, including investigations of their dimensionality dependence, that will further our understanding of nonequilibrium dynamical situations.

Figure 1

Experimental setup. (a) Particles are trapped in a harmonic potential with trapping frequency ${\omega }_0$. The coupling of the particles to the quantum vacuum causes energy shifts of the particles’ motional states. For a free particle, it leads to a mass increase, which would manifest itself to a smaller energy difference between the harmonic trap levels, as shown in the second column of the inset. For a trapped particle, the effective mass approximation breaks down and the energy shift has to be calculated from the self-energy in the confined geometry. The difference between the complete calculation and the effective mass approximation is termed phononic Lamb shift, and it can be so substantial that the interaction with the vacuum actually leads to an increased energy difference compared to the bare case, as shown in the third column of the inset. (b) Scheme of the experimental setup. The two atomic species are trapped in a single dipole trap (red beams). The lithium atoms, which act as the particles in this setup, experience additionally a strong confinement by a deep small-angle optical lattice (green beams), whose minima can be described by a harmonic potential with trapping frequency ${\omega }_0$. Inset: This setup results in a multitude of independent lithium clouds (blue) embedded in a large sodium Bose-Einstein condensate (yellow), which acts as the quantum vacuum.

Figure 2

Motional Ramsey spectroscopy. (a) The sequence of total time $t$ consists of a $\pi /2$ pulse creating an equal superposition between the ground and the excited state and a subsequent free evolution during which a phase $\varphi$ is accumulated. A second $\pi /2$ pulse maps the phase shift on the population $\eta$. (b) Shaking the optical lattice couples the motional states of the lithium particles and allows the preparation of a superposition of the two lowest motional states. (c) Changing the evolution time $t$ results in a Ramsey fringe in the population of the excited state, which is read out via band mapping on the first and second Brillouin zone. Typical optical density (OD) profiles of the resulting absorption images are depicted with a field of view of $1176\mu \mathrm{m}$. The dashed lines mark the limit of the Brillouin zones. (d) An exemplary interference fringe of ${}^6\mathrm{Li}$ after an evolution time of 1.1 ms is shown. The green and the blue dot correspond to the profiles that are presented above. In the case of an evolution with the BEC present, the amplitude is reduced due to dephasing and the fringe is shifted, showing an increased energy difference. The shaded area corresponds to the $1\sigma$ confidence levels of the phase estimation.

Figure 3

Observation of the energy shift. The residual modulation of the density of the background BEC leads to an additional energy shift of the particle energy level. This can be identified by altering the ratio $V_V/V_P$ through a change of the detuning of the optical lattice while keeping the potential depth for the particle constant. A clear linear dependence on $V_V/V_P$ is observed. Interpolating the measurement data to $V_V=0$ allows for extraction of the energy shift. The positive offset is a direct signature of the phononic Lamb shift, since it contradicts the negative energy shift expected from the increased effective mass. The shaded areas show the theoretically predicted shift in the Fröhlich scenario, taking into account the uncertainties of the experimental parameters. We find perfect agreement without free parameters for fermionic particles (blue). In the case of a noncondensed bosonic particle cloud (green squares), no shift is observable in agreement with the theoretical expectation. For a condensed bosonic particle cloud (red diamonds), we observe a significant change of the energy shift. The theoretical expectation, not taking into account thermal excitations ($T=0$), is depicted as the red shaded area.

Figure 4

Population dependence of the energy shift. The energy shift depends on the relative state population $\eta$ as ground and excited state experience different energy shifts. This can be observed in a Ramsey sequence with an unequal superposition during free time evolution, keeping the other relevant parameters constant. Experiments with bosonic particles are shown by red diamonds. From these data we extract a critical ${\eta }_{0,\mathrm{expt}}=0.81(7)$, in good agreement with theoretical predictions of ${\eta }_{0,\text{theor}}=0.85(2)$. For fermionic particles (blue circles) we do not observe any dependence on $\eta$, as expected from our calculations.