Amplification of optical or microwave fields is often achieved by strongly driving a medium to induce population inversion such that a weak probe can be amplified through stimulated emission. Here we strongly couple a superconducting qubit, an artificial atom, to the field in a semi-infinite waveguide. When driving the qubit strongly on resonance such that a Mollow triplet appears, we observe a 7% amplitude gain for a weak probe at frequencies in between the triplet. This amplification is not due to population inversion, neither in the bare qubit basis nor in the dressed-state basis, but instead results from a four-photon process that converts energy from the strong drive to the weak probe. We find excellent agreement between the experimental results and numerical simulations without any free fitting parameters. Since our device consists of a single two-level artificial atom, the simplest possible quantum system, it can be viewed as the most fundamental version of a four-wave-mixing parametric amplifier.

Virtual photons can mediate interaction between atoms, resulting in an energy shift known as a collective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscured by radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits in a transmission line terminated by a mirror, which suppresses decay. We measure a collective Lamb shift reaching 0.8% of the qubit transition frequency and exceeding the transition linewidth. We also show that the qubits can interact via the transmission line even if one of them does not decay into it.Introduction. In 1947, when attempting to pinpoint the fine structure of the hydrogen atom, Lamb and Retherford [1] discovered a small energy difference between the levels 2S 1/2 and 2P 1/2 , which were thought to be degenerate according to Dirac's theory of electrons. This energy difference between the two levels can be understood when vacuum fluctuations are included in the picture, as was verified later by self-energy calculations in the framework of quantum field theory [2][3][4]. Briefly put, a hydrogen atom will emit photons which are instantaneously reabsorbed; while these "virtual" photons are not detectable by themselves, they leave their traces in the Lamb shift.The hydrogen atoms that Lamb and Retherford used for their experiment were obtained from molecular hydrogen through tungsten catalyzation. Since the conversion rate for this process was very low, the 2S 1/2 level was only populated in a few atoms. Hence, the observable effects of virtual photon processes were limited to selfinteraction; exchanges of virtual photons between atoms could not be detected. However, it was later realized that atom-atom interaction mediated by virtual photons also gives rise to an energy shift, referred to as a collective, or cooperative, Lamb shift [5][6][7][8][9]. The atom-atom interaction also underpins the collective decay known as Dicke superradiance [10,11].There are several obstacles impeding the experimental observation of the collective Lamb shift. The shift can be enhanced by using many atoms, but, if these atoms are too close together, direct atom-atom interactions (not via virtual photons) can obscure the effect. Furthermore, the interaction giving rise to the collective Lamb shift is relatively weak in three dimensions, and the shift can also be hidden by the radiative linewidth (e.g., due to the collective decay). Despite these obstacles, there have been a few experimental demonstrations of collective Lamb shifts: in xenon gas [12], iron nuclei [13], rubidium vapor [14], strontium ions [15], cold rubidium atoms [16], and potassium vapor [17]. Mostly, these experiments used developments in atomic trapping and cooling [18] that have enabled higher densities of atomic ensembles, leading to a strong coupling between atomic condensates and cavity fields [19,20]. An improved theoretical understanding [21-23] of collective Dicke states also aided some of the experiments.With the single exception of Ref.[15], these previous expe...

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Loading quantum information deterministically onto a quantum node is an important step toward a quantum network. Here, we demonstrate that coherent-state microwave photons with an optimal temporal waveform can be efficiently loaded onto a single superconducting artificial atom in a semi-infinite one-dimensional (1D) transmission-line waveguide. Using a weak coherent state (the number of photons (N) contained in the pulse ≪1) with an exponentially rising waveform, whose time constant matches the decoherence time of the artificial atom, we demonstrate a loading efficiency of 94.2% ± 0.7% from 1D semifree space to the artificial atom. The high loading efficiency is due to time-reversal symmetry: the overlap between the incoming wave and the time-reversed emitted wave is up to 97.1% ± 0.4%. Our results open up promising applications in realizing quantum networks based on waveguide quantum electrodynamics.