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.
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.
Quantum theory predicts that empty space is not truly empty. Even in the absence of any particles or radiation, in pure vacuum, virtual particles are constantly created and annihilated. In an electromagnetic field, the presence of virtual photons manifests itself as a small renormalization of the energy of a quantum system, known as the Lamb shift. We present an experimental observation of the Lamb shift in a solid-state system. The strong dispersive coupling of a superconducting electronic circuit acting as a quantum bit (qubit) to the vacuum field in a transmission-line resonator leads to measurable Lamb shifts of up to 1.4% of the qubit transition frequency. The qubit is also observed to couple more strongly to the vacuum field than to a single photon inside the cavity, an effect that is explained by taking into account the limited anharmonicity of the higher excited qubit states.
Abstract Vacuum fluctuations fundamentally affect an atom by inducing a finite excited state lifetime along with a Lamb shift of its transition frequency. Here we report the reverse effect: modification of vacuum modes by a single atom in circuit quantum electrodynamics. Our one-dimensional vacuum is a long section of a high wave impedance (comparable to resistance quantum) superconducting transmission line. It is directly wired to a transmon qubit circuit. Owing to the combination of high impedance and galvanic connection, the transmon’s spontaneous emission linewidth can greatly exceed the discrete transmission line modes spacing. This condition defines a previously unexplored superstrong coupling regime of quantum electrodynamics where many frequency-resolved vacuum modes hybridize with a single atom. We establish this regime by observing the spontaneous emission line of the transmon, revealed through the mode-by-mode measurement of the vacuum’s density of states. The linewidth as well as the atom-induced dispersive photon−photon interaction are accurately described by a physically transparent Caldeira−Leggett model, with the transmon’s quartic nonlinearity treated as a perturbation. Nonperturbative modification of vacuum, including inelastic scattering of single photons, can be enabled by the superstrong coupling regime upon replacing the transmon by more anharmonic qubits, with broad implications for simulating quantum impurity models of many-body physics.
We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ([Formula: see text] CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency [Formula: see text] for the coupling between the FQ and the modes in the CPWR, where the coupling grows as [Formula: see text] for [Formula: see text] and decreases as [Formula: see text] for [Formula: see text]. Here [Formula: see text] is the frequency of the nth mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above [Formula: see text] at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off.
According to quantum field theory, the vacuum is filled with virtual particles which can be turned into real ones under the influence of external perturbations. Phenomena of this kind are commonly referred to as quantum vacuum phenomena. Several quantum vacuum phenomena related to the peculiar nature of the quantum vacuum have been predicted, some of which, such as the Lamb shift and the Casimir effect, have been experimentally found. Other examples of quantum vacuum phenomena include the Unruh effect, the dynamical Casimir effect and the dynamical Lamb effect. The dynamical Lamb effect was first predicted by considering the situation of an atom passing through a cavity at relativistic speed. In this case, the excitation of the atom and the generation of photons is thought to arise because of the sudden change of Lamb shift of the atom. If multiple atoms concurrently experience the dynamical Lamb effect, they can become entangled. A theoretical framework to investigate the time-evolution of a system of N qubits, modeling atoms, coupled to a harmonic oscillator, modeling a cavity, with time-dependent coupling is developed. For the case of a closed system, three different analytical approaches are presented. First, a perturbative approach in the time-domain. This is the standard approach in the solution of Schrödinger equation. Second, a perturbative approach in the Laplace-domain. This can simplify the solution of Schrödinger equation with time-dependent Hamiltonians. However, both approaches require the interaction part of the Hamiltonian to be small compared to the unperturbed Hamiltonian. Third, the framework of Floquet theory is introduced. This can be used in the case of periodic Hamiltonians regardless of the interaction strength. For the case of an open system, the dynamics is investigated by numerically solving the Lindblad master equation. These approaches are then applied to the study of the time-evolution of the quantum entanglement between two and three qubits coupled to a harmonic oscillator. It is demonstrated that different measures of entanglement show a different level of detail of the latter and that time-dependent Greenberger-Horne-Zeilinger states can be created even in the presence of dissipation.The dynamical Lamb effect is predicted to arise in superconducting circuits when the coupling of a superconducting qubit with a resonator is periodically switched “on” and “off”nonadiabatically. We show that by using a superconducting circuit which allows one to switch between longitudinal and transverse coupling of a qubit to a resonator, it is possible to observe the dynamical Lamb effect. The switching between longitudinal and transverse coupling can be achieved by modulating the magnetic flux through the circuit loops. Superconducting circuits of this kind are routinely used as quantum information processors. These devices take advantage of quantum properties for mathematical computations, improving their computational capabilities compared to regular computers. We have implemented Shor’s factoring algorithm, in a simplified form, on one such device made available through the cloud by IBM. We study the results of the algorithm on the ibmqx5 superconducting chip, for the particular case of factoring N= 15, 21, and 35. Positive results are obtained for factoring15 and 21, although the device fails to compute the factors of 35 due to the errors introduced by the presence of noise.
The quantum properties of electromagnetic, mechanical or other harmonic oscillators can be revealed by investigating their strong coherent coupling to a single quantum two level system in an approach known as cavity quantum electrodynamics (QED). At temperatures much lower than the characteristic energy level spacing the observation of vacuum Rabi oscillations or mode splittings with one or a few quanta asserts the quantum nature of the oscillator. Here, we study how the classical response of a cavity QED system emerges from the quantum one when its thermal occupation-or effective temperature-is raised gradually over 5 orders of magnitude. In this way we explore in detail the continuous quantum-to-classical crossover and demonstrate how to extract effective cavity field temperatures from both spectroscopic and time-resolved vacuum Rabi measurements.
\abstract{The Lamb shift, one of the most fundamental interactions in atomic physics, arises from the interaction of H atoms with the electromagnetic fluctuations of the quantum vacuum. The energy shift has been computed in a variety of ways. The energy shift, as Feynman and Power demonstrated, equals the change in the vacuum energy in the volume containing the H atoms due to the change in the index of refraction arising from the presence of the H atoms. By using this result and a group theoretical calculation of the contribution to the Lamb shift from each frequency of the vacuum fluctuations, we can obtain an expression for the size of the region of vacuum energy for each frequency \texorpdfstring{ω}{ω} around the H atom due to the Lamb shift. The ground state atom is surrounded by a region of positive vacuum energy that extends well beyond the atom for low frequencies. This region can be described as a steady state cloud of virtual quanta. For energies \texorpdfstring{E=ℏω}{E=hbar omega} eV less than 1 eV, the radius of the positive energy region is approximately 14. 4/E Angstroms. For a vacuum fluctuation of wavelength \texorpdfstring{λ}{lambda} the radius is \texorpdfstring{(α/2π)λ}{(alpha/2pi) lambda}. Thus, for long wavelengths, the region has macroscopic dimensions. The energy-time Uncertainty Relation predicts a maximum possible radius that is larger than this by a factor of \texorpdfstring{1/4α}{1/(4 alpha)}.}
\keyword{Bethe; radiative shift; shift spectral density; spectral volume; vacuum fluctuations; vacuum field; Lamb shift; QED; energy field, renormalization, zero point fluctuations; hydrogen atom}.