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arXiv:1610.04665 · Physical Review A

Quantum entanglement for two qubits in a nonstationary cavity

Oleg L. Berman, Roman Ya. Kezerashvili, Yu. E. Lozovik

Published Nov 7, 2016• 10 citations
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Abstract

The quantum entanglement and the probability of the dynamical Lamb effect for two qubits caused by nonadiabatic fast change of the boundary conditions are studied. The conditional concurrence of the qubits for each fixed number of created photons in a nonstationary cavity is obtained as a measure of the dynamical quantum entanglement due to the dynamical Lamb effect. We discuss the physical realization of the dynamical Lamb effect, based on superconducting qubits.

DOI
10.1103/physreva.94.052308

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The quantum entanglement and the probability of the dynamical Lamb effect for two qubits caused by nonadiabatic fast change of the boundary conditions are studied. The conditional concurrence of the qubits for each fixed number of created photons in a nonstationary cavity is obtained as a measure of the dynamical quantum entanglement due to the dynamical Lamb effect. We discuss the physical realization of the dynamical Lamb effect, based on superconducting qubits.
We investigate the tunable quantum entanglement and the probabilities of excitations in a system of three qubits in a nonstationary cavity due to the dynamical Lamb effect, caused by nonadiabatic fast change of the boundary conditions of the cavity. The transition amplitudes and the probabilities of excitation of qubits due to the dynamical Lamb effect have been evaluated. The conditional concurrence and the conditional residual tangle for each fixed amount of created photons are introduced and calculated as measures of the pairwise or three-way dynamical quantum entanglement of the qubits. We also give a prescription on how to increase the values of those quantities by controlling the frequency of the cavity photons. A physical realization of the system with three superconducting qubits, coupled to a coplanar waveguide entangled due to the nonadiabatic fast change of boundary conditions of the cavity is proposed.
It has been shown that superconducting qubit systems, having high tunability, can be used as a platform for the experimental study of various effects of nonstationary quantum electrodynamics in a cavity. In particular, the dynamic Lamb effect can be implemented owing to a nonadiabatic change in the effective coupling between the subsystem of qubits and a cavity. This effect is manifested in the excitation of a qubit (atom) at the change in the Lamb shift of its levels. It is remarkable that the effect of energy dissipation in such parametrically excited systems can be very nontrivial: dissipation in one of the subsystems of the hybrid system can enhance quantum effects in the other subsystem. This refers to various phenomena such as parametric qubit excitation, generation of photons from vacuum, and creation and confinement of finite entanglement of qubits.
Superconducting circuits provide a new platform for study of nonstationary cavity QED phenomena. An example of such a phenomenon is the dynamical Lamb effect, which is the parametric excitation of an atom due to nonadiabatic modulation of its Lamb shift. This effect was initially introduced for a natural atom in a varying cavity, while we suggest its realization in a superconducting qubit-cavity system with dynamically tunable coupling. In the present paper, we study the interplay between the dynamical Lamb effect and the energy dissipation, which is unavoidable in realistic systems. We find that despite naive expectations, this interplay can lead to unexpected dynamical regimes. One of the most striking results is that photon generation from vacuum can be strongly enhanced due to qubit relaxation, which opens another channel for such a process. We also show that dissipation in the cavity can increase the qubit excited-state population. Our results can be used for experimental observation and investigation of the dynamical Lamb effect and accompanying quantum effects.
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.
A theoretical framework to investigate the time evolution of the quantum entanglement due to the dynamical Lamb effect between two and three superconducting qubits coupled to a coplanar waveguide in the presence of different sources of dissipation is developed. Guidelines on how to proceed in the NNN-qubit case are also given. We quantitatively analyze the case of single switching of the coupling and absence of dissipation within a perturbative approach and show that it is a good approximation to the case of periodic switching of the coupling for high frequencies of switching. The same systems are analyzed for the general case of periodic switching of the coupling at any frequency and in the presence of dissipation via numerical calculations. Different measures of entanglement compatible with mixed states are adopted. It is demonstrated that the different measures show a different level of detail of the latter. The concurrence and the negativity are obtained in the two-qubit case; the three-$\ensuremath{\pi}$ and the negativity are obtained in the three-qubit case. It is shown that time-dependent Greenberger-Horne-Zeilinger states can be created even in the presence of dissipation. To maximize the quantum entanglement between the qubits, the effects of tuning several parameters of the system are investigated.
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. By solving the Schr\"odinger equation for a qubit coupled to a resonator, we calculate the time evolution of the number of excitations in the qubit and the resonator due to the dynamical Lamb effect. The number of excitations created in the system is maximum when the coupling is periodically switched between longitudinal and transverse using a square-wave or sinusoidal modulation of the magnetic flux with frequency equal to the sum of the average qubit and photon transition frequencies.