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arXiv:2006.16769 · New Journal of Physics

Nonclassicality of open circuit QED systems in the deep-strong coupling regime

Tomohiro Shitara, Motoaki Bamba, Fumiki Yoshihara, Tomoko Fuse, Sahel Ashhab, Kouichi Semba, Kazuki Koshino

Published Sep 20, 2021• 9 citations
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Abstract

We investigate theoretically how the ground state of a qubit–resonator (Q–R) system in the deep-strong coupling (DSC) regime is affected by the coupling to an environment. We employ as a variational ansatz for the ground state of the Q–R–environment system a superposition of coherent states displaced in qubit-state-dependent directions. We show that the reduced density matrix of the Q–R system strongly depends on how the system is coupled to the environment, i.e. capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator–waveguide (R–W) coupling. In contrast, when the two couplings are of the same type (for instance, both are inductive), by increasing the R–W coupling strength, the average number of virtual photons increases and the quantum superposition realized in the Q–R entangled ground state is partially degraded. Since the superposition becomes more fragile with increasing the Q–R coupling, there exists an optimal coupling strength to maximize the nonclassicality of the Q–R system.

DOI
10.1088/1367-2630/ac2850

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We investigate theoretically how the ground state of a qubit–resonator (Q–R) system in the deep-strong coupling (DSC) regime is affected by the coupling to an environment. We employ as a variational ansatz for the ground state of the Q–R–environment system a superposition of coherent states displaced in qubit-state-dependent directions. We show that the reduced density matrix of the Q–R system strongly depends on how the system is coupled to the environment, i.e. capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator–waveguide (R–W) coupling. In contrast, when the two couplings are of the same type (for instance, both are inductive), by increasing the R–W coupling strength, the average number of virtual photons increases and the quantum superposition realized in the Q–R entangled ground state is partially degraded. Since the superposition becomes more fragile with increasing the Q–R coupling, there exists an optimal coupling strength to maximize the nonclassicality of the Q–R system.
We investigate theoretically how the ground state of a qubit-resonator system in the deep-strong coupling (DSC) regime is affected by the coupling to an environment. We employ a superposition of coherent states displaced in the qubit-state-dependent directions as a variational ansatz for the ground state of the qubit-resonator-environment system. We show that the reduced density matrix of the qubit-resonator system strongly depends on types of the resonator-waveguide and resonator-qubit coupling, i.e., capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator-waveguide coupling. In contrast, when the types of two couplings are the same (for instance, both are inductive), by increasing the resonator-waveguide coupling strength, the average number of virtual photons increases and the quantum superposition realized in the qubit-resonator entangled ground state is partially degraded. Since the superposition becomes more fragile when the qubit-resonator coupling strength gets large, there exists an optimal strength of the qubit-resonator coupling to maximize the nonclassicality of the qubit-resonator system.
Interaction of the vacuum field of an optical mode with an electronic two-level system can drive non-perturbative light-matter dynamics, resulting in an exotic quantum ground state containing a sizeable population of virtual excitations. In the deep-strong coupling (DSC) regime, where the Rabi frequency,ΩR\Omega_{\mathrm{R}}ΩR​, exceeds the eigen-frequencies of the coupled systems, this extreme ground state can result in spectacular effects of cavity-QED, such as the expected release of the virtual photon ground state population as vacuum radiation by non-adiabatic modulation of the coupling mechanism [1], [2]. However, for resonant coupling of two single modes [3], the finite oscillator strength of resonant electronic excitation sets a natural limit to the achievable coupling strength, hindering further exploration of this exciting field.
Cavity quantum electrodynamics (QED) investigates the interaction between resonator-confined radiation and natural atoms. Similar phenomena can be explored using superconducting artificial atoms coupled to microwave resonators. Here, we examine incoherent and coherent spectra in a circuit QED system consisting of a flux qubit coupled to an LC resonator, considering different coupling configurations to the output transmission line. Although such systems can be effectively described by the quantum Rabi model, we demonstrate that the choice of the coupling between the resonator and the input-output channel can significantly affect the emission properties, particularly in the ultrastrong coupling regime. Specifically, we show that when the resonator is capacitively coupled with the input-output transmission line, the spectral features at zero flux offset can be mapped onto those of the cavity QED model. However, this correspondence breaks down for the mutual inductive coupling case. Extending our analysis to the parity-symmetry broken regime of the flux qubit, we further highlight how capacitive and inductive coupling mechanisms lead to distinct spectral features, emphasizing the necessity of a careful input-output treatment for an accurate description of the emission processes.
We analyze a coupling scheme for qubits in different cavities of circuit-QED architecture. In contrast to the usual scheme where the cavities are coupled by an interface capacitance we employ a bridge qubit connecting cavities to mediate two-qubit coupling. This active-coupling scheme makes it possible to switch on or off and adjust the strength of qubit-qubit coupling, which is essential for scalability of quantum circuits. By transforming the Hamiltonian we obtain an exact expression of two-qubit coupling in the rotating-wave approximation. For the general case of nnn qubits the Hamiltonian can produce the WWW state as an eigenstate of the system. We calculate the decay rate of the coupled qubit-resonator system to find that it is viable in real experiments.
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.
Under the non-rotating wave approximation, the quantum evolution of entanglement property of a two-qubit and oscillator coupling system is accurately investigated by the method of coherent-state orthogonalization expansion. The property of the ground state for qubit-oscillator system and the difference between qubit-oscillator entanglement and qubit-qubit entanglement when resonant vibration occurs are discussed. The calculation results show that when the external field is not taken into consideration, the qubit-qubit entanglement reduces from 1 to 0 rapidly with the increase of coupling strength, indicating the strong sensitivity of the entanglement to the coupling strength. On the contrary, with the increase of the coupling the qubit-oscillator entanglement rises from 0, but does not reach the maximum value 2. At the beginning, when the two qubits do not entangle, the vacuum field does not lead to the entanglement in weak coupling. However, the strong coupling can induce the sudden appearance of the entanglement.