We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for strong drive (classical phase) it presents an oscillatory behaviour as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.The Rabi Hamiltonian -describing a two-level system coupled to a cavity (resonator) mode -is a paradigmatic model in quantum physics. In the rotating-wave approximation (RWA) it leads to the well-known JaynesCummings (JC) model. In the dispersive limit this model predicts the appearance of ac-Stark shifts in the energy levels of both the qubit and the cavity. The inclusion of counter-rotating terms produces an additional displacement of the energy levels. This Bloch-Siegert (BS) shift [1] is usually very small on standard experimental platforms since it depends on the ratio between the coupling and the sum of the Larmor and cavity frequencies. Recently however, significant experimental effort has been put into increasing the coupling to values comparable with the Larmor frequency [2-4], most notable in semiconductor dots [5,6] and superconducting circuits [7][8][9][10][11][12]. Other approaches for observing the Bloch-Siegert shift include the detailed analysis of two-level Landau-Zener spectra of Rydberg atoms [13,14] and Cooper-pair boxes [15][16][17], as well as the simulation of the Rabi model in rotating frames [18,19].In this work we take a different route. We recognize that the counter-rotating terms do not conserve the excitation number. Therefore, the natural framework for their experimental demonstration is that of driven-dissipative systems [20,21]. Guided by this intuition, we realize a setup consisting of a transmon [22] dispersively coupled to a cavity, where the cavity is driven at a fixed off-resonance microwave tone, while at the same time the spectrum is scanned by a comparatively weaker probe field, see Fig. 1. At low driving powers, we observe the expected vacuum ac-Stark shift [23][24][25][26][27]. This is followed by a transition regime dominated by nonlinear effects as the power is increased. For the Jaynes-Cummings model, such transition has been predicted and studied in the resonant qubit-cavity case [28][29][30][31]. In the dispersive limit, the transition region is no longer abrupt, but it is We drive the cavity with a detuned drive frequency ω d < ω c . The spectrum of the system is monitored by a weak probe with frequency ω p , which is swept within the window [4.36, 4.39] GHz. (d) In the T 1 measu...
We study numerically the onset of higher-level excitations and resonance frequency shifts in the generalized multilevel Rabi model with dispersive coupling under strong driving. The response to a weak probe is calculated using the Floquet method, which allows us to calculate the probe spectrum and extract the resonance frequency. We test our predictions using a superconducting circuit consisting of transmon coupled capacitively to a coplanar waveguide resonator. This system is monitored by a weak probe field, and at the same time driven at various powers by a stronger microwave tone. We show that the transition from the quantum to the classical regime is accompanied by a rapid increase of the transmon occupation and, consequently that the qubit approximation is valid only in the extreme quantum limit. Keywords Quantum Rabi model · Multilevel effects · Floquet method 1 IntroductionThe interaction between light and matter has been traditionally studied in terms of the Jaynes-Cummings (JC) model [1]. In the quantum optical nomen-I. Pietikäinen
We investigate the driven quantum phase transition between the oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. We refer to this as the Josephson-Rabi model. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperiodicity originating in the discrete energy spectrum of the bound states. We calculate the photon number in the resonator and show that its dependence on the drive power is nonlinear. In addition, the resulting multi-photon blockade phenomenon is sensitive to the truncation of the number of states in the transmon, which limits the applicability of the standard Jaynes-Cummings model as an approximation for the pendulum-oscillator system. We calculate the nth order correlation functions of the blockaded microwave photons and observe the differences between the rotating-wave approximation and the full multilevel Josephson-Rabi Hamiltonian with the counterrotating terms included. Finally, we compare two different approaches to dissipation, namely the Floquet-Born-Markov and the Lindblad formalisms. arXiv:1901.05655v2 [cond-mat.mes-hall]
Cavity optomechanics and electromechanics form an established field of research investigating the interactions between electromagnetic fields and the motion of quantum mechanical resonators. In many applications, linearised form of the interaction is used, which allows for the system dynamics to be fully described using a Lyapunov equation for the covariance matrix of the Wigner function. This approach, however, is problematic in situations where the Hamiltonian becomes time dependent as is the case for systems driven at multiple frequencies simultaneously. This scenario is highly relevant as it leads to dissipative preparation of mechanical states or backaction-evading measurements of mechanical motion. The time-dependent dynamics can be solved with Floquet techniques whose application is, nevertheless, not straightforward. Here, we describe a general method for combining the Lyapunov approach with Floquet techniques that enables us to transform the initial timedependent problem into a time-independent one, albeit in a larger Hilbert space. We show how the lengthy process of applying the Floquet formalism to the original equations of motion and deriving a Lyapunov equation from their time-independent form can be simplified with the use of properly defined Fourier components of the drift matrix of the original time-dependent system. We then use our formalism to comprehensively analyse dissipative generation of mechanical squeezing beyond the rotating wave approximation. Our method is applicable to various problems with multitone driving schemes in cavity optomechanics, electromechanics, and related disciplines. CONTENTS
The Mach-Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states-such as coherent states or single photons-can reach the standard quantum limit of phase estimation while more complicated states can be used to reach Heisenberg scaling; the latter, however, require complex states at the input of the interferometer which are difficult to prepare. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient.Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on ancilla qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We perform detailed analysis of ancilla errors, showing that biasing the ancilla towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity.
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