Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition

Abstract

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...

Limitations of rotating-wave approximation in magnetic resonance: characterization and elimination of the Bloch–Siegert shift in magneto-optics

“…We fit those residual signals with small absorptive resonances with predefined positions (ω r /2 and 2ω r ) and a given width. The residual fields cause also a small mixing between two lock-in channels (in-phase and quadrature), which is taken into account by fitting the experimental data with a combination of the resonance shapes given by equations (11). In case of the rotating-field-induced resonance, residual fields are exclusively responsible for the mixing.…”

“…Experimental studies of the effect were performed with different electron-spin [5,6], as well as, nuclear-spin systems [7,8]. Recently, the effect was revisited for novel applications in quantum metrology [9] and quantum information processing [10][11][12]. In the latter case, the interest is motivated by the need to perform coherent control of a quantum system with possibly short electromagnetic pulses to reach high bandwidth of Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.…”

“…Consequently, for strong driving fields, when the shift becomes comparable with the eigenfrequency of the system (the Larmor frequency in our case), significant departures from RWA hinder coherent control of the qubits because of complex, nonharmonic Rabi oscillations. Recent work with superconducting circuits [10,11] and nitrogen-vacancy centers in diamonds [13,14] demonstrate that this difficulty can be successfully alleviated. The BSS appears important also for experiments with cold atoms in dressed and adiabatic potentials [15][16][17].…”

Probabilistic motional averaging

“…And when the power of driving is strong enough, in the strong-driving regime (ii), the spectral lines converge into a peak centered at the bare cavity and qubit transition frequencies. We note that the transition between these regimes can also be referred to as the driven quantum-to-classical transition [45][46][47] .…”

Quantum systems under frequency modulation

“…The rotating wave approximation generally becomes insufficient as the probe amplitude g P is increased. As a consequence, the resonance locations are shifted due to the enhanced contributions of the counterrotating (Bloch-Siegert shift [78]) and other multi-mode (generalized Bloch-Siegert [79,80]) terms. However, in the case of direct probing of modulated transition energy, one can typically assume that the probe amplitude is weak and, thus, the RWA is valid.…”