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arXiv:quant-ph/0212121

Fundamental limitation on qubit operations due to the Bloch-Siegert Oscillation

M. S. Shahriar, Prabhakar Pradhan

Published Dec 20, 2002• 5 citations
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Abstract

We show that if the Rabi frequency is comparable to the Bohr frequency so that the rotating wave approximation is inappropriate, an extra oscillation is present with the Rabi oscillation. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-frequency transitions. We present a scheme for observing this effect in an atomic beam.

DOI
10.48550/arxiv.quant-ph/0212121

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We show that if the Rabi frequency is comparable to the Bohr frequency so that the rotating wave approximation is inappropriate, an extra oscillation is present with the Rabi oscillation. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-frequency transitions. We present a scheme for observing this effect in an atomic beam.
The rotation of a quantum bit (qubit) is an important step in quantum computation. The rotation is generally performed using a Rabi oscillation. In a direct two-level qubit system, if the Rabi frequency is comparable to its resonance frequency, the rotating wave approximation is not valid, and the Rabi oscillation is accompanied by the so-called Bloch–Siegert oscillation (BSO) that occurs at twice the frequency of the driving field. One implication of the BSO is that for a given interaction time and Rabi frequency, the degree of rotation experienced by the qubit depends explicitly on the initial phase of the driving field. If this effect is not controlled, it leads to an apparent fluctuation in the rotation of the qubit. Here we show that when an off-resonant lambda system is used to realize a two-level qubit, the BSO is inherently negligible, thus eliminating this source of potential error.
In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance frequency, the rotating wave approximation is not appropriate. In this case, the Rabi oscillation is accompanied by another oscillation at twice the frequency of the driving field, the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the driving field. This oscillation may restrict the precise rotation of a qubit made of direct two-level system. Here, we show that in case of an effectively two-level lambda system, the BSO is inherently negligible, implying a greater precision for rotation of a qubit made of such a lambda system when compared to a direct two-level qubit in a strong driving field.
The need to maximize the number of operations of a quantum bit within its decoherence time may require the ratio of Rabi frequency to transition frequency to be large enough to invalidate the rotating-wave approximation. The state of the quantum bit under any initial condition then depends explicitly on the phase of the driving field, resulting in driver-phase-correlated fluctuations and a violation of the rule that the degree of excitation depends only on the pulse area. This is due to the interference of the excitations caused by the corotating and counterrotating fields, and is a significant source of error, correctable only by controlling the driver phase. We present a scheme for observing this effect under currently realizable parameters.
We present investigations of radio-frequency (RF) resonances observed in an optically pumped rubidium vapor. By measuring the systematic shifts (the Bloch–Siegert shifts) of RF resonances in low magnetic fields, we demonstrate limitations of the rotating-wave approximation in the case of angular momentum F ≥ 1. The resonance shifts and deformations are characterized in a wide range of parameters and it is shown that the observed behavior is far more complex than in a standard two-level system. It is also demonstrated that the shifts can be controllably turned on or off by switching between the oscillating and rotating magnetic field. Experimental results are supported with numerical calculations, reproducing all features of the observed signals. Besides fundamental aspect of the research, application of rotating magnetic field helps to suppress/evaluate spectroscopic-measurement and precise-metrology systematic errors. The reported study has also important implications for quantum metrology and information processing beyond RWA and standard two-state qubit dynamics.
Abstract We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to the destructive interference of multiple photon processes within the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch–Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch–Siegert oscillation. In this case, triplets in the Fourier spectra of the coherent response are transformed into doublets with the splitting between the lines equal to twice the Bloch–Siegert shift. These unusual properties are demonstrated in the conditions of experiments with a nitrogen vacancy center in the diamond.
The dynamics of a two-level spin system dressed by bichromatic radiation is studied under the conditions of double resonance when the frequency of one (microwave) field is equal to the Larmor frequency of the spin system and the frequency of the other (radio-frequency) field ω_{rf} is close to the Rabi frequency ω_{1} in a micro-wave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency εare accompanied by higher-frequency oscillations at frequencies nω_{rf} and nω_{rf}\pm ε, where n = 1, 2, .... The most intense among these are the signals corresponding to n = 1. The counter-rotating (antiresonance) components of the RF field give rise to a shift of the dressed-state energy, i.e., to a frequency shift similar to the Bloch-Siegert shift. In particular, this shift is manifested as the dependence of the Rabi-oscillation frequency εon the sign of the detuning ω{1} -ω{rf} from resonance. In the case of double resonance, the oscillation amplitude is asymmetric; i.e., the amplitude at the sum frequency ω_{rf} +εincreases, while the amplitude at the difference frequency ω_{rf} -εdecreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.