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arXiv:2404.12236 · Physical Review A

Qubit dynamics driven by smooth pulses of finite duration

Ivo S. Mihov, Nikolay V. Vitanov

Published Nov 18, 2024• 4 citations
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Abstract

In this study, we investigate the dynamics of a qubit driven by a pulsed field of finite duration, focusing on smooth pulse shapes that start and end linearly in time. All of these pulse shapes, such as a sine function sampled between two adjacent nodes, provide smooth alternatives to the commonly used rectangular pulse. Owing to its continuity, the sine pulse exhibits weaker power broadening and leads to the rapid vanishing of the wings in the excitation line profile, significantly reducing the sidebands. Additionally, these pulse shapes avert the spurious truncation effects that arise with infinite-duration shapes, such as the Gaussian, where truncation is inevitable. We derive two approximate analytic solutions which describe the ensuing quantum dynamics. Both approximations assume that the field changes linearly at the beginning and end of the driving pulse, and adiabatically in between. The first approximation matches the linear and adiabatic parts at an appropriate instant of time and is expressed in terms of Weber's parabolic cylinder functions. The second, much simpler, approximation uses asymptotics of the Weber function to replace it with simpler functions, along with additional transformations. Both approximations were shown to be highly accurate, with near-complete agreement between theory and observations on two IBM Quantum processors, further validated by measurements of the predicted weaker power broadening and highly suppressed sidebands.

DOI
10.1103/physreva.110.052609

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In this study, we investigate the dynamics of a qubit driven by a pulsed field of finite duration, focusing on smooth pulse shapes that start and end linearly in time. All of these pulse shapes, such as a sine function sampled between two adjacent nodes, provide smooth alternatives to the commonly used rectangular pulse. Owing to its continuity, the sine pulse exhibits weaker power broadening and leads to the rapid vanishing of the wings in the excitation line profile, significantly reducing the sidebands. Additionally, these pulse shapes avert the spurious truncation effects that arise with infinite-duration shapes, such as the Gaussian, where truncation is inevitable. We derive two approximate analytic solutions which describe the ensuing quantum dynamics. Both approximations assume that the field changes linearly at the beginning and end of the driving pulse, and adiabatically in between. The first approximation matches the linear and adiabatic parts at an appropriate instant of time and is expressed in terms of Weber's parabolic cylinder functions. The second, much simpler, approximation uses asymptotics of the Weber function to replace it with simpler functions, along with additional transformations. Both approximations were shown to be highly accurate, with near-complete agreement between theory and observations on two IBM Quantum processors, further validated by measurements of the predicted weaker power broadening and highly suppressed sidebands.
We consider the interaction of a two-state quantum system with a class of pulses of finite temporal duration. The pulse shape function f(t)f(t)f(t) of such a pulse is a nonanalytic function of time, with discontinuous derivatives at the turn-on and turn-off times. The excitation line width---the excited-state population versus the detuning---is determined primarily by the magnitude of the jumps of the derivative f(n)(t){f}^{(n)}(t)f(n)(t) at the points of nonanalyticity, where nnn is the order of the first discontinuous derivative; this nonanalyticity shows up in the nnnth superadiabatic basis. The excitation line width for such pulses exhibits weak power broadening---it scales up as ${\ensuremath{\Omega}}_{0}^{1/(n+1)}$, where ${\ensuremath{\Omega}}_{0}$ is the peak Rabi frequency of the transition: $\ensuremath{\Omega}(t)={\ensuremath{\Omega}}_{0}f(t)$. As a specific example, we consider the power-of-sine class $f(t)={\mathrm{sin}}^{n}(\ensuremath{\pi}t/T)$ ($0\ensuremath{\leqq}t\ensuremath{\leqq}T$) and a truncated Gaussian pulse, and we compare their excitation line widths with the well-known excitation profile of the rectangular pulse (the Rabi formula). We find that, because of the reduced power broadening, the sinn{\mathrm{sin}}^{n}sinn and truncated Gaussian pulses may accelerate manipulation of qubits compared to rectangular pulses. The reason is that the lower power broadening allows one to use higher Rabi frequency, and hence shorter pulse duration, without affecting significantly other closely lying states.
We present a study of the coherent interaction of a qubit with a pulse-shaped external field of a constant carrier frequency. We explore, theoretically and experimentally, the transition line profile (the dependence of the transition probability on the detuning) for five different pulse shapes: Rectangular, Gaussian, hyperbolic-secant, squared hyperbolic-secant, and exponential. The theoretical description for all cases but sech2{\mathrm{sech}}^{2}sech2 is based on the analytical solutions to the Schr\"odinger equation or accurate approximations available in the literature. For the sech2{\mathrm{sech}}^{2}sech2 pulse we derive an analytical expression for the transition probability using the Rosen-Zener conjecture, which proves very accurate. In addition, the same conjecture turns out to provide very accurate approximations for the Gaussian and exponential pulses. The experimental results are obtained with one of IBMQ's quantum processors. An excellent agreement between theory and experiment is observed, demonstrating some pulse-shape-dependent fine features of the transition probability profile. The mean absolute error (a measure of the accuracy of the fit) features an improvement by a factor of 4 to 7 for the analytic models compared to the commonly used Lorentzian fits. Moreover, the uncertainty of the qubit's resonance frequency is reduced by a factor of 4 for the analytic models compared to the Lorentzian fits. These results demonstrate both the accuracy of the analytic modeling of quantum dynamics and the excellent coherent properties of IBMQ's qubit.
Power broadening-the broadening of the spectral line profile of a two-state quantum transition as the amplitude of the driving field increases-is a well-known and thoroughly examined phenomenon in spectroscopy. It typically occurs in continuous-wave driving when the intensity of the radiation field increases beyond the saturation intensity of the transition. In pulsed-field excitation, linear power broadening occurs for a pulse of rectangular temporal shape. Pulses with smooth shapes are known to exhibit much less power broadening, e.g., logarithmic for a Gaussian pulse shape. It has been predicted, but never experimentally verified, that pulse shapes which vanish in time as ∼|t|^{-λ} should exhibit the opposite effect-power narrowing-in which the postpulse transition line width decreases as the amplitude of the driving pulse increases. In this Letter, power narrowing is demonstrated for a class of powers-of-Lorentzian pulse shapes on the IBM Quantum processor ibmq_manila. Reduction of the line width by a factor of over 10 is observed when increasing the pulse area from π to 7π, in a complete reversal of the power broadening paradigm. Moreover, thorough study is conducted on the truncation of the pulse wings which introduces a (small) power-broadened term which prevents power narrowing from reaching extreme values. In the absence of other power broadening mechanisms, Lorentzian pulses truncated at sufficiently small values can achieve as narrow line profiles as desired.
We present an analytically exactly solvable two-state quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The exact solution is expressed in terms of associated Legendre functions. An interesting feature of this model is that the excitation probability does not vanish, except for zero pulse area or zero detuning; this feature is attributed to the asymmetric pulse shape. Two limiting cases are considered. When the coupling rises very slowly, it is nearly linear and the tanh model reduces to the shark model introduced earlier. When the coupling rises very quickly, the tanh model reduces to the Rabi model, which assumes a rectangular pulse shape and hence a sudden switch on. Because of its practical significance, we have elaborated the asymptotics of the solution in the Rabi limit, and we have derived the next terms in the asymptotic expansion, which deliver the corrections to the amplitude and the phase of the Rabi oscillations due to the finite rise time of the coupling.
Power broadening refers to the widening of the spectral line profile in a two-state quantum transition as the strength of the driving field increases. This phenomenon commonly arises in continuous-wave driving when the radiation field's intensity exceeds the transition's saturation intensity and it has been extensively studied in spectroscopy. For pulsed-field excitation, the spectral response of the quantum system may differ significantly: while a rectangular-shaped pulse leads to a linear power broadening, pulses with smooth shapes show significantly reduced power broadening, for instance, logarithmic for the Gaussian shape and none for the hyperbolic-secant shape. Recently [Phys. Rev. Lett. 132, 020802 (2024)], in a dramatic paradigm shift, we have demonstrated experimentally that for Lorentzian-shaped pulses, the opposite effect - power narrowing - takes place: the width of the spectral profile decreases when the driving pulse amplitude increases, with a narrowing factor of as much as 10 observed. While in high-resolution spectroscopy the push is for eliminating or even inverting the power broadening, there are applications where it is used to an advantage for it facilitates off-resonance excitation. Here, we present a number of shaped pulses that exhibit power broadening much greater than that of the rectangular pulse of the same pulse area. They are grouped in two families of pulse shapes. In particular, in regard to the width of the second Rabi oscillation maximum, the quadratic pulse family shows an increase by a factor of 3.3 whereas the even-exponent pulse family exhibits an increase by a factor of more than 3.5.
Power broadening refers to the widening of the spectral line profile in a two-state quantum transition as the strength of the driving field increases. This phenomenon commonly arises in continuous-wave driving when the radiation field's intensity exceeds the transition's saturation intensity and it has been extensively studied in spectroscopy. For pulsed-field excitation, the spectral response of the quantum system may differ significantly: while a rectangular-shaped pulse leads to a linear power broadening, pulses with smooth shapes show significantly reduced power broadening, for instance, logarithmic for the Gaussian shape and none for the hyperbolic-secant shape. Recently [Phys. Rev. Lett. 132, 020802 (2024)], in a dramatic paradigm shift, we have demonstrated experimentally that for Lorentzian-shaped pulses, the opposite effect---power narrowing---takes place: the width of the spectral profile decreases when the driving pulse amplitude increases, with a narrowing factor of as much as 10 observed. While in high-resolution spectroscopy the push is for eliminating or even inverting the power broadening, there are applications where it is used to an advantage for it facilitates off-resonance excitation. Here, we present a number of shaped pulses that exhibit power broadening much greater than that of the rectangular pulse of the same pulse area. They are grouped in two families of pulse shapes. In particular, in regard to the width of the second Rabi oscillation maximum, the quadratic pulse family shows a spectral increase by a factor of 3.3 whereas the power-law pulse family exhibits an increase by a factor of more than 3.5.