The Hamiltonian of a linearly driven two level system, or qubit, in a rotating frame contains non-commuting terms that oscillate at twice the drive frequency, $\omega$. This makes the task of analytically finding the qubit's time evolution nontrivial. For near-resonant drives, the application of the rotating wave approximation (RWA), which is suitable only for drives whose amplitude $H_1$ is small compared to $\omega$ and varies slowly on the time scale of $1/\omega$, yields a simple Hamiltonian that can be integrated easily. We present a series of corrections to the RWA Hamiltonian in $1/\omega$, resulting in an effective Hamiltonian whose time evolution is accurate also for strong and time-dependent drives, assuming $H_1(t) \lesssim \omega$. As a result of the envelope $H_1$ being time dependent, our effective Hamiltonian is a function not only of $H_1(t)$ itself but also of its time derivatives. Our effective Hamiltonian is obtained through a recurrence relation which we derive using a combination of a Magnus expansion of the original rotating-frame Hamiltonian and a Taylor series of $H_1$, called the Magnus-Taylor expansion. Using the same tool, we further derive kick operators that complete our effective Hamiltonian treatment for non-analyticities of the drive envelope. The time evolution generated by the effective Hamiltonian agrees with the exact time evolution at periodic points in time. For the most important correction (first order in $1/\omega$), we find that besides the well-known Bloch-Siegert shift there are two competing terms that depend on the first derivative of the envelope, $\dot H_1$, and on the detuning, $\Delta$.

中文翻译：

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精确的旋转波近似

旋转坐标系中线性驱动的两级系统或量子位的哈密顿量包含以两倍驱动频率 $\omega$ 振荡的非交换项。这使得分析发现量子比特的时间演化的任务变得非常重要。对于近谐振驱动器，旋转波近似（RWA）的应用，仅适用于振幅$H_1$与$\omega$相比较小并且在$1/\omega$的时间尺度上缓慢变化的驱动器，产生一个简单的哈密顿量，可以很容易地积分。我们对 $1/\omega$ 中的 RWA 哈密顿量进行了一系列修正，假设 $H_1(t) \lesssim \omega$，则产生了一个有效的哈密顿量，其时间演化对于强驱动和依赖时间的驱动也是准确的。由于信封 $H_1$ 是时间相关的，我们的有效哈密顿量不仅是 $H_1(t)$ 本身的函数，也是其时间导数的函数。我们的有效哈密顿量是通过递推关系获得的，我们使用原始旋转坐标系哈密顿量的马格努斯展开和$H_1$ 的泰勒级数（称为马格努斯-泰勒展开）的组合推导出该递推关系。使用相同的工具，我们进一步推导出踢算子，完成我们对驱动包络的非解析性的有效哈密顿处理。有效哈密顿量产生的时间演化与周期时间点的准确时间演化一致。对于最重要的修正（$1/\omega$ 中的一阶），我们发现除了众所周知的 Bloch-Siegert 位移之外，还有两个相互竞争的项取决于包络的一阶导数 $\dot H_1$，以及在失谐上，$\Delta$。