Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR

Comprehending the dynamical behavior of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalization of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion [Commun. Pure Appl. Math. 7, 649 (1954)], no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here on an entirely different nonperturbative approach, termed path sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems, and is invariant under scale transformations of the quantum state space. Path sum can be combined with any state-space reduction technique and can exactly reconstruct the dynamics of a many-body quantum system from the separate, isolated, evolutions of any chosen collection of its subsystems. As examples of application, we solve analytically for the dynamics of all two-level systems as well as of a many-body Hamiltonian with a particular emphasis on nuclear magnetic resonance applications: Bloch-Siegert effect, coherent destruction of tunneling, and $N$-spin systems involving the dipolar Hamiltonian and spin diffusion.

Figure 1

The path-sum continued fraction for the exact calculation of the entries of $\mathsf{U}(t^{\prime },t)$ is always of finite depth and breadth. (a) Illustrative example of a $2\times 2$ time-dependent Hamiltonian $\mathsf{H}\left(t\right)$, involved for instance in spin dynamics. (b) Dynamical graph ${\mathcal{G}}_t=K_2$ with adjacency matrix $\mathsf{H}\left(t\right)$. Circles correspond to self-loops [diagonal terms of $\mathsf{H}\left(t\right)$] and directed edges to off-diagonal terms. The associated weights are the entries of $\mathsf{H}\left(t\right)$. (c) Evolution operator $\mathsf{U}(t^{\prime },t)$ as seen by path sum, with integrals of the $G_{K_2,ii}$ quantities. (d) Step-by-step evaluation of $G_{K_2,11}(t^{\prime },t)$ (dashed rectangle) showing the finite character of the continued fraction. The sum is performed on the simple cycles ($\mathcal{C}$) of lengths 1 and 2 (these are indicated in red and other edges are indicated in dashed gray lines). At each step of the continued fraction, a vertex is removed (gray cross) and we work on subgraphs of less and less complexity. The calculation of entry ${\mathsf{U}}_{21}$ from $G_{K_2,11}$ is also illustrated; it includes a single term with two $*$ products as the graph has a single simple path ($\mathcal{P}$) from 2 to 1 (red arrow). (e) A pictorial representation of the “descending ladder principle.” The calculation starts at the top of the ladder with each $*$ inverse leading to the step below and ending in all cases on the ground. For $2\times 2$ Hamiltonians there are only two steps on the ladder, i.e., all continued fractions stop at depth 2. For all $3\times 3$ and $4\times 4$ Hamiltonians, the continued fractions stop at depth 3 and 4, respectively.

Figure 2

Graph illustrating the use of path sum.

Figure 3

Bloch-Siegert dynamics: resonant ${\omega }_0=\omega$ transition probability $P_{\uparrow \mapsto \downarrow }\left(t\right)$ as a function of time in the weak- to ultrastrong-coupling regimes, with (a) $\beta /\omega =0.5$, (b) $\beta /\omega =0.7$, (c) $\beta /\omega =0.9$, (d) $\beta /\omega =1.2$, (e) $\beta /\omega =1.6$, (f) $\beta /\omega =2$, (g) $\beta /\omega =3.5$, (h) $\beta /\omega =5$, and (i) $\beta /\omega =15$. Shown here are the numerical solution (dashed black line) and the fully analytical formulas for the Neumann expansions of the exact path-sum solution $P_{\uparrow \to \downarrow }^{\left(3\right)}\left(t\right)$ (dotted red line), $P_{\uparrow \to \downarrow }^{\left(7\right)}\left(t\right)$ (dotted blue line), and $P_{\uparrow \to \downarrow }^{\left(13\right)}\left(t\right)$ (solid blue line) (see Appendix pp2). As seen here, each of these formulas are equally valid throughout the coupling regimes, from weak to ultrastrong. Whenever longer times are desired, higher orders of the path-sum solution are readily available analytically. Also shown in (a) and (b) are the second-order Floquet theory [11] (solid green line and green points). The Floquet result is not shown in subsequent figures, where it is wildly inaccurate. Parameters: two-level system driven by the Bloch-Siegert Hamiltonian of Eq. (8) [11, 12] starting in the $|\uparrow \rangle$ state at $t=0$.

Figure 4

Bloch-Siegert dynamics: off-resonance ${\omega }_0\ne \omega$ transition probability $P_{\uparrow \mapsto \downarrow }\left(t\right)$ as a function of time in the weak- to strong-coupling regimes, with (a) ${\omega }_0=2\omega$, $\beta /\omega =0.7$, (b) ${\omega }_0=2\omega$, $\beta /\omega =1.6$, (c) ${\omega }_0=2\omega$, $\beta /\omega =3.5$, (d) ${\omega }_0=8\omega$, $\beta /\omega =0.7$, (e) ${\omega }_0=8\omega$, $\beta /\omega =1.6$, and (f) ${\omega }_0=8\omega$, $\beta /\omega =3.5$. Shown here are the numerical solution (dashed black line) and the fully analytical formula for the Neumann expansion of the exact path-sum solution at the fourth order $P_{\uparrow \to \downarrow }^{\left(4\right)}\left(t\right)$ (solid blue line). Parameters: two-level system driven by the Bloch-Siegert Hamiltonian of Eq. (8) [11, 12] starting in the $|\uparrow \rangle$ state at $t=0$.

Figure 5

Bloch-Siegert dynamics: resonant ${\omega }_0=\omega$ spin-flip duration $t_{sf}$, at which $P_{\uparrow \to \downarrow }\left(t\right)$ first peaks at or close to 1, as a function of the coupling strength $\beta$. Shown here are the exact formula of Eq. (11) (solid red line) and fully numerical results (blue dots). Discontinuities in the numerical results are due to small oscillations of $P_{\uparrow \to \downarrow }\left(t\right)$, which make $t_{sf}$ undergo discrete jumps as one wins over the others. These are well captured analytically by a more advanced analysis including the $S_{2k}$ and $C_{2k>0}$ functions. Insets: two examples of time evolution of $P_{\uparrow \to \downarrow }\left(t\right)$, the straight red lines are the predictions of Eq. (11).

Figure 6

Bloch-Siegert dynamics: resonant ${\omega }_0=\omega$ transition probability $P_{\uparrow \to \downarrow }\left(t\right)$ in the ultraweak-coupling regime $\beta /\omega =0.05$ for short times. Shown here are the order-0 formula $P_{\uparrow \to \downarrow }^{\left(0\right)}\left(t\right)$ of Eq. (12) (solid blue line), second-order Floquet theory [11] (solid green line and green points), and the numerical solution (dashed black line).

Figure 7

Coherent destruction of tunneling. Top line: return probability $P_{\uparrow \to \uparrow }\left(t\right)$ in the ultrastrong-coupling regime $\beta /{\omega }_0=30$ for (a) $\omega =4{\omega }_0$; (b) $\omega =20{\omega }_0$; and (c) $\omega =100{\omega }_0$. Shown here are $P_{\uparrow \to \uparrow }^{(acc,0)}\left(t\right)$ as given by Eq. (13) (solid blue line), the numerical solution (dashed black line), and its predicted time-average equation (14) (solid red straight line, indistinguishable from the numerically computed time average). Middle line: transition probability $P_{{\psi }_{-}\to {\psi }_{+}}\left(t\right)$ for a system starting in the $|{\psi }_{-}\rangle$ state at $t=0$ with (d) $4\beta /\omega =2.404...$, first zero of $J_0(4\beta /\omega )$; (e) $4\beta /=11.79...$, fourth zero of $J_0(4\beta /\omega )$; and (f) $4\beta /\omega =27.49...$, ninth zero of $J_0(4\beta /\omega )$. Note the changes of scales. Shown here are the formula of Eq. (15) (solid blue line) and the numerical solution (dashed black line), these two being completely indistinguishable. Bottom line: far off-resonance $\omega =100{\omega }_0$ expectation value of ${\sigma }_x$ for a system starting in the $|\uparrow \rangle$ state at $t=0$ with (g) $4\beta /\omega =2.404...$, first zero of $J_0(4\beta /\omega )$; (h) $4\beta /\omega =11.79...$, fourth zero of $J_0(4\beta /\omega )$; and (i) $4\beta /\omega =27.49...$, ninth zero of $J_0(4\beta /\omega )$. Note the changes of scales in $\langle {\sigma }_x\rangle$. Shown here are the formula of Eq. (16) (solid blue line) and the numerical solution (dashed black line), these two being indistinguishable.

Figure 8

Analytical spin diffusion on a cationic tin oxo cluster with $N=42$ protons (shown in inset) submitted to the time-dependent high-field dipolar Hamiltonian under MAS (rotor angular velocity ${\omega }_r=2\pi \times 10\mathrm{kHz}$). The figure shows the time evolution of the probability ${|\langle \psi \left(t\right)|\uparrow }_{z,i}{\rangle |}^2$ of finding a spin up along $z$ on proton $i$ for three protons: a hydroxyl proton H1 (solid red line), on which the excitation starts; a nearby hydroxyl proton H2 (dashed blue line); and a methyl proton H3 (dotted-dashed black line).

Figure 9

Return probability for the spin excitation on the initial hydroxyl proton H1 as a function of ${\omega }_R$ (one plot point every 20 Hz): (a) after a fixed time of $t=0.05\mathrm{ms}$, a situation exhibiting numerous peaks for small ${\omega }_R$ values that are not all resolved on this picture; and (b) after two rotor periods $t=2\times (2\pi /{\omega }_R)$. (c) Probability of finding the spin excitation on hydroxyl protons H1 (solid red line) or H2 (dashed blue line) as a function of time for ${\omega }_R=10\mathrm{kHz}$ and with a very strong offset of roughly 30 ppm at 1.5 GHz on all protons except H1 and H2. The total probability of being either on H1 or H2 (dotted black line) never goes below $\simeq 0.94$ over three rotor periods.

Figure 10

Building the path sum on the cationic tin oxo cluster. (a) Probability of return of the spin excitation on a hydroxyl proton H1 shown on Fig. 8 as a function of time for ${\omega }_r=2\pi \times 60\mathrm{kHz}$ over three rotor periods: (i) solution with no cutoff (solid black line, identical with $\mathrm{\Lambda }>100$), (ii) analytical approximation with low interaction cutoff $\mathrm{\Lambda }=40$ (dashed red line), and (iii) further approximation obtained upon setting ${\mathrm{\Sigma }}_5$ to zero (red points). (b) Discrete structure ${\tilde{\mathcal{G}}}_t$ of the quantum state space as seen by path sum when $\mathrm{\Lambda }=40$ and corresponding to the equations given in the text for ${\mathsf{U}}_{{\text{(OH)}}_3}$. Edges and self-loops correspond to intergroup and intragroup interactions, respectively. The adjacency matrix of this graph is the $14\times 14$ Hamiltonian with $3\times 3$ matrix-valued entries evoked in the text. Thus, the shape of the graph is essentially that of the molecule at the methyl and group of 3 hydroxyls level. It comprises two disconnected pathways for spin diffusion corresponding to the opposite sides of the cationic tin oxo cluster which become connected for higher cutoff values $\mathrm{\Lambda }>42$.