Interaction-induced long-time tail of a nonlinear ac absorption in a localized system: a relay-race mechanism

TL;DR

This paper investigates how weak dipole-dipole interactions induce a long-time tail in the nonlinear ac absorption of localized systems, revealing a relay-race mechanism of multi-generation Rabi oscillations.

Contribution

It introduces a novel relay-race mechanism explaining long-time oscillations in ac absorption due to interaction-induced second-generation Rabi oscillations.

Findings

Long-time oscillations occur at frequencies much lower than the Rabi frequency.

Second-generation Rabi oscillations are driven by fields generated during initial Rabi cycles.

Non-diagonal dipole-dipole interactions facilitate the formation of these long-time tails.

Abstract

In conventional solid-state electron systems with localized states the ac absorption is linear since the inelastic widths of the energy levels exceeds the drive amplitude. The situation is different in the systems of cold atoms in which phonons are absent. Then even a weak drive leads to saturation of the ac absorption within resonant pairs, so that the population of levels oscillates with the Rabi frequency. We demonstrate that, in the presence of weak dipole-dipole interactions, the response of the system acquires a long-time component which oscillates with frequency much smaller than the Rabi frequency. The underlying mechanism of this long-time behavior is that the fields created in the course of the Rabi oscillations serve as resonant drive for the second-generation Rabi oscillations in pairs with level spacings close to the Rabi frequency. The frequency of the second-generation oscillations is of the order of interaction strength. As these oscillations develop, they can initiate the next-generation Rabi oscillations, and so on. Formation of the second-generation oscillations is facilitated by the non-diagonal component of the dipole-dipole interaction tensor.

Figures

Equations

$$\begin{gathered} \hat{H}=\frac{B_a}{2}{S}_z^a+\frac{B_b}{2}{S}_z^b+2B_1\left(S_x^a+S_x^b\right)\cos \omega t \\ +2B_1\left(S_y^a+S_y^b\right)\sin \omega t-D\frac{3({\mathbf{S}}^a\cdot \mathbf{r})({\mathbf{S}}^b\cdot \mathbf{r})-({\mathbf{S}}^a\cdot {\mathbf{S}}^b)r^2}{r^5}, \end{gathered}$$

$${\Omega }_a={\left[B_1^2+(\omega -B_a{)}^2\right]}^{1/2}.$$

$$\frac{d{S}_x^a}{dt}$$

$$\frac{d{S}_y^a}{dt}$$

$$\frac{d{S}_z^a}{dt}$$

$$\frac{d{S}_x^b}{dt}$$

$$\frac{d{S}_z^b}{dt}$$

$$\frac{dS_{z}^{a}}{dt}=-B_{1}^{2}\int\limits_{0}^{t}dt^{\prime}S_{z}^{a}(t^{\prime})\times\cos\Big{[}(\omega-B_{a})(t-t^{\prime})-D\int\limits_{t^{\prime}}^{t}dt^{\prime\prime}S_{x}^{b}(t^{\prime\prime})\Big{]}.$$

$$\frac{d{S}_z^b}{dt}=-D^2\underset{0}{\overset{t}{\int }}d{t}^{\prime }{S}_z^b(t^{\prime })\left[S_z^a(t)S_z^a(t^{\prime })\right]\cos B_b(t-t^{\prime }).$$

$$S_x^b(t)=-\frac{D}{B_b}(1-\cos B_{b}t)+D\underset{0}{\overset{t}{\int }}d{t}^{\prime }{S}_z^a(t^{\prime })S_z^b(t^{\prime })\sin B_b(t-t^{\prime }).$$

$$S_z^a(t)=\frac{{\left(\omega -B_a\right)}^2}{{\Omega }_a^2}+\frac{B_1^2}{{\Omega }_a^2}\cos {\Omega }_{a}t,$$

$$\begin{gathered} S_z^a(t)S_z^a(t^{\prime })=\frac{{\left(\omega -B_a\right)}^4}{{\Omega }_a^4}+\frac{B_1^4}{{\Omega }_a^4}\cos {\Omega }_{a}t\cos {\Omega }_a{t}^{\prime } \\ +\frac{{\left(\omega -B_a\right)}^2{B}_1^2}{{\Omega }_a^4}\left(\cos {\Omega }_{a}t+\cos {\Omega }_a{t}^{\prime }\right) \end{gathered}$$

$$\frac{d{S}_z^b}{dt}=-\frac{D^2{B}_1^4}{4{\Omega }_a^4}\underset{0}{\overset{t}{\int }}d{t}^{\prime }{S}_z^b(t^{\prime })\cos \left[\left({\Omega }_a-B_b\right)(t-t^{\prime })\right].$$

$$S_z^b=\frac{({\Omega }_a-B_b{)}^2}{{\Omega }_b^2}+\frac{B_2^2}{{\Omega }_b^2}\cos {\Omega }_{b}t,$$

$$B_2=\frac{D{B}_1^2}{2{\Omega }_a^2},{\Omega }_b={\left[B_2^2+({\Omega }_a-B_b{)}^2\right]}^{1/2}.$$

$$1-\langle S_{z}^{b}(t)\rangle=g\int\limits_{a}^{\infty}dr4\pi r^{2}\int\limits_{0}^{\infty}dB_{b}\frac{\left(\frac{D(r)B_{1}^{2}}{4\Omega_{a}^{2}}\right)^{2}}{\left(\frac{D(r)B_{1}^{2}}{4\Omega_{a}^{2}}\right)^{2}+(\Omega_{a}-B_{b})^{2}}\\ \times 2\sin^{2}\Bigg{\{}\Bigg{[}\left(\frac{D(r)B_{1}^{2}}{4\Omega_{a}^{2}}\right)^{2}+\left(\Omega_{a}-B_{b}\right)^{2}\Bigg{]}^{1/2}\hskip 8.53581pt\frac{t}{2}\hskip 2.84526pt\Bigg{\}},$$

$$D(r)=D_0{\left(\frac{a}{r}\right)}^3,$$

$$r={\left(\frac{D_0{B}_1^2{a}^3}{8{\Omega }_a^{2}u}t\right)}^{1/3},B_b={\Omega }_a-\frac{2v}{t}.$$

$$1-⟨S_z^b(t)⟩=\frac{2\pi g{a}^3{D}_0{B}_1^2}{3{\Omega }_a^2}\underset{0}{\overset{D_{0}t/2}{\int }}du\underset{-{\Omega }_{a}t}{\overset{\infty }{\int }}dv\frac{{\sin }^2{\left(u^2+v^2\right)}^{1/2}}{u^2+v^2}.$$

$$1-\langle S_{z}^{b}(t)\rangle\Bigg{|}_{3D}=\frac{4\pi ga^{3}D_{0}B_{1}^{2}}{3\Omega_{a}^{2}}\int\limits_{0}^{D_{0}t/2}\frac{du}{u}\Phi(u),$$

$$\Phi (u)=\underset{0}{\overset{\pi /2}{\int }}d\psi {\sin }^2\left(\frac{u}{\sin \psi }\right).$$

$$\displaystyle 1-\langle S_{z}^{b}(t)\rangle\Bigg{|}_{1D}=\frac{4gaD_{0}}{3\left(\frac{D_{0}t}{2}\right)^{2/3}}\left(\frac{B_{1}^{2}}{4\Omega_{a}^{2}}\right)^{1/3}\int\limits_{0}^{D_{0}t/2}\frac{du}{u^{1/3}}\Phi(u),\hskip 5.69054pt$$

$$\displaystyle 1-\langle S_{z}^{b}(t)\rangle\Bigg{|}_{2D}=\frac{8\pi ga^{2}D_{0}}{3\left(\frac{D_{0}t}{2}\right)^{1/3}}\left(\frac{B_{1}^{2}}{4\Omega_{a}^{2}}\right)^{2/3}\int\limits_{0}^{D_{0}t/2}\frac{du}{u^{2/3}}\Phi(u).\hskip 8.53581pt$$

$$\Phi(u)\Big{|}_{u\ll 1}=\frac{\pi u}{2}.$$

$$\Phi(u)\Big{|}_{u\gg 1}=\frac{\pi}{4}-\left(\frac{\pi}{8}\right)^{1/2}\frac{\cos\left(2u+\frac{\pi}{4}\right)}{u^{1/2}}.$$

$$1-\langle S_{z}^{b}(t)\rangle=\frac{8\pi ga^{3}D_{0}B_{1}^{2}}{\Omega_{a}^{2}}\times\left\{\begin{array}[]{ll}D_{0}t,{}{}&\frac{D_{0}t}{2}\ll 1,\\ \frac{1}{2}\ln(D_{0}t),{}{}&D_{0}t\gg 1.\end{array}\right.$$

$$1-⟨S_z^b(t)⟩=\frac{4ga{D}_0}{3{\left(\frac{D_{0}t}{2}\right)}^{2/3}}{\left(\frac{B_1^2}{4{\Omega }_a^2}\right)}^{1/3}F(D_{0}t),$$

$$F(D_{0}t)=\underset{0}{\overset{D_{0}t/2}{\int }}\frac{du}{u^{1/3}}\Phi (u).$$

$$F(D_{0}t)=\frac{3\pi }{8}{\left(\frac{D_{0}t}{2}\right)}^{2/3}+\underset{0}{\overset{D_{0}t/2}{\int }}\frac{du}{u^{1/3}}\left(\Phi (u)-\frac{\pi }{4}\right).$$

$$-{\left(\frac{\pi }{8}\right)}^{1/2}\frac{\sin \left(D_{0}t+\frac{\pi }{4}\right)}{2{\left(\frac{D_{0}t}{2}\right)}^{5/6}}.$$

$$S_z^a(t)S_z^b(t)=\frac{{\left(\omega -B_a\right)}^2{\left({\Omega }_a-B_b\right)}^2}{{\Omega }_a^2{\Omega }_b^2}+\frac{B_1^2{B}_2^2}{{\Omega }_a^2{\Omega }_b^2}\cos {\Omega }_{a}t\cos {\Omega }_{b}t+\frac{{\left({\Omega }_a-B_b\right)}^2{B}_1^2}{{\Omega }_a^2{\Omega }_b^2}\cos {\Omega }_{a}t+\frac{{\left(\omega -B_a\right)}^2{B}_2^2}{{\Omega }_a^2{\Omega }_b^2}\cos {\Omega }_{b}t.$$

$$\omega =B_b\left(1+\frac{D^2}{B_b^2}\right).$$

Full text

I Introduction

A transparent physical picture of absorption of the ac electric field in a system with localized electron states was proposed by N. F. Mott.Mott According to Mott, absorption takes place within pairs of states with energy spacing $\hbar\omega$, where $\omega$ is the driving frequency. Frequency dependence of the ac conductivity within this picture is $\sigma(\omega)\propto\omega^{2}\ln^{2}\omega$ (in one dimension), where one power of $\omega$ comes from the photon energy, while the other comes from the restriction that the pair is singly occupied. Finally, $\ln\omega$ comes from the overlap integral between initial and final states. Later, Mott’s formula was rigorously derived from the Kubo linear-response formalism by V.  L. BerezinskiiBerezinskii .

The condition of applicability of the linear response is that inelastic widths of the localized levels are much bigger than the absorption matrix element. This condition is satisfied in conventional solid-state systems where inelastic widths are due to the phonon emission.

Regime of strong ac drive, opposite to the linear response, can be realized in cold-atom systems, where phonons are absent. The ac drive in these systems is implemented by the synchronous modulation of the intensity of laser beams which create a quasi-random 1D on-site energy profile.NaturePhysics

Possibility to realize the regime of strong drive in a localized system without thermal bath raises a number of conceptual questions which, with rare exceptions,GefenThouless ; Kravtsov ; Kottos were not addressed in earlier studies. These questions can be conventionally divided into three groups:

(i) On the single-particle level, SondhiNaturePhysics ; French ; Bhatt ; AA ; Chalker the fundamental question is: does the localization persist in the presence of strong drive, when electron states evolve into the Floquet eigenstates? Anderson localization is the result of interference of the backscattering amplitudes in the course of multiple scatteringBerezinskii . Floquet states can be viewed as combination of satellites with energies separated by $n\hbar\omega$. Development of satellites upon increasing drive leads to the new channels of interference, and thus suppresses the localization, like in multichannel wires.

(ii) Another physical mechanism relevant for nonlinear ac response of localized non-interacting systems is the adiabatic Landau-Zener transitions.Bhatt ; Chalker This mechanism comes into play when the drive is strong and slow. In this limit, the effect of drive can be viewed as periodic modulation of energies of the localized states.Wilkinson1988 As the levels corresponding to neighboring states slowly pass by each other, an electron can adiabatically change the level. This, in turn, can lead to the long-time component of the ac absorption.Demler1 Spreading of electron due to the level crossings illustrates the tendency of drive to suppress the localization.

(iii) The third group of papers is the most numerous, see e.g. Refs.Demler1, ; Huse1, ; AbaninDriven, ; AbaninDriven1, ; Moessner, ; Bukov, ; Moessner1, ; Lindner, , and addresses the physics of ac driven localized interacting systems. They are focused on the dynamics of heating and on the long-times properties of non-equilibrium state. In particular, the question of interest is whether or not the long-time behavior of interacting many-body system is sensitive to its initial state.

When the drive amplitude is much smaller than the drive frequency, $\omega$, resonant pairs get saturated after the time of the order of the inverse Rabi period. Higher harmonics in the pair dynamics are small in this regime.Shirley Landau-Zener transitions also do not take place when the drive is fast. It is argued in Ref. Demler1, that long-time dynamics in this limit is due to interaction between the pairs. Namely, a group of $n$ interacting pairs can be engaged into collective Rabi oscillations, whose frequency is proportional to $n$-th power of drive.

In the present paper we propose an alternative mechanism of long-time dynamics in a system of weakly-interacting pairs under a weak drive. Namely, a saturated pair, executing the Rabi oscillations, creates a field which plays the role of drive for a distant pair, thus causing the second-generation Rabi oscillations. At resonance, the level spacing of the second-generation pair is equal to the Rabi frequency of the first-generation pair. If this frequency is much smaller than the pair-pair interaction, then the second-generation Rabi oscillations are slow. We perform statistical averaging analytically and find the slow component of the absorption.

II Dynamics of two interacting driven spins

To illustrate the proposed mechanism we employ the simplest model. Namely, as is common in the literature, we employ the language of spins to describe two-level systems and, correspondingly, the ac magnetic field to describe the drive. Consider two spins, $a$ and $b$, subject to magnetic fields $B_{a}$ and $B_{b}$, respectively. Since the drive amplitude, $B_{1}$, is much smaller than $\omega$, the rotating-wave approximation applies. Then the Hamiltonian of the pair reads

[TABLE]

where we have incorporated the dipole-dipole interaction with magnitude, $D$, while ${\bf r}$ is the vector-distance between $a$ and $b$.

In the absence of interaction, only spin $a$ is in resonance with the drive $|\omega-B_{a}|\ll\omega$, while spin $b$ is off resonance, $B_{b}\ll\omega$, and does not respond to the drive. Components $S_{a}^{x}$ and $S_{a}^{y}$ of the driven spin $a$ oscillate with frequencies close to $\omega$. Thus, the fields produced by these components on spin $b$ via dipole-dipole interaction, do not induce the dynamics of $b$. On the other hand, the $z$-component of spin $a$ oscillates with much smaller frequency

[TABLE]

These slow oscillations of $S_{a}^{z}$ translate into the field acting on $b$. The field generated by the $z$-$z$ component of the interaction is also inefficient, since spin $b$ is already directed along $z$. Spin $b$ can be set in motion via the $z$-$x$ and $z$-$y$ components of the dipole-dipole interaction when $\Omega_{a}$ is close to $B_{b}$. This is why we will keep only the $z$-$x$ component.

Summarizing, the relevant components of the field acting on $a$ are $(B_{1}\cos\omega t,B_{1}\sin\omega t,B_{a}+DS_{x}^{b})$, while the field acting on $b$ has an $x$-component, equal to $DS_{z}^{a}$, and $z$-component, $B_{b}$.

The equations of motion for the projections of $a$ which follow from $\frac{d\bf{S}}{dt}=\bf{B}\times\bf{S}$, read

[TABLE]

while the equations of motion for the components of spin $b$ have the form

[TABLE]

To analyze the coupled equations of motion for $a$ and $b$ it is convenient to cast them into the integral form. First, we express $S_{x}^{a}$ and $S_{y}^{a}$ in terms of $S_{z}^{a}$ and $S_{x}^{b}$. Substituting the result into the equation for $S_{z}^{a}$ and taking into account the initial condition $S_{z}^{a}(0)=1$, we get

[TABLE]

Similarly, we express $S_{x}^{b}$ and $S_{y}^{b}$ in terms of $S_{z}^{a}$ and $S_{z}^{b}$ and, using $S_{z}^{b}(0)=1$, substitute them into the equation for $S_{z}^{b}$. This yields

[TABLE]

To get the closed system, we also invoke the expression for $S_{x}^{b}$ obtained in the course of solving the system (6).

[TABLE]

Three equations (8), (9), and (10) describe fully the dynamics of both spins.

For $D=0$, spin $b$ points along $z$, while spin $a$ executes the Rabi nutations. In course of these nutations $S_{z}^{a}(t)$ follows the seminal Rabi formula

[TABLE]

which also follows from equation (8).

For a finite $D$ spin $b$ is also set into motion. This motion causes a “feedback” on spin $a$, reflected by the term proportional to $D$ in the argument of cosine. Most importantly, comparison of (8) and (9) quantifies our main message that the motion of spin $a$ plays the role of drive for the spin $b$; the role of the driving field is played by $DS_{z}^{a}(t)$.

For spin $a$ the resonant drive corresponds to the frequency $\omega=B_{a}$. Under this condition, $S_{z}^{a}$ oscillates with frequency $B_{1}$. Thus, for spin $b$, the resonant condition is $B_{1}=B_{b}$. We also expect that, within a factor, the nutation frequency of spin $b$ at resonance is equal to $D$, as illustrated in Fig. 2. Note that the nutation frequency, $D$, of the second-generation Rabi oscillations does not depend on $B_{1}$. However, this is valid only at exact resonance. We will see below that, for any small deviation from resonance, both the amplitude and the frequency of the second-generation Rabi oscillations acquire the $B_{1}$-dependence. In particular, the amplitude vanishes in the limit $B_{1}\rightarrow 0$.

Upon substituting (11) into (9), the product $S_{z}^{a}(t)S_{z}^{a}(t^{\prime})$ assumes the form

[TABLE]

The second term of (12) generates the sum $\cos\Omega_{a}(t+t^{\prime})+\cos\Omega_{a}(t-t^{\prime})$. It is the second cosine that acts as a resonant drive for the spin $b$. Keeping only $\cos\Omega_{a}(t-t^{\prime})$-term in (9), we get

[TABLE]

This equation has a solution

[TABLE]

where the “second-generation” drive and the second-generation Rabi frequency are defined as

[TABLE]

Equations (14) and (15) constitute the main result of the present paper. We analyze this result below.

III Statistical averaging, long-time tail of the ac absorption

The energy absorbed is proportional to averaged $S_{z}^{b}(t)$. We should average $S_{z}^{b}(t)$ over $B_{a}$, $B_{b}$ and over $D$, which is equivalent to averaging over distances to the neighbors. We notice that $B_{a}$ enters into the $S^{b}_{z}(t)$ only in combination $(B_{a}-\omega)^{2}+B_{1}^{2}$, so that $B_{a}$ does not affect the result of averaging.

We now average $S_{z}^{b}(t)$, from equation (14), assuming the density of spin states, $g$, to be constant. The averaging amounts to the two-fold integral

[TABLE]

where $a$ is the distance to the neighboring spin. Angular averaging is not important, so we choose

[TABLE]

where $D_{0}$ is the dipole-dipole interaction between two neighbors.

To proceed further, it is convenient to introduce, instead of variables $r$ and $B_{b}$, new variables $u$ and $v$ defined as

[TABLE]

Then the integral assumes the form

[TABLE]

In the long-time limit $D_{0}t\gg 1$, $\Omega_{a}t\gg D_{0}t$, we can replace $\Omega_{a}t$ with infinity in the lower limit of the $v$-integral. In order to see the asymptotic behavior of $\langle S_{z}^{b}(t)\rangle$, we substitute $v$ as, $v=\frac{u}{\tan\psi}$. With new variable, $\psi$, the integral (19) can be rewritten as

[TABLE]

with $\Phi(u)$ defined as

[TABLE]

In one and two dimensions, the corresponding expressions for $1-\langle S_{z}^{b}(t)\rangle$ are similar to (20) and read

[TABLE]

The function $\Phi(u)$ in (21) can be calculated analytically in two limits. For small $u\ll 1$, only small $\psi\sim u$ contribute to the integral. This allows to replace $\sin\psi$ by the argument and extend the integration to infinity. The resulting integral can be calculated explicitly, and one gets

[TABLE]

For large $u\gg 1$, the typical argument of $\sin^{2}$ is big, so that $\sin^{2}$ can be replaced by $1/2$, leading to \Phi(u)\big{|}_{u\gg 1}\approx\frac{\pi}{4}. The leading $u$-dependent correction comes from the vicinity of $\psi=\frac{\pi}{2}$, and thus oscillates with $u$. The asymptote has the form

[TABLE]

Using (24) and (25), we find the behavior of $\langle S_{z}^{b}(t)\rangle$ in three dimensions

[TABLE]

Note that, at $D_{0}t\ll 1$, the average $\langle S_{z}^{b}(t)\rangle$ decreases linearly with time. This is despite the fact that, for any given spin $b$, the time deviation of $S_{z}$ from $S_{z}=1$ is quadratic. The reason is that, if one expands (16) at small $t$, then the integral over $B_{b}$ will diverge. Overall, the characteristic time-scale for the change of $\langle S_{z}^{b}(t)\rangle$ is $D_{0}^{-1}$.

In derivation of the expression for $S_{z}^{b}(t)$, we have already assumed that $D_{0}\ll B_{1}$ when we neglected the feedback of $b$ on $a$. Now we see that the same assumption insures that the the evolution of the ensemble-averaged $S_{z}^{b}(t)$ is slow.

In Fig. 3, we show $\langle S_{z}^{b}(t)\rangle$ calculated numerically from the equation (20). Logarithmic behavior is evident. One can also distinguish weak oscillations on the background of log profile. These oscillations become more pronounced in lower dimensions.

For example, in one dimension, the equation (22) can be written in the following form:

[TABLE]

where

[TABLE]

If we substitute the leading asymptote, $\Phi(u)=\frac{\pi}{4}$, we will find that $\langle S_{z}^{b}(t)\rangle$ is time-independent. In order to capture the time dependence, we add and subtract $\pi/4$ from $\Phi(u)$. Then (28) takes the form

[TABLE]

At long times, the first term describes the leading contribution, while the second term saturates. This saturation is accompanied by the oscillations. To establish the form of this oscillating correction, it is convenient to present the integral $\int\limits_{0}^{D_{0}t/2}du$ as the difference of integrals $\int\limits_{0}^{\infty}du$ and $\int\limits_{D_{0}t/2}^{\infty}du$. Then, in the integral, $\int\limits_{D_{0}t/2}^{\infty}du$, we can use the oscillating term from the large-$u$ asymptote in (25). This generates the following correction to $F(D_{0}t)$:

[TABLE]

Substituting this correction into (27) indicates that the amplitude of oscillations in $\langle S_{z}^{b}(t)\rangle$ falls off as $t^{-3/2}$. Numerical plots for $d=1$ and $d=2$ in Fig. 3 confirm the saturation of $\langle S_{z}^{b}(t)\rangle$ at long times, which is accompanied by slow-decaying oscillations. As follows from analytics and numerics, the small time behavior of $1-\langle S_{z}^{b}(t)\rangle$ is linear in one and two dimensions as well.

IV The role of feedback

Unconventionally, we find that the coupling of two spins via dipole-dipole interaction is “unidirectional”: spin $a$ drives spin $b$, while the feedback effect of $b$ on $a$ is negligible under the condition $D\ll B_{1}$. On the other hand, it is the domain $D\ll B_{1}$, which is of interest, since it is in this domain where the long-time tail of the ac absorption develops. If we take into account that spin $b$ is dipole-dipole coupled to spin “$c$”, see Fig. 1, then $b$ will drive $c$ under the resonant condition, but with negligible feedback. This is why we identify this spin dynamics with relay-race.

To estimate the effect of feedback, we take the expression for $S_{z}^{a}(t)S_{z}^{b}(t)$ obtained in the lowest order and substitute it into (10). The expression for $S_{z}^{a}(t)S_{z}^{b}(t)$ has the form similar to the expression for $S_{z}^{a}(t)S_{z}^{a}(t^{\prime})$ in (12) with $t^{\prime}=t$

[TABLE]

As seen from (10), $S_{x}^{b}$ contains a “free precession” term, $\frac{D}{B_{b}}(1-\cos B_{b}t)$, and the “drive-induced” term. The role of the free precession term is the shift of resonance $\omega=B_{a}$. Indeed, substituting this term into the argument of cosine in (8), and assuming that $B_{b}t$ is big (or, equivalently, that $\Omega_{a}t$ is big), results in replacement of $(\omega-B_{b})$ by $(\omega-B_{b}-\frac{D^{2}}{B_{b}})$, i.e. the corrected resonance condition is

[TABLE]

Since the relevant value of $B_{b}$ is the Rabi frequency, $B_{1}$, we conclude that the shift is relatively small under the condition $B_{1}\gg D$, which coincides with the condition that the second-generation Rabi oscillations are slow.

We neglected $\cos B_{b}t$ in free precession term because it leads to the oscillating contribution, $\frac{D^{2}}{B_{b}^{2}}\left(\sin B_{b}t-\sin B_{b}t^{\prime}\right)$ in the argument of cosine in (8). This oscillating contribution results to effective renormalization of the drive amplitudeGlenn1 $B_{1}\rightarrow B_{1}J_{0}\left(\frac{D^{2}}{B_{b}^{2}}\right)$, where $J_{0}$ is the Bessel function. This renormalization is small by virtue of the same condition, $D\ll B_{b}\sim B_{1}$.

We now turn to the effect of feedback from the drive-induced term. As we have established in the course of statistical averaging, the second-generation Rabi oscillations essentially saturate at times $\sim\frac{1}{D}$. On the other hand, we do not expect significant feedback at times smaller that the period of the first-generation Rabi oscillations. This simplifies our task by restricting the time to the interval

[TABLE]

. For further simplification, we consider the most “dangerous” situation $\Omega_{a}=B_{b}$. Under this condition, spin $b$ is resonantly driven, so that the expected feedback is the strongest. Setting $\Omega_{a}=B_{b}$ in (30) we find that the first and the third terms vanish. In the two remaining terms it is sufficient to set $\cos\Omega_{b}t=1$, since $\Omega_{b}$ is of the order of $D$. Also, with $\Omega_{a}=B_{b}$, we have $\Omega_{b}=B_{2}$. After that, (30) simplifies to

[TABLE]

which is nothing but simply $S_{z}^{a}(t)$. Still, the behavior of $S_{x}^{b}(t)$ emerging upon substitution of (32) into (10) is nontrivial due to the beating of $\cos\Omega_{a}t$ and $\sin B_{b}(t-t^{\prime})$. This beating generates a contribution to $S_{x}^{b}$ equal to

[TABLE]

We see that this contribution exceeds the free-precession contribution and at $t\sim 1/D$ becomes of the order of $1$, which could be expected under the resonant condition $\Omega_{a}=B_{b}$. Our main point is that, even under this condition, the feedback of $S_{x}^{b}$ on the first-generation Rabi oscillations remains small. Indeed, performing the integration $\int\limits_{t^{\prime}}^{t}dt^{\prime\prime}$ in the argument of cosine in (8) generates the correction to this argument equal to

[TABLE]

While this correction grows with $t^{\prime}$ it does not exceed $1$ as long as $t^{\prime}$ is smaller than $1/D$. Thus. we conclude that when the drive exceeds the interaction magnitude, the feedback effect is negligible.

V Discussion

i. In a driven system of non-interacting spins in a random magnetic field (random $B_{a}$), only resonant spins respond to the drive. With Rabi frequencies depending on $B_{a}$, Rabi oscillations of different spins average out, so that the average $\langle S_{z}^{a}(t)\rangle$ approaches a constant. We note that this approach is accompanied by slow-decaying oscillations. Indeed, for a given spin, the oscillating part $S_{z}^{a}(t)$ has the form

[TABLE]

Assuming the homogeneous distribution of $B_{a}$ in the interval $0<B_{a}<\Delta$, the disorder-average of (33) has the form

[TABLE]

At long times, $B_{1}t\gg 1$, only the resonant spins contribute to the integral. This allows to expand the argument of cosine as

[TABLE]

We see that the relevant domain of $\left(\omega-B_{a}\right)$ is $\sim\left(\frac{B_{1}}{t}\right)^{1/2}\ll B_{1}$. This allows us to set $B_{a}=\omega$ in the denominator of (34) and to extend the integration domain over $(\omega-B_{a})$ to $(-\infty,\infty)$. Performing the Gaussian integration, we get

[TABLE]

In Fig. 2 the result of numerical calculation of $\langle S_{z}^{a}(t)\rangle$ for a certain parameter set is shown. Numerics confirms the presence of slow-decaying periodic oscillations in average $\langle S_{z}^{a}\rangle$. The amplitude of these oscillations of average $S_{z}(t)$ coming from the sparse resonant spins should be compared to the $S_{z}^{b}$ coming from the typical second-generation Rabi oscillations (26). The reasonable choice of $\Delta$ is $\omega$. Characteristic $t$ in (26) is $\sim 1/D_{0}$. Then $S_{z}^{b}$ is $\sim ga^{3}D_{0}$, while the oscillating part of $\langle S_{z}^{a}(t)\rangle$ can be presented as $\sim\frac{B_{1}}{\omega}\left(\frac{D_{0}}{B_{1}}\right)^{1/2}$, which is the product of two small parameters. On the other hand, the fact that we considered the interaction of spin $a$ with only one spin $b$, requires that the product $ga^{3}D_{0}$ is also small.

ii. Spin $b$ can induce even slower Rabi nutations in spin $c$, see Fig. 1. The corresponding Rabi frequency of these third-generation oscillations will be

[TABLE]

where $B_{3}$ is given by

[TABLE]

[TABLE]

If we are away from resonance at each step, then drive amplitude at $n$-th step will be

[TABLE]

By contrast, if we are at resonance in each step, then the drive amplitude for the $n$-th step will depend only on the dipole-dipole interaction between $n$-th spin and the $n-1$-th spin. For example, the drive amplitude for $n=3$ is $D_{bc}/2$, and the drive amplitude for $n=2$ is $D_{ab}/2$, see Fig. 1.

Suppose that spin $a$ is not in resonance with the drive, $(\omega-B_{a})>B_{1}$. Then the amplitude of the first-generation Rabi oscillations is small. Still, spin $b$ can oscillate with big amplitude, $\sim 1$, provided that $B_{b}$ is equal to $\Omega_{a}\approx(\omega-B_{a})$. At the same time, the frequency of the oscillations of spin $b$ will be approximately $\frac{D_{ab}B_{1}^{2}}{2(\omega-B_{a})^{2}}$, which is much smaller than $D_{ab}$.

iii. It follows from Eq. (26) that, while the contribution of the second-generation Rabi oscillations to the absorption is a slow function of time, the magnitude of this slow component contains a small parameter $ga^{3}D_{0}$. Thus, as the dipole-dipole interactions increases, the amplitude of the slow component grows linearly with $D_{0}$. On the other hand, the characteristic time before they saturate drops as $1/D_{0}$.

iv. For slow second-generation Rabi oscillations to develop the drive amplitude should be bigger than the interaction strength. On the other hand, the drive is assumed to be weak, $B_{1}\ll\omega$, which, in classification of Ref. Chalker, , corresponds to the linear absorption regime. Such a weak drive cannot affect the overall many-body localized regime.PRL2 ; PRL1 ; Science1 ; Science2 ; Science3

v. In spirit, the relay-race mechanism considered in the present paper bares some similarity to the mechanism of delocalization of eigenmodes of dipole-dipole coupled oscillators or of the ensemble of two-level systems. Levitov1 ; Levitov2 ; Levitov3 ; Kagan In Refs. Levitov1, ; Levitov2, ; Levitov3, ; Kagan, two undriven oscillators or two spins get hybridized when the corresponding frequencies match each other within the interaction magnitude. This hybridization can be mediated by $z$-$z$ component of the interaction. In our notations, the frequencies of two hybridized oscillators can be expressed as

[TABLE]

We see that, even at resonance $B_{a}=B_{b}$, hybridization does not result in a slow motion. By contrast, in our situation, the resonance is dictated by drive and hybridization takes place when $\Omega_{a}$ is close to $B_{b}$. In other words, the motion of $a$ in the “rotated” frame is in resonance with $b$ in the lab frame.

vi. We introduced the relay-race mechanism using the language of spins driven by ac magnetic field. In Refs. Chalker, , Demler1, , and Bhatt, the ac absorption of electric field by localized electrons has been studied. The main difference between the two scenarios is that we considered the fields $B_{a}$ to be random, but parallel to $z$. In the case of the ac electric field, $\bm{{\cal E}}\cos\omega t$, the Hamiltonian describing the drive has the form $\bm{P\cdot{\cal E}}\cos\omega t$, where $\bm{P}$ is the dipole matrix element between the ground and excited states. In spin language, randomness of the directions of $\bm{P}$ translates into the randomness of the directions of $B_{a}$. In a general case when ${\bf B}_{a}={\bf n}_{a}B_{a}$, ${\bf B}_{1}={\bf n}_{1}B_{1}$, where ${\bf n}_{a}$ and ${\bf}n_{1}$ are the unit vectors, the drive amplitude in the above expressions should be modified as

[TABLE]

It is important to note that when the directions of the fields ${\bf B}_{a}$ are random, we do not need the non-diagonal component of dipole-dipole interaction to induce the second-generation Rabi oscillations.

vii. There is a similarity between the relay-race mechanism considered above and the Rabi-vibronic resonance studied in Ref. Glenn, . In the latter case,= the Rabi oscillations are resonantly coupled to a vibronic mode rather than to the neighboring spin.

Acknowledgements

The work was supported by the Department of Energy, Office of Basic Energy Sciences, Grant No. DE- FG02-06ER46313.