Dressed Rabi oscillation in a crystalline organic radical

TL;DR

This paper reports on the observation of frequency shifts during Rabi oscillations in a crystalline organic radical, attributed to spin-spin interactions, modeled with a nonlinear semi-classical approach, advancing understanding of spin dynamics without cavity effects.

Contribution

It introduces the concept of dressed Rabi oscillation in a crystalline organic radical and provides a semi-classical model that explains the observed frequency shifts.

Findings

Frequency shifts up to 11 MHz observed during Rabi oscillations.

Nonlinear semi-classical model reproduces experimental results.

Frequency shifts attributed to spin-spin interactions and demagnetizing fields.

Abstract

Free electron laser-powered pulsed electron paramagnetic resonance (EPR) experiments performed at 240 GHz / 8.56 T on the crystalline organic radical BDPA reveal a tip-angle dependent resonant frequency. Frequency shifts as large as 11 MHz (45 ppm) are observed during a single Rabi oscillation. We attribute the frequency shifts to a "dressing" of the nutation by spin-spin interactions. A nonlinear semi-classical model which includes a temperature- and sample-geometry-dependent demagnetizing field reproduces experimental results. Because experiments are performed without a cavity, radiation damping -- the most common nonlinear interaction in magnetic resonance -- is negligible in our experiments.

Figures

Equations

$$\displaystyle\frac{d}{dt}\mathbf{M}=\gamma\mathbf{M}\times\mu_{0}\big{(}\mathbf{H}_{0}-N\cdot\mathbf{M}\big{)}$$

$$\frac{d}{dt}{M}_x$$

$$\frac{d}{dt}{M}_y$$

$$\frac{d}{dt}{M}_z$$

$$\omega (M_z)={\mu }_0\gamma (H_0-{\theta }_d{M}_z)$$

$$\frac{\Delta \omega }{{\omega }_L}={\mu }_{0}n\frac{(\gamma \hslash {)}^2}{4k_{B}T}$$

Full text

Understanding the magnetic properties of systems with unpaired electron spins is at the heart of much of modern condensed matter physics. Over the past decades, extensive efforts have focused on understanding systems at thermal equilibrium. As a result, ordered ground states such as the ferromagnet (FM) and antiferromagnet (AFM) are now well understood. Much current research on magnetism in thermal equilibrium focuses on the search for more exotic, highly entangled ground states such as quantum spin liquids in frustrated magnets Anderson (1973); Senthil and Fisher (2000); Moessner and Sondhi (2001); Kitaev (2006); Balents (2010); Powell and McKenzie (2011); Savary and Balents (2016); Vojta (2018). The behavior of large numbers of interacting spins far from equilibrium, however, remains challenging to study despite intense theoretical Else et al. (2016) and experimental investigations Choi et al. (2017).

Crystalline organic radicals—crystals composed of organic molecules that each contain an unpaired electron spin—are promising candidates for hosting interesting phases of strongly interacting spins driven far from equilibrium. Spins are sufficiently close to each other that the dominant interaction is exchange, but, because of small spin-orbit coupling, spins are sufficiently isolated from the lattice that a non-equilibrium spin state decays much more slowly than the time scales associated with spin-spin interactions.

BDPA (1,3-bisdiphenylene-2-phenylallyl), also known as the Koelsch radical, is an organic spin-1/2 radical that has been the subject of much study Duffy et al. (1972); Azuma et al. (1994); Mitchell et al. (2011); Yamauchi and Deguchi (1977); Hamilton and Pake (1963) and is widely used as a standard sample in electron paramagnetic resonance (EPR) Bennati et al. (1999); Goldfarb et al. (2008); Durkan and Welland (2002) and as a polarizing agent in dynamic nuclear polarization (DNP) enhanced nuclear magnetic resonance (NMR) experiments Dane and Swager (2010); Giraudeau et al. (2009). Crystallized 1:1 complexes of BDPA with benzene (BDPA-Bz, C39H27) are well described by a quasi-one dimensional Heisenberg AFM linear chain model, with an exchange integral $J/k_{B}=-4.4$ K along the chain Duffy et al. (1972); Yamauchi and Deguchi (1977), isosceles type spin frustration Azuma et al. (1994), and the onset of antiferromagnetic order at 1.695 K Duffy et al. (1972). Well above the ordering temperature, the exchange interaction remains important. Thermal fluctuations in the exchange fields narrow the EPR line to a value much smaller than one would expect based on dipole-dipole interactions Anderson and Weiss (1953); Misra (2011).

In this Letter, we report the observation of a novel dynamical phenomenon we call dressed Rabi oscillations. Rabi oscillations occur when an ensemble of spins is placed in a static magnetic field $B_{0}$ and driven by a transverse magnetic field $B_{1}$ oscillating near the Larmor frequency $\omega_{0}=g\mu_{B}B_{0}/\hbar$. If the Rabi frequency $\omega_{1}=g\mu_{B}B_{1}/\hbar$ is larger than the linewidth $\Gamma$ of the magnetic resonance, the spin ensemble undergoes nutation and the direction of the spin ensemble magnetization vector traces out circles on the surface of the Bloch sphere in the rotating frame. We report on Rabi oscillation experiments performed on grains of crystalline BDPA with $\Gamma/2\pi=4-6$ MHz at $B_{0}=8.56$ T, where the electron spin Larmor frequency is 240 GHz, driven by pulses of resonant radiation from the UCSB mm-wave free electron laser with Rabi frequencies $\omega_{1}$ as large as 25 MHz. We find that Rabi oscillations are dressed by spin-spin interactions and exhibit nonlinear dynamics which we model semi-classically with a Bloch equation modified to include the effects of sample magnetization.

Individual BDPA-Bz grains were mounted on a silver-coated mirror and placed at the end of a corrugated waveguide which tapers to a diameter of 5 mm, then loaded into the center of a tunable (0 to 12.5 T) superconducting magnet. Each grain was $\sim$300 to $\sim$500 $\mu$m across, significantly smaller than the 1.25 mm wavelength 240 GHz radiation. Linearly polarized 240 GHz radiation generated by the UCSB mm-FEL in “long” pulses of 1 to 3 $\mu$s were “sliced” into excitation pulses of well-defined duration by light-activated silicon switches Hegmann and Sherwin (1996). The silvered mirror below the BDPA-Bz reflected both the linearly polarized excitation pulse and the circularly polarized EPR signal to a superheterodyne receiver. A light-activated silicon isolation switch placed before the receiver was turned on only after the excitation pulse had passed. For short excitation pulses $<40$ ns, the FEL cavity dump coupler Takahashi et al. (2009) was used to boost the excitation pulse power. When longer excitation pulses were required, the cavity dump coupler was not activated. Single-frequency operation of the FEL was achieved through injection-locking Takahashi et al. (2007). For details of the FEL-EPR spectrometer, see Takahashi et al. (2012); Edwards et al. (2013); Wilson et al. (2018).

Figure 1a shows the response of a single BDPA-Bz grain placed in a 8.56 T field to a 2 ns long, resonant excitation pulse with a Rabi frequency $\omega_{1}/2\pi=25$ MHz. The short, resonant pulse tips the sample magnetization away from thermal equilibrium, which subsequently precesses at 240 GHz and emits circularly polarized magnetic dipole radiation proportional to the magnetization’s projection onto the transverse plane. After a $\sim$75 ns delay, the silicon isolation switch activates and a free induction decay (FID) is acquired by the receiver, mixed down to an intermediate frequency (IF = 500 MHz), and digitized. The complex FID signal is then Fourier transformed to extract the Fourier transform-EPR (FT-EPR) lineshape (Figure 1b). Figure 1c shows the integrated FT-EPR power spectrum as a function of pulse length. As the pulse length increases, the magnetization rotates further in the Bloch sphere, undergoing Rabi oscillations Rabi (1937). At the first signal maximum, occurring for a 10 ns excitation pulse duration, the magnetization has rotated by $\pi/2$, while at the first minimum, occurring at around 20 ns, the magnetization has nominally been inverted. However, the fact that this minimum does not correspond to an integrated FT-EPR intensity near zero indicates inversion is far from complete.

Analysis of the FT-EPR signal in frequency space reveals the frequency of the FID, as measured by the FT-EPR signal, changes as a function of pulse length (Figure 1d). This indicates that the EPR frequency $\omega_{0}$, typically given by the Larmor condition $\omega_{L}=\gamma B$ where $\gamma=g\mu_{B}/\hbar$, changes as a function of pulse length, a situation unexpected in conventional EPR experiments. The driven change in EPR frequency $\omega_{0}$ can be attributed to the mean field exerted on one spin by all the other spins in the sample. The dressed Rabi oscillation phase can be modeled in a mean-field sense by modifying the semiclassical Bloch equations to include the demagnetizing field $H_{M}$ of the BDPA-Bz grain itself, by analogy to ferromagnetic resonance (FMR) Kittel (1948). For an ellipsoid, $\mathbf{H}_{M}=-N\cdot\mathbf{M}$ where $\mathbf{M}$ is the magnetization and $N$ is the demagnetization tensor Zangwill (2013). Writing the total magnetic field $\mathbf{B}=\mu_{0}(\mathbf{H}_{0}+\mathbf{H}_{M}+\mathbf{M})$, the Bloch equations in the absence of an excitation pulse and neglecting relaxation become

[TABLE]

where $\mathbf{H}_{0}=\mathbf{B}_{0}/\mu_{0}$ is the externally applied magnetic field, $\gamma=g\mu_{B}/\hbar$ is the electron gyromagnetic ratio, and $\mathbf{M}\times\mathbf{M}$ identically vanishes.

Modeling the BDPA-Bz grain as an ellipsoid, the freely precessing magnetization $\mathbf{M}$ obeys the equations of motion

[TABLE]

where $\delta_{x},\delta_{x},\delta_{z}$ are the principal values of $N$, $M_{0}$ is the equilibrium magnetization, and $T_{1}$ and $T_{2}$ are the phenomenological Bloch spin-spin and spin-lattice relaxation times Bloch (1946). Assuming axial symmetry so that $\delta_{x}=\delta_{y}=\delta_{\perp}$ and $\delta_{z}=\delta_{\parallel}$, the magnetization precesses at a frequency $\omega(M_{z})$ which is a function of the $z$-component of $\mathbf{M}$,

[TABLE]

where $\theta_{d}=\delta_{\parallel}-\delta_{\perp}$. Two special cases are of particular interest: for a very thin, flat disk, $\theta_{d}=1$, while for a sphere, $\theta_{d}=0$ and the nonlinear term vanishes.

Equations 2 and 3 predict that the small tip-angle EPR frequency is shifted from the Larmor condition by an amount $\Delta\omega=-\mu_{0}\gamma\theta_{d}M_{z}$, neglecting higher order terms of order $|M_{z}^{2}/H_{0}|$. The dependence of the small tip-angle frequency on the magnetization, through the demagnetizing field, is analogous to the FMR condition which appears in the Kittel equations Kittel (1948). However, unlike in the FMR case where changes in $M_{z}$ due to resonant driving fields are negligible, for a paramagnet the full magnetization $\mathbf{M}$ can be rotated into the transverse plane or even inverted by a resonant microwave pulse. From Equations 2 and 3, a near-resonance microwave excitation pulse shifts the precession frequency by an amount proportional to the change in $M_{z}$, with the maximum precession frequency shift $\Delta F=2\gamma\mu_{0}\theta_{d}M_{0}/2\pi$ occurring after the initial magnetization $M_{0}$ is inverted. For BDPA-Bz, the maximum frequency shift can be estimated at room temperature in terms of the equilibrium magnetization $M_{0}=M_{eq}=270A/m$ Duffy et al. (1972) to be $\Delta F\sim\theta_{d}\times 19$ MHz, which is on the order of the observed frequency shifts and several times the observed linewidth (Figure 1).

A grain of BDPA-Bz with a geometry close to an axially-symmetric ellipsoid (Figure 2a), which had an aspect ratio of roughly 7:1 for $\theta_{d}=0.65$, was selected for further experiments. Figure 2b shows a nutation experiment performed on this BDPA-Bz grain, with the colorbar indicating the FT-EPR signal strength and with the FT-EPR frequency indicated on the vertical axis. The magnetic field was chosen so the excitation pulses are resonant with the bare Larmor frequency, rather than with the small tip-angle frequency. As a consequence, the spin system initially has a precession frequency below the excitation frequency, but moves on to resonance for a $\pi/2$ pulse, moves to a frequency above the excitation frequency for a $\pi$ pulse, and then returns to resonance for a $3\pi/2$ pulse. Rabi oscillations performed in this way achieve nearly complete magnetization inversion, as evidenced by the minimum being almost zero in Figure 2c.

Numerical simulations of the nonlinear modified Bloch equations with excitation pulses of varying lengths were carried out assuming an oblate spheroid geometry with an aspect ratio of 7:1, for an initial magnetization $M_{0}=270$ A/m. The phenomenological Bloch relaxation times $T_{1}$ and $T_{2}$ were independently measured to be $270\pm 40$ ns and $120\pm 30$ ns, respectively at 8.56 T (see Supplemental Material SI ), in good agreement with values measured at low field Mitchell et al. (2011). Simulations of the FID frequency dependence on pulse length over a full Rabi oscillation were in excellent agreement with experimental results, as shown in Figure 2d. The width of the simulated frequency shift indicates a 95% confidence interval, taking into account uncertainties in $T_{1}$, $T_{2}$, the Rabi frequency $\omega_{1}$, and the FEL detuning (see Supplemental Material SI ).

Magnetization evolution was probed over many Rabi cycles by varying the pulse length from 0 - 700 ns in steps of 10 ns, using an on-resonance Rabi frequency $\omega_{1}/2\pi=8$ MHz (Figure 3). At least five Rabi cycles were observed. The width of the simulation indicates a 95% confidence interval, and is in excellent agreement with experimental results. The decay in the observed Rabi oscillations is primarily related to spin-lattice ($T_{1}$) and spin-spin ($T_{2}$) relaxation.

The first condition for resolving a tip-angle dependent frequency shift in the dressed Rabi oscillation phase is that the ratio $|\Delta\omega/\Gamma|\geq$ 1, where $\Delta\omega=-\mu_{0}\gamma\theta_{d}M_{0}$ is the demagnetizing field-induced frequency shift and $\Gamma$ is the EPR linewidth. For BDPA-Bz grains at 8.56 T, $\Gamma\simeq$ 4 - 6 MHz. For $\theta_{d}=0.65$ used in simulations described above, we have $|\Delta\omega/\Gamma|\simeq 1.3$. The second condition is that the ratio $\omega_{1}/\Gamma\geq$ 1 , where $\omega_{1}$ is the Rabi frequency. For all previous EPR studies we are aware of, the combination of a sufficiently large equilibrium magnetization, narrow linewidth, and large Rabi frequency has been difficult to achieve, $|\Delta\omega/\Gamma|\ll 1$, and resolving magnetization-dependent frequency shifts has been difficult or impossible.

The temperature dependence of the fractional frequency shift $|\Delta\omega/\omega_{L}|$ in the dressed Rabi oscillation phase can be estimated for a paramagnet consisting of spin-1/2 electrons with density $n$ at thermal equilibrium, where $M_{0}=\frac{g\mu_{B}}{2}n\tanh\frac{\mu_{0}\gamma H_{0}}{2k_{B}T}$. When the high temperature approximation $k_{B}T\gg\mu_{0}\gamma\hbar H_{0}$ is valid,

[TABLE]

Figure 4 shows contour plots of nutation experiments performed at room temperature, 229 K, and 190 K. The temperature dependence of the maximum observed frequency shift $\Delta F$ is consistent with the predicted temperature dependence (Figure 4a). However, as the temperature is lowered, evidence of more complicated dynamics emerges in the nutation experiment: small additional frequency components are observed in FIDs for pulse durations shorter than 20 ns, as shown in Figures 4b and 4c.

Demagnetization field effects are commonly encountered in ferromagnetic resonance Kittel (1948), where sample geometry plays an important role in determining the resonance condition at very small microwave fields and tip angles. However, as microwave fields are increased, even in spherical samples, spin wave instabilities destroy the state with spatially-uniform magnetization before significant flip angles can be achieved Suhl (1956). Demagnetization field effects are also encountered in nuclear magnetic resonance (NMR) in highly concentrated spin systems Levitt (1996). Nonlinear magnetization-dependent effects in NMR can manifest in the generation of multiple spin-echos by a pair of radio frequency pulses Bowtell et al. (1990); Deville et al. (1979); Bedford et al. (1991) or in anomalous frequency correlations appearing in correlation spectroscopy experiments Warren et al. (1993); Richter et al. (1995); Jeener (2000). Taking $\theta_{d}=1$, corresponding to a flat disk geometry, Equation 4 predicts the 1H NMR resonance measured in room-temperature water should shift by $\sim 4$ ppb, corresponding to a frequency shift of 1.6 Hz in a 9.4 T magnetic field. In 1990, Edzes reported an anomalous 1H NMR frequency shift of $\sim 2.5$ ppb in protonated solvents, consistent with Equation 4 for $\theta_{d}\sim 0.6$ Edzes (1990). Outside of this example, demagnetization field-induced frequency shifts are not commonly observed in NMR, which can be explained by the fact that the gyromagnetic ratio of electrons is 657 times larger than that of protons, together with the $\gamma^{2}$ dependence of Equation 4.

Another nonlinear, magnetization-dependent effect encountered in magnetic resonance experiments is radiation damping Bloembergen and Pound (1954). Radiation damping typically manifests as magnetization and tip-angle dependent line broadening Augustine (2002) in magnetic resonance experiments which employ resonators with $Q\gg 1$, with high filling factors Benner (1977); Prisner et al. (1992). The effects of radiation damping are small in our experiments, since we do not employ a resonator cavity and our effective filling factor $\eta\ll 1$, but may be responsible for some line broadening (see Supplemental Material SI ).

Direct radiative losses due to magnetic dipole radiation may also contribute to the observed EPR linewidths, especially at low temperatures. The instantaneous power $P$ radiated by magnetic dipole radiation at frequency $\omega$ by a sample of volume $V$ with uniform magnetization $M_{0}$ is given by $P=\frac{\mu_{0}}{6\pi c^{3}}\omega^{4}M_{0}^{2}V^{2}\sin^{2}\theta$, where $\theta$ is the magnetization tip-angle. Radiative losses lead to magnetization decay with characteristic time constant $\tau_{rad}=(\frac{\mu_{0}}{6\pi c^{3}}\gamma\omega^{3}M_{0}V)^{-1}$ (see Supplemental Material SI ). For the grain used in experiments presented in Figure 2, a lower limit of $\tau_{rad}\simeq 360$ ns was calculated, which is longer than $T_{2}$, $T_{2}^{*}=1/\pi\Gamma$, or $T_{1}$, but which may become important to the linewidth at low temperatures.

Nonlinear spin-spin interactions like those reported in this Letter are essential for developing the quantum-mechanical correlations between spins necessary to generate squeezed spin states Kitagawa and Ueda (1993). Schemes for realizing squeezed states of electron spin ensembles have been proposed involving phonon-induced interactions between nitrogen-vacancy (NV) centers in diamond Bennett et al. (2013), or by coupling a NV center ensemble to a magnetic tip attached to a nanomechanical resonator Ma et al. (2016). Demagnetization field effects offer an alternative method of engineering nonlinear spin-spin interactions. The nonlinear interaction strength can be tuned by adjusting the magnetization magnitude through temperature changes, as shown in Figure 2f, as well as by optimizing the sample geometry, as illustrated by Equation 2. The long-range magnetic interactions that dress Rabi oscillations in this work may also be useful for coupling distant spin qubits Trifunovic et al. (2013).

In this Letter, we have shown that, when both the microwave field and the demagnetization field are larger than the line width of a magnetic resonance, the non-interacting spin dynamics may be visibly “dressed” by each spin’s interaction with the mean field generated by all the other spins in the sample. The signatures of dressed Rabi oscillations are free-induction decays whose frequencies depend on tip angle, and anomalous non-circular trajectories of the sample magnetization direction on the surface of the Bloch sphere. Because the experiments are carried out without a resonator, the “radiation damping” effects that usually arise in the spin dynamics of concentrated spin systems are negligible. We think of dressed Rabi oscillation as a transient dynamical phase in a system of interacting spins driven far from thermal equilibrium. Dressed Rabi oscillation can be modeled using classical equations of motion. As the temperature is lowered, in addition to the strength of the “dressing” increasing, the thermal fluctuations in the exchange fields that are responsible for narrowing the EPR line near room temperature will be quenched, and signatures of new manifestly quantum states of interacting spins far from thermal equilibrium may emerge.

I References