## Abstract

Spin-photon interactions such as the Faraday effect provide techniques for measuring coherent spin dynamics in semiconductors. In contrast to typical ultrafast pulsed laser techniques, which measure spin dynamics in the time domain with an intense, spectrally broad probe pulse, we demonstrate a frequency-domain spin-photon resonance effect using modulated continuous-wave lasers which enables measurement of GHz-scale coherent spin dynamics in semiconductors with minimal spectral linewidth. This technique permits high-resolution spectroscopic measurements not possible with ultrafast methods. We have employed this effect to observe coherent spin dynamics in CdSe nanocrystals using standard diode lasers. By fitting the results to the expected model, we extract electron g-factors, and spin coherence and dephasing times in agreement with time-domain measurements.

©2012 Optical Society of America

The coherent dynamics of electron spins in semiconductor quantum dots have proven to be a rich source of fascinating physics, with promise for new types of spin-based electronics and quantum information processing devices [1, 2]. Combining optical measurements based on effects such as Faraday rotation or ellipticity with ultrafast pump-probe techniques has been a powerful method to uncover the (typically GHz-scale) coherent dynamics of the spin states [1]. However, these ultrafast pump-probe techniques give rise to several disadvantages, discussed below. Here, we present an effect, Fourier transform spin resonance (FTSR), which occurs when a sinusoidally modulated optical probe is in resonance with coherently evolving electron spins. By using the FTSR effect to optically probe GHz-scale coherent spin dynamics in semiconductors, the problems arising from ultrafast pump-probe techniques may be overcome.

The method we present here overcomes problems caused by 1. the broad spectrum and 2. the high peak intensity of ultrafast optical pulses. For an optical pulse of duration ∆*t*, the uncertainty principle imposes a minimum width Δ*E* ~*h/*Δ*t* of the pulse’s energy spectrum, and therefore sets a limit on the resolution of spectroscopic measurements. Optical transitions in semiconductor quantum dots give rise to spectral features on the scale of µeV [3], including spin-dependent features used to probe spin in single dots [4, 5]. Typical ultrafast optical pulses with duration Δ*t* ~ps have Δ*E* ~meV, three orders of magnitude greater than the spectral features of interest. Clearly, a much smaller Δ*E* ~µeV is required in order to efficiently detect spins in individual quantum dots, which would yield (Δ*t*)^{−1} ~GHz, a typical frequency range for coherent spin dynamics [6]. Second, in order to provide enough time-averaged power to overcome noise in the detector, an ultrafast probe pulse must have very high peak intensity. Such high peak intensity can give rise to unwanted effects, such as the optical Stark effect, which causes a shift in the transition energies [7]. Also, because the Faraday effect is a maximum when the probe energy is nonresonant with optical transitions, Faraday rotation may be used to perform non-destructive or non-demolition measurements [5]. To do this, it is necessary to avoid unintentional excitations which may be caused by either a broad energy spectrum, or high peak intensity.

In FTSR, the sinusoidally modulated optical probe optimizes the trade-off between time and energy bandwidth, and also avoids the high peak intensities necessary in pulsed measurements. As described below, a non-zero FTSR signal occurs when a frequency component of the spin dynamics is resonant with the laser modulation. Essentially, in a rotating reference frame, spin dynamics at frequency Ω are statically pumped and probed by optical signals modulated at frequency Ω. Rotating-frame effects can occur in ultrafast pulsed measurements of spin dynamics, for example in resonant spin amplification (RSA) [8, 9] and all-optical NMR [10], where the pulse repetition frequency is resonant with the electron and nuclear spin dynamics, respectively. The FTSR effect can be viewed as RSA in the ultimate limit of narrow laser linewidth, or in other words, as RSA with all but the lowest non-zero frequency component of the pump and probe stripped away. Of course, in standard RSA and all-optical NMR the disadvantages of ultrafast measurements mentioned above are still present. Refs [11, 12]. and [13] provide an alternative approach to probing spin dynamics with a narrow linewidth laser by using a cw probe detected with high-bandwidth, high-throughput electronics. In contrast, here we are not limited by the bandwidth and speed of the detection electronics because the signal is shifted to near zero frequency for measurement by standard low-bandwidth detectors.

FTSR can be understood via a general description of spins pumped and probed by periodic optical signals. Consider a system in which spins can be optically pumped into some coherent state at a rate_{$g\left(t\right)=\gamma {I}_{1}\left(t\right),$} where *γ* is a constant, and_{${I}_{1}\left(t\right)=\mathrm{Re}{\displaystyle {\sum}_{n=0}^{\infty}{\alpha}_{n}{e}^{in\Omega t}}$}is the intensity of a pump laser with period 2π/Ω. (The *α _{n}* are complex Fourier coefficients.) The absorption of a pump photon at time

*t*= 0 initializes a spin into a particular coherent state, which then evolves on the Bloch sphere in some way. Averaging over many spins initialized simultaneously (or repeated measurements of a single spin) yields the spin dynamics

**s**(

*t*) = (

*s*(

_{x}*t*),

*s*(

_{y}*t*),

*s*(

_{z}*t*)), where

**(**

*s**t*) = 0 for

*t*< 0 and

*t*→∞. At time

*t*, the total spin polarization

_{$\u3008s\left(t\right)\u3009$}is given by the convolution of the pump rate

*g*(

*t*) with the spin dynamics

**(**

*s**t*):

Because ** s**(

*t*) = 0 for

*t*< 0, the lower limit of integration can be extended to –∞, and the integral yields the Fourier transform

_{$\tilde{s}\left(n\Omega \right)$}of

**s**(

*t*) at frequency

*n*Ω:

_{$\u3008{s}_{z}\left(t\right)\u3009$}can then be measured by a probe laser (say, via Faraday rotation), yielding signal

_{$\Theta \left(t\right)=\theta {I}_{2}\left(t\right)\u3008{s}_{z}\left(t\right)\u3009,$}where θ is a constant and

_{${I}_{2}\left(t\right)=\mathrm{Re}{\displaystyle {\sum}_{n=0}^{\infty}{\beta}_{n}{e}^{in\Omega t}}$}is the periodic intensity of the probe laser. Making use of Eq. (2) and the fact that Re(

*z*) = (

*z**+

*z*)/2, we obtain

We then measure the signal with a low-bandwidth detector, eliminating any component with frequency ω ≥ Ω, and retaining just the DC component of the signal

_{${\tilde{s}}_{z}\left(0\right)$},

*α*

_{0}, and

*β*

_{0}to be real.

Now let_{${\alpha}_{n}={\beta}_{n}{e}^{in\varphi}.$} That is, the pump and probe have the same time-dependence but are shifted by a time delay Δ*t* = *ϕ*/Ω. In a standard pulsed pump-probe measurement, the pump and probe are approximated by a series of delta-function pulses (*α _{n}* = 1 for all

*n*), and

**s**(

*t*) decays to zero much faster than the period between pulses. Equation (4) then reduces to Θ

_{0}(Δ

*t*) ∝

*s*(Δ

_{z}*t*). If

**s**(

*t*) remains nonzero for longer than the period between pulses, then Eq. (4) yields the resonant spin amplification effect [8, 9]. As the pulses approach true delta functions, the problems with ultrafast pump-probe measurement become increasingly significant: both the peak intensity and the spectral bandwidth of the lasers diverge.

Here, instead of using a series of pump and probe pulses, we modulate the pump and probe lasers sinusoidally such that *α*_{0} = *α*_{1} = 1 and all other *α _{n}* = 0. For

*ϕ*= 0 or π/2, this yields

_{0}vs. Ω, with

*ϕ*= 0 and π/2, we can obtain both the real and imaginary part of the Fourier transform of

*s*(

_{z}*t*) – that is, we can obtain the same information as in the standard ultrafast pump-probe measurement up to the maximum achievable Ω. The difference here is that a laser with linewidth

_{$\Delta E=\hslash \Omega $}that measures spin dynamics with frequency Δ

*t*

^{−1}= Ω/2π optimally satisfies the uncertainty relation Δ

*E*Δ

*t*~

*h*, and the peak intensity is only a factor of two greater than the average intensity. In practice, it is convenient to fix Ω and sweep an external magnetic field

*B*to avoid the need to compensate for the non-constant frequency response of the RF electronics. The resulting curve yields information about the

*B*-dependence of

*s*(

_{z}*t*).

Figure 1
illustrates the calculated FTSR spectrum for three common coherent spin behaviors. The time-domain spin dynamics *s _{z}*(

*B*,

*t*) at a fixed

*B*is shown in column (i), and corresponding FTSR spectra are shown in columns (ii)-(iv). The spectra in columns (ii) and (iii) are calculated from Eq. (5), and display FTSR as a function of Ω at fixed

*B*, and as a function of

*B*at fixed Ω, respectively. Column (iv) shows FTSR vs.

*B*, calculated from Eq. (5) modified to include higher harmonics in the pump and probe modulation:

*c*are the Fourier coefficients of the modulated pump and probe. For the spectra in Fig. 1, column (iv), we have chosen

_{n}*c*= 1/

_{n}*n*.

Figure 1(a) shows spins precessing at a single frequency, with an exponential decoherence given by_{${s}_{z}\left(B,t\right)=A\mathrm{cos}\left(g{\mu}_{B}Bt/\hslash \right)\mathrm{exp}\left(-t/\tau \right),$} with *g* = 1.2, *B* = 140 mT, and τ = 5 ns. In Fig. 1(aii), the FTSR spectrum with *ϕ* = 0 (*ϕ* = π/2) shows an even (odd) Lorentzian peak centered at the precession frequency _{$g{\mu}_{B}B/\hslash ,$} and with width proportional to (*gτ*)^{−1}. While in principle, the spectra with *ϕ* = 0 (*ϕ* = π/2) are related by the Kramers-Kronig relation, the ability to measure both spectra permits confirmation that measured features actually arise from FTSR. With Ω fixed and *B* varied [Fig. 1(aiii)], a central peak occurs due to the DC component of the pump and probe modulation, and a pair of satellite features occur when the modulated pump and probe are resonant with the precessing spins at_{$B=\pm \hslash \Omega /\left(g{\mu}_{B}\right).$} Again, the positions of the satellite features allow the determination of *g*, and with this information, the feature width yields *τ*. Unlike in RSA measurements, FTSR data as a function of *B* show only one set of resonance features, since the pump and probe contain just a single non-zero frequency component. If the pump and probe are not perfect sinusoids, then additional satellite peaks arise at integer multiples of_{$B=\pm \hslash \Omega /\left(g{\mu}_{B}\right)$} with amplitude proportional to the Fourier coefficient of the corresponding harmonic.

In an ensemble of nanocrystals, inhomogeneity in size and shape give rise to a distribution of g-factors *P*(*g*-*g*_{0}) centered about *g*_{0}, which causes inhomogeneous dephasing. This behavior is shown in Fig. 1(b), with

_{$\tilde{P}$}is the Fourier transform of

*P*with respect to

*g*. In Fig. 1(bi), we plot

*s*(

_{z}*B*,

*t*) from Eq. (7) with a Gaussian g-factor distribution. This results in a faster decay envelope with a shape determined by the exponential decay and

_{$\tilde{P}.$}Correspondingly, the FTSR spectrum in Fig. 1(bii), shows a broadened resonance, with position related to

*g*, and shape determined by the Lorentzian decoherence and inhomogeneous dephasing. Note that if the inhomogeneous dephasing dominates over the exponential decoherence, then the shape of the FTSR spectrum with

*ϕ*= 0 directly reflects the shape of the g-factor distribution

*P*(

*g*). Figure 1(biii) also shows broadened satellite features, with some distortion arising from the increase in inhomogeneous dephasing with magnetic field. This distortion of the features provides a means to separate the exponential decay from the inhomogeneous dephasing. The ability to separate these two contributions is further improved with contributions from higher harmonics in the pump/probe modulation [Fig. 1(biv)]. Here, successive satellite peaks show increasing broadening due to increasing inhomogeneous dephasing. From the positions and widths of the peaks, we can determine the g-factor, exponential decay time, and g-factor distribution width. The amplitude of the features depends on the Fourier coefficients

*c*, as well as

_{n}*τ*and

*P*(

*g*). Because the positions and widths of the features are sufficient to determine

*τ*and

*P*(

*g*), the amplitudes yield information about the non-harmonicity of the pump/probe modulation.

FTSR is also useful for probing more complex spin dynamics, for example, dynamics arising from a bimodal g-factor distribution yielding dephasing centered at two distinct g-factors, *g*_{1} and *g*_{2}. Figure 1(c) displays such dynamics calculated from the sum of two terms from Eq. (7):

*η*may have different lifetimes and g-factor distributions

*τ*

_{1,2}and

*P*

_{1,2}. Figure 1(ci) shows Eq. (8) with

*τ*

_{1}=

*τ*

_{2}= 5 ns,

*g*

_{1}= 1.2,

*g*

_{2}= 1.44,

*η*= 1.4 and

*P*

_{1}and

*P*

_{2}given by the same Gaussian distribution. In the time domain, the bimodal g-factor distribution is evidenced by the slight beating that is visible. In the FTSR data in Fig. 1(cii)-1(civ), the bimodal distribution can be seen directly in the double-peak structure.

To demonstrate FTSR, we have measured room-temperature coherent spin dynamics in an ensemble of CdSe nanocrystal quantum dots (NN-Labs, CSE620). The nanocrystals, with diameter ~6 nm, were suspended in toluene in a quartz cell. This system provides a good test case for this technique, as previous time-domain Faraday rotation measurements [14] have found non-trivial coherent spin dynamics, arising from a bimodal distribution of electron g-factors within the ensemble.

A schematic of the experimental setup is shown in Fig. 2
. The experiment includes no moving parts (e.g. optical choppers or delay lines). The pump and probe lasers were both diode lasers (Hitachi HL6320G) with wavelength λ = 635 nm. Low frequency modulation with 100% depth was applied to both lasers by varying the drive current (*f*_{1} = 20 kHz for the pump, *f*_{2} = 520 Hz for the probe) to allow lock-in detection techniques. The high frequency modulation at ω = Ω of the lasers was accomplished by sending the output of an RF signal generator (Rohde & Schwarz SMB100A) through an RF splitter (Pasternack PE2003), and then adding it to both laser drive currents via a bias tee (built into Thorlabs LM9LP), with the power set to modulate between 0 and 100% of the low-frequency-modulated signals. Smaller modulation depth could also be used, at the cost of smaller FTSR signal. Since the laser intensity is always positive, a zero-frequency (DC) component is always present, comparable to the component at frequency Ω. Because the laser output is nonlinear with the drive current, higher harmonics of Ω are also seen. A more pure harmonic modulation could be achieved using electro-optic modulators. Both lasers travel equal distance to the sample so that the RF pump and probe modulation are in phase (thus measuring the real part of_{${\tilde{s}}_{z}\left(B,\Omega \right)$}) and the lasers are focused to overlapping spots within the sample. The remainder of the experimental setup follows [14]. An electromagnet provides a magnetic field *B* at the sample perpendicular to the laser propagation direction. The pump laser, which is circularly polarized by a quarter waveplate, serves to excite spin polarized electron/hole pairs in the CdSe nanocrystals. Through the Faraday effect, the linearly polarized probe experiences a rotation of its plane of polarization through an angle Θ as it passes through the sample, proportional to the projection of the total spin state in the probe propagation direction. After passing through the sample, the pump is blocked and the probe polarization is measured using a balanced photodiode bridge circuit (Thorlabs PDB150A) with a bandwidth of 300 kHz. The detection is therefore not sensitive to components of the signal at frequency ω ≥ Ω. Only the signal given in Eq. (5) is measured, modulated at the sum and difference of the pump and probe frequencies, *f*_{1} ± *f*_{2}. The Faraday rotation signal is measured using a bandpass filter from 10 kHz to 30 kHz, and two cascaded lock-in amplifiers with reference frequencies *f*_{1} and *f*_{2}. The final Faraday rotation signal Θ_{0} is obtained by taking the difference between Θ at opposite pump helicities to ensure that the signal arises from optically pumped spins, and subtracting a small linear drift to restore symmetry about *B* = 0.

To use FTSR for a Faraday rotation measurement of coherent spin dynamics, one must have sufficient signal strength, and must be able to modulate the lasers at sufficiently high frequency to observe the dynamics of interest. The strength of the Faraday effect depends significantly on probe laser wavelength, and spin pumping efficiency typically depends on the pump laser wavelength. Here, we have chosen the size of the CdSe nanocrystals to match the absorption edge to the laser wavelength where both spin pumping and Faraday rotation are efficient. Furthermore, one must consider the difference in spin pumping from a pulsed pump vs. a modulated cw pump. A pulsed measurement where the spin lifetime is less than the pulse repetition period will yield a larger initial spin polarization than an FTSR measurement with the same time-averaged pump power. To use the FTSR effect, one must ensure that the spin pumping is sufficient to yield a measurable Faraday rotation. A final limitation of the FTSR technique that must be considered is the maximum RF modulation frequency *f*_{max} that can be achieved. In the demonstration here, we are limited to *f*_{max} = Ω_{max}/2π ~1 GHz. Above this frequency, the RF signal generator lacks sufficient power to overcome the impedance mismatch between the RF electronics and the laser diode. If the laser modulation were implemented using electro-optic modulators, *f*_{max} could be increased to tens of GHz.

Plots of Faraday rotation Θ_{0} versus *B* are shown in Fig. 3
. In Fig. 3(a), no RF modulation is applied to the pump and probe lasers, resulting in a typical Hanle-type measurement [15]. At *B* = 0, a steady-state spin polarization builds up yielding a central peak. As |*B*| increases, the spins precess with increasing frequency. As the precession period becomes less than the spin lifetime, Θ_{0} falls to near zero. Such time-averaged, steady-state measurements have been widely used to explore spin dynamics in semiconductors [15]. However, from these data, only the product of the spin lifetime and the g-factor can be extracted, and not the two parameters individually. Furthermore, nontrivial spin dynamics, such as multiple g-factors or magnetic-field-dependent dephasing may only appear as slight changes in the lineshape, making interpretation ambiguous. By contrast, the data in Fig. 3(b) shows Θ_{0} vs. *B* with Ω = 5.65 GHz. Here, the resonances between the frequency components of the spin dynamics and the pump and probe signals give rise to the peaks seen in the data. The DC component of the lasers reproduces the Hanle peak shown in Fig. 3(a). The first set of satellite peaks, at |*B*| ≈50 mT, arise from coherent spin dynamics in the CdSe nanocrystals at ω = 5.65 GHz. Two peaks are visible, at *B*_{1} = 39 mT and *B*_{2} = 51 mT. These arise from two distinct precession frequencies, as seen previously in these nanocrystals [14]. Smaller peaks occur at higher |*B*| due to higher harmonics in the pump and probe modulation, appearing at integer multiples of *B*_{1} and *B*_{2} indicated by solid and dashed arrows respectively.

To extract information about the spin dynamics, we start with a model function *s _{z}*(

*B*,

*t*), then compute the Fourier transform, and fit the data to Eq. (6). In practice, we only need the first several

*c*to obtain good agreement with the data. As in previous time-domain measurements in CdSe nanocrystals, we obtain the best fit from a bimodal distribution of g-factors [Eq. (8)]. This may arise from two subpopulations of nanocrystals with different charge states [14, 16]. It is particularly simple to choose a g-factor distribution that is the sum of two Lorentzians with half-widths Δ

_{n}*g*

_{1}and Δ

*g*

_{2}, ratio of amplitudes

*η*, and spin coherence times

*τ*

_{1}and

*τ*

_{2}for the two components. The Fourier coefficients

*c*of the pump and probe modulation are varied fit parameters, as well as an overall constant offset. In general, we find that

_{n}*c*

_{0}~

*c*

_{1}with higher

*c*falling to zero. The values of

_{n}*c*fall off somewhat more slowly than expected given the measured spectrum of the pump/probe modulation, possibly arising from magnetic-field dependence of the spin pumping efficiency.

_{n}With Lorentzian g-factor distributions in Eq. (8), we obtain the model function

_{0}(

*B*, Ω) given in Eq. (6).

A fit to the FTSR data as a function of *B* at Ω = 5.65 GHz is shown in Fig. 3(b). FTSR data obtained at Ω = 1.88 GHz, Ω = 3.77 GHz, and Ω = 5.65 GHz are shown in Fig. 4
, with associated fits. All the data in Fig. 4 were fit simultaneously, with the same values of *τ*_{1,2}, *g*_{1,2}, Δ*g*_{1,2}, and *η*. The Fourier coefficients were allowed to vary between different data sets, as the pump and probe modulation spectrum changes with frequency due to variation in the transfer function of the RF electronics. The linear dependence between frequency and peak magnetic field, highlighted by the dashed gray lines, confirms that the peaks arise from Zeeman-induced spin precession. The inset to Fig. 4 further displays the linear shift of the FTSR peaks with Ω, showing the evolution of the first set of satellite peaks as Ω is changed from 0.69 GHz to 5.72 GHz.

The best-fit parameters for the fits in Fig. 3(b) and Fig. 4 are given in Table 1
. The fits from Fig. 3(b) and Fig. 4 yield the same results within the confidence bounds, with smaller bounds for the parameters from the Fig. 4 fit. The two g-factors *g*_{1} = 1.21 and *g*_{2} = 1.62 are in excellent agreement with previous time-domain measurements. In prior measurements, values for *τ*_{1} and *τ*_{2} were difficult to distinguish from magnetic-field-induced dephasing [14]. Here, we obtain values of *τ*_{1} ≈0.8 ns and *τ*_{2} ≈3 ns. In agreement with [14], we find Δ*g*_{2} > Δ*g*_{1}. The magnitude of Δ*g*_{2} is comparable to that in [14], while Δ*g*_{1} is too small to be determined accurately by the fit. The result of a fit with a bimodal Gaussian distribution of g-factors (with the Fourier transform computed numerically) does not yield a statistically significant difference in the goodness of fit.

We have described here a spin-photon resonance effect, FTSR, and demonstrated a technique based on it for measuring coherent spin dynamics using Faraday rotation at GHz frequencies using sinusoidally-modulated diode lasers. By using integrated optical modulators, this technique may be extended to frequencies of tens of GHz. In addition to being a simple alternative to ultrafast pulsed laser systems, this technique employs minimal laser linewidth and low peak intensity allowing high-resolution spin spectroscopy, of particular interest for probing and manipulating spins in single quantum dots [5, 17] or in quantum dots coupled to high-finesse optical resonators.

## Acknowledgments

We acknowledge support from the Air Force Office of Scientific Research under award No. FA9550-12-1-0277.

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