## Abstract

Multidimensional Coherent Optical Photocurrent Spectroscopy (MD-COPS) is implemented using unstabilized interferometers. Photocurrent from a semiconductor sample is generated using a sequence of four excitation pulses in a collinear geometry. Each pulse is frequency shifted by a unique radio frequency through acousto-optical modulation; the Four-Wave Mixing (FWM) signal is then selected in the frequency domain. The interference of an auxiliary continuous wave laser, which is sent through the same interferometers as the excitation pulses, is used to synthesize reference frequencies for lock-in detection of the photocurrent FWM signal. This scheme enables the partial compensation of mechanical fluctuations in the setup, achieving sufficient phase stability without the need for active stabilization. The method intrinsically provides both the real and imaginary parts of the FWM signal as a function of inter-pulse delays. This signal is subsequently Fourier transformed to create a multi-dimensional spectrum. Measurements made on the excitonic resonance in a double InGaAs quantum well embedded in a p-i-n diode demonstrate the technique.

© 2013 Optical Society of America

## 1. Introduction

Multi-Dimensional Coherent Spectroscopy (MDCS) is an ultra-fast spectroscopy technique that unfolds spectral information onto several dimensions to correlate absorption, mixing, and emission frequencies [1]. Originally developed for nuclear magnetic resonance experiments [2], MDCS was brought to the optical domain [3]. This powerful technique has been very successfully applied to molecules [4–6], photosynthetic complexes [7], semiconductor quantum wells [8–10], or atomic vapors [11, 12]. MDCS enables the separation of homogeneous and inhomogeneous line widths of a resonance [10, 13], the characterization of single and two-quantum coherences [14, 15], and the study of the role played by many-body interactions [8]. It is also an ideal tool to study coupling between transitions, such as excitons confined in separated QWs [16,17] or Quantum Dots (QDs) [18–20]. However, conventional MDCS techniques rely on the wave-vector selection of the FWM signal based on phase matching. This selection only applies in 2D or 3D systems such as atomic vapors, semiconductor QWs, or dense ensembles of nano-objects. Thus, conventional techniques cannot be applied to sub-wavelength structures where the translational symmetry is broken, such as individual (or small ensembles of) QDs, or other single nano-objects. In this case, radiation by point sources prevents the formation of a well defined FWM beam in the phase-matched direction. So far, only a sophisticated heterodyne mixing technique has been demonstrated to provide access to FWM of single QDs [19, 20], relying on assumptions made about the phase evolution of a reference resonance in order to phase individual one-dimensional spectra.

Here we present a MDCS technique that enables measurement of FWM signals from nanostructures without relying on wave-vector selection of the signal. Using a collinear geometry, non-linear signals are isolated in the frequency domain. The new approach measures the FWM signal via photocurrent readout and is suitable for any sample with which electrical contacts can be made, including single nano-objects. Advantages of the method include the intrinsic measurement of the FWM amplitude and phase, and a natural reduction of the impact of mechanical fluctuations, allowing us to achieve sufficient phase stability without active stabilization. As a proof of principle, 2D photocurrent spectra of excitons confined in a double InGaAs QW are presented.

## 2. Experiment

MDCS is an extension of FWM experiments. In conventional MDCS a sequence of three pulses is sent to the sample, and the radiated FWM is heterodyned with a local oscillator for phase resolution and detected by a spectrometer. Inter-pulse delays are precisely stepped, and the data is then Fourier transformed with respect to these delays to obtain multi-dimensional spectra. This scheme has been implemented in various geometries. A common one is the box geometry [6, 21, 22]. In this geometry, rephasing and non-rephasing contributions to the FWM signal can be recorded using different pulse sequences [8, 23]. The box geometry allows the characterization of two-quantum coherences [14, 15], and can provide long phase-locked inter-pulse delays if active stabilization is used [22]. However, MDCS in the box geometry requires additional procedures to obtain real and imaginary parts of the multidimensional spectra [24–27]. Another common configuration is the pump-probe geometry, where the phase-locked delay between first and second pulses can be conveniently achieved using pulse-shaping methods [28], and the signal is self-heterodyned with the transmitted probe serving as the local oscillator. In the pump-probe method, purely absorptive 2D spectra (i.e. the sum of real parts of the rephasing and non-rephasing components [23]) are directly recorded. Rephasing and non-rephasing components can be separated using phase-cycling algorithms [29,30]. The drawbacks of generating pulse pairs using pulse-shaping methods is that delays are limited to about 10 ps (shorter than coherence times in some semiconductor materials). Moreover, as mentioned above, conventional non-collinear geometries cannot be used to study single or few zero-dimensional objects.

A possible approach to perform FWM experiments on sub-wavelength structures is to use methods that isolate FWM signals without relying on the phase-matching condition of a radiated field [31]. Such solutions can be implemented in a collinear geometry, where isolation of the non-linear signal can be based either on phase cycling algorithms [31–34], selection of the FWM signal in the frequency domain [35], or subtraction of single-pulse and two-pulse contributions in the THz range [9]. Phase cycling can also be used in non-collinear geometries to improve signal-to-noise by suppressing scattering from single pulses [22, 29, 30].

In the present work, we use a collinear geometry where four pulses are sent to the sample. The role of the fourth pulse is to convert the third order polarization into a fourth order population that we detect in the form of a photocurrent signal. The FWM signal is isolated in the frequency domain, using an approach similar to that developed by Tekavec *et al*, where the fourth order population generated in an atomic vapor was detected in the form of a fluorescence signal [35]. In our case, we choose to detect a photocurrent signal, since we expect the collection of an incoherent, isotropic luminescence from the semiconductor sample to be rather inefficient. Moreover, electrical detection of the signal provides a realistic approach to study the non-linear response of optoelectronic devices such as photo-detectors or solar cells. The four excitation pulses are denoted A, B, C, and D (in the chronological order at which they hit the sample) and the corresponding inter-pulse delays are denoted *τ*, *T* and *t*. (Fig. 1(a)). The four-pulse sequence is generated by sending the beam of a mode-locked Ti:Sapph oscillator (Coherent MIRA −76 MHz repetition rate [36]) into a set of two Mach-Zehnder interferometers nested within a larger Mach-Zehnder interferometer (Fig. 1(b)). Three translation stages are used to control inter-pulse delays *τ*, *T* and *t*. Each interferometer arm contains an Acoustic Optical Modulator (AOM) (Isomet 1205c-1) driven by a phase-locked direct digital synthesizer (Analog Devices AD9959). In each arm the 0* ^{th}* diffraction order of the AOM is spatially blocked and the 1

*order beam is used. Each AOM is driven with a unique radio frequency*

^{st}*ω*, with

_{i}*i*= {

*A*,

*B*,

*C*,

*D*}. By scattering on the acoustic grating, the optical carrier frequency

*ω*

_{0}of the pulse gets shifted by an amount that equals the frequency of the acoustic wave. Each excitation pulse is thus “tagged” by oscillating at a uniquely shifted optical frequency

*ω′*=

_{i}*ω*

_{0}+

*ω*. Although much smaller than the laser pulse bandwidth, the frequency shift

_{i}*ω*does impact the laser beam when its effect is considered on a train of pulses (whose interference produces a frequency comb [37]), as developed in Section 3. The non-linear signals resulting from multiple pulse contributions oscillate at radio frequencies given by sums and differences of the AOM frequencies.

_{i}As a simple example, the photocurrent signal obtained from a combination of pulses A and B
oscillates at the difference frequency (beat note) *ω _{AB}* =

*ω*−

_{A}*ω*. Dual phase lock-in detection is used to isolate and demodulate this signal. A continuous wave (cw) laser (external cavity diode laser,

_{B}*λ*= 941.5 nm) runs through the same optics as the excitation laser, but is vertically offset from the path of the excitation laser. The beat note resulting from its interference, recorded by reference photodetectors (REF1 for arms A and B on Fig. 1(b)) serves as a reference for lock-in detection of the signal. This detection scheme provides several advantages: (1) The bandpass filtering around the beat note frequency achieved by the lock-in ensures that the detected signal results from a wave-mixing process where pulses A and B contribute once each. Other contributions (single pulse contributions, noise at mechanical vibration frequencies, dark current) are filtered out by the lock-in. (2) The phase of the signal is correlated to the phase of the excitation pulses, providing a direct, intrinsic access to the complex signal Z = X + iY from the dual phase lock-in amplifier output. (3) When stepping delay

*τ*, the signal phase at the lock-in output does not evolve at an optical frequency, but at a reduced frequency

*ν*

^{*}= |

*ν*−

_{sig}*ν*| given by the difference between the signal and cw laser optical frequencies. This “physical undersampling” of the signal results in a smaller number of points required to sample the signal, and in an improvement of the signal-to-noise ratio: the impact of a mechanical fluctuation in the setup, corresponding to a delay fluctuation

_{cw}*δτ*, scales as

*δτ*·

*ν*

^{*}[35, 38]. In other words, the cw laser senses and partially compensates for phase noise resulting from optical path fluctuations occurring in the setup. As a consequence, it is important to choose a cw laser wavelength as close as possible to the signal wavelength. With this technique, large phase-locked inter-pulse delays are achievable (limited only by the coherence length of the cw laser or, as in our case, by the length of the delay stages). This is particularly useful for the spectroscopy of semiconductor nanostructures such as QDs, where coherence times can be of the order of the nanosecond [39, 40]. The use of an auxiliary cw laser contrasts with the experiment of Tekavec

*et al*[35], where the lock-in reference beat note was generated from the interference of the excitation pulses themselves, after temporally broadening them using monochromators–providing phase-controlled delays limited by the monochromator resolution to about 10 ps. We show in Fig. 2 the measurement to determine zero delay between pulses A and B. For this, we substitute for the sample with a broadband commercial detector (Thorlabs DET10A Si detector [36]). Stepping the delay

*τ*, we perform a field auto-correlation of the excitation pulse. The signal, oscillating in real time at the frequency

*ω*, is demodulated by the lock-in detection scheme. Figure 2(a) shows the in-phase and in-quadrature components of the signal, provided by the lock-in outputs X and Y, respectively, and the signal amplitude R. X and Y, shifted by 90°, evolve with the stepping of

_{AB}*τ*at the reduced frequency

*ν*

^{*}= |

*ν*−

_{sig}*ν*| ∼ 3.5 THz. Since the phase and amplitude of the signal are known, a Fourier transform with respect to

_{cw}*τ*provides us with the one-dimensional power spectrum of the excitation laser (Fig. 2(b)).

In a similar way, the four pulses can be used to generate a FWM signal, and the delays *τ*, *T* and *t* can be stepped to record 2D or 3D data, which can be Fourier transformed with respect to each delay to produce multi-dimensional spectra. The most typical 2D spectrum in conventional MDCS is obtained by stepping delays *τ* and *t* while keeping *T* at a constant value. To produce the appropriate lock-in reference for the FWM signal, we detect the beat notes *ω _{AB}* and

*ω*on two separate photodetectors (REF1 and REF2 in Fig. 1(b)). These beat frequencies are mixed digitally using a digital signal processor (DSP) (SigmaStudios ADAU1761Z [36]). The mixing algorithm (in-quadrature mixing) is as follows: the beat notes cos(

_{CD}*ω*

_{AB}t^{*}) and cos(

*ω*

_{CD}t^{*}) (with

*t*

^{*}being the real time) are input into the DSP, where they independently undergo a Hilbert transform [41]. For a general function, the Hilbert transform shifts the phase of positive frequency components by $\frac{\pi}{2}$. E.g., the Hilbert transform of cos(

*ω*

_{AB}t^{*}) is:

After performing the Hilbert transform we then have both the in-quadrature cos(*ω _{AB}t*

^{*}) and in-phase sin(

*ω*

_{AB}t^{*}) components of the beat note (and the same thing is done for cos(

*ω*

_{CD}t^{*})). These components are then appropriately multiplied with each other, and we make use of the identity

This process provides reference frequencies

*k⃗*= −

_{FWM}*k⃗*+

_{A}*k⃗*+

_{B}*k⃗*and

_{C}*k⃗*=

_{FWM}*k⃗*−

_{A}*k⃗*+

_{B}*k⃗*of non-collinear MDCS, corresponding to the so-called rephasing (or

_{C}*S*) and non-rephasing (or

_{I}*S*) pulse sequences, respectively. In the box geometry,

_{II}*S*and

_{I}*S*need to be recorded separately since they correspond to a different pulse sequence, while with frequency domain selection

_{II}*S*and

_{I}*S*can be recorded simultaneously on two separate lock-in amplifiers. It is also possible to detect two-quantum (or

_{II}*S*) signals that oscillate at the radio frequency

_{III}*ω*

_{SIII}=

*ω*+

_{A}*ω*−

_{B}*ω*−

_{C}*ω*. The reference frequency that allows demodulation of such a signal cannot be generated from reference detectors REF1 and REF2. It can be extracted from detector REF3 at the output of the nested Mach-Zehnder interferometer (see Fig. 1), after appropriate frequency mixing and filtering by the DSP.

_{D}## 3. Analogy with phase-cycling

We would like to point out the conceptual link between the frequency selection scheme used in this work and phase cycling. In phase cycling methods, several measurements are taken applying different phase shifts on the excitation beams, in order to suppress single pulse and pairwise contributions and enhance the FWM signal [31–34]. The frequency selection scheme that we use can be seen as a phase modulation scheme, or, in other words, as a dynamic (or “pulse-to-pulse”) phase cycling, when considering carrier-envelope phase effects on a train of pulses. At the output of the mode-locked laser, the pulse train carries an initial carrier-envelope offset frequency *f _{CE}* [37], which is linked to a shift of the carrier-envelope phase from pulse to pulse
$\mathrm{\Delta}{\varphi}_{CE}=2\pi \frac{{f}_{CE}}{{f}_{\mathit{rep}}}$ (with

*f*the laser repetition rate). The nested interferometers create four copies of the original pulse train. However, due to the acousto-optical modulation, each of the four pulse trains is frequency-shifted by a unique radio frequency

_{rep}*ω*(without altering the repetition rate

_{i}*f*). Thus, each beam acquires an

_{rep}*additional*, unique carrier-envelope offset frequency, resulting in a pulse-to-pulse carrier-envelope phase shift that is different for each pulse train. In this way, the phase difference between the pulses of trains A and B is cycled pulse-to-pulse. Explicitly, the electric field of the

*n*pulse in train

^{th}*i*as a function real time

*t*

^{*}can be written as

*a*(

*t*

^{*}) is the pulse envelope and ${T}_{rep}=\frac{1}{{f}_{rep}}$. The shifted optical carrier frequency can be decomposed as

*ω′*=

_{i}*ω*

_{0}+

*ω*= 2

_{i}*π*(

*N*×

*f*+

_{rep}*f*) +

_{CE}*ω*, where

_{i}*ω*

_{0}is the original carrier frequency and

*N*is an integer. Since

*ω*and

_{i}*f*are much smaller than the pulse bandwidth, the frequency shift results in a pulse-to-pulse carrier-envelope phase shift for beam

_{CE}*i*that is

If we set the carrier-envelope offset to be zero for the pulses *n* = 0, as in Fig. 3, then we can write the phase difference between the *n ^{th}* pulses of train

*i*and

*j*as Δ

*ϕ*= (

_{i,j}*ω*−

_{i}*ω*)(

_{j}*nT*). In our situation, we obtain

_{rep}The phase difference between beams A and B, and between C and D, is thus cycled at a rate given by the difference frequencies *ω _{AB}* and

*ω*, respectively. The principle can be visualized in Fig. 3, for the simple situation where the delays

_{CD}*τ*,

*T*, and

*t*are equal to 0. This dynamic, pulse-to-pulse phase-cycling results in an evolution of the phase of the FWM signals

*S*and

_{I}*S*as

_{II}*ϕ*

_{SI}= Δ

*ϕ*− Δ

_{C,D}*ϕ*and

_{A,B}*ϕ*

_{SII}= Δ

*ϕ*+ Δ

_{C,D}*ϕ*. This phase cycling leads to the oscillation of the FWM signal amplitudes at the precise frequencies that are selected by the lock-in detection scheme detailed in the previous section. Let us note that this picture of the dynamic phase-cycling does not need a stable

_{A,B}*f*to be valid: fluctuations of

_{CE}*f*are duplicated in all four pulse trains, and cancel as we measure the phase difference between two pulse trains.

_{CE}## 4. Results

We demonstrate the working principle of our setup on a double In_{0.2}Ga_{0.8}As/GaAs QW, embedded within the intrinsic region of a p-i-n diode. The double QW consists in a 4.8 nm thick QW and a 8 nm thick QW, separated by a barrier of 4 nm thickness. An Au-Ni-Ge bottom contact was deposited on the n-doped substrate. A top contact (5nm Ti and 200nm Au) was deposited on part of the sample surface. The sample was kept at a temperature of 15.5K in a cold finger liquid Helium cryostat. In order to deal with the sample capacitance, a “bootstrap” trans-impedance circuit [42, 43] was used to convert and amplify the photocurrent from the sample into a voltage that is read by the lock-in amplifiers (Stanford Research SRS830 [36]) inputs (Fig. 4).

The AOM frequencies that we use are *ω _{A}* = 80.109 MHz,

*ω*= 80.104 MHz,

_{B}*ω*= 80.019 MHz,

_{C}*ω*= 80 MHz. The beat notes recorded by reference detectors REF1 and REF2 as a result of the cw laser interference are then

_{D}*ω*= 5 kHz and

_{AB}*ω*= 19 kHz, respectively. As a result of in-quadrature mixing by the DSP, the reference frequencies provided to the lock-in amplifiers for the

_{CD}*S*and

_{I}*S*FWM signal are

_{II}*ω*

_{SI}= 14kHz and

*ω*

_{SII}= 24kHz.

Let us underline that thanks to the “physical undersampling” phenomenon mentioned above, we do not need to implement active stabilization, a major advantage compared to conventional MDCS techniques [22].

Figure 5 shows 2D spectra that were recorded from the sample. A forward bias *V _{b}* = 0.5 V, for which we obtain the strongest FWM signal, was applied through the bootstrap circuit (see Fig. 4). The laser spectrum was the same as shown in Fig. 2(b), exciting the lowest energy excitonic resonance of the double QW structure. The total excitation power (four pulses) was 250

*μW*. The pulse sequence was focused on the sample using a microscope lens (Nikon EPI ELWD CF Plan 20x, NA = 0.4 [36]), providing an excitation spot of ∼5

*μ*m diameter. While delays

*τ*and

*t*were stepped, delay

*T*was kept at 200 fs. A fast Fourier transform with respect to delays

*τ*and

*t*provides 2D spectra as a function of

*h̄ω*and

_{τ}*h̄ω*. Non-rephasing (

_{t}*S*- Fig. 5(a)–(b)) and rephasing ((

_{II}*S*- Fig. 5(c)–(d)) spectra were obtained simultaneously from two separate lock-in amplifiers.

_{I}The absolute value of *S _{II}* and

*S*spectra (Fig. 5(a) and (c)) show a peak on the diagonal corresponding to the lowest energy exciton of the double QW. In the

_{I}*S*spectrum the peak is slightly elongated along the diagonal, a sign of slight inhomogeneous broadening of the excitonic resonance. Thanks to the intrinsic phase resolution of the technique, real and imaginary parts of the data are directly obtained as well. To determine the absolute phase offset of the signal, we set the phase of the time domain data to be zero at zero

_{I}*τ*and

*t*delays [35]:

*arg*{

*Z*(

*τ*= 0,

*T*= 200

*fs*,

*t*= 0)} = 0. We show in Fig. 5(b) and (d) the real parts of the

*S*and

_{I}*S*spectra, respectively, exhibiting a typical absorptive line shape.

_{II}## 5. Conclusion

We have implemented and demonstrated a robust, versatile platform for the MDCS of nanostructures. In a collinear geometry, the FWM signal is detected as a photocurrent and isolated in the frequency domain. This method provides an intrinsic phase resolution of the signal, and enables the recording of rephasing and non-rephasing FWM signals simultaneously. An auxiliary cw laser runs through the same optics as the excitation laser to partially compensate for mechanical fluctuations occurring in the setup, enabling optical MDCS without the need of active stabilization. Long phase-locked inter-pulse delays can be achieved (only limited by the coherence length of the cw laser or the length of the delay stages), particularly adapted to the long coherence and population times of excitations in semiconductor materials. Demonstrated on an InGaAs double QW structure, the technique can be extended to the MDCS of single nano-objects, since it does not rely on the wave-vector selection of the FWM signal. Considering that photocurrent measurements have been reported in single QDs [44,45], carbon nanotubes [46], and nanowires [47–49], the Multidimensional Coherent Optical Photocurrent Spectroscopy (MD-COPS) technique can be applied on all of these types of single nanostructures. Let us finally note that, while we have shown 2D rephasing and non-rephasing spectra, the MD-COPS setup can realize 3D spectra [12, 16], as well as two-quantum 2D spectra, without any technical modification.

## Acknowledgments

We acknowledge Terry Brown, Andrej Grubisic and David Alchenberger for technical help and fruitful discussions, and Richard Mirin for the epitaxial growth of the QW sample. G.N. acknowledges support by the Swiss National Science Foundation (SNSF).

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