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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28617–28627
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Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure

Gaël Nardin, Travis M. Autry, Kevin L. Silverman, and S. T. Cundiff  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28617-28627 (2013)
http://dx.doi.org/10.1364/OE.21.028617


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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

1. S. T. Cundiff and S. Mukamel, “Optical multidimensional coherent spectroscopy,” Phys. Today 66, 44 (2013).

]. Originally developed for nuclear magnetic resonance experiments [2

2. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions (Oxford Uni. Press, London/New York, 1987).

], MDCS was brought to the optical domain [3

3. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54, 425–463 (2003). [CrossRef] [PubMed]

]. This powerful technique has been very successfully applied to molecules [4

4. M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. USA 97, 8219–8224 (2000). [CrossRef]

6

6. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Coherent 2D IR spectroscopy: molecular structure and dynamics in solution,” J. Phys. Chem. A 107, 5258–5279 (2003). [CrossRef]

], photosynthetic complexes [7

7. T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434, 625–628 (2005). [CrossRef] [PubMed]

], semiconductor quantum wells [8

8. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. 96, 057406 (2006). [CrossRef] [PubMed]

10

10. R. Singh, T. M. Autry, G. Nardin, G. Moody, H. Li, K. Pierz, M. Bieler, and S. T. Cundiff, “Anisotropic homogeneous linewidth of the heavy-hole exciton in (110)-oriented GaAs quantum wells,” Phys. Rev. B 88, 045304 (2013). [CrossRef]

], or atomic vapors [11

11. X. Dai, A. D. Bristow, D. Karaiskaj, and S. T. Cundiff, “Two-dimensional fourier-transform spectroscopy of potassium vapor,” Phys. Rev. A 82, 052503 (2010). [CrossRef]

, 12

12. H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, “Unraveling quantum pathways using optical 3D fourier-transform spectroscopy,” Nat. Commun. 4, 1390 (2013). [CrossRef] [PubMed]

]. MDCS enables the separation of homogeneous and inhomogeneous line widths of a resonance [10

10. R. Singh, T. M. Autry, G. Nardin, G. Moody, H. Li, K. Pierz, M. Bieler, and S. T. Cundiff, “Anisotropic homogeneous linewidth of the heavy-hole exciton in (110)-oriented GaAs quantum wells,” Phys. Rev. B 88, 045304 (2013). [CrossRef]

, 13

13. M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, “Resonance lineshapes in two-dimensional fourier transform spectroscopy,” Opt. Express 18, 17699–17708 (2010). [CrossRef] [PubMed]

], the characterization of single and two-quantum coherences [14

14. K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324, 1169–1173 (2009). [CrossRef] [PubMed]

, 15

15. D. Karaiskaj, A. D. Bristow, L. Yang, X. Dai, R. P. Mirin, S. Mukamel, and S. T. Cundiff, “Two-quantum many-body coherences in two-dimensional fourier-transform spectra of exciton resonances in semiconductor quantum wells,” Phys. Rev. Lett. 104, 117401 (2010). [CrossRef] [PubMed]

], and the study of the role played by many-body interactions [8

8. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. 96, 057406 (2006). [CrossRef] [PubMed]

]. It is also an ideal tool to study coupling between transitions, such as excitons confined in separated QWs [16

16. J. A. Davis, C. R. Hall, L. V. Dao, K. A. Nugent, H. M. Quiney, H. H. Tan, and C. Jagadish, “Three-dimensional electronic spectroscopy of excitons in asymmetric double quantum wells,” J. Chem. Phys 135, 044510 (2011). [CrossRef] [PubMed]

,17

17. G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F. Morier-Genoud, and S. T. Cundiff, “Coherent excitonic coupling in an asymmetric double InGaAs quantum well,” arXiv e-print 1308.1689 (2013).

] or Quantum Dots (QDs) [18

18. G. Moody, M. E. Siemens, A. D. Bristow, X. Dai, D. Karaiskaj, A. S. Bracker, D. Gammon, and S. T. Cundiff, “Exciton-exciton and exciton-phonon interactions in an interfacial GaAs quantum dot ensemble,” Phys. Rev. B 83, 115324 (2011). [CrossRef]

20

20. F. Albert, K. Sivalertporn, J. Kasprzak, M. Strauss, C. Schneider, S. Höfling, M. Kamp, A. Forchel, S. Reitzenstein, E. A. Muljarov, and W. Langbein, “Microcavity controlled coupling of excitonic qubits,” Nat. Commun. 4, 1747 (2013). [CrossRef] [PubMed]

]. 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

19. J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2011). [CrossRef]

, 20

20. F. Albert, K. Sivalertporn, J. Kasprzak, M. Strauss, C. Schneider, S. Höfling, M. Kamp, A. Forchel, S. Reitzenstein, E. A. Muljarov, and W. Langbein, “Microcavity controlled coupling of excitonic qubits,” Nat. Commun. 4, 1747 (2013). [CrossRef] [PubMed]

], 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

6. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Coherent 2D IR spectroscopy: molecular structure and dynamics in solution,” J. Phys. Chem. A 107, 5258–5279 (2003). [CrossRef]

, 21

21. T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, “Phase-stabilized two-dimensional electronic spectroscopy,” J. Chem. Phys. 121, 4221–4236 (2004). [CrossRef] [PubMed]

, 22

22. A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80, 073108 (2009). [CrossRef] [PubMed]

]. In this geometry, rephasing and non-rephasing contributions to the FWM signal can be recorded using different pulse sequences [8

8. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. 96, 057406 (2006). [CrossRef] [PubMed]

, 23

23. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Obtaining absorptive line shapes in two-dimensional infrared vibrational correlation spectra,” Phys. Rev. Lett. 90, 047401 (2003). [CrossRef] [PubMed]

]. The box geometry allows the characterization of two-quantum coherences [14

14. K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324, 1169–1173 (2009). [CrossRef] [PubMed]

, 15

15. D. Karaiskaj, A. D. Bristow, L. Yang, X. Dai, R. P. Mirin, S. Mukamel, and S. T. Cundiff, “Two-quantum many-body coherences in two-dimensional fourier-transform spectra of exciton resonances in semiconductor quantum wells,” Phys. Rev. Lett. 104, 117401 (2010). [CrossRef] [PubMed]

], and can provide long phase-locked inter-pulse delays if active stabilization is used [22

22. A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80, 073108 (2009). [CrossRef] [PubMed]

]. However, MDCS in the box geometry requires additional procedures to obtain real and imaginary parts of the multidimensional spectra [24

24. S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A 103, 10489–10505 (1999). [CrossRef]

27

27. E. H. G. Backus, S. Garrett-Roe, and P. Hamm, “Phasing problem of heterodyne-detected two-dimensional infrared spectroscopy,” Opt. Lett. 33, 2665–2667 (2008). [CrossRef] [PubMed]

]. 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

28. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15, 16681–16689 (2007). [CrossRef] [PubMed]

], 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

23. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Obtaining absorptive line shapes in two-dimensional infrared vibrational correlation spectra,” Phys. Rev. Lett. 90, 047401 (2003). [CrossRef] [PubMed]

]) are directly recorded. Rephasing and non-rephasing components can be separated using phase-cycling algorithms [29

29. J. A. Myers, K. L. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express 16, 17420–17428 (2008). [CrossRef] [PubMed]

,30

30. S.-H. Shim and M. T. Zanni, “How to turn your pumpprobe instrument into a multidimensional spectrometer: 2D IR and vis spectroscopies via pulse shaping,” Phys. Chem. Chem. Phys. 11, 748–761 (2009). [CrossRef] [PubMed]

]. 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

31. M. Aeschlimann, T. Brixner, A. Fischer, C. Kramer, P. Melchior, W. Pfeiffer, C. Schneider, C. Strüber, P. Tuchscherer, and D. V. Voronine, “Coherent two-dimensional nanoscopy,” Science 333, 1723–1726 (2011). [CrossRef] [PubMed]

]. 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

31. M. Aeschlimann, T. Brixner, A. Fischer, C. Kramer, P. Melchior, W. Pfeiffer, C. Schneider, C. Strüber, P. Tuchscherer, and D. V. Voronine, “Coherent two-dimensional nanoscopy,” Science 333, 1723–1726 (2011). [CrossRef] [PubMed]

34

34. C. Li, W. Wagner, M. Ciocca, and W. S. Warren, “Multiphoton femtosecond phase-coherent two-dimensional electronic spectroscopy,” J. Chem. Phys. 126, 164307 (2007). [CrossRef] [PubMed]

], selection of the FWM signal in the frequency domain [35

35. P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

], or subtraction of single-pulse and two-pulse contributions in the THz range [9

9. W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Two-dimensional terahertz correlation spectra of electronic excitations in semiconductor quantum wells,” J. Phys. Chem. B 115, 5448–5455 (2011). [CrossRef]

]. Phase cycling can also be used in non-collinear geometries to improve signal-to-noise by suppressing scattering from single pulses [22

22. A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80, 073108 (2009). [CrossRef] [PubMed]

, 29

29. J. A. Myers, K. L. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express 16, 17420–17428 (2008). [CrossRef] [PubMed]

, 30

30. S.-H. Shim and M. T. Zanni, “How to turn your pumpprobe instrument into a multidimensional spectrometer: 2D IR and vis spectroscopies via pulse shaping,” Phys. Chem. Chem. Phys. 11, 748–761 (2009). [CrossRef] [PubMed]

].

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

35. P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

]. 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

36. Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

]) 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 0th diffraction order of the AOM is spatially blocked and the 1st order beam is used. Each AOM is driven with a unique radio frequency ωi, with i = {A, B, C, D}. By scattering on the acoustic grating, the optical carrier frequency ω0 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 = ω0 + ωi. Although much smaller than the laser pulse bandwidth, the frequency shift ωi does impact the laser beam when its effect is considered on a train of pulses (whose interference produces a frequency comb [37

37. S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003). [CrossRef]

]), 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.

Fig. 1 (a) Pulse sequence used in the experiment. The four pulses form a collinear sequence. In a simple picture, the black line shows the first order polarization, second order population, and third order polarization generated by pulses A, B, and C, respectively. (b) Scheme of the experimental setup (described in text). Note: the cw laser is actually vertically offset, but shown horizontally offset in the figure for clarity.

As a simple example, the photocurrent signal obtained from a combination of pulses A and B oscillates at the difference frequency (beat note) ωAB = ωAωB. Dual phase lock-in detection is used to isolate and demodulate this signal. A continuous wave (cw) laser (external cavity diode laser, 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* = |νsigνcw| 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 δτ, scales as δτ · ν* [35

35. P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

, 38

38. P. F. Tekavec, T. R. Dyke, and A. H. Marcus, “Wave packet interferometry and quantum state reconstruction by acousto-optic phase modulation,” J. Chem. Phys. 125, 194303 (2006). [CrossRef] [PubMed]

]. 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

39. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001). [CrossRef] [PubMed]

, 40

40. D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence in self-assembled quantum dots,” Phys. Rev. Lett. 87, 227401 (2001). [CrossRef] [PubMed]

]. The use of an auxiliary cw laser contrasts with the experiment of Tekavec et al [35

35. P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

], 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

36. Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

]). Stepping the delay τ, we perform a field auto-correlation of the excitation pulse. The signal, oscillating in real time at the frequency ωAB, 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 τ at the reduced frequency ν* = |νsigνcw| ∼ 3.5 THz. Since the phase and amplitude of the signal are known, a Fourier transform with respect to τ provides us with the one-dimensional power spectrum of the excitation laser (Fig. 2(b)).

Fig. 2 (a) Result of a pulse-pulse correlation between pulses A and B, recorded on a broadband detector. X and Y are the in-phase and in-quadrature outputs of the lock-in amplifier, corresponding to the real and imaginary parts of the two-wave mixing signal Z = X + iY. R=X2+Y2 is the field amplitude. X and Y oscillate with τ at the reduced frequency ν* ∼ 3.5 THz (see text). (b) Fast Fourier transform (FFT) of the complex signal. As the detector is broadband, |FFT(Z)| (red curve) provides the power spectrum of the laser. The relative spectral phase is given by arg(FFT(Z)) (blue dots). A normalized spectrum of the excitation laser, recorded with an Ocean Optics USB 4000 spectrometer [36], is plotted for comparison (green circles).

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 ωAB and ωCD 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

36. Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

]). The mixing algorithm (in-quadrature mixing) is as follows: the beat notes cos(ωABt*) and cos(ωCDt*) (with t* being the real time) are input into the DSP, where they independently undergo a Hilbert transform [41

41. F. W. King, Hilbert Transforms, vol. 1 & 2 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2009).

]. For a general function, the Hilbert transform shifts the phase of positive frequency components by π2. E.g., the Hilbert transform of cos(ωABt*) is:
H[cos(ωABt*)]=sin(ωABt*).
(1)

After performing the Hilbert transform we then have both the in-quadrature cos(ωABt*) and in-phase sin(ωABt*) components of the beat note (and the same thing is done for cos(ωCDt*)). These components are then appropriately multiplied with each other, and we make use of the identity
cos(ωABt*±ωCDt*)=cos(ωABt*)cos(ωCDt*)sin(ωABt*)sin(ωCDt*).
(2)

This process provides reference frequencies
ωSI=ωCDωAB=ωA+ωB+ωCωD
(3)
ωSII=ωCD+ωAB=ωAωB+ωCωD
(4)
for the lock-in detection of the photocurrent FWM signal. These frequencies are analogous to the phase-matched directions k⃗FWM = − k⃗A + k⃗B + k⃗C and k⃗FWM = k⃗Ak⃗B + k⃗C of non-collinear MDCS, corresponding to the so-called rephasing (or SI) and non-rephasing (or SII) pulse sequences, respectively. In the box geometry, SI and SII need to be recorded separately since they correspond to a different pulse sequence, while with frequency domain selection SI and SII can be recorded simultaneously on two separate lock-in amplifiers. It is also possible to detect two-quantum (or SIII) signals that oscillate at the radio frequency ωSIII = ωA + ωBωCωD. 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.

3. Analogy with phase-cycling

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 nth pulses of train i and j as Δϕi,j = (ωiωj)(nTrep). In our situation, we obtain
ΔϕA,B=ωAB(nTrep)
(7)
ΔϕC,D=ωCD(nTrep).
(8)

Fig. 3 Illustration of how the phase modulation by AOM’s can be seen as a dynamic, pulse-to-pulse phase cycling between the four pulse trains. The four pulse trains (A,B,C,D) are represented as a function of real time t*. For simplicity, delays τ, T and t are set to 0. Each beam is modulated by a separate AOM, and thus shifted by a unique radio frequency. This leads to an effective carrier-envelope offset frequency that is different for every beam. Δϕi is the pulse-to-pulse carrier-envelope phase shift for the pulse train i. Phase differences ΔϕA,B and ΔϕC,D are shown (modulo 2π) for pulse n = 2.

4. Results

We demonstrate the working principle of our setup on a double In0.2Ga0.8As/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

42. J. G. Graeme, Photodiode Amplifiers: OP AMP Solutions (McGraw Hill Professional, 1996).

, 43

43. See also application note from http://cds.linear.com/docs/en/datasheet/6244fb.pdf.

] 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

36. Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

]) inputs (Fig. 4).

Fig. 4 Scheme of the electronic circuitry used to detect photocurrent from the sample and to generate lock-in references for rephasing (SI) and non-rephasing (SII) signals. MO: microscope objective. REF: reference photodetector. DSP: digital signal processor.

The AOM frequencies that we use are ωA = 80.109 MHz, ωB = 80.104 MHz, ωC = 80.019 MHz, ωD = 80 MHz. The beat notes recorded by reference detectors REF1 and REF2 as a result of the cw laser interference are then ωAB = 5 kHz and ωCD = 19 kHz, respectively. As a result of in-quadrature mixing by the DSP, the reference frequencies provided to the lock-in amplifiers for the SI and SII FWM signal are ωSI = 14kHz and ωSII = 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

22. A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80, 073108 (2009). [CrossRef] [PubMed]

].

Figure 5 shows 2D spectra that were recorded from the sample. A forward bias Vb = 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

36. Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

]), providing an excitation spot of ∼5 μm 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 h̄ωτ and h̄ωt. Non-rephasing (SII - Fig. 5(a)–(b)) and rephasing ((SI - Fig. 5(c)–(d)) spectra were obtained simultaneously from two separate lock-in amplifiers.

Fig. 5 (a) Absolute value, and (b) real part of the 2D spectrum recorded on the double InGaAs QW sample, using a non-rephasing pulse sequence (SII). The spectra are plotted as a function of h̄ωτ and h̄ωt. (c) Absolute value, and (d) real part of the 2D spectrum recorded using a rephasing pulse sequence (SI). The data to produce (a), (b), (c) and (d) were collected simultaneously.

The absolute value of SII and SI spectra (Fig. 5(a) and (c)) show a peak on the diagonal corresponding to the lowest energy exciton of the double QW. In the SI 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 τ and t delays [35

35. P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

]: arg{Z(τ = 0, T = 200 fs, t = 0)} = 0. We show in Fig. 5(b) and (d) the real parts of the SI and SII spectra, respectively, exhibiting a typical absorptive line shape.

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

44. A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, and G. Abstreiter, “Coherent properties of a two-level system based on a quantum-dot photodiode,” Nature 418, 612–614 (2002). [CrossRef] [PubMed]

,45

45. M. Zecherle, C. Ruppert, E. C. Clark, G. Abstreiter, J. J. Finley, and M. Betz, “Ultrafast few-fermion optoelectronics in a single self-assembled InGaAs/GaAs quantum dot,” Phys. Rev. B 82, 125314 (2010). [CrossRef]

], carbon nanotubes [46

46. W. Bao, M. Melli, N. Caselli, F. Riboli, D. S. Wiersma, M. Staffaroni, H. Choo, D. F. Ogletree, S. Aloni, J. Bokor, S. Cabrini, F. Intonti, M. B. Salmeron, E. Yablonovitch, P. J. Schuck, and A. Weber-Bargioni, “Mapping local charge recombination heterogeneity by multidimensional nanospectroscopic imaging,” Science 338, 1317–1321 (2012). [CrossRef] [PubMed]

], and nanowires [47

47. J. Wang, M. S. Gudiksen, X. Duan, Y. Cui, and C. M. Lieber, “Highly polarized photoluminescence and photodetection from single indium phosphide nanowires,” Science 293, 1455–1457 (2001). [CrossRef] [PubMed]

49

49. P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. Fontcuberta i Morral, “Single-nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013). [CrossRef]

], 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

12. H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, “Unraveling quantum pathways using optical 3D fourier-transform spectroscopy,” Nat. Commun. 4, 1390 (2013). [CrossRef] [PubMed]

, 16

16. J. A. Davis, C. R. Hall, L. V. Dao, K. A. Nugent, H. M. Quiney, H. H. Tan, and C. Jagadish, “Three-dimensional electronic spectroscopy of excitons in asymmetric double quantum wells,” J. Chem. Phys 135, 044510 (2011). [CrossRef] [PubMed]

], 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).

References and links

1.

S. T. Cundiff and S. Mukamel, “Optical multidimensional coherent spectroscopy,” Phys. Today 66, 44 (2013).

2.

R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions (Oxford Uni. Press, London/New York, 1987).

3.

D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54, 425–463 (2003). [CrossRef] [PubMed]

4.

M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. USA 97, 8219–8224 (2000). [CrossRef]

5.

S. Woutersen and P. Hamm, “Nonlinear two-dimensional vibrational spectroscopy of peptides,” J. Phys.: Condens. Matter14, R1035 (2002). [CrossRef]

6.

M. Khalil, N. Demirdöven, and A. Tokmakoff, “Coherent 2D IR spectroscopy: molecular structure and dynamics in solution,” J. Phys. Chem. A 107, 5258–5279 (2003). [CrossRef]

7.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434, 625–628 (2005). [CrossRef] [PubMed]

8.

X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. 96, 057406 (2006). [CrossRef] [PubMed]

9.

W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Two-dimensional terahertz correlation spectra of electronic excitations in semiconductor quantum wells,” J. Phys. Chem. B 115, 5448–5455 (2011). [CrossRef]

10.

R. Singh, T. M. Autry, G. Nardin, G. Moody, H. Li, K. Pierz, M. Bieler, and S. T. Cundiff, “Anisotropic homogeneous linewidth of the heavy-hole exciton in (110)-oriented GaAs quantum wells,” Phys. Rev. B 88, 045304 (2013). [CrossRef]

11.

X. Dai, A. D. Bristow, D. Karaiskaj, and S. T. Cundiff, “Two-dimensional fourier-transform spectroscopy of potassium vapor,” Phys. Rev. A 82, 052503 (2010). [CrossRef]

12.

H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, “Unraveling quantum pathways using optical 3D fourier-transform spectroscopy,” Nat. Commun. 4, 1390 (2013). [CrossRef] [PubMed]

13.

M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, “Resonance lineshapes in two-dimensional fourier transform spectroscopy,” Opt. Express 18, 17699–17708 (2010). [CrossRef] [PubMed]

14.

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324, 1169–1173 (2009). [CrossRef] [PubMed]

15.

D. Karaiskaj, A. D. Bristow, L. Yang, X. Dai, R. P. Mirin, S. Mukamel, and S. T. Cundiff, “Two-quantum many-body coherences in two-dimensional fourier-transform spectra of exciton resonances in semiconductor quantum wells,” Phys. Rev. Lett. 104, 117401 (2010). [CrossRef] [PubMed]

16.

J. A. Davis, C. R. Hall, L. V. Dao, K. A. Nugent, H. M. Quiney, H. H. Tan, and C. Jagadish, “Three-dimensional electronic spectroscopy of excitons in asymmetric double quantum wells,” J. Chem. Phys 135, 044510 (2011). [CrossRef] [PubMed]

17.

G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F. Morier-Genoud, and S. T. Cundiff, “Coherent excitonic coupling in an asymmetric double InGaAs quantum well,” arXiv e-print 1308.1689 (2013).

18.

G. Moody, M. E. Siemens, A. D. Bristow, X. Dai, D. Karaiskaj, A. S. Bracker, D. Gammon, and S. T. Cundiff, “Exciton-exciton and exciton-phonon interactions in an interfacial GaAs quantum dot ensemble,” Phys. Rev. B 83, 115324 (2011). [CrossRef]

19.

J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2011). [CrossRef]

20.

F. Albert, K. Sivalertporn, J. Kasprzak, M. Strauss, C. Schneider, S. Höfling, M. Kamp, A. Forchel, S. Reitzenstein, E. A. Muljarov, and W. Langbein, “Microcavity controlled coupling of excitonic qubits,” Nat. Commun. 4, 1747 (2013). [CrossRef] [PubMed]

21.

T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, “Phase-stabilized two-dimensional electronic spectroscopy,” J. Chem. Phys. 121, 4221–4236 (2004). [CrossRef] [PubMed]

22.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum. 80, 073108 (2009). [CrossRef] [PubMed]

23.

M. Khalil, N. Demirdöven, and A. Tokmakoff, “Obtaining absorptive line shapes in two-dimensional infrared vibrational correlation spectra,” Phys. Rev. Lett. 90, 047401 (2003). [CrossRef] [PubMed]

24.

S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A 103, 10489–10505 (1999). [CrossRef]

25.

T. Zhang, C. Borca, X. Li, and S. Cundiff, “Optical two-dimensional fourier transform spectroscopy with active interferometric stabilization,” Opt. Express 13, 7432–7441 (2005). [CrossRef] [PubMed]

26.

A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express 16, 18017–18027 (2008). [CrossRef] [PubMed]

27.

E. H. G. Backus, S. Garrett-Roe, and P. Hamm, “Phasing problem of heterodyne-detected two-dimensional infrared spectroscopy,” Opt. Lett. 33, 2665–2667 (2008). [CrossRef] [PubMed]

28.

E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15, 16681–16689 (2007). [CrossRef] [PubMed]

29.

J. A. Myers, K. L. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express 16, 17420–17428 (2008). [CrossRef] [PubMed]

30.

S.-H. Shim and M. T. Zanni, “How to turn your pumpprobe instrument into a multidimensional spectrometer: 2D IR and vis spectroscopies via pulse shaping,” Phys. Chem. Chem. Phys. 11, 748–761 (2009). [CrossRef] [PubMed]

31.

M. Aeschlimann, T. Brixner, A. Fischer, C. Kramer, P. Melchior, W. Pfeiffer, C. Schneider, C. Strüber, P. Tuchscherer, and D. V. Voronine, “Coherent two-dimensional nanoscopy,” Science 333, 1723–1726 (2011). [CrossRef] [PubMed]

32.

D. Keusters, H.-S. Tan, and W. S. Warren, “Role of pulse phase and direction in two-dimensional optical spectroscopy,” J. Phys. Chem. A 103, 10369–10380 (1999). [CrossRef]

33.

P. Tian, D. Keusters, Y. Suzaki, and W. S. Warren, “Femtosecond phase-coherent two-dimensional spectroscopy,” Science 300, 1553–1555 (2003). [CrossRef] [PubMed]

34.

C. Li, W. Wagner, M. Ciocca, and W. S. Warren, “Multiphoton femtosecond phase-coherent two-dimensional electronic spectroscopy,” J. Chem. Phys. 126, 164307 (2007). [CrossRef] [PubMed]

35.

P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127, 214307 (2007). [CrossRef] [PubMed]

36.

Mention of commercial products is for information only ; it does not imply NIST recommendation or endorsment, nor does it imply that the products mentioned are necessarily the best available for the purpose.

37.

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003). [CrossRef]

38.

P. F. Tekavec, T. R. Dyke, and A. H. Marcus, “Wave packet interferometry and quantum state reconstruction by acousto-optic phase modulation,” J. Chem. Phys. 125, 194303 (2006). [CrossRef] [PubMed]

39.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001). [CrossRef] [PubMed]

40.

D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence in self-assembled quantum dots,” Phys. Rev. Lett. 87, 227401 (2001). [CrossRef] [PubMed]

41.

F. W. King, Hilbert Transforms, vol. 1 & 2 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2009).

42.

J. G. Graeme, Photodiode Amplifiers: OP AMP Solutions (McGraw Hill Professional, 1996).

43.

See also application note from http://cds.linear.com/docs/en/datasheet/6244fb.pdf.

44.

A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, and G. Abstreiter, “Coherent properties of a two-level system based on a quantum-dot photodiode,” Nature 418, 612–614 (2002). [CrossRef] [PubMed]

45.

M. Zecherle, C. Ruppert, E. C. Clark, G. Abstreiter, J. J. Finley, and M. Betz, “Ultrafast few-fermion optoelectronics in a single self-assembled InGaAs/GaAs quantum dot,” Phys. Rev. B 82, 125314 (2010). [CrossRef]

46.

W. Bao, M. Melli, N. Caselli, F. Riboli, D. S. Wiersma, M. Staffaroni, H. Choo, D. F. Ogletree, S. Aloni, J. Bokor, S. Cabrini, F. Intonti, M. B. Salmeron, E. Yablonovitch, P. J. Schuck, and A. Weber-Bargioni, “Mapping local charge recombination heterogeneity by multidimensional nanospectroscopic imaging,” Science 338, 1317–1321 (2012). [CrossRef] [PubMed]

47.

J. Wang, M. S. Gudiksen, X. Duan, Y. Cui, and C. M. Lieber, “Highly polarized photoluminescence and photodetection from single indium phosphide nanowires,” Science 293, 1455–1457 (2001). [CrossRef] [PubMed]

48.

L. Cao, J. S. White, J.-S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8, 643–647 (2009). [CrossRef] [PubMed]

49.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. Fontcuberta i Morral, “Single-nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013). [CrossRef]

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(230.5590) Optical devices : Quantum-well, -wire and -dot devices
(300.6250) Spectroscopy : Spectroscopy, condensed matter
(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

ToC Category:
Spectroscopy

History
Original Manuscript: September 17, 2013
Revised Manuscript: November 1, 2013
Manuscript Accepted: November 2, 2013
Published: November 13, 2013

Citation
Gaël Nardin, Travis M. Autry, Kevin L. Silverman, and S. T. Cundiff, "Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure," Opt. Express 21, 28617-28627 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28617


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References

  1. S. T. Cundiff and S. Mukamel, “Optical multidimensional coherent spectroscopy,” Phys. Today66, 44 (2013).
  2. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions (Oxford Uni. Press, London/New York, 1987).
  3. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem.54, 425–463 (2003). [CrossRef] [PubMed]
  4. M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. USA97, 8219–8224 (2000). [CrossRef]
  5. S. Woutersen and P. Hamm, “Nonlinear two-dimensional vibrational spectroscopy of peptides,” J. Phys.: Condens. Matter14, R1035 (2002). [CrossRef]
  6. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Coherent 2D IR spectroscopy: molecular structure and dynamics in solution,” J. Phys. Chem. A107, 5258–5279 (2003). [CrossRef]
  7. T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature434, 625–628 (2005). [CrossRef] [PubMed]
  8. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett.96, 057406 (2006). [CrossRef] [PubMed]
  9. W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Two-dimensional terahertz correlation spectra of electronic excitations in semiconductor quantum wells,” J. Phys. Chem. B115, 5448–5455 (2011). [CrossRef]
  10. R. Singh, T. M. Autry, G. Nardin, G. Moody, H. Li, K. Pierz, M. Bieler, and S. T. Cundiff, “Anisotropic homogeneous linewidth of the heavy-hole exciton in (110)-oriented GaAs quantum wells,” Phys. Rev. B88, 045304 (2013). [CrossRef]
  11. X. Dai, A. D. Bristow, D. Karaiskaj, and S. T. Cundiff, “Two-dimensional fourier-transform spectroscopy of potassium vapor,” Phys. Rev. A82, 052503 (2010). [CrossRef]
  12. H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, “Unraveling quantum pathways using optical 3D fourier-transform spectroscopy,” Nat. Commun.4, 1390 (2013). [CrossRef] [PubMed]
  13. M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, “Resonance lineshapes in two-dimensional fourier transform spectroscopy,” Opt. Express18, 17699–17708 (2010). [CrossRef] [PubMed]
  14. K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science324, 1169–1173 (2009). [CrossRef] [PubMed]
  15. D. Karaiskaj, A. D. Bristow, L. Yang, X. Dai, R. P. Mirin, S. Mukamel, and S. T. Cundiff, “Two-quantum many-body coherences in two-dimensional fourier-transform spectra of exciton resonances in semiconductor quantum wells,” Phys. Rev. Lett.104, 117401 (2010). [CrossRef] [PubMed]
  16. J. A. Davis, C. R. Hall, L. V. Dao, K. A. Nugent, H. M. Quiney, H. H. Tan, and C. Jagadish, “Three-dimensional electronic spectroscopy of excitons in asymmetric double quantum wells,” J. Chem. Phys135, 044510 (2011). [CrossRef] [PubMed]
  17. G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F. Morier-Genoud, and S. T. Cundiff, “Coherent excitonic coupling in an asymmetric double InGaAs quantum well,” arXiv e-print 1308.1689 (2013).
  18. G. Moody, M. E. Siemens, A. D. Bristow, X. Dai, D. Karaiskaj, A. S. Bracker, D. Gammon, and S. T. Cundiff, “Exciton-exciton and exciton-phonon interactions in an interfacial GaAs quantum dot ensemble,” Phys. Rev. B83, 115324 (2011). [CrossRef]
  19. J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics5, 57–63 (2011). [CrossRef]
  20. F. Albert, K. Sivalertporn, J. Kasprzak, M. Strauss, C. Schneider, S. Höfling, M. Kamp, A. Forchel, S. Reitzenstein, E. A. Muljarov, and W. Langbein, “Microcavity controlled coupling of excitonic qubits,” Nat. Commun.4, 1747 (2013). [CrossRef] [PubMed]
  21. T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, “Phase-stabilized two-dimensional electronic spectroscopy,” J. Chem. Phys.121, 4221–4236 (2004). [CrossRef] [PubMed]
  22. A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional fourier-transform spectroscopy,” Rev. Sci. Instrum.80, 073108 (2009). [CrossRef] [PubMed]
  23. M. Khalil, N. Demirdöven, and A. Tokmakoff, “Obtaining absorptive line shapes in two-dimensional infrared vibrational correlation spectra,” Phys. Rev. Lett.90, 047401 (2003). [CrossRef] [PubMed]
  24. S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A103, 10489–10505 (1999). [CrossRef]
  25. T. Zhang, C. Borca, X. Li, and S. Cundiff, “Optical two-dimensional fourier transform spectroscopy with active interferometric stabilization,” Opt. Express13, 7432–7441 (2005). [CrossRef] [PubMed]
  26. A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express16, 18017–18027 (2008). [CrossRef] [PubMed]
  27. E. H. G. Backus, S. Garrett-Roe, and P. Hamm, “Phasing problem of heterodyne-detected two-dimensional infrared spectroscopy,” Opt. Lett.33, 2665–2667 (2008). [CrossRef] [PubMed]
  28. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express15, 16681–16689 (2007). [CrossRef] [PubMed]
  29. J. A. Myers, K. L. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express16, 17420–17428 (2008). [CrossRef] [PubMed]
  30. S.-H. Shim and M. T. Zanni, “How to turn your pumpprobe instrument into a multidimensional spectrometer: 2D IR and vis spectroscopies via pulse shaping,” Phys. Chem. Chem. Phys.11, 748–761 (2009). [CrossRef] [PubMed]
  31. M. Aeschlimann, T. Brixner, A. Fischer, C. Kramer, P. Melchior, W. Pfeiffer, C. Schneider, C. Strüber, P. Tuchscherer, and D. V. Voronine, “Coherent two-dimensional nanoscopy,” Science333, 1723–1726 (2011). [CrossRef] [PubMed]
  32. D. Keusters, H.-S. Tan, and W. S. Warren, “Role of pulse phase and direction in two-dimensional optical spectroscopy,” J. Phys. Chem. A103, 10369–10380 (1999). [CrossRef]
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