Coupling Map

Photoluminescence excitation (PLE) spectroscopy is a common tech-nique wherein the excitation laser scans over excited states while the photo-luminescence (PL) from the ground state is monitored (c.f. [94–96]). In this fashion, the energy level structure of a particular QD can be mapped. In the past, energies very close to the ground state could not be probed due to laser scattering overwhelming the ground state PL [96]. With the side-excitation technique, however, ability to resonantly probe states without the detector be-coming saturated by laser scattering allows unprecedented measurements on states very close to the excitation energy. Further, the ground state itself can be resonantly excited and emission from higher-energy states can sometimes be observed (see Chap. 5 for more details).

With laser scattering minimized, a PLE spectrum does not have to halt at the ground state and the energy level strucure of multiple dots can be mapped at once. If multiple emission spectra are recorded as the excitation laser is scanned over the wavelength range of interest and the spectra are stacked vertically according to the corresponding excitation wavelength, a 2-dimensional map of energy level coupling can be formed such as the one shown in Fig. 3.11. Here color indicates intensity of emission on a logarithmic scale.

Peaks along the diagonal correspond to resonance fluorescence of different states, i.e. excitation and emission at the same energy. The off-diagonal peaks show couplings between states: those on the lower right are down-conversion from excited states to lower energy states; those on the upper left are actually up-conversion to higher energy states, a phenomenon explored more fully in Chap. 5. The dotted and dashed lines in Fig. 3.11 highlight the couplings between states and clearly show two sets of states which are not coupled to each other. A 2-D coupling map is a vivid presentation of the sometimes quite complex structure of quantum dot energy levels and can be very useful in separating the structure of different dots which are close both spatially and energetically.

Figure 3.11 bears some superficial resemblance to the maps created using 2-D nonlinear optical spectroscopy [53, 55–58], but is different in both the kind of system probed and the data gleaned. Nonlinear spectroscopy in general depends on the polarization response of an ensemble to create the emission signal and allows detection of both the phase and amplitude of the polarization.

Figure 3.11: 2-D Coupling map of multiple quantum dots; the color represents the emission intensity on a logarithmic scale. The diagonal is resonance flu-orescence, and the off-diagonal peaks are up- or down-conversion. The lines connect coupled states; the dotted and dash-dotted lines correspond to two uncoupled sets of states.

The 2-D coupling map presented here only contains amplitude information, although only one or two quantum dots are probed and the ensemble behavior is irrelevant. In this sense it is similar to a recent form of transient nonlinear spectroscopy that was implemented by Langbein and Patton [59] to detect the nonlinear polarization of a single excitonic transition.

Chapter 4

Resonance Fluorescence

Resonance fluorescence is resonant emission from a resonantly excited two-level quantum system, which is the basic system described by the optical Bloch equations in Chap. 2. First described by Mollow in 1969 [97], resonance fluorescence was observed decades ago in atoms [69–72] using prependicular excitation and detection directions. In solid-state systems, however, resonance fluorescence has been impossible to observe because the scattered laser light is at the same frequency as the fluorescence and thus cannot be differentiated using a spectrometer as in above-band or quasi-resonant excitation. With the waveguide-mode excitation technique described in Chap. 3, however, we have observed the resonance fluorescence from a single quantum dot.

Coherent control of the exciton state is required for numerous applica-tions, and has been demonstrated in many previous works showing Rabi split-ting [98], Rabi oscillations [46, 99, 100], Autler-Townes splitsplit-ting and Mollow absorption [43], although these do not resonantly detect the photon emission.

Besides pump/probe type approaches, non-resonant exciton generation via ex-cited QD states or a continuum has previously been used for discriminating the photoluminescence from the strong laser background, but at the cost of

sacrificing coherence. For single photon sources, such incoherent excitation has also been shown to result in a trade-off between the indistinguishability of the emitted photons and the quantum efficiency of the emission [19]. Resonant excitation, however, does not suffer from this drawback and, combined with the ability to simultaneously collect the emission, will be critical for future applications in indistinguishable single photon sources and quantum informa-tion processing. Although simultaneous resonant excitainforma-tion and emission is routinely done in single atoms or ions [69–72], it has been difficult to achieve in the solid-state because scattered laser light overwhelms the weak single pho-ton emission. Recent demonstration of resonance fluorescence from single QDs in a microcavity [42] and single molecules under a solid immersion lens [45], however, will allow explorations of coherent control and emission in the solid-state. Of these two methods, the microcavity approach used here achieves orders of magnitude better background suppression and allows measurement of the Mollow emission spectrum with virtually no contamination from the excitation laser.

4.1 Basics

In our sample we focus on QDs coupled to a cavity mode centered around 915 nm, with a quality factor of about 250. When the laser is frequency-scanned over the excitonic ground-state of a single QD, the resonance fluores-cence is observed as a bright peak in the CCD images, localized both spectrally and spatially. In contrast, the remaining background laser light appears as a

-4 -2 2 4

Figure 4.1: (a) Spatially (ordinates) and spectrally (abscissas) resolved fluores-cence images for a single resonantly excited quantum dot. The residual laser appears as a faint line. (b) Spectral line profile of dot in (a) fit by a Lorentzian as a function of detuning between the laser and quantum dot resonance. (c) Linewidth temperature dependence fit by Eqn. (4.1). (d) Second-order corre-lation measurement of another single quantum dot in resonance fluorescence fit by Eqn. (4.2) (see section 4.3.1 for more detail).

faint vertical (i.e. spatially delocalized) line. In Fig. 4.1(a), a series of such CCD images at increasing excitation energy are shown. The laser bandwidth is less than 25 MHz, narrow enough that the total integrated intensity as a function of detuning measures the excitation linewidth of the ground state transition, as plotted explicitly in Fig. 4.1(b). For this particular dot we

ob-tain a full width at half maximum (FWHM) of 2.8 µeV (T₂ = 470 ps) at 4.7 K. A strong dependence on temperature is observed in Fig. 4.1(c) and is fit with the function

F W HM = aT + b 1

exp(~ω^LO/kBT )− 1, (4.1) where the linear term is due to acoustic phonon scattering and the second term is due to optical phonons [101]. All subsequent measurements are performed at 10 K as a compromise between reducing dephasing effects and minimizing liquid helium used to maintain a low temperature. Moreover, second-order correlation measurements, performed on single peaks using a Hanbury-Brown and Twiss setup (HBT) [102], reveal a pronounced anti-bunching dip shown in Fig. 4.1(d), confirming their single emitter nature (see section 4.3.1). The correlation is fit by the function

g⁽²⁾(τ ) = 1− T₁

which is the low intensity limit of Eqn. (4.10) (see section 4.3 and Appendix 1).

At low intensity, a harmonic driving field (amplitude E₀), which may be detuned from the transition frequency of the two-level system by an amount

∆ω = ω− ω⁰, initially increases the population of the upper state. When the field is so strong that the Rabi frequency Ω = µE₀ exceeds the total decoher-ence rate 1/T₂ in the system, however, the probability to find it in the upper state reaches a maximum before it decreases again. Here µ denotes the dipole

moment of the transition with resonance frequency ω₀. In fact, both the pop-ulations and the coherences of the system then oscillate at the Rabi frequency, which in the language of quantum computation corresponds to quantum bit ro-tations. Written out explicitly in the rotating wave approximation, the Bloch equations for the upper and lower state populations, n(t) = T r{ρ(t) |1i h1|}

and m(t) = T r{ρ(t) |0i h0|}, and for the coherence, α(t) = T r{ρ(t) |0i h1|},

Here ρ(t) is the density operator, and T₁ and T₂ denote the diagonal and off-diagonal phenomenological damping constants, respectively, and n(t)+m(t) = 1. The quasi steady-state solutions of Eqs. (4.3) and (4.4) are obtained as:

n∞(∆ω) = 1 and describe well-known saturation phenomena which are directly observed in the experiments with single dots, since the time-averaged fluorescence inten-sity is proportional to n∞(∆ω) . Specifically, one can see that (i) the total integrated fluorescence at resonance (∆ω = 0) saturates once the square Rabi frequency substantially exceeds the quantity (T₁T₂)⁻¹, (ii) that the linewidth of the Lorentzian in Eq. (4.6) increases slowly with the square root of intensity,

a phenomenon known as “power broadening”, and (iii) that the low intensity limit of the linewidth equals 2/T₂.

4.2 First-Order Correlation Function

More interesting is the actual shape of the fluorescence spectrum, which goes beyond the straightforward steady-state solutions. In fact, while the optical Bloch equations are directly borrowed from nuclear magnetic resonance theory, a comprehensive theoretical description of resonance fluorescence was only given in 1969 by Mollow [97]. He first obtained the two-time (first-order) correlation function g(t, τ ) =b^†(t)b(t + τ ) of the field emitted by the system, where b and b^† are the field operators which, when produced by spontaneous emission, are proportional to the atomic dipole operators |0i h1| and |1i h0|.

The complete resonance fluorescence spectrum was then derived as the Fourier transform of g(t, τ ) and results in the well-known “Mollow triplet”. Reduction to single time expectation values is done with the quantum regression theorem [103, 104]. Here we use a Michelson interferometer to measure the correlation function directly, obtained as:

g(τ ) =|α^∞(0)|²+n∞(0)

2 e^{−τ /T2}+n_∞(0)e^{−τ (1/T1+1/T2)/2}[N cos(Ω⁰τ )+M sin(Ω⁰τ )]

(4.8) based on the derivations in [97] and [61] but extended to include a constant de-phasing time T₂. The generalized Rabi frequency is Ω⁰ =pΩ²− (1/T1− 1/T2)²/4

and the constants N and N are

N = 1 2

Ω² − (1/T1− 1/T2)/T₁

Ω²+ 1/T₁T₂ (4.9a)

M = − 1 4Ω⁰

Ω²(1/T₂− 3/T¹) + (T₁− T²)²/T₁³T₂² Ω²+ 1/T1T2

(4.9b)

When Ω 1/T², then g(τ ) reduces to a simple exponential decay, with decay constant T2, corresponding to a Lorentzian spectral line profile of FWHM 2/T2. On the other hand, when Ω  1/T², the system is in the strong excitation regime and g(τ ) is oscillatory. Note that T₁ and T₂ are related through T₂⁻¹ = (2T₁)⁻¹ + γ, where γ denotes pure dephasing (i.e. loss of coherence without population decay).

4.2.1 Michelson Interferometer

The first-order correlation function of the emission is measured using a Michelson interferometer to interfere the emitted single photons with them-selves. The length of one interferometer arm, and thus the corresponding time delay, is varied coarsely over a number of points with a highly stable transla-tion stage, and finely using a piezo actuator to record the fringe contrast at each point. The contrast is defined as the difference between the maximum and minimum of the interference signal divided by their sum, and as a function of time delay corresponds to g(τ ). Such a technique, borrowed from Fourier spectroscopy [105–108], can routinely provide an equivalent spectral resolution of about 1 µeV, much smaller than is available with conventional grating-based spectrometers. The recorded fringe contrast is proportional to g(τ ). Figure

M1

M2

M5 M6

BS To Spectrometer CCD

Delay Arm Stationary Arm

From Objective M3

M4

M7

Figure 4.2: Interferometer schematic

4.2 shows a schematic of the interferometer and its relationship to the rest of the experimental setup. The spectrometer CCD camera was used as a very sensitive detector to record the intensity of the interfered light.

The variable arm of the interferometer had a translation range of 90 mm, and since the light must traverse that distance twice this results in a time delay variation of 600 ps. The time delay variation was placed such that the absolute time delay between the two arms ranged from -100 ps to +500 ps.

Since the fringe contrast is expected to be symmetric around zero time delay, this configuration allows the instrument to scan about 1.5 times T₂.

The alignment of the interferometer’s mirrors must be very precise to ensure proper operation and not cause an artificial decrease in the fringe con-trast at large time delays. It is fairly simple to align the output beams of the two arms at one given time delay, but maintaining that alignment as the time delay varies requires special attention to the colinearity of the beam and

the translation axis of the variable arm. A CW laser was coupled into an optical fiber and the output was placed at the focus of the objective lens.

This provided a bright light source with which to accomplish the preliminary alignment. An aperture was positioned on the translation stage in place of the mirrors and M1 was used to align the beam so that it passed through the aperture. Then the translation stage was moved to the largest possible time delay and M2 was used to align the beam to pass through the aperture.

Iterations of these two steps will eventually make the beam colinear with the axis of motion of the translation stage.

The next alignment step is to align the beam reflected back from the mirrors on the translation stage such that it is also colinear with the stage axis. The mirror pair was placed on the translations state in the path of the beam and the back-reflected beam was directed to a wall several meters away.

While the stage was at zero time delay, the location of the beam spot on the wall was marked, then the stage was translated to the largest time delay and the movement of the beam spot on the wall was noted. The second mirror of the pair, M4, was adjusted such that the beam spot on the wall did not move when the stage was translated. The combination of aligning the input and output beams to the stage axis made the spot location on the beamsplitter remain stationary when the stage was translated.

The final, and most complex, process of the alignment was to ensure that the beam from the other, stationary arm of the interferometer would overlap the first beam. This required that the beams not only hit the CCD

detector at the same place, but that they be colinear to provide the maximum interference. The quality of the alignment could be judged using the strong fiber light source and an IR-sensitive video camera to which the interferometer output could be directed. Blocking one of the intereferometer arms resulted in a single spot seen on the camera; blocking the other arm showed the sec-ond spot. The secsec-ond mirror on the stationary arm, M6, was used to overlap these two spots. When both beams were unblocked, there could be seen super-imposed on the two beam spots, an intensity modulation due to interference fringes. When the beams were far from colinear, the fringes were linear and could easily be seen to move when the translation stage was gently pushed to slightly change the time delay. When closer to colinear, the fringes had a distinct curvature, indicating that they were actually circular fringes observed far from the center of curvature. To bring the beams closer to colinear, the first mirror on the stationary arm, M5, was angled to bring the center of curva-ture of the fringes closer to the beam spots. This, of course, moved the spots farther from overlap, but iteration of the two processes of spot overlap and fringe centering gradually aligned the two beams such that the fringe center was on top of the beam spots when they were overlapped. As the fringe center got closer to the beam spots, the fringe spatial wavelength increased until the point at which pushing on the translation stage caused the entire intensity distribution to rise and fall as a unit. This is the point of perfect colinearity and interference of the two beams in the interferometer.

Figure 4.3 shows the fringe contrast as a function of path length

dif--1000 0 100 200 300 0.2

0.4 0.6 0.8 1

Path Length Difference (mm)

Fringe Contrast

Student Version of MATLAB

Figure 4.3: Characterization interferogram of the ring laser. The points are the fringe contrast as a function of coarse time-delay between the interferometer arms and are fit by a Gaussian resulting in a coherence length of 323± 18 mm.

ference for the ring laser measured using the interferometer. Since the laser will have a very long coherence length compared to the maximum path length difference in the interferometer, it provides a good measure of the quality of the instrument. The maximum fringe contrast was 93% and the measured co-herence length was 323 mm, giving a coco-herence time of 1.1 ns. From linewidth measurements of the laser light the predicted coherence time is be 6.4 ns, so the measured decrease in contrast at longer delay times must be due to the instrument itself. The limiting time of the instrument response, however, is longer than the expected decoherence times of the QDs, which are usually

< 400 ps. Therefore the interferometer is suitable for measuring the first-order correlation function of single photon emission from a single QD.

4.2.2 Analysis

For the same dot as in Fig. 4.1(a), the excitation linewidth is plotted as a function of intensity in Fig. 4.4(a) to illustrate power broadening. The broadening is an intrinsic feature of the resonantly driven two-level system described by Eq. (4.6). This is in contrast to the excitation-induced dephasing that often arises in solid-state spectroscopy whereby the spectral line width increases due to an increase in the non-resonant generation of carriers in the semiconductor matrix. As mentioned in the discussion of Eq. (4.6), the value of T₂ can be obtained from the low intensity limit of the linewidth; here T₂ = 380 ps. If T₁ is also known, these measurements provide the proportionality constant between the excitation intensity and the square Rabi frequency. At zero temperature there would be no dephasing due to phonons, and the low intensity resonant pump should not cause spectral broadening by creating nearby transient charges. Therefore T1 can be taken from the T → 0 limit of the linewidth in Fig. 4.1(c), which means T1 = 290 ps. With this extrapolated T₁ and measured T₂, we plot our data as a function of Ω² directly. The emission intensity as a function of excitation intensity is plotted in Fig. 4.4(b) and clearly shows the population saturation behavior predicted by Eq. (4.6) at intensities such that Ω (T¹T₂)^−1/2. Noted on the graphs of Fig. 4.4 are the thresholds for reaching the strong excitation regime.

For the same dot as in Figs. 4.1 and 4.4, we examine the resulting fringe contrast as a function of time delay; it is plotted in Fig. 4.5(a)-(e) for increasing excitation intensities, i.e. Rabi energies. Fits to the data are plotted

0 10 20 30 40

Figure 4.4: (a) Linewidth as a function of square Rabi energy. The resonances were recorded by scanning the laser over the quantum dot transition, and monitoring the total emitted intensity. (b) Resonance amplitude versus square Rabi energy extracted from the same data. The solid lines are obtained from the theoretically predicted steady state population as a function of detuning following Eq. (4.6). The precise functional form of the fits are noted on the plots.

on top of the data points using Eq. (4.8) and the above values of T₁ and T₂. The Rabi frequency, Ω, as well as an offset due to laser background and a renormalization are chosen to best fit the data; Rabi energies up to 13.3 µeV are found. Note that because the sample was cleaved along [110] or [110], only one of the fine-structure states is being excited, the emission is polarized, and we do not observe beatings in the correlation function such as reported in Ref.

[16].

When the intensity is weak, i.e. Ω 1/T², a single exponential decay is obtained. But with increasing intensity, the fringe contrast develops an os-cillatory feature at frequency Ω⁰, defined above, which approximately equals Ω if Ω 1/T². The oscillations can be understood as an amplitude modulation

When the intensity is weak, i.e. Ω 1/T², a single exponential decay is obtained. But with increasing intensity, the fringe contrast develops an os-cillatory feature at frequency Ω⁰, defined above, which approximately equals Ω if Ω 1/T². The oscillations can be understood as an amplitude modulation