Theoertical investigation of quantum waveform shaping for single photon emitters

Leno M. Pedrotti; Imad Agha

doi:10.1364/OE.24.016687

1. Introduction

In optics-based quantum information processing [1, 2], there is a strong need for systems capable of lossless frequency conversion, as different physical systems that constitute a complete quantum network (sources, circuits, memories, detectors) are often optimal at disparate wavelengths [3]. While atom-based quantum memories [4–6] and single-photon avalanche photodiodes [7, 8] are optimal in the visible and near-infrared, quantum circuits [9, 10] (especially fiber-based or silicon-based CMOS-compatible ones) often need to be in the the telecommunications band, where losses are low and optoelectronic components are readily available [11]. While quantum frequency conversion renders the central wavelengths of different elements compatible [12, 13], often an incompatibility resides in the the spectro-temporal properties [14, 15] (linewidth, spectro-temporal profile, coherence). For example, a triggered InAs self-assembled quantum dot emits a wavepacket in the near-IR with a natural lifetime near ≈ 1 ns, characterized by a mono-exponential decaying waveform. On the other hand, modern telecommunications networks and components are far better suited to Gaussian/flat-top pulses [16] that are far shorter in duration. Additionally, even if the photon emitter is a 2/3–level system, it may not be suited to act as a quantum memory (e.g. quantum dot), and a different species at a different color and with a different bandwidth is required. One possible solution to waveform shaping resides in systems based on cold atoms that allow shaping of narrow-band single photon wavepackets via three-level transitions and control of the input pump. However, such systems offer a limited bandwidth and require complex cooling and trapping mechanisms. On the other hand, ultra-broadband quantum states are amenable to classical dispersive pulse shaping techniques. Unfortunately, many quantum emitters, such as quantum dots and nitrogen-vacancy centers, have bandwidths that fall in between these two extremes, and a two step process that includes both spectral broadening and pulse shaping (phase manipulation) is optimal. Recently, further interest in waveform shaping has come from the quantum information community, with regards to a new form of information encoding on the temporal and spatial modes of a photon wavepacket, as opposed to encoding on its polarization state [17]. Temporal and spatial encoding of quantum states [17, 18] allow for higher dimensional encoding of quantum states and hence more information on a photon, yet require additonal elements to manipulate the qubits. Much like a polarization rotator is used in manipulating the polarization state of a single/entangled photon wavepacket, a waveform converter is indispensable when the quantum state/correlations are encoded in orthogonal temporal modes. While the time-lens approach was recently proposed and implemented as means to manipulate single photons through spectro-temporal compression/expansion, that approach does not carry the ability to convert the actual shape of the photon wavepacket [19, 20].

In order to preserve the quantum coherence and statistics of quantum states of light, both the frequency conversion and phase manipulation components (necessary in quantum waveform conversion (QWC)) have to be noise-free. Additionally, losses have to be kept at a minimum as any amplification/gain adds noise and corrupts the quantum state. The QWC process described in this work starts with temporal phase manipulation. This is readily achievable via nonlinear mixing with a chirped pump, whereby the temporal phase on the pump is imprinted on the signal, while the central frequency of the pump (via sum or difference-frequency generation) shifts the quantum state’s frequency. The second and final step is a spectral phase compensation step, which is again readily achieved via dispersive fibers, spatial-light modulators-based setups or phase plates.

In this paper we show that QWC via temporal phase manipulation and spectral phase compensation is capable of both temporally compressing Gaussian pulses, as well as converting mono-exponentially decaying pulses (characteristic of triggered single-photon emitters) to symmetric Lorentizian pulses of shorter duration (that have high degree of overlap with Gaussian wavepackets), through uniquely lossless phase operations. As this work involves phase-only operations, it preserves the photon statistics and coherence of single photons [15, 19, 20]. The simplicity of the phase functions we derive and the ready availability of technology to implement them makes this approach both optimal and highly realistic for current state-of-the-art single-photon emitters [21–24]. Our approach complements recent experimental work on spectral/temporal compression/broadening [19, 20], amplitude modulation [25], Purcell enhancement [26], and extends recent theoretical calculations and experimental demonstrations that were based on mathematical approximations and fairly complicated phase functions [14, 15]. We highlight the experimental requirements needed for complete spectro-temporal QWC for various systems in light of current technology and realistic experimental considerations.

2. Theory

Consider the general problem of lossless pulse reshaping. We consider an input field of the form Ψ_in(t) and a targeted output waveform Ψ_out(t). The output form is obtained by the following sequence of operations on the input waveform. A quadratic temporal phase factor characterized as φ(t) = Ft² is added to the input field yielding

Ψ_{1} (t) = Ψ_{in} (t) e^{i φ (t)} = Ψ_{in} (t) e^{i F t^{2}} .

The Fourier transform of this field can be written as

{\tilde{Ψ}}_{1} (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} Ψ_{in} (t) e^{i F t^{2} (t)} e^{- i ω t} d t .

A quadratic spectral phase of the form γ(ω) = Gω² can be added to Ψ̃₁(ω) to produce the inverse Fourier transform of the desired output waveform, that is,

{\tilde{Ψ}}_{out} (ω) = {\tilde{Ψ}}_{1} (ω) e^{i G ω^{2}},

and

Ψ_{out} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\tilde{Ψ}}_{1} (ω) e^{i G ω^{2}} e^{i ω t} d ω .

Note that this process is lossless since only phase transformations are made. In the following subsections we show, for that first time, that this process can be applied to convert a Gaussian waveform to another Gaussian waveform of arbitrarily narrow width and to convert a one-sided exponential waveform to a Lorentzian waveform of arbitrarily narrow width. We note that these transformations are exact–no mathematical approximations are made– and the fidelity of these transformations is limited only by experimental considerations.

2.1. Gaussian to Gaussian transformation

Take the input waveform to be a Gaussian of width μ and the desired output waveform to be a Gaussian of width σ < μ:

Ψ_{in} = η e^{- \frac{t^{2}}{2 μ}} .

Using this form in Equation (2) gives

{\tilde{Ψ}}_{1} (ω) = η \frac{μ}{\sqrt{1 - 2 i μ^{2} F}} e^{- \frac{μ^{2} ω^{2}}{2 (1 - 2 i μ^{2} F)}} .

Choosing

F = - \frac{\sqrt{μ^{2} - σ^{2}}}{2 σ μ^{2}}

leads to

{\tilde{Ψ}}_{1} (ω) = η \sqrt{μ σ} e^{- i θ / 2} e^{- i σ \sqrt{μ^{2} - σ^{2}} ω^{2} / 2} e^{- σ^{2} ω^{2} / 2},

where

θ = {tan}^{- 1} (\sqrt{μ^{2} - σ^{2}} / 2)

is an unimportant phase factor. Performing the spectral phase transformation indicated in Equation (3) with

G = i σ \sqrt{μ^{2} - σ^{2}} / 2

gives the desired output Gaussian waveform of width σ,

{\tilde{Ψ}}_{out} (ω) = η \sqrt{μ σ} e^{- i θ / 2} e^{- σ^{2} ω^{2} / 2},

and,

Ψ_{out} (t) = η \sqrt{\frac{μ}{σ}} e^{- i θ / 2} e^{- \frac{t^{2}}{2 σ^{2}}} .

Note that since only phase transformations are made, the process is lossless. That is,

\int_{- \infty}^{\infty} {| Ψ_{in} (t) |}^{2} d t = \int_{- \infty}^{\infty} {| Ψ_{out} (t) |}^{2} d t = {| η |}^{2} \sqrt{π} μ .

The development given above extends the work of Wisemann et. al [14] who made a stationary phase approximation to show that an approximate Gaussian to Gaussian transformation could be made if σ ≪ μ. We have developed the recipe for an exact Gaussian to Gaussian transformation valid so long as σ < μ.

2.2. One-sided exponential to Lorentzian transformation

The case of converting a non-symmetric single photon to a symmetric Lorentzian/Gaussian is of practical significance, as symmetric pulses with smooth Gaussian-like shapes are a) compatible with standard optical communications components and b) are optimal for quantum information processing [27]. Take the input waveform to be a one-sided exponential of the form

Ψ_{in} (t) = {\begin{array}{l} 0 & t < 0 \\ η e^{- t / τ} & t > 0 \end{array} .

Using this form in Equation (2) gives

{\tilde{Ψ}}_{1} (ω) = - \sqrt{i} η \frac{erfc (\frac{\sqrt{- i} (i - ω τ)}{2 τ \sqrt{F}}) e^{- \frac{ω}{2 τ F} + \frac{i}{4 τ^{2} F} - \frac{i ω^{2}}{4 F}}}{2 \sqrt{2 F}} .

Here, erfc(x) is the complimentary error function. Performing the spectral phase transformation indicated in Equation (3) with

G = \frac{1}{4 F},

leads to

{\tilde{Ψ}}_{out} (ω) = - \sqrt{i} η \frac{erfc (\frac{\sqrt{- i} (i - ω τ)}{2 τ \sqrt{F}}) e^{- \frac{ω}{2 τ F} + \frac{i}{4 F τ^{2}}}}{2 \sqrt{2 F}},

and

{\tilde{Ψ}}_{out} (t) = - \sqrt{i} η \sqrt{\frac{τ}{2 π}} \frac{1 / \sqrt{2 F τ}}{i t - \frac{1}{2 F τ}} e^{- i t^{2} F - \frac{i}{4 τ^{2} F}} .

Note that the magnitude-squared of the output waveform is a simple Lorentzian of width 1/(Fτ). The width of the Lorentzian can be made arbitrarily small by appropriate choice of the free parameter F. Again, we emphasize that this reshaping process is lossless in the sense that

\int_{- \infty}^{\infty} {| Ψ_{in} (t) |}^{2} d t = \int_{- \infty}^{\infty} {| Ψ_{out} (t) |}^{2} d t = {| η |}^{2} \frac{τ}{2} .

3. Experimental considerations

In techniques involving pulse shaping and compression, it is vital for any theory to match experimental realities, especially with regards to the requirements on the temporal/spectral phase compensation. In particular, imprinting a quadratic phase can be achieved by mixing the signal with a phase-modulated pump that in itself possesses a quadratic phase. We note here that, contrary to approach taken in [14, 15], the phase functions derived in this paper are both exact and simpler (quadratic) in nature, and hence can be implemented in a straighforward manner via off-the-shelf sinewave generators and dispersive fibers (It is important to note that this technique only works when the pulse is coherent across its entire temporal/spectral width, otherwise any effect of phase manipulation will only work on the phase-coherent portion of the pulse.)

Using a sinewave generator would be the most straight-forward manner of imparting a quadratic phase as the sine function can approximate a quadratic function very well up to 1/2 a period [28] as shown in Fig. 1. Using standard sinewave generators and RF amplifiers, it is reasonable to attain the regime of 20–30 ps as a lower bound for pulse compression. In the following example, we demonstrate the requirements on compressing a 100 ps pulse down to 10 ps. For such a compression, the requirements on the phase modulation are fairly modest: a 20 Ghz sinewave generator, and a phase modulator capable of generating ≈ 3π in modulation depth as seen in Fig. 2(a). After applying the temporal phase, the intermediate spectrum expands, while simultaneously a residual unwanted spectral phase is imparted on the pulse [15], as seen in Fig. 2(b).

Fig. 1 Comparison between sinewave and quadratic functions, as a function of time (in sinewave period units). The two functions match nearly perfectly up to half the period’s length

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Fig. 2 a) Input pulse (green) and necessary temporal phase (blue). The temporal phase can be imparted directly via a phase modulator, or by mixing the input with a chirped pump via four-wave or three-wave mixing. b) Intermediate pulse spectrum and necessary spectral phase compensation. The spectral phase can be achieved by passing the intermediate pulse through a dispersion-compensating module

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Recovering a transform limited pulse requires adding a spectral phase to de-chirp the output pulse. This can be readily achieved using a dispersion compensating fiber module, as group-velocity dispersion acts as a quadratic spectral phase that is imparted on the pulse [29]. Figure 2(b) shows both the intermediate spectrum as well as the necessary spectral phase that needs to be added to de-chirp the pulse. Such a phase corresponds to a G-parameter (Eq. (3), Eq. (9)) ≈ 1.2x10⁻²²s². At a wavelength of operation near 1550 nm, such a phase parameter can be achieved by passing the pulse through 1 km of readily available dispersion compensation modules (DCM) possessing a dispersion parameter of ≈ 50 ps/nm.km. After passing through 1 km of DCM, the output pulse is rendered Fourier-transform limited and is temporally compressed, as shown in Fig. 3.

Fig. 3 Input pulse (red) and output pulse (blue). The output pulse is temporally compressed and maintains the Gaussian nature of the input.

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The case of exponentially decaying pulses is more interesting as it illustrates the ability of this technique to not only temporally compress, but also shape the optical pulses themselves. Mono-exponential decays are characteristic of single photon emitters, such as diamond NV centers and semiconductor quantum dots, and, as such are of strong interest in the quantum information community. As the pulses are characterized by a short rise-time and long tail, they are generally not suited for off-the-shelf optoelectronic components, nor for telecommuications components that work best with symmetric waveforms that approach the Gaussian wavepacket limit. Moreover, temporal compression is attractive as a reduced duty cycle allows for temporal multiplexing and increased data rates. Additionally, a reduced overlap between the tail of one single photon emission and the start of the next helps in improving g⁽²⁾ detection using a Hanburry-Brown and Twiss setup by reducing the spill-over between conesecutive pulse time-bins [25].

As shown in the theory section, it is readily possible to convert such waveforms into a Lorentzian (in time) symmetric wavepacket. In fact, while the spectral phase imparted is still quadratic and on the order of 1 – 3π, the G-parameter (Eq. (14)) is slightly lower for similar pulse compression values. In fact, to convert a 500 ps exponentially decaying pulse, which is on the order of the width of a self-assembled semiconductor quantum dot emission, to a 100 ps FWHM Lorentzian, a quadratic temporal phase on the order of 2π is sufficient as can be seen from Fig. 4(a), while a G-parameter of 1.25x10⁻²⁰s² is necessary. While the same technique for quadratic temporal phase addition is employed, 10–20 meters of dispersion compensation fiber is sufficient in this case. It is important to stress here that, beyond what can be done with a time-lens setup, this case presents not only pulse compression or expansion, but also waveform shaping as the spectro-temporal waveform is fundametally changed, as shown in Fig. 4(b). This shows that it is possible, in principle, to shape pulses using phase-only operations in a regime inaccessible to standard pulse shaping equipent that rely on strong dispersive broadening in the femto-second regime. Additonally, being strictly a post-processing technique, our method does not require modification of the source/pumping scheme itself [26, 30] to achieve linewidth/temporal width modification.

Fig. 4 a) Input pulse (red) and necessary temporal phase (blue). b) Input pulse (red) and output pulse (blue). The output pulse is a symmetric Lorentzian of shorter duration.

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The ultimate experimental limits on possible waveform shapes and pulse widths will come from both the speed and depth of modulation that we can achieve with current state-of-the-art components, in addition to the loss accumulated in dispersion compensation modules. Both 40 GHz sinewave generators and electro-optic modulators are readily available, allowing us to manipulate pulses down to ∼ 10 ps, while a maximum depth of modulation of ∼ 4π allows for a maximum temporal compression ratio of ≈ 10. (Recent developments of thin-film ferroelectrics [31] could allow us to reach much higher phase shifts and, consequently, compression ratios.) Another limitation for quantum information is loss; mixing with the pump in a highly nonlinear fiber/nonlinear crystal will incur ∼ 1–2 dB of loss [12, 15, 20], while dispersion compensation with a fiber up to 20 km in length, will add another 2 dB (using the standard 0.2 dB/km number), for a total loss of ∼ 3–4 dB. This number is acceptable and compares favorably with pulse shaping systems that employ lossy gratings [19].

4. Conclusion

We have described a simple technique for pulse shaping that relies on simple quadratic phase operations that can be readily implemented using phase modulation, nonlinear wave mixing and dispersive fibers, and which constitutes a first step towards arbitrary waveform shaping operations. This technique is most useful in the regime where it is impossible to apply standard pulse shaping techniques that require the large bandwidth of femtosecond pulses. Additionally, this technique goes beyond the time-lens method, which can magnify/compress the spectrum/temporal-length without changing the actual shape; the simplest example being a conversion of a monoexponentially-decaying pulse to a Lorentzian symmetric pulse. Such a waveform conversion would be well suited to quantum emitters, such as semiconductor quantum dots and nitrogen-vacancy centers, as it renders the emission closer to a Gaussian, and hence better suited to standard optical components. This technique can also temporally compress the emission, hence reducing the photon-to-photon overlap and improving the single photon purity of the quantum wavepacket. Our next steps will be to implement the current scheme experimentally, verify its quantum-preserving nature, and extend the theory to encompass completely arbitrary waveform shaping using numerical phase extraction techniques.

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Theoertical investigation of quantum waveform shaping for single photon emitters

Abstract

1. Introduction

2. Theory

2.1. Gaussian to Gaussian transformation

2.2. One-sided exponential to Lorentzian transformation

3. Experimental considerations

4. Conclusion

References and links

Cited By

Figures (4)

Equations (18)

Optics Express