## Abstract

We demonstrate the potential of semiconductor quantum dot nanomaterials for solid-state based controllable quantum memories in which losses may be compensated by gain. The dynamic photonic quantum-coherence present in a quantum dot ensemble and generated by a coherent signal pulse is influenced and controlled by disorder, spectral detuning and the power of the pulse. We show that the high coupling of spatial and temporal degrees of freedom is a key requirement for coherence transfer and/or storage.

©2008 Optical Society of America

## 1. Introduction

Light is a promising candidate for carrying information in both, classical and quantum communication systems. However, not only the ability to transmit a photonic qbit but also to maintain and store quantum coherence is key in the realization of quantum information technology. Various experiments have therefore focused on an exploration of the coupling of light and matter with the aim to preserve and store quantum coherence. In principle, the realization of such a “quantum memory” system requires precise knowledge of the coherence properties of the particular material employed as well as the system itself. So far, concepts and realizations of quantum memory elements have typically been based on atom and ion ensembles and various investigations have concentrated on atomic vapors or ensembles of atoms [1, 2, 3, 4, 5] and, indeed, the transfer of a quantum state between matter and light has been achieved [6]. The stopping and storage of a slow light pulse and the transfer of information through conversion of the light pulse into a travelling matter wave could also be demonstrated for a Bose-Einstein condensate [7]. Exploiting the characteristic properties of tapered negative-index metamaterial heterostructures we have recently shown on the basis of two independent classical theories that light can be efficiently slowed down and eventually trapped while each frequency component of a wave packet is stopped at a different layer thickness, leading to the spatial separation of the packets spectrum and the formation of a ‘trapped rainbow’ [8].

Here, we demonstrate a controlled storage and transfer of photonic space-time quantum coherence realized by a quantum dot nano-medium subjected to a coherent tunable light signal. This system has a large potential for the realization of a solid-state based controllable quantum memory: Quantum dots can be positioned in a controlled way, addressed electrically and/or optically, embedded into active devices [9] or integrated into larger structures, e.g. arrays or micro cavities to improve the efficiency. They are attractive for quantum experiments, e.g. for single photon sources [10]. The ability to deterministically transform quantum states of light to quantum states of matter makes them a key tool in the development of light-based quantum technologies.

## 2. Theoretical description

We focus on a model system consisting of a quantum dot layer embedded in a vertical cavity that has been optically excited by a short (500 fs) pulse (see insert of Fig. 2). The quantum dot ensemble (material system InGaAs) is organized in a two-dimensional layer of 10*µ*m×0*µ*m that is optically excited by a light signal of 5*µ*m diameter - the carrier of quantum information. Relating a state at point **r** in the centre of the excited area to another state near the rim (**r**′) gives a measure for the data stored or transferred after a given time. The injection of a light pulse (here: Gaussian shaped spatial beam profile, width 5 *µ*m) high above the band gaps of the semiconductor quantum dots generates a (partially incoherent) excited electron-hole plasma which then leads to a hierarchy of carrier relaxation processes within the bands followed by (low-momentum) radiative carrier recombination that can be observed as (photo-)luminescence.

The coherence properties of the emitted radiation are then directly reflected in the dynamics of the expectation value of the field-field correlation operator 〈*b̂*
^{†}
_{R}
*b̂*
_{R′}〉 and the expectation value of the field-dipole correlation operator 〈*b̂*
^{†}
_{R}
*ĉ*
_{R′}
*d̂*
_{R″}〉 where *b̂*
_{R} (*b̂*
^{†}
_{R}) is the (spatially dependent) annihilation (creation) operator for photons, *ĉ*
_{R} (*ĉ*
^{†}
_{R}) and *d̂*
_{R} (*d̂*
^{†}
_{R}) are the annihilation (creation) operators for electrons and holes, respectively, where the discrete positions **R** correspond to the lattice sites of the Bravais lattice describing the semiconductor crystal. Each lattice site actually represents the spatial volume *ν*
_{0} of the Wigner-Seitz cell of the lattice.

On the basis of these operators we now derive equations of motion for the expectation values of the carriers (〈*ĉ*
^{†}
_{R}
*ĉ*
_{R′}〉 and 〈*d̂*
^{†}
_{R}
*d̂*
_{R′}〉), dipoles (〈*ĉ*
_{R}
*d̂*
_{R′}〉), fields (〈*b̂*
_{R}〉) and, in particular, equations for the expectation values of the operators describing field-field (〈*b̂*
^{†}
_{R}
*b̂*
_{R′}〉) and field-dipole correlations (〈*b̂*
^{†}
_{R}
*ĉ*
_{R′}
*d̂*
_{R″}〉). We then define the field-field and field-dipole correlation variables on a quasi-continuous length scale, *ρ ^{I}*(

**r**=

**R**;

**R**′=

**R**′)=

*ν*

_{0}

^{−1}〈

*b̂*

^{†}

_{R}

*b̂*

_{R′}〉, Θ

*(*

^{corr.}**r**=

**R**;

**r**′=

**R**′,

**r**″=

**R**″)=

*ν*

_{0}

^{−3/2}〈

*b̂*

^{†}

_{R}

*ĉ*

_{R′}

*d̂*

_{R″}〉 and transform the variables into Wigner functions by $I(\mathbf{r};\mathbf{r}\prime ,{k}_{e},{k}_{h})=\int {d}^{3}\mathbf{r}\prime {e}^{-i{\mathbf{k}}_{\mathbf{eh}}\mathbf{r}\prime}{\rho}^{I}\left(\mathbf{r}-\frac{\mathbf{r}\prime}{2},\mathbf{r}+\frac{\mathbf{r}\prime}{2}\right)$ , $C(\mathbf{r};\mathbf{r}\prime ,{k}_{e},{k}_{h})=\int {d}^{3}\mathbf{r}\u2033{e}^{-i{\mathbf{k}}_{\mathbf{eh}}\mathbf{r}\u2033}{\Theta}^{\mathrm{corr}.}\left(\mathbf{r};\mathbf{r}\prime -\frac{r\u2033}{2},\mathbf{r}\prime +\frac{\mathbf{r}\u2033}{2}\right)$ where

**k**is the wave vector of the electron hole pair. Using the local approximation finally leads to the following equations of motion for the field-field correlation

_{eh}*I*and the field-dipole correlation

*C*(

*r*and

*r*′ refer to two spatial locations in the two-dimensional numerical grid)

$$-i\frac{{\mathbf{\omega}}_{0}}{2{k}_{0}^{2}{\epsilon}_{r}}\left({\mathbf{\Delta}}_{\mathbf{r}}-{\mathbf{\Delta}}_{\mathbf{r}\mathbf{\prime}}\right){I}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\mathbf{\prime},{k}_{e},{k}_{h})$$

$$-i{g}_{0}\frac{\sqrt{{v}_{0}}}{4{\pi}^{2}}[{C}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})-{C}_{\mathrm{ji}}^{*}(\mathbf{r}\prime ,\mathbf{r},{k}_{e},{k}_{h})$$

$$+\frac{1}{{k}_{0}^{2}{\epsilon}_{r}}\left(\frac{{\partial}^{2}}{\partial {r\prime}_{j}{r\prime}_{k}}{C}_{\mathrm{ik}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})-\frac{{\partial}^{2}}{\partial {r}_{i}{r}_{k}}{C}_{\mathrm{jk}}^{*}(\mathbf{r}\prime ,\mathbf{r},{k}_{e},{k}_{h})\right)]$$

$$-\left({\kappa}_{i}-i\delta {\omega}_{i}\right){C}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})-i\frac{{\omega}_{0}}{2{k}_{0}^{2}{\epsilon}_{r}}{\Delta}_{\mathbf{r}}{C}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})$$

$$+i{g}_{0}\sigma \sqrt{{v}_{0}}\left({f}^{e}\left(\mathbf{r}\prime ,{k}_{e},{k}_{h})\right)+{f}^{h}(\mathbf{r}\prime ,{k}_{e},{k}_{h})-1\right){U}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})$$

$$+i{g}_{0}\sigma \sqrt{{v}_{0}}\delta \left(\mathbf{r}-\mathbf{r}\prime \right){\delta}_{\mathrm{ij}}({f}^{e}(\mathbf{r},{k}_{e},{k}_{h})\xb7{f}^{h}(\mathbf{r},{k}_{e},{k}_{h})$$

$$+p(\mathbf{r},{k}_{e},{k}_{h})\xb7{p}^{*}(\mathbf{r},{k}_{e},{k}_{h}))$$

The subscripts *i* and *j* denote the index for the two (*x* and *y*) spatial dimensions, respectively, i.e. *i* = 1, …*nx*, *j* = 1, …*ny* where *nx* and *ny* are the number of grid points in *x* and *y* direction. *κ* is the cavity loss rate. The birefringence term *δω* includes the difference between the bandgap frequency and the longitudinal frequencies of the confined light for the two polarization directions. Δ_{r}=∑* _{k}*∂

^{2}/∂

*r*

_{k}^{2}and

*g*

_{0}is the coupling constant characterizing the interaction of carriers and photons.

*∑*is the confinement factor in

*z*direction.

*ω*

_{0}and

*k*

_{0}are the central frequency and wavenumber of the light fields, respectively and

*ε*is the relative permittivity. Further details on the general approach can be found in [11]. For a consideration of the carrier potential modified by many-body effects we have introduced in Eq. (2) the local field ${U}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{k}_{e},{k}_{h})={I}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}{\prime}_{,}{k}_{e},{k}_{h})+{\int}_{{{k}_{e}}^{\prime},{{k}_{h}}^{\prime}}{V}_{{k}_{e},{k}_{h},{{k}_{e}}^{\prime},{{k}_{h}}^{\prime}}{I}_{\mathrm{ij}}(\mathbf{r},\mathbf{r}\prime ,{{k}_{e}}^{\prime},{{k}_{h}}^{\prime})$ . which consists of the field-field correlations and a term describing the renormalization due to the Coulomb interactions.

_{r}*f*(

^{e}**r**,

*k*,

_{e}*k*),

_{h}*f*(

^{h}**r**,

*k*,

_{e}*k*) and

_{h}*p*(

**r**,

*k*,

_{e}*k*) are the (dynamically calculated) electron and hole distributions and the inter-level polarisation with

_{h}*k*and

_{e}*k*being the wavenumbers of the electron and hole states, respectively.

_{h}*k*

_{0}is the vacuum wavevector corresponding to the central frequency

*ω*

_{0}and

*ε*is the relative permittivity. ${\Gamma}_{{k}_{e},{k}_{h}}=\left({\Gamma}_{{k}_{e},{k}_{h}}^{e,\mathrm{out}}+{\Gamma}_{{k}_{e},{k}_{h}}^{e,\mathrm{in}}+{\Gamma}_{{k}_{e},{k}_{h}}^{h,\mathrm{out}}+{\Gamma}_{{k}_{e},{k}_{h}}^{h,\mathrm{in}}\right)$ Includes in and out scattering processes as discussed in [11]. Many particle effects such as band gap renormalization or the Coulomb enhancement lead to corrections in the carrier energies and carrier potential [12]. The corrections in carrier energies are considered via the momentum dependent frequency ${\Omega}_{{k}_{e},{k}_{h}}$ .

_{r}## 3. Radial signal propagation and coherence trapping

To explore the potential of quantum dot nanomaterials for retaining quantum information we will consider quantum photonic input-signals at resonance as well as input-signals tuned off-resonant. In each case the field-field (*I*) and field-dipole (*C*) correlations as introduced above directly contain the full information on all correlated light-matter interactions affecting spacetime coherence and memory effects. Note that due to this continuous accessibility to all spatio-temporal information we are not required to additionally model the injection of a pump and a delayed probe beam as necessary in a typical experiment [6].

Figure 1 shows snapshots of the carrier distribution (a), field-field correlation (b) and field-dipole correlation (c) in the quantum dot layer following a resonant input-pulse. The maximum of the injected pulse (duration 500 fs) enters the active area in the first time step (left column). The time between successive plots is 1 ps. In our example we chose a pulse with a high intensity so that long-lived responses can be expected in the carrier ensemble and pre-excite the quantum dot nanomaterial slightly above transparency. The plots refer to the spectrally integrated values at the output facet. For each grid point the spatial correlations of all points have been summed up, i.e.
${\tilde{I}}^{\mathrm{eh}}={\int}_{\mathbf{r}\prime}{\sum}_{{k}_{e},{k}_{h}}I(r\prime ;r,{k}_{e},{k}_{h})\mathrm{dr}\prime $
. We then plot the real quantity *I ^{eh}* = Re(

*Ĩ*)

^{eh}^{2}+Im(

*Ĩ*)

^{eh}^{2}.

The signal input-pulse locally depletes the carrier inversion in the quantum dot layer, creating characteristic excited states. These relax via a hierarchy of ultrafast (fs … ps) intraband- and interband scattering mechanism leading to characteristic local oscillations. In combination with spontaneous and induced recombination processes these oscillations induce a characteristic spatio-temporal dynamics in the correlations. Thereby, the dynamics of the field-dipole correlations is shifted in time compared to the field-field correlations. Depending on the power level of the input signal and on the injection density the refilling of the spatio-spectral hole induced by the pulse may lead to strong damping in the correlation dynamics or excite a fast oscillatory wave-like spreading of quantum correlation in the radial direction. The spatial spreading is particularly efficient and strong if the injected pulse has a high intensity. The saturation of the inversion at the centre of the probe in combination with the spatio-temporal coupling of light and matter then induces the radial movement in the field and correlation dynamics that can be seen in Fig. 1. This effect is reinforced if a partial refilling of the levels in the centre is realized by a high carrier injection level. The resulting changes in the carrier distribution keeps the changes in radial direction at a high level leading to an additional pushing of the radial dynamics.

A common method to quantify the quality of a quantum computing device is to define a figure of merit (FOM) [6, 13, 14]. In analogy to the FOM definition we here define a correlation figure of merit for the field-field and field-dipole correlation at the centre and at the rim of the active area: for this purpose we integrate the correlations to all other spatial grid points and normalize them to their initial values. This directly reveals the quantum memory storage time [15] of the active nanomaterial as well as the dynamics of coherence transfer. In other words, the FOM provides information on the (time-dependent) quality of the storage or the transfer of a coherent signal (here: the mixed light-matter state induced by the light pulse). We would like to note that our definition of the correlation FOM includes – similar to previous definitions of the FOM on the basis of e.g. probability distributions – the full dependence of the storage and/or transfer properties (affected by e.g. dephasing and propagation times) of a medium. It can thus be used to characterize the operation of a real system. The application of our space-time simulation to the measurement process of single qubits represented by e.g. single photon pulses typically investigated with FOM measurements, however, will be done in future work. Here, the focus is the demonstration and interpretation of space-time changes that cannot be obtained within the frame of previous theoretical approaches.

Figure 2 summarizes the dynamics of the field-dipole FOM and the field-field FOM for resonant (a) and off-resonant (b) excitation. The left column refers to the centre of the probe whereas the right column displays the dynamics of the correlations relative to a point at the rim. In the case of the resonant injection (Fig. 2(a)) the field-dipole FOM is characterized by a significant decrease in magnitude. This orignates from the fact that the localized excitatation of the dipoles (excited states in the nanomaterial) is followed their local decay. Nevertheless, approximately 10 % of the correlation has been spatially translated to the rim (via the mutual interplay of light and matter during the lateral expansion of the fields) just while they are decaying in the centre. The field-field FOM is characterized by a slight reshaping of the curve. The second pulse that can be seen in the central distribution (left column) is a ’shadow pulse’ arising from the strong excitation of the medium: The pattern imprinted in the mdium by the excitation pulse induces the light-dipole coupling which then feeds back to the light-light coupling. This effect is a direct consequence of the coupling of the correlations. Furthermore, one can see that the light-light FOM at the rim is of similar magnitude as in the centre. This directly demonstrates the fact that the dynamic field correlations are free to advance in space. Indeed, this effect can only occur because of the non-vanishing contribution in the field-dipole correlation. Although the coherent dipole density alone (being partially bound to individual quantum dots) is significantly more localized in space it nevertheless mediates the radial transfer of the coherence and memory properties of light.

A spectral detuning between the spectral maximum of the excited states of the quantum dot nanomaterial and the central wavelength of the signal pulse (here: 6 nm towards absorption) leads to a different response of the medium to light (Fig. 2(b)). The correlations at the centre (left column) show a characteristic local oscillation that does not appear at the rim. This directly reflects the influence of the intra-level dynamics in active quantum dots on the response of the whole nanomaterial: The pulse induces a complex carrier relaxation and scattering of the high-energy carriers. As a result, the correlation distributions show characteristic modulations and the light-light FOM at the rim is significantly decreased as compared to the initial value near the centre. An off-resonant injection thus impedes the formation of the wave-like expansion of coherence and may lead to a spatial trapping of photonic space-time coherence. A spectral detuning may thus be a means of controlling memory effects.

The influence of the spectral detuning can be further inferred from lateral cuts of the correlation functions. Figure 3 compares the dynamics of the real part of the field-field correlation function and the imaginary part of the diagonal elements of the field-dipole correlation function for a detuning of 6 nm (a) and 3 nm (b) to resonant signal (c). A strong detuning (Fig. 3(a)) leads to slight dynamic modulations on a ps time scale in the space-time correlations. These coherence oscillations are a direct footprint of the dynamic spatio-temporal interplay of fields and dipoles and do not originate from round-trip effects of a cavity into which the nanomaterial is embedded.

An off-resonant light signal field leads to a delayed response of the medium, i.e. the light-dipole correlations increase only after the first maximum of the field-field correlation. The induced changes in the dipole density, on the other hand, have a delayed influence on the light fields leading to a second maximum in the field-field correlation. As a result, the field-field correlations have high values whenever the light-dipole correlations have minima and vice versa. We would like to note that, similarly, the imaginary part of the diagonal elements of the field-field correlation is strongly related to the real part of the diagonal elements of the field-dipole correlation function. The situation is changed in the case of a small detuning (Fig. 3(b)). In this case, the field-field correlation shows a slow and intense response. However, the field-dipole correlation still shows fast oscillations reflecting the energetic reshuffling of the carriers by the light fields. Finally, for a resonant excitation Fig. 3(c)) reveals a prompt reaction of all correlation distributions to the excitation process. It is now evident that a thorough tuning of the excitation frequency may play a key role for the spatio-temporal transfer of coherence properties in optically excited active semiconductor media.

## 4. Influence of disorder

Another key property of active nanomaterials is the degree of disorder as represented by material inhomogeneities (e.g. in size, energy levels and dipole matrix elements). Figure 4 shows an example demonstrating the influence of disorder (here: spatial variation of 5% in all material parameters) on the spatio-temporal coherence properties (*N ^{eh}*,

*I*and

^{eh}*C*) for an input power of 0.2

^{eh}*P*(left) and 0.7

_{s}*P*(right) and

_{s}*t*= 1 ps after the input of the signal (

*P*denoting the power that is required to reduce the inversion to transparency). With the quantum dot nanomaterial electrically (i.e. incoherently) pre-excited to slightly above threshold the additional resonant signal leads, particularly and at low input signal levels, to pronounced spatio-temporal modulations in all distributions. The disorder thereby disturbs the spatial and temporal coherence and eventually leads to a reduced transfer of coherence. In the correlation FOM the disorder leads for every time step to correspondingly lower values of the integrated correlations. In the case of a higher power signal the coherence properties of the injected light (here: with ideal coherence properties) are via the dynamic light-matter coupling to a higher degree transferred to the medium leading to smaller modulations. In the correlation FOM this is reflected in a more moderate decrease in the correlation FOM compared to the undisturbed system. The pure material variations alone thus are only partially responsible for the coherence properties of a quantum dot nanomaterial. In addition, it is the excitation and the resulting interplay of light and matter that influences the spatio-temporal dynamics and memory properties of a given device based on quantum dot nanomaterials. Disturbances induced by inhomogeneities of individual quantum dots may thus even be counterbalanced by excitation conditions. The absolute value of the correlation FOM thereby may be a measure of the degree of the coherence transferable in a given system.

_{s}## 5. Conclusion

In conclusion, we have explored the controlled storage and transfer of photonic space-time quantum coherence in active quantum dot nanomaterials. Our analysis of spatial coherence patterns that directly profits from the full information represented in the distributions of field-field and field-dipole correlations demonstrates a strong dependence of storage and transfer of quantum coherence on both optical and electrical excitation conditions. Furthermore, the quantum correlations reveal the influence of intradot hot-carrier relaxation and shed light on the importance of disorder in the dynamic response and coupling of light and matter. Clearly, such studies will become increasingly important for the development and design of novel functional nano-materials with quantum functionalities such as spatially selective signal processing, ultra-fast electro-optical switching elements, spatio-spectral light trapping and quantum information processing.

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