## Controlled storage and transfer of photonic space-time quantum-coherence in active quantum dot nanomaterials

Optics Express, Vol. 16, Issue 6, pp. 3744-3752 (2008)

http://dx.doi.org/10.1364/OE.16.003744

Acrobat PDF (522 KB)

### 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

1. A. T. Black, J. K. Thompson, and V. Vuletic, “On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency,” Phys. Rev. Lett. **95**, 133601 (2005). [CrossRef] [PubMed]

2. M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping Quantum Pulses of Light Via Coherent Atomic Memory,” Phys. Rev. Lett. **93**, 233602 (2004). [CrossRef] [PubMed]

3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature **423**, 731–734 (2003). [CrossRef] [PubMed]

4. C. W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, “Single-Photon Generation from Stored Excitation in an Atomic Ensemble.” Phys. Rev. Lett. **92**, 213601 (2004). [CrossRef] [PubMed]

5. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature **432**, 482–486 (2004). [CrossRef] [PubMed]

6. D. N. Matsukevich and A. Kuzmich, “Quantum State transfer between Matter and Light,” Science **306**, 663–666 (2004). [CrossRef] [PubMed]

7. N. S. Ginsberg, S. R. Garner, and L. V. Hau, “Coherent control of optical information with matter wave dynamics,” Nature **445**, 623–626 (2007). [CrossRef] [PubMed]

8. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature **450**, 397–401 (2007). [CrossRef] [PubMed]

9. M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature **432**, 81–84 (2004). [CrossRef] [PubMed]

10. C. Santori, D. Fattal, D, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistiguishable photons from a single-photon device,” Nature **419**, 594–597 (2002). [CrossRef] [PubMed]

## 2. Theoretical description

*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.

*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.

*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

**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*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

_{r}11. E. Gehrig and O. Hess, *Spatio-Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers* (Springer, Heidelberg, 2003). [CrossRef]

*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.

_{r}11. E. Gehrig and O. Hess, *Spatio-Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers* (Springer, Heidelberg, 2003). [CrossRef]

12. M. Kira, F. Jahnke, and S. W. Koch, “Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence,” Phys. Rev. Lett. **81**, 3263–3266 (1998). [CrossRef]

## 3. Radial signal propagation and coherence trapping

*I*= Re(

^{eh}*Ĩ*)

^{eh}^{2}+Im(

*Ĩ*)

^{eh}^{2}.

6. D. N. Matsukevich and A. Kuzmich, “Quantum State transfer between Matter and Light,” Science **306**, 663–666 (2004). [CrossRef] [PubMed]

13. J. Fiurasek, “Optimal probabilistic estimation of quantum states,” New J. Phys. **8**, 192–199 (2006). [CrossRef]

15. T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature **438**, 83 (2005). [CrossRef]

## 4. Influence of disorder

*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

## References and links

1. | A. T. Black, J. K. Thompson, and V. Vuletic, “On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency,” Phys. Rev. Lett. |

2. | M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping Quantum Pulses of Light Via Coherent Atomic Memory,” Phys. Rev. Lett. |

3. | A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature |

4. | C. W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, “Single-Photon Generation from Stored Excitation in an Atomic Ensemble.” Phys. Rev. Lett. |

5. | B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature |

6. | D. N. Matsukevich and A. Kuzmich, “Quantum State transfer between Matter and Light,” Science |

7. | N. S. Ginsberg, S. R. Garner, and L. V. Hau, “Coherent control of optical information with matter wave dynamics,” Nature |

8. | K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature |

9. | M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature |

10. | C. Santori, D. Fattal, D, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistiguishable photons from a single-photon device,” Nature |

11. | E. Gehrig and O. Hess, |

12. | M. Kira, F. Jahnke, and S. W. Koch, “Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence,” Phys. Rev. Lett. |

13. | J. Fiurasek, “Optimal probabilistic estimation of quantum states,” New J. Phys. |

14. | C. W. Gardiner and P. Zoller, |

15. | T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(140.5960) Lasers and laser optics : Semiconductor lasers

(270.1670) Quantum optics : Coherent optical effects

(160.4236) Materials : Nanomaterials

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 22, 2008

Revised Manuscript: March 3, 2008

Manuscript Accepted: March 3, 2008

Published: March 6, 2008

**Citation**

E. Gehrig and O. Hess, "Controlled storage and transfer of photonic space-time quantum-coherence in active quantum dot nanomaterials," Opt. Express **16**, 3744-3752 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3744

Sort: Year | Journal | Reset

### References

- A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005). [CrossRef] [PubMed]
- M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004). [CrossRef] [PubMed]
- A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003). [CrossRef] [PubMed]
- C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004). [CrossRef] [PubMed]
- B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004). [CrossRef] [PubMed]
- D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004). [CrossRef] [PubMed]
- N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007). [CrossRef] [PubMed]
- K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007). [CrossRef] [PubMed]
- M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004). [CrossRef] [PubMed]
- C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002). [CrossRef] [PubMed]
- E. Gehrig and O. Hess, Spatio-Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers (Springer, Heidelberg, 2003). [CrossRef]
- M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998). [CrossRef]
- J. Fiurasek, "Optimal probabilistic estimation of quantum states," New J. Phys. 8, 192-199 (2006). [CrossRef]
- C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000).
- T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.