Abstract

A vital requirement for a quantum computer is the ability to locally address, with high fidelity, any of its qubits without affecting their neighbors. We propose an addressing method using composite sequences of laser pulses that dramatically reduces the addressing error in a lattice of closely spaced atoms or ions and at the same time significantly enhances the robustness of qubit manipulations. To this end, we design novel (to our knowledge) high- fidelity composite pulses for the most important single-qubit operations. In principle, this method allows one to beat the diffraction limit, for only atoms situated in a small spatial region around the center of the laser beam are excited, well within the laser beam waist.

© 2011 Optical Society of America

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References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  2. http://qist.lanl.gov/qcomp_map.shtml.
  3. T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
    [PubMed]
  4. We follow the usual NMR terminology, which is related to the features of the excitation profile, rather than the radiation.
  5. E. L. Hahn, Phys. Rev. 80, 580 (1950).
  6. M. H. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979).
  7. R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).
  8. H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
    [PubMed]
  9. M. H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).
  10. R. Freeman, Spin Choreography (Spektrum, 1997).
  11. S. Wimperis, J. Magn. Reson. 109, 221 (1994).
  12. H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).
  13. Allowing for different pulse areas, and thereby seemingly for more free parameters, does not seem to be advantageous. We have found through extensive examination that the total pulse area (and the pulse duration) of the composite sequence is not reduced.
  14. S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
    [PubMed]
  15. F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
    [PubMed]
  16. N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).
  17. I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
    [PubMed]
  18. I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  19. C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

2009 (1)

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

2008 (3)

H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

2003 (2)

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

2000 (1)

I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
[PubMed]

1994 (1)

S. Wimperis, J. Magn. Reson. 109, 221 (1994).

1986 (2)

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

M. H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).

1980 (1)

R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).

1979 (1)

M. H. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979).

1950 (1)

E. L. Hahn, Phys. Rev. 80, 580 (1950).

Becher, C.

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Blatt, R.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Bloch, I.

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
[PubMed]

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Cheneau, M.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Cho, H. M.

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

Chuang, I. L.

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Chwalla, M.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Dalibard, J.

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

Deuschle, T.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Elman, V.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Endres, M.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Eschner, J.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

Esslinger, T.

I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
[PubMed]

Freeman, R.

R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).

M. H. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979).

R. Freeman, Spin Choreography (Spektrum, 1997).

Fukuhara, T.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Glaser, S.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Guckenheimer, J.

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

Gulde, S.

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Häffner, H.

H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Hahn, E. L.

E. L. Hahn, Phys. Rev. 80, 580 (1950).

Hänsch, T. W.

I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
[PubMed]

Hennrich, M.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Hünsel, W.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Johanning, M.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Kempsell, S. P.

R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).

Kim, K.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Kuhr, S.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Lancaster, G. P. T.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

Levitt, M. H.

M. H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).

R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).

M. H. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979).

Monz, T.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Neuhauser, W.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Pines, A.

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

Riebe, M.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Roos, C. F.

H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

Schauß, P.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Schindler, P.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Schmidt-Kaler, F.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

Sherson, J. F.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Timoney, N.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Tycko, R.

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

Villar, A. S.

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Weiss, C.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Weitenberg, C.

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Wimperis, S.

S. Wimperis, J. Magn. Reson. 109, 221 (1994).

Wunderlich, C.

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Zwerger, W.

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

J. Magn. Reson. (3)

M. H. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979).

R. Freeman, S. P. Kempsell, and M. H. Levitt, J. Magn. Reson. 38, 453 (1980).

S. Wimperis, J. Magn. Reson. 109, 221 (1994).

Nature (3)

S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
[PubMed]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003).
[PubMed]

I. Bloch, T. W. Hänsch, and T. Esslinger, Nature 403, 166 (2000).
[PubMed]

Phys. Rep. (1)

H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155(2008).

Phys. Rev. (1)

E. L. Hahn, Phys. Rev. 80, 580 (1950).

Phys. Rev. A (1)

N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Phys. Rev. A 77, 052334 (2008).

Phys. Rev. Lett. (2)

H. M. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Phys. Rev. Lett. 56, 1905 (1986).
[PubMed]

T. Monz, K. Kim, W. Hünsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 102, 040501 (2009).
[PubMed]

Prog. Nucl. Magn. Reson. Spectrosc. (1)

M. H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).

Rev. Mod. Phys. (1)

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

Other (6)

C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, “Single-spin addressing in an atomic Mott insulator,” arXiv:1101.2076v1 (2011).

Allowing for different pulse areas, and thereby seemingly for more free parameters, does not seem to be advantageous. We have found through extensive examination that the total pulse area (and the pulse duration) of the composite sequence is not reduced.

R. Freeman, Spin Choreography (Spektrum, 1997).

We follow the usual NMR terminology, which is related to the features of the excitation profile, rather than the radiation.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

http://qist.lanl.gov/qcomp_map.shtml.

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Figures (2)

Fig. 1
Fig. 1

Top frames: excitation probability p in a laser field with a Gaussian spatial profile (gray shaded) with FWHM of Rabi frequency ξ versus the distance from its center for various composite pulses from Table 1: N 5 ( π ) , P 7 ( π ) , N 21 ( π ) , and P 17 ( π ) (left) and N 7 ( π / 2 ) , P 9 ( π / 2 ) , and P 17 ( π / 2 ) (right). The excitation profiles of single π and π / 2 pulses are shown too. Bottom frames: deviation p 0 p , with p 0 = 1 (left) and p 0 = 0.5 (right) being the desired excitation probability.

Fig. 2
Fig. 2

Phase deviation | ϕ 3 π / 2 | versus pulse area deviation of the composite pulses N 7 ( π , 3 π 2 ) and P 11 ( π , 3 π 2 ) , shown in Table 2. The dashed curves are for frequency mismatch Δ = 0.001 / T . The horizontal line is the threshold above which the infidelity exceeds 10 4 .

Tables (2)

Tables Icon

Table 1 Phases ϕ k (in Units π) for Some NB ( N N ) and PB ( P N ) Sequences of N = 2 n + 1 Phased Resonant Pulses of Area A: A 0 A ϕ 2 A ϕ 3 A ϕ n + 1 A ϕ 3 A ϕ 2 A 0 a

Tables Icon

Table 2 Phases ϕ k (in Units π) for Some NB ( N N ( A , φ ) ) and PB ( P N ( A , φ ) ) Sequences of N = 2 n + 1 Phased Resonant Pulses of Area A: A 0 A ϕ 2 A ϕ 3 A ϕ n + 1 A ϕ 3 A ϕ 2 A 0 That Produce Phased Rotations of Angle φ at Area A a

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

U = [ a b b * a * ] .
U ϕ = [ a b e i ϕ b * e i ϕ a * ] .
U ( N ) = U ( A ϕ N ) U ( A ϕ N 1 ) U ( A ϕ 1 ) .
[ U 11 ( N ) ] A = A = cos ( A / 2 ) ,
[ A k U 11 ( N ) ] A = 0 = 0 ( k = 2 , 4 , , 2 n 1 ) ,
[ A k U 11 ( N ) ] A = A = 0 ( k = 1 , 2 , , n 2 ) ,
[ arg U 21 ( N ) ] A = A = φ .
[ arg A k U 21 ( N ) ] A = A = 0 ( k = 1 , 2 , , n 3 ) .

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