Abstract

Optical trapping of atoms employs high-intensity fields that necessarily alter atomic level structure. The calculation of light shifts by perturbation theory fails for scenarios that arise, for example, when the trapping light is near an excited-state transition or for polychromatic fields. We show here that non-perturbative methods based on Floquet’s theorem elegantly handle such scenarios. We compare our calculation to precision absorption spectroscopy on cold 87Rb atoms in a bichromatic optical dipole trap at 1560 + 1529 nm. Proximity to excited-state resonances induces highly nonlinear level shifts, providing a strong test of theory. The good theory-experiment agreement suggests a new method for accurate measurements of excited-state electric-dipole matrix elements and a precision tool for engineering custom atomic level structures.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  3. C. D. Herold, V. D. Vaidya, X. Li, S. L. Rolston, J. V. Porto, and M. S. Safronova, “Precision measurement of transition matrix elements via light shift cancellation,” Phys. Rev. Lett. 109, 243003 (2012).
    [Crossref]
  4. I. Novikova, A. B. Matsko, V. L. Velichansky, M. O. Scully, and G. R. Welch, “Compensation of ac Stark shifts in optical magnetometry,” Phys. Rev. A 63, 063802 (2001).
    [Crossref]
  5. D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3, 227–234 (2007).
    [Crossref]
  6. R. Jiménez-Martínez, S. Knappe, and J. Kitching, “An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening,” Rev. Sci. Int. 85, 045124 (2014).
  7. J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
    [Crossref]
  8. H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
    [Crossref] [PubMed]
  9. A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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  10. T. Zanon-Willette, E. de Clercq, and E. Arimondo, “Probe light-shift elimination in generalized hyper-Ramsey quantum clocks,” Phys. Rev. A 93, 042506 (2016).
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  18. F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
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    [Crossref]
  21. A. Haffmans, R. Blümel, P. M. Koch, and L. Sirko, “Prediction of a new peak in two-frequency microwave “ionization” of excited hydrogen atoms,” Phys. Rev. Lett. 73, 248–251 (1994).
    [Crossref] [PubMed]
  22. P. F. Griffin, K. J. Weatherill, S. G. MacLeod, R. M. Potvliege, and C. S. Adams, “Spatially selective loading of an optical lattice by light-shift engineering using an auxiliary laser field,” New J. Phys. 8, 11 (2006).
    [Crossref]
  23. R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).
  24. T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
    [Crossref] [PubMed]
  25. P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402 (2017).
    [Crossref]
  26. G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, “Ultracold atoms in a tunable optical kagome lattice,” Phys. Rev. Lett. 108, 045305 (2012).
    [Crossref] [PubMed]
  27. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
    [Crossref] [PubMed]
  28. M. S. Safronova and U. I. Safronova, “Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in 87Rb,” Phys. Rev. A 83, 052508 (2011).
    [Crossref]
  29. E. Fortson and L. Lewis, “Atomic parity nonconservation experiments,” Phys. Rep. 113, 289–344 (1984).
    [Crossref]
  30. M. C. Noecker, B. P. Masterson, and C. E. Wieman, “Precision measurement of parity nonconservation in atomic cesium: A low-energy test of the electroweak theory,” Phys. Rev. Lett. 61, 310–313 (1988).
    [Crossref] [PubMed]
  31. B. K. Sahoo, L. W. Wansbeek, K. Jungmann, and R. G. E. Timmermans, “Light shifts and electric dipole matrix elements in Ba+ and Ra+,” Phys. Rev. A 79, 052512 (2009).
    [Crossref]
  32. S.-I. Chu, “Recent developments in semiclassical Floquet theories for intense-field multiphoton processes,” Advances in Atomic and Molecular Physics 21, 197–253 (1985).
  33. A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.3), [Online]. Available: http://physics.nist.gov/asd [2016, September 12]. National Institute of Standards and Technology, Gaithersburg, MD. (2015).
  34. We obtained all the matrix elements except three from [28], elements between the 4d and 4f states were obtained directly in a private communication from M. S. Safronova of U. Delaware. All of the elements we used are available online [36].
  35. J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Addison-Wesley, 2011), 2nd ed.
  36. https://github.com/simocop/LightShiftCalculator .
  37. W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
    [Crossref]
  38. G. Reinaudi, T. Lahaye, Z. Wang, and D. Guéry-Odelin, “Strong saturation absorption imaging of dense clouds of ultracold atoms,” Opt. Lett. 32, 3143–3145 (2007).
    [Crossref] [PubMed]
  39. C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “Detection of small atom numbers through image processing,” Phys. Rev. A 82, 061606 (2010).
    [Crossref]
  40. Quantitative prediction of Ci is feasible but would require use of the optical Bloch equations to solve for atomic dynamics in the presence of the 1560 nm beam, the single probe beam at 780 nm, and repump light also at 780 nm which is emitted from six directions toward the centre of the trap.
  41. We estimate the uncertainties on the fitted values as the square root of the diagonal elements in the p × p matrix (JT J)−1RT R/(Npts − p). Where J is the Jacobian of the model fit function, R is a vector of the fit residuals, Npts is the number of measurements, and p the number of fitting parameters. In this case p = 9: the light intensity I, quadrature phase ϕ, and the seven peak amplitudes Ci.
  42. A. Fallon and C. Sackett, “Obtaining atomic matrix elements from vector tune-out wavelengths using atom interferometry,” Atoms 4(2) 12 (2016).
    [Crossref]

2017 (1)

P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402 (2017).
[Crossref]

2016 (6)

A. Fallon and C. Sackett, “Obtaining atomic matrix elements from vector tune-out wavelengths using atom interferometry,” Atoms 4(2) 12 (2016).
[Crossref]

A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
[Crossref]

T. Zanon-Willette, E. de Clercq, and E. Arimondo, “Probe light-shift elimination in generalized hyper-Ramsey quantum clocks,” Phys. Rev. A 93, 042506 (2016).
[Crossref]

F. Schmidt, D. Mayer, M. Hohmann, T. Lausch, F. Kindermann, and A. Widera, “Precision measurement of the 87Rb tune-out wavelength in the hyperfine ground state f = 1 at 790 nm,” Phys. Rev. A 93, 022507 (2016).
[Crossref]

G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
[Crossref]

F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
[Crossref]

2015 (3)

R. H. Leonard, A. J. Fallon, C. A. Sackett, and M. S. Safronova, “High-precision measurements of the 87Rb D-line tune-out wavelength,” Phys. Rev. A 92, 052501 (2015).
[Crossref]

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87, 637–701 (2015).
[Crossref]

R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).

2014 (1)

R. Jiménez-Martínez, S. Knappe, and J. Kitching, “An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening,” Rev. Sci. Int. 85, 045124 (2014).

2013 (3)

F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, “Dynamical polarizability of atoms in arbitrary light fields: general theory and application to cesium,” Eur. Phys. J. D 67, 1–16 (2013).
[Crossref]

T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
[Crossref] [PubMed]

W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
[Crossref]

2012 (2)

G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, “Ultracold atoms in a tunable optical kagome lattice,” Phys. Rev. Lett. 108, 045305 (2012).
[Crossref] [PubMed]

C. D. Herold, V. D. Vaidya, X. Li, S. L. Rolston, J. V. Porto, and M. S. Safronova, “Precision measurement of transition matrix elements via light shift cancellation,” Phys. Rev. Lett. 109, 243003 (2012).
[Crossref]

2011 (1)

M. S. Safronova and U. I. Safronova, “Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in 87Rb,” Phys. Rev. A 83, 052508 (2011).
[Crossref]

2010 (2)

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref] [PubMed]

C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “Detection of small atom numbers through image processing,” Phys. Rev. A 82, 061606 (2010).
[Crossref]

2009 (1)

B. K. Sahoo, L. W. Wansbeek, K. Jungmann, and R. G. E. Timmermans, “Light shifts and electric dipole matrix elements in Ba+ and Ra+,” Phys. Rev. A 79, 052512 (2009).
[Crossref]

2008 (1)

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

2007 (3)

B. Arora, M. S. Safronova, and C. W. Clark, “Determination of electric-dipole matrix elements in K and Rb from Stark shift measurements,” Phys. Rev. A 76, 052516 (2007).
[Crossref]

D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3, 227–234 (2007).
[Crossref]

G. Reinaudi, T. Lahaye, Z. Wang, and D. Guéry-Odelin, “Strong saturation absorption imaging of dense clouds of ultracold atoms,” Opt. Lett. 32, 3143–3145 (2007).
[Crossref] [PubMed]

2006 (1)

P. F. Griffin, K. J. Weatherill, S. G. MacLeod, R. M. Potvliege, and C. S. Adams, “Spatially selective loading of an optical lattice by light-shift engineering using an auxiliary laser field,” New J. Phys. 8, 11 (2006).
[Crossref]

2003 (1)

H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
[Crossref] [PubMed]

2002 (1)

A. Kaplan, M. Fredslund Andersen, and N. Davidson, “Suppression of inhomogeneous broadening in rf spectroscopy of optically trapped atoms,” Phys. Rev. A 66, 045401 (2002).
[Crossref]

2001 (1)

I. Novikova, A. B. Matsko, V. L. Velichansky, M. O. Scully, and G. R. Welch, “Compensation of ac Stark shifts in optical magnetometry,” Phys. Rev. A 63, 063802 (2001).
[Crossref]

2000 (1)

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).
[Crossref]

1994 (1)

A. Haffmans, R. Blümel, P. M. Koch, and L. Sirko, “Prediction of a new peak in two-frequency microwave “ionization” of excited hydrogen atoms,” Phys. Rev. Lett. 73, 248–251 (1994).
[Crossref] [PubMed]

1990 (1)

1988 (1)

M. C. Noecker, B. P. Masterson, and C. E. Wieman, “Precision measurement of parity nonconservation in atomic cesium: A low-energy test of the electroweak theory,” Phys. Rev. Lett. 61, 310–313 (1988).
[Crossref] [PubMed]

1985 (1)

S.-I. Chu, “Recent developments in semiclassical Floquet theories for intense-field multiphoton processes,” Advances in Atomic and Molecular Physics 21, 197–253 (1985).

1984 (1)

E. Fortson and L. Lewis, “Atomic parity nonconservation experiments,” Phys. Rep. 113, 289–344 (1984).
[Crossref]

1965 (1)

J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979–B987 (1965).
[Crossref]

Adams, C. S.

P. F. Griffin, K. J. Weatherill, S. G. MacLeod, R. M. Potvliege, and C. S. Adams, “Spatially selective loading of an optical lattice by light-shift engineering using an auxiliary laser field,” New J. Phys. 8, 11 (2006).
[Crossref]

Aldegunde, J.

P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402 (2017).
[Crossref]

Arimondo, E.

T. Zanon-Willette, E. de Clercq, and E. Arimondo, “Probe light-shift elimination in generalized hyper-Ramsey quantum clocks,” Phys. Rev. A 93, 042506 (2016).
[Crossref]

Arora, B.

B. Arora, M. S. Safronova, and C. W. Clark, “Determination of electric-dipole matrix elements in K and Rb from Stark shift measurements,” Phys. Rev. A 76, 052516 (2007).
[Crossref]

Aspect, A.

R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).

T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
[Crossref] [PubMed]

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

Becher, C.

H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
[Crossref] [PubMed]

Bernon, S.

T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
[Crossref] [PubMed]

Bertoldi, A.

R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).

T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
[Crossref] [PubMed]

Blackmore, J. A.

P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402 (2017).
[Crossref]

Blatt, R.

H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
[Crossref] [PubMed]

Blümel, R.

A. Haffmans, R. Blümel, P. M. Koch, and L. Sirko, “Prediction of a new peak in two-frequency microwave “ionization” of excited hydrogen atoms,” Phys. Rev. Lett. 73, 248–251 (1994).
[Crossref] [PubMed]

R. Blümel and U. Smilansky, “Ionization of hydrogen Rydberg atoms in strong monochromatic and bichromatic microwave fields,” J. Opt. Soc. Am. B 7, 664–679 (1990).
[Crossref]

Bourdel, T.

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

Bouyer, P.

R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).

T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, S. Bernon, A. Aspect, A. Landragin, and P. Bouyer, “Feedback control of trapped coherent atomic ensembles,” Phys. Rev. Lett. 110, 210503 (2013).
[Crossref] [PubMed]

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

Boyd, M. M.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87, 637–701 (2015).
[Crossref]

Bransden, B. H.

B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Prentice Hall, 2003), 2nd ed.

Brantut, J. P.

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

Budker, D.

D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3, 227–234 (2007).
[Crossref]

Calosso, C. E.

F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
[Crossref]

Camparo, J.

F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
[Crossref]

Camparo, J. C.

G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
[Crossref]

Cantin, E.

R. Kohlhaas, A. Bertoldi, E. Cantin, A. Aspect, A. Landragin, and P. Bouyer, “Phase locking a clock oscillator to a coherent atomic ensemble,” Phys. Rev. X 5, 021011 (2015).

Chu, S.-I.

S.-I. Chu, “Recent developments in semiclassical Floquet theories for intense-field multiphoton processes,” Advances in Atomic and Molecular Physics 21, 197–253 (1985).

Clark, C. W.

B. Arora, M. S. Safronova, and C. W. Clark, “Determination of electric-dipole matrix elements in K and Rb from Stark shift measurements,” Phys. Rev. A 76, 052516 (2007).
[Crossref]

Clément, J. F.

J. P. Brantut, J. F. Clément, M. R. de Saint Vincent, G. Varoquaux, R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer, “Light-shift tomography in an optical-dipole trap for neutral atoms,” Phys. Rev. A 78, 031401 (2008).
[Crossref]

Cornish, S. L.

P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402 (2017).
[Crossref]

Costanzo, G. A.

G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
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A. Kaplan, M. Fredslund Andersen, and N. Davidson, “Suppression of inhomogeneous broadening in rf spectroscopy of optically trapped atoms,” Phys. Rev. A 66, 045401 (2002).
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E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
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R. H. Leonard, A. J. Fallon, C. A. Sackett, and M. S. Safronova, “High-precision measurements of the 87Rb D-line tune-out wavelength,” Phys. Rev. A 92, 052501 (2015).
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A. Kaplan, M. Fredslund Andersen, and N. Davidson, “Suppression of inhomogeneous broadening in rf spectroscopy of optically trapped atoms,” Phys. Rev. A 66, 045401 (2002).
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F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
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G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
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A. Haffmans, R. Blümel, P. M. Koch, and L. Sirko, “Prediction of a new peak in two-frequency microwave “ionization” of excited hydrogen atoms,” Phys. Rev. Lett. 73, 248–251 (1994).
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A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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B. K. Sahoo, L. W. Wansbeek, K. Jungmann, and R. G. E. Timmermans, “Light shifts and electric dipole matrix elements in Ba+ and Ra+,” Phys. Rev. A 79, 052512 (2009).
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A. Kaplan, M. Fredslund Andersen, and N. Davidson, “Suppression of inhomogeneous broadening in rf spectroscopy of optically trapped atoms,” Phys. Rev. A 66, 045401 (2002).
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F. Schmidt, D. Mayer, M. Hohmann, T. Lausch, F. Kindermann, and A. Widera, “Precision measurement of the 87Rb tune-out wavelength in the hyperfine ground state f = 1 at 790 nm,” Phys. Rev. A 93, 022507 (2016).
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R. Jiménez-Martínez, S. Knappe, and J. Kitching, “An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening,” Rev. Sci. Int. 85, 045124 (2014).

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R. Jiménez-Martínez, S. Knappe, and J. Kitching, “An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening,” Rev. Sci. Int. 85, 045124 (2014).

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A. Haffmans, R. Blümel, P. M. Koch, and L. Sirko, “Prediction of a new peak in two-frequency microwave “ionization” of excited hydrogen atoms,” Phys. Rev. Lett. 73, 248–251 (1994).
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H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
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A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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R. H. Leonard, A. J. Fallon, C. A. Sackett, and M. S. Safronova, “High-precision measurements of the 87Rb D-line tune-out wavelength,” Phys. Rev. A 92, 052501 (2015).
[Crossref]

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G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
[Crossref]

F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
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E. Fortson and L. Lewis, “Atomic parity nonconservation experiments,” Phys. Rep. 113, 289–344 (1984).
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F. Schmidt, D. Mayer, M. Hohmann, T. Lausch, F. Kindermann, and A. Widera, “Precision measurement of the 87Rb tune-out wavelength in the hyperfine ground state f = 1 at 790 nm,” Phys. Rev. A 93, 022507 (2016).
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G. A. Costanzo, S. Micalizio, A. Godone, J. C. Camparo, and F. Levi, “ac Stark shift measurements of the clock transition in cold Cs atoms: Scalar and tensor light shifts of the D2 transition,” Phys. Rev. A 93, 063404 (2016).
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F. Levi, J. Camparo, B. Francois, C. E. Calosso, S. Micalizio, and A. Godone, “Precision test of the ac Stark shift in a rubidium atomic vapor,” Phys. Rev. A 93, 023433 (2016).
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A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
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A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
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M. C. Noecker, B. P. Masterson, and C. E. Wieman, “Precision measurement of parity nonconservation in atomic cesium: A low-energy test of the electroweak theory,” Phys. Rev. Lett. 61, 310–313 (1988).
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W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
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C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “Detection of small atom numbers through image processing,” Phys. Rev. A 82, 061606 (2010).
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A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87, 637–701 (2015).
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C. D. Herold, V. D. Vaidya, X. Li, S. L. Rolston, J. V. Porto, and M. S. Safronova, “Precision measurement of transition matrix elements via light shift cancellation,” Phys. Rev. Lett. 109, 243003 (2012).
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P. F. Griffin, K. J. Weatherill, S. G. MacLeod, R. M. Potvliege, and C. S. Adams, “Spatially selective loading of an optical lattice by light-shift engineering using an auxiliary laser field,” New J. Phys. 8, 11 (2006).
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F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, “Dynamical polarizability of atoms in arbitrary light fields: general theory and application to cesium,” Eur. Phys. J. D 67, 1–16 (2013).
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Reitz, D.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
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A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
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H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
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C. D. Herold, V. D. Vaidya, X. Li, S. L. Rolston, J. V. Porto, and M. S. Safronova, “Precision measurement of transition matrix elements via light shift cancellation,” Phys. Rev. Lett. 109, 243003 (2012).
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R. H. Leonard, A. J. Fallon, C. A. Sackett, and M. S. Safronova, “High-precision measurements of the 87Rb D-line tune-out wavelength,” Phys. Rev. A 92, 052501 (2015).
[Crossref]

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R. H. Leonard, A. J. Fallon, C. A. Sackett, and M. S. Safronova, “High-precision measurements of the 87Rb D-line tune-out wavelength,” Phys. Rev. A 92, 052501 (2015).
[Crossref]

C. D. Herold, V. D. Vaidya, X. Li, S. L. Rolston, J. V. Porto, and M. S. Safronova, “Precision measurement of transition matrix elements via light shift cancellation,” Phys. Rev. Lett. 109, 243003 (2012).
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[Crossref] [PubMed]

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B. K. Sahoo, L. W. Wansbeek, K. Jungmann, and R. G. E. Timmermans, “Light shifts and electric dipole matrix elements in Ba+ and Ra+,” Phys. Rev. A 79, 052512 (2009).
[Crossref]

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J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Addison-Wesley, 2011), 2nd ed.

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F. Schmidt, D. Mayer, M. Hohmann, T. Lausch, F. Kindermann, and A. Widera, “Precision measurement of the 87Rb tune-out wavelength in the hyperfine ground state f = 1 at 790 nm,” Phys. Rev. A 93, 022507 (2016).
[Crossref]

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A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87, 637–701 (2015).
[Crossref]

Schmidt, R.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref] [PubMed]

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H. Häffner, S. Gulde, M. Riebe, G. Lancaster, C. Becher, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor,” Phys. Rev. Lett. 90, 143602 (2003).
[Crossref] [PubMed]

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F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, “Dynamical polarizability of atoms in arbitrary light fields: general theory and application to cesium,” Eur. Phys. J. D 67, 1–16 (2013).
[Crossref]

Scully, M. O.

I. Novikova, A. B. Matsko, V. L. Velichansky, M. O. Scully, and G. R. Welch, “Compensation of ac Stark shifts in optical magnetometry,” Phys. Rev. A 63, 063802 (2001).
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Smith, J.

A. C. Lee, J. Smith, P. Richerme, B. Neyenhuis, P. W. Hess, J. Zhang, and C. Monroe, “Engineering large Stark shifts for control of individual clock state qubits,” Phys. Rev. A 94, 042308 (2016).
[Crossref]

Spreeuw, R. J. C.

C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “Detection of small atom numbers through image processing,” Phys. Rev. A 82, 061606 (2010).
[Crossref]

Stamper-Kurn, D. M.

G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, “Ultracold atoms in a tunable optical kagome lattice,” Phys. Rev. Lett. 108, 045305 (2012).
[Crossref] [PubMed]

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W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
[Crossref]

Stroescu, I.

W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovič, D. B. Hume, and M. K. Oberthaler, “Optimized absorption imaging of mesoscopic atomic clouds,” Appl. Phys. B 113169–73 (2013).
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C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “Detection of small atom numbers through image processing,” Phys. Rev. A 82, 061606 (2010).
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G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, “Ultracold atoms in a tunable optical kagome lattice,” Phys. Rev. Lett. 108, 045305 (2012).
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https://github.com/simocop/LightShiftCalculator .

Quantitative prediction of Ci is feasible but would require use of the optical Bloch equations to solve for atomic dynamics in the presence of the 1560 nm beam, the single probe beam at 780 nm, and repump light also at 780 nm which is emitted from six directions toward the centre of the trap.

We estimate the uncertainties on the fitted values as the square root of the diagonal elements in the p × p matrix (JT J)−1RT R/(Npts − p). Where J is the Jacobian of the model fit function, R is a vector of the fit residuals, Npts is the number of measurements, and p the number of fitting parameters. In this case p = 9: the light intensity I, quadrature phase ϕ, and the seven peak amplitudes Ci.

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Figures (4)

Fig. 1
Fig. 1

87Rb energy levels used for calculations in this paper. We performed several representative calculations with many more matrix elements up to n = 10 and found that these made a < 1 MHz contribution to the calculated light shifts under our experimental conditions, which is less than the uncertainty in our measurement. The inset shows the hyperfine splitting of the 5P3/2 levels. We included hyperfine splitting for all levels except the 4F levels, for which we were unable to find hyperfine constants in the literature.

Fig. 2
Fig. 2

Beam and atom cloud geometry in the experiment. A constant-power optical dipole trap at 1560.492 nm confines a cloud of 105 atoms while a mode-matched beam around 1529 nm with adjustable power induces strong light shifts in the atoms. A probe beam with adjustable frequency at 780 nm is used for measuring absorption of the cloud as a function of frequency.

Fig. 3
Fig. 3

Relative optical depth of atoms in the dipole trap around the light-shifted F = 2 → F′ = 3 transition with only 1560 nm light. Blue dots show measured optical depth in a small transverse slice of the dipole trap, extracted from absorption images. Each point is from a single experiment run. The red line is a fit using Eq. (12). The x-axis is relative to the free-space 5S1/2, F = 2 → 5P3/2, F = 3 transition. Free parameters in the fit were peak amplitudes, trap depth, and the ellipticity of trap light. Arrows show positions of resonances at maximum trap depth, “×2" indicates 2 resonances within the width of the arrow.

Fig. 4
Fig. 4

Relative optical depth with λ1 = 1560.492 nm and λ2 given in the respective caption. The probe frequency is relative to the free-space 5S1/2, F = 2 → 5P3/2, F = 3 transition. The black lines show calculated energy levels of dressed states. Shading shows measured relative optical depth of the atomic cloud in arbitrary units with the scale shown in the colour bar on the right. Each column is scaled to have the same maximum value. After using the data shown in Fig. 3 as a calibration of the experimental parameters, the only fitting parameter here is the calibration of the 1529 nm beam power. The arrows point to lines showing the calculated light shifts for a change in wavelength of the 1529 nm laser by ±0.001 nm, for a particular level.

Equations (15)

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i t ψ ( t ) = H ( t ) ψ ( t )
U ( t N , t 0 ) = U ( t N , t N 1 ) U ( t 2 , t 1 ) U ( t 1 , t 0 ) .
U ( T , 0 ) k = 0 N 1 e i H ( t k ) T / ( N )
n J F M | H 0 | n J F M = n J | H 0 | n J + 1 2 A n J K + B n J 3 K ( K + 1 ) / 2 2 I ( I + 1 ) J ( J + 1 ) 2 I ( 2 I 1 ) 2 J ( 2 J 1 )
V ( t ) = E ( t ) d
n J F M | d z | n J F M = n J | | e r | | n J ( 1 ) M + J + I ( 2 F + 1 ) ( 2 F + 1 ) × ( F 1 F M 0 M ) { J J 1 F F I } ,
d x = e i F y π / 2 d z e i F y π / 2 d y = e i F x π / 2 d z e i F x π / 2
n J F M | F ± | n J F ' M ' = ( F M + 1 ) ( F ± M ) δ n J F , n J F δ M , M ± 1 .
E π ( t ) = cos ( ω t ) z ^
E σ ± ( t ) = 2 [ cos ( ω t ) x ^ ± sin ( ω t ) y ^ ] .
E ( t ) = 1 cos ( ω 1 t ) n ^ 1 + 2 cos ( ω 2 t ) n ^ 2 .
A ( δ ) = i = 1 7 C i 0 u 2 e u 2 d u 1 + 4 ( δ + v i t i u 2 ) 2
E 1 ( t ) = 1 2 ( cos ( ω 1 t ) x ^ + cos ( ω 1 t + ϕ ) y ^ ) ,
I = ϵ 0 c 2 | | 2
E ( t ) = E 1 ( t ) + 2 2 [ cos ( ω 2 t ) x ^ cos ( ω 2 t ) y ^ ]

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