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Squeezing Two Mean Field Modes of a Bose–Einstein Condensate

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Spin Squeezing and Non-linear Atom Interferometry with Bose-Einstein Condensates

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Abstract

Bose–Einstein condensation has been predicted in 1924/1925 by Satyendra Nath Bose and Albert Einstein. The Nobel prize 2001 was awarded to Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman for the first experimental observation of Bose–Einstein condensation in dilute gases of laser cooled alkali atoms in 1995. Almost 15 years later a whole new sub field of atomic physics developed dealing with Bose–Einstein condensates and degenerate Fermi gases. A lot of effort has been made, both experimentally and theoretically, to explore the basic physics of ultracold quantum degenerate gases. Extraordinary experimental control over the trapped quantum gases and the possibility to measure and adjust almost all relevant parameters directly (e.g. interaction strength, relative phases, ...) opens up a new route in atomic physics. The quantum gases can be used to engineer specific Hamiltonians that map for example to problems in solid state physics where some measurements are hard to perform and many parameters are not controllable. Ultracold quantum gases are promising candidates for quantum simulators of solid state systems. In the field of quantum metrology degenerate gases have been proposed to be one experimental system that allows for a precision beyond the “classical” projection noise limit in atom interferometry. Controllable many-body entanglement can be used as a resource to beat the standard quantum limit.

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Notes

  1. 1.

    \(2\pi\hbar\) is Planck’s constant and m the atomic mass.

  2. 2.

    The exact value depends on the density of states.

  3. 3.

    \(\phi_i\) is the normalized \(i{\text {th}}\) eigenfunction of the single particle Hamiltonian, \(V(\mathbf{r} - \mathbf{r}^{\prime})\) is the interatomic interaction potential, later approximated as a contact interaction \(V(\mathbf{r} - \mathbf{r}^{\prime}) \propto \delta(\mathbf{r} - \mathbf{r}^{\prime}).\)

  4. 4.

    The beam waist is \(5.1\,\mu{\text{m}}.\)

  5. 5.

    The barrier height is tunable between \(V_0 = 2\pi \cdot 250 \) Hz and \(V_0 = 2\pi \cdot 3{,}000 \) Hz.

  6. 6.

    A dependence of \(E_J(n)\) from the occupation number difference is omitted here, i.e. we assume \(n \ll N.\) This term is included in the discussion presented in Sect. 3.1.3.

  7. 7.

    \({n}_{l,r}(\mathbf{r})\) is the atomic density of the left (right) mode and \({n}_{l,r} = \int_{-\infty, 0}^{0,\infty} \hbox{d}\mathbf{r} {n}_{l,r}(\mathbf{r})\) the mode occupation number.

  8. 8.

    The authors of reference [39] show that the Heisenberg limit can be reached within a factor of two.

  9. 9.

    In the experiment we typically have \(T/{\rm 2}E_J,max\approx {\rm 10^{-2}}.\)

  10. 10.

    \(\hat{n}\) and \(\hat{\varphi}\) are symmetric variables in Eq.  (3.14). The fluctuations in the relative phase \(\Updelta{\hat{\varphi}}^2 \approx T/{E_{J}}\) are obtained in the same way as described for \(\Updelta{\hat{n}}^2 = \langle{\hat{n}^{2}}\rangle\), but replacing the matrix element by \(\langle{k}|{\hat{\varphi}^{2}}|{k}\rangle=( k+1/2)\sqrt{E_{C}/{E_{J}}}.\)

  11. 11.

    For our parameters \(E_{C,f}\approx E_{C,i} = E_C\) holds.

  12. 12.

    The frequency of this beam differs from the frequency of the main dipole beam by 30 MHz to average their interference pattern.

  13. 13.

    The thermal expansion coefficient of alluminium is ca. \(23\,\times\,10^{-6}\)/K at room temperature, leading to a temperature stability requirement of 10 mK over a few hours, the typical duration of the experiment.

  14. 14.

    We ramp from \(V_{0,i}=2\pi\cdot430\) Hz for all end values \(V_0\geq2\pi\cdot430\) Hz and from \(V_{0,i}= 2\pi\cdot250\) Hz for all other end values.

  15. 15.

    In the double-well case two more data points not shown in Fig. 3.18 between \(V_0=2\pi\cdot1{,}650\) Hz and \(V_0=2\pi \cdot1{,}800 \)Hz contribute to the averaging.

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  1. University of Heidelberg, Heidelberg, Germany

    Christian Groß

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  1. Christian Groß
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Groß, C. (2012). Squeezing Two Mean Field Modes of a Bose–Einstein Condensate. In: Spin Squeezing and Non-linear Atom Interferometry with Bose-Einstein Condensates. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25637-0_3

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  • DOI: https://doi.org/10.1007/978-3-642-25637-0_3

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