Localization transition in the presence of cavity backaction

Katharina Rojan, Rebecca Kraus, Thomás Fogarty, Hessam Habibian, Anna Minguzzi, and Giovanna Morigi

Phys. Rev. A 94, 013839 – Published 20 July 2016

Abstract

We study the localization transition of an atom confined by an external optical lattice in a high-finesse cavity. The atom-cavity coupling yields an effective secondary lattice potential, whose wavelength is incommensurate with the periodicity of the optical lattice. The cavity lattice can induce localization of the atomic wave function analogously to the Aubry-André localization transition. Starting from the master equation for the cavity and the atom we perform a mapping of the system dynamics to a Hubbard Hamiltonian, which can be reduced to the Harper's Hamiltonian in appropriate limits. We evaluate the phase diagram for the atom's ground state and show that the transition between extended and localized wave function is controlled by the strength of the cavity nonlinearity, which determines the size of the localized region and the behavior of the Lyapunov exponent. The Lyapunov exponent, in particular, exhibits resonancelike behavior in correspondence with the optomechanical resonances. Finally we discuss the experimental feasibility of these predictions.

Received 4 May 2016

DOI:https://doi.org/10.1103/PhysRevA.94.013839

Physics Subject Headings (PhySH)

Atoms, ions, & molecules in cavities Cavity quantum electrodynamics Cold gases in optical lattices Optomechanics

Ultracold gases

Atomic, Molecular & Optical

Authors & Affiliations

Katharina Rojan^1,2,3, Rebecca Kraus¹, Thomás Fogarty^1,4, Hessam Habibian^1,*, Anna Minguzzi^2,3, and Giovanna Morigi¹

¹Theoretische Physik, Universität des Saarlandes, D-66123 Saarbrücken, Germany
²Université Grenoble-Alpes, LPMMC, Boîte Postale 166, F-38042 Grenoble, France
³CNRS, LPMMC, Boîte Postale 166, F-38042 Grenoble, France
⁴Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, Japan

^*Currently at Accenture S.L.U., 08174 Sant Cugat del Vallés, Barcelona, Spain.

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Vol. 94, Iss. 1 — July 2016

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Images

Figure 1
(a) A single atom is confined by an optical lattice, sketched as blue (light gray) line, with wave number $k_{0} = 2 π / λ_{0}$ and frequency $ω_{0}$ within a standing-wave resonator. Its motion optomechanically interacts with a high-finesse mode, shown in red (dark gray), at frequency $ω_{c}$ and wave number $k = 2 π / λ$ , whose wavelength $λ$ is incommensurate with the optical lattice periodicity $λ_{0} / 2$ . The depth of the cavity lattice is determined by the balance between a pump, with strength $η$ and frequency $ω_{p}$ , and the losses at rate $κ$ . We study the localization transition in this setup, where the nonlinearity due to strong coupling with the cavity (given by the cooperativity $C$ ) modifies the effective incommensurate potential. The optomechanical potential is illustrated in panel (b) as a function of $x$ for three different values of the cooperativity $C$ and when the laser is resonant with the cavity. The limit $| C | ≪ 1$ corresponds to the Aubry-André model.
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Figure 2
(a) IPR, Eq. (6), as a function of $v_{0}$ (in units of $t$ ) for $δ_{c}^{'} = 0$ and $C = - 0.5, - 2$ , and $- 4$ (see legend). (b) Contour plot of the IPR as a function of $v_{0}$ (in units of $t$ ) and of $C$ , for $δ_{c}^{'} = 0$ . The red (gray) solid lines correspond to Eq. (5). For calculating $δ ε_{n}$ in Eq. (4) we used the Wannier function for a confining potential of depth $W_{0} = - 29 E_{r}$ , with $E_{r} = ℏ^{2} k_{0}^{2} / (2 m)$ being the recoil energy.
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Figure 3
Lyapunov exponent as a function of $v_{0}$ , in units of the critical depth $v_{c}$ , which we extract from the numerical behavior of the IPR in Fig. 2, for $δ_{c}^{'} = 0$ and for (a) $C = - 0.5, - 2$ , and $- 4$ and (b) $C = 0.5$ , 2, and 4. The black dashed line corresponds to the functional behavior of the Lyapunov exponent in Aubry-André's model, Eq. (8). The insets display the probability densities as a function of $x$ for the corresponding values of $C$ of the curves in the onset and for $v_{0} / v_{c} = 2$ . Panels (c) and (d) display the Lyapunov exponent as a function of $C$ for the fixed ratio $v_{0} / v_{c} = 1.2$ ; the horizontal line indicates the value predicted by Eq. (8) $[v_{c}$ depends on $C$ , for each value of $C$ it is extracted from the curves of the IPR as in Fig. 2]. For calculating $δ ε_{n}$ in Eq. (4) we used the Wannier function for a confining potential of depth $W_{0} = - 29 E_{r}$ , with $E_{r} = ℏ^{2} k_{0}^{2} / (2 m)$ being the recoil energy.
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Figure 4
Contour plot of the IPR as a function of $v_{0}$ (in units of $t$ ) and of $C$ , for $δ_{c}^{'} = - 2$ . The red (gray) solid lines correspond to Eq. (5). The other parameters are the same as those in Fig. 3.
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Figure 5
Panels (a) and (b) display the Lyapunov exponent as a function of $C$ for $v_{0} / v_{c} = 1.2$ . The black horizontal dashed line indicates the value predicted by Eq. (8). The insets display probability densities as a function of $x$ for different values of $C$ and for the fixed ratio $v_{0} / v_{c} = 2$ . The other parameters are the same as in those Fig. 3.
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Figure 6
(a) Inverse participation ratio, Eq. (6), and (b) mean intracavity photon number $\bar{n}$ , Eq. (26), as a function of the parameters $η$ and $C = U_{0} / κ$ in the setup where the resonator is driven and for detuning $δ_{c} = - 5.5 κ$ . Here, $η$ is the strength of the laser and $U_{0}$ is the strength of the optomechanical coupling. In panels (a) and (b) the potential depth is fixed to $W_{0} = - 15 E_{r}$ , where $E_{r}$ is the recoil energy associated with the $D$ line of ${}^{87}{Rb}$ atoms. Panels (c) and (d) show the IPR and $\bar{n}$ as a function of $η$ and $δ_{c}$ (in units of $κ$ ) for $C = - 1$ and $W_{0} = - 14 E_{r}$ . The other parameters are the number of sites $L = 233$ and $β = \frac{\sqrt{5} - 1}{2}$ . For the parameters of Refs. [28, 29], where $κ \approx E_{r} / ℏ$ , the time-scale separation at the basis of our model is warranted when the detuning $| δ_{c} | > E_{r} / ℏ$ and the atom's temperature $T < 1 μ K$ .
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