Dynamical tunneling with ultracold atoms in magnetic microtraps

The study of dynamical tunneling in a periodically driven anharmonic potential probes the quantum-classical transition via the experimental control of the effective Planck's constant for the system. In this paper we consider the prospects for observing dynamical tunneling with ultracold atoms in magnetic microtraps on atom chips. We outline the driven anharmonic potentials that are possible using standard magnetic traps and find the Floquet spectrum for one of these as a function of the potential strength, modulation, and effective Planck's constant. We develop an integrable approximation to the nonintegrable Hamiltonian and find that it can explain the behavior of the tunneling rate as a function of the effective Planck's constant in the regular region of parameter space. In the chaotic region we compare our results with the predictions of models that describe chaos-assisted tunneling. Finally, we examine the practicality of performing these experiments in the laboratory with Bose-Einstein condensates.

Figure 1

Poincaré sections for the classical dynamics of the atom-chip Hamiltonian, Eq. (2) at $t=0$, illustrating the influence of the potential strength $\kappa$ and modulation amplitude $\epsilon$ on the classical dynamics. We show results for the parameters: (a) $\kappa =1.2$, $\epsilon =0.1$; (b) $\kappa =1.2$, $\epsilon =0.6$; (c) $\kappa =1.8$, $\epsilon =0.1$; (d) $\kappa =1.8$, $\epsilon =0.6$.

Figure 2

Distance in phase-space between the period-one resonances at $t=0$ as a function of potential strength $\kappa$ for three values of modulation strength $\epsilon$. The black curves are determined from the Poincaré section for the full Hamiltonian Eq. (2). The cyan (gray) curves are the corresponding result derived from the integrable approximation to the full Hamiltonian, Eq. (9), as introduced in Sec. 3b. These give an indication of the validity of the integrable approximation. Solid lines: $\epsilon =0.1$. Dashed lines: $\epsilon =0.4$. Dash-dot lines: $\epsilon =0.8.$

Figure 3

Quasienergy splitting $\Delta E$ of the tunneling Floquet states as a function of the inverse effective Planck's constant $1/{\hslash }_{\mathrm{eff}}$ and the modulation strength $\epsilon$ for the atom-chip Hamiltonian Eq. (2) for two different potential strengths. (a) $\kappa =1.2$; (b) $\kappa =2.0$.

Figure 4

Quasienergy splitting $\Delta E$ of the tunneling Floquet states as a function of the inverse effective Planck's constant $1/{\hslash }_{\mathrm{eff}}$ and the modulation strength $\epsilon$ for the (a) quartic oscillator with Hamiltonian given by Eq. (7) and $\kappa =2$; (b) nonlinear pendulum with Hamiltonian given by Eq. (8) and $\kappa =1.5$.

Figure 5

Comparison of the Floquet spectrum of the full Hamiltonian $H_m$, Eq. (2), and the energy spectrum of the integrable approximation $H_i$, Eq. (9), for $\kappa =1.2$ and $\epsilon =0.1$. (a) The quasienergy spectrum of the full Hamiltonian as a function of $1/{\hslash }_{\mathrm{eff}}$. The black points are for the even Floquet states, and the cyan (gray) points are for the odd Floquet states. The points corresponding to the tunneling states are joined with thick solid lines of the same color. (b) The energy spectrum of the integrable Hamiltonian as a function of $1/{\hslash }_{\mathrm{eff}}$, with the same color scheme as for (a). The structures here are very similar to those in (a), indicating that the integrable approximation captures the important features of the full Hamiltonian in this regime. (c) A plot of the quasienergy difference of the even and odd tunneling states as a function of $1/{\hslash }_{\mathrm{eff}}$. Black curve: full Hamiltonian. Cyan (gray) curve: integrable approximation.

Figure 6

Comparison of the Poincaré sections for the full Hamiltonian $H_m$, Eq. (2), and the integrable approximation $H_i$, Eq. (9) at $t=0$, for (a) $\kappa =1.3$ and $\epsilon =0.1$, (b) $\kappa =1.1$ and $\epsilon =0.6$, (c) $\kappa =1.3$ and $\epsilon =0.6$. Although the presence of chaos for $H_m$ in this latter regime makes the Poincaré sections of $H_m$ and $H_i$ look rather different, we find good agreement for the regular regions. The quantum behavior of the two systems agree for relatively large values of ${\hslash }_{\mathrm{eff}}$ as shown in Sec. 3b.

Figure 7

(a) Energy versus inverse action for the trajectories of the integrable approximation $H_i$ that are not located on the islands (solid black curve). The gap occurs due to the presence of the islands. The inset shows selected trajectories of $H_i$. The energy of the trajectories in cyan (gray) are denoted in the main figure by cyan (gray) points as indicated by the arrows. (b) Energy of the semiclassical and quantum states as a function of $1/{\hslash }_{\mathrm{eff}}$. The thin curves are the EBK quantization of the trajectories of $H_i$, Eq. (9). The thin solid black curves correspond to even parity states, and the thin dashed cyan (gray) curves to odd parity states. The thick black curve is the energy of the even tunneling state, and the thick cyan (gray) curve is the energy of the odd tunneling state, both found via the numerical diagonalization of $H_i$, Eq. (9). The avoided crossings of the quantum tunneling states with other states of the same parity are clearly visible.

Figure 8

Quasienergy splitting of the tunneling Floquet states versus $1/{\hslash }_{\mathrm{eff}}$ for $\kappa =1.2$, $\epsilon =0.75$. Thin solid black line: Exact result. Thick cyan (gray) solid line: Numerical diagonalization of the integrable approximation. Dashed red line: Result of the Podolskiy-Narimanov theory with the proportionality coefficient of Eq. (12) as a fitting parameter. Inset: Poincaré sections of the full Hamiltonian (right) and of its integrable approximation (left). The tunneling behavior of the full Hamiltonian $H_m$ [Eq. (2)] agrees with the integrable approximation $H_i$ [Eq. (9)] as long as the quantum coarse-graining effect prevents the quantum particle from seeing the layers of chaos. For smaller values of ${\hslash }_{\mathrm{eff}}$, tunneling is enhanced by the presence of chaos in the underlying classical phase space according to the Podolskiy-Narimanov model [24].

Figure 9

Quasienergy splitting of the tunneling Floquet states as a function of ${\hslash }_{\mathrm{eff}}$ for $\kappa =2$, $\epsilon =15/32$. Solid black line: Exact numerical result. Red dashed line: Podolskiy-Narimanov theory (one fitting parameter). Blue dash-dot line: Eltschka-Schlagheck theory (no fitting parameter). Inset: Poincaré section centered on ${\mathcal{I}}_{+}$. One secondary resonances chain is clearly dominant. For large values of ${\hslash }_{\mathrm{eff}}$, the quantum coarse-graining prevents the system from seeing these small phase-space structures, and its behavior is then well described by Eq. (12). For smaller values of ${\hslash }_{\mathrm{eff}}$ the Eltschka-Schlagheck model [21], which takes the island chain into account, shows much better agreement with the numerical data.

Figure 10

Schematic of the arrangement of condensate, confining potentials, and atom-chip surface (yellow) with characteristic $z$-shaped wire. The atom-chip potential has the form Eq. (2) along $x=q$ (red) and is weakly harmonic along $z$ (potential not shown). Superimposed is an optical lattice potential (blue), freezing out the dynamics along $y$. The gray shaded volume is the resulting isodensity surface of a trapped BEC. Embedded density profiles along the $x$, $y$, $z$, coordinate directions are also sketched.

Figure 11

(a) Poincaré section at $t=0$ for the proposed experimental parameters $\kappa =1.01$, ${\hslash }_{\mathrm{eff}}=1/73$, $\epsilon =0.48$. Note that no chaos is present between the islands. (b) Husimi function for the even tunneling state. (c) Husimi function for the odd tunneling state.

Figure 12

[(a)–(c)] Dynamical tunneling signatures from condensate mean-field theory according to Eq. (14) for parameters as in Fig. 11 and $U_{1\mathrm{D}}=1\times {10}^{-4}$. (a) Husimi function (full width at half-maximum) of the initial state located within the Poincaré section: (thick red, solid) initial Floquet state $|{\varphi }_{+}\rangle$, (thick blue, dashed) experimentally accessible Gaussian approximation as explained in the text, $|{\phi }_{+}\rangle$. (b) Dynamical tunneling in momentum space from initial Floquet state and (c) from initial Gaussian approximation as explained in the text. [(d)–(f)] The same as (a)–(c) but for parameters $\kappa =1.3$, ${\hslash }_{\mathrm{eff}}=0.5$, $\epsilon =0.2$, and $U_{1\mathrm{D}}=0.01$, with $p_0=0.65$ and ${\kappa }_{\text{ini}}=0.4$.