# An Optomechanical Elevator: Transport of a Bloch Oscillating Bose–Einstein Condensate up and down an Optical Lattice by Cavity Sideband Amplification and Cooling

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## Abstract

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## 1. Introduction

**Figure 1.**(

**a**) An optical lattice is created by pumping a Fabry–Perot cavity. The bias force F tilts the lattice potential and causes a trapped Bose–Einstein condensate (green) to undergo Bloch oscillations. Atomic backaction leads to a time-modulation of the lattice amplitude, which in turn induces coherent directed transport of the condensate, provided the total detuning from resonance is not zero. The transport corresponds to atoms climbing or descending the ladder of Wannier–Stark states, where each state is separated by the energy $\hslash {\omega}_{B}$. (

**b**) The archetype of cavity optomechanics is a cavity with one end mirror suspended on a spring. The motional states of the mirror correspond to the excitations of a harmonic oscillator. Cavity amplification or cooling of the mirror moves it up or down the ladder of oscillator states. Bloch oscillation dynamics in a cavity can be mapped onto standard cavity optomechanics, but with the harmonic oscillator ladder replaced by the Wannier–Stark ladder.

## 2. Bloch Oscillation and Transport in a Cavity

**Figure 2.**Backaction-induced atomic transport and breathing dynamics in a cavity with ${U}_{0}=-2\pi \times 1$ Hz, $\kappa =-N{U}_{0}=2\pi \times 1$ kHz and Bloch frequency ${\omega}_{B}=2\pi \times 744.5$ Hz. (

**a**) Condensate centroid position as a function of time showing uphill (blue, ${\Delta}_{c}-N{U}_{0}=1.3\kappa $) and downhill (red, ${\Delta}_{c}-N{U}_{0}=-0.7\kappa $) transport for an initial atomic wave packet delocalised over 20 lattice sites; (

**b**) breathing dynamics of the condensate density at the Bloch period for ${\Delta}_{c}-N{U}_{0}=-0.7\kappa $ and an initial atomic wave packet localised within one lattice site.

**Figure 3.**Backaction-induced modulation of the lattice depth (in units of the recoil energy ${E}_{r}=\hslash {\omega}_{r}$) as a function of time during intracavity Bloch oscillations. The blue (red) curve corresponds to parameters giving uphill (downhill) transport in Figure 2a. The black curve corresponds to the breathing dynamics plotted in Figure 2b.

**Figure 4.**Power spectral density of the cavity field and force-displacement plots for the atomic dynamics. The parameters are the same as in Figure 2a. (

**a**) Power spectral density of $\alpha \left(t\right)$, obtained by a fast Fourier transform at resolution ${\omega}_{B}/100$, showing asymmetric sidebands at multiples of $\pm {\omega}_{B}$ with more power in the red upper (blue lower) side band corresponding to downhill (uphill) transport in Figure 2a. Note that the frequency origin corresponds to the driving laser frequency, which is different for the red and blue points. (

**b**) Average of the force applied by the dynamical intracavity optical lattice as a function of the atomic wave packet centroid position. During uphill (downhill) transport of the atoms, the blue (red) curve is traversed in the clockwise (anti-clockwise) direction, corresponding to positive (negative) work done on the atoms and accounted for by the down-conversion (up-conversion) of the blue-detuned (red-detuned) pump laser photons (see the inset). The insets also show the dominant terms in the effective optomechanical Hamiltonian, which we derive in Section 3, with the operator ${\widehat{b}}_{M}$ annihilating a quantum of excitation from the Bloch oscillator. The area enclosed by the loops gives the work done on the atoms by the lattice.

## 3. Mapping to an Optomechanical Hamiltonian

## 4. Comparison to Standard Cavity Optomechanics

## 5. Transport as a Manifestation of Cavity Amplification or Cooling

**Figure 5.**Comparison of the transport velocity as a function of the cavity-driving laser detuning calculated from the numerical simulation and using the analytical expression Equation (34). System parameters are as introduced in Figure 2a with η varied to maintain an initial lattice depth of $-3{E}_{r}$.

## 6. Metrology

## 7. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Prasanna Venkatesh, B.; O’Dell, D.H.J.; Goldwin, J.
An Optomechanical Elevator: Transport of a Bloch Oscillating Bose–Einstein Condensate up and down an Optical Lattice by Cavity Sideband Amplification and Cooling. *Atoms* **2016**, *4*, 2.
https://doi.org/10.3390/atoms4010002

**AMA Style**

Prasanna Venkatesh B, O’Dell DHJ, Goldwin J.
An Optomechanical Elevator: Transport of a Bloch Oscillating Bose–Einstein Condensate up and down an Optical Lattice by Cavity Sideband Amplification and Cooling. *Atoms*. 2016; 4(1):2.
https://doi.org/10.3390/atoms4010002

**Chicago/Turabian Style**

Prasanna Venkatesh, B., Duncan H.J. O’Dell, and Jonathan Goldwin.
2016. "An Optomechanical Elevator: Transport of a Bloch Oscillating Bose–Einstein Condensate up and down an Optical Lattice by Cavity Sideband Amplification and Cooling" *Atoms* 4, no. 1: 2.
https://doi.org/10.3390/atoms4010002