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

Round  1

Reviewer 1 Report

This is a nice contribution to the special issue of cold atoms in optical cavities. The auhtors investigate the motion of atoms in a periodic potential inside he cavity taking into account the effect of continuous measurement of photons leaking out of the cavity.

I support publication, but I have a few comments that the authors hopefully can address before publication.

The link to the sensing of weak forces (in this case gravity) is interesting, but I am not convinced whether this is a practical approach. The estimated accuracy is $10^6$ that does not sound great. Have the effects of radiation pressure forces, etc., taken into account? What is the role of excitations to higher Bloch bands?

In my opinion, the main interest of the work is in the cold atoms and cavity dynamics, continuous observation, and in the specific optomechanical dynamics. The role of sensing may in the end be less important and could be downplayed a bit in the manuscript.

There are several other theoretical studies of the continuous quantum measurement of many-atom/cavity systems and condensate systems more generally that could be cited. 

A condenste-cavity optomechanical  study with continuous observation

http://dx.doi.org/10.1103/PhysRevA.90.023628  

Other examples of many-atom-cavity continuous observation papers:

W. Niedenzu, S. Schütz, H. Habibian, G. Morigi, and H. Ritsch, Phys. Rev. A 88, 033830 (2013).

I. B. Mekhov and H. Ritsch, Phys. Rev. Lett. 102, 020403 (2009).

J. F. Corney and G. J. Milburn, Phys. Rev. A 58, 2399 (1998).

I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, Phys. Rev. Lett. 104, 073602 (2010).

There are other papers that link BECs, sensing and continuous quantum measurement process (in the absence of a cavity)

M. Saba, T. A. Pasquini, C. Sanner, Y. Shin, W. Ketterle, and D. E. Pritchard, Science 307, 1945 (2005).

 M. D. Lee, S. Rist, and J. Ruostekoski, New J. Phys. 14, 073057 (2012).

The effect of continuous measument process of scattered light on condensate dynamics:

J. Ruostekoski and D. F. Walls, Phys. Rev. A 58, R50 (1998).

D. A. R. Dalvit, J. Dziarmaga, and R. Onofrio, Phys. Rev. A 65, 053604 (2002).

J. Javanainen and J. Ruostekoski, New J. Phys. 15, 013005 (2013). 

Author Response

We have added all the new references suggested by Referee 2 on page 3 of the new manuscript. These includethe ones mentioned in our reply to Referee 1’s point 4 above, and all also the references:
Corney98,Mekhov09,leroux10,Mekhov12,Niedenzu13,Lee14. Concerning the referee’s other point, about the suitability of this system to measure weak forces, we first note that the use of Bloch oscillations without a cavity is proving to be an excellent way to measure gravity (e.g. the experiments by G.M. Tino’s group). The sensitivity of their measurements, after many years of development, are only one order of magnitude above our estimated sensitivity, and their measurements take one hour whereas ours should take one second which means it might have applications in rather different situations (e.g. local gravity mapping while flying). Our work is very much still in progress and this paper will not, we hope, be our final word on this subject. Indeed, we point out a possible way forward in the last section that builds on the insight gained from the optomechanics analogy. It seems likely that the homodyne phase measurement regime, where backaction can be reduced, will likely be more favourable when it comes to absolute sensitivity than photon counting type measurement we focus on at the moment. For these reasons we would like to keep the measurement aspects of the paper intact: we would like to advertise them and, perhaps, be a little provocative.

Concerning the question of whether we have taken the effects of higher Bloch bands into effect, our numerical calculations do take this into account. The question of higher Bloch bands is a little subtle when it comes to our analytical calculation done in the Wannier-Stark picture. In the presence of the force, the true eigenstates of the system are no longer the Bloch states but the localized Wannier-Stark states. In the analytical calculation we restrict ourselves to just the ground band Wannier-Stark states. However, even with this approximation we find very good agreement between the numerical and analytical calculations. Concerning the question of radiation
pressure forces, we take them into account on the atoms but not on the mirrors.

Reviewer 2 Report

The authors present a new interpretation to study Bloch oscillations in a cavity sustained optical lattice and the potential use of this system for very sensitive force measurement.

The idea is closely related to previous work as Ref. 33/34 and the results are similar. Nevertheless the presentaion sheds new light onto the underlying microscopic dynamics and connects the dynamics to proven optomechanical models of cavity BEC dynamics.

This work is nicely presented with well chosen examples and graphs.  Hence, in principle I consider it suitable for publication.  

There are however a few things to be clarified before publication:

(1) It should be somehow mentioned more clearly (maybe in the abstract) that the idea is not completely new but work presents a new and more intutive picture of the system dynamics.

(2) It is very helpful to connect cavity output spectra and forces. To my knowledge this was first discussed in: Gangl, M., EPJD, 8(1), 29-40. This should be mentioned.

(3) The authors claim there should be no backaction of the field on the oscillation frequencies. This is somehow in contradiction with the eigenfrequency calculations (see Fig.4)  in Horak, P.,Physical Review A, 61(3), 033609, 2000.  Also the BEC will lead to an effective change of the cavity length and lattice constant, which should renormalize the wavelength and thus om_recoil.

(4) the cavity is an open quantum system. This has two consequences:

(a) any measurement of the output field will generate backaction onto the condensate (Mekhov, J.Phys. B: Atomic, Molecular and Optical Physics, 45(10), 102001). This will generate extra noise, but might also help as it reduces atom number uncertainty in the cavity.

(b) There will be heating limiting the available measurement time. In particular for the blue detuned case not only the average momentum but also the momentum spread will grow. as the time scale can be faster than om_recoil, it could limit the measurement time to a fairly short period. Some rough estimate on the time scales should be added.

Author Response

The authors present a new interpretation to study Bloch oscillations in a cavity sustained optical lattice and the potential use of this system for very sensitive force measurement.
The idea is closely related to previous work as Ref. 33/34 and the results are similar. Nevertheless the presentaion sheds new light onto the underlying microscopic dynamics and connects the dynamics to proven optomechanical models of cavity BEC dynamics.
This work is nicely presented with well chosen examples and graphs. Hence, in principle I consider it suitable for publication.
There are however a few things to be clarified before publication:
(1) It should be somehow mentioned more clearly (maybe in the abstract) that the idea is not completely new but work presents a new and more intutive picture of the system dynamics.

Done. We have re-written the first part of the abstract as: “In this paper we give a new description, in terms of optomechanics, of previous work on the problem of an atomic Bose-Einstein condensate interacting with the optical lattice inside a laser-pumped optical cavity and subject to a bias force such as gravity \cite{Ven09,Ped09,Gol14}. An atomic wave packet in a tilted lattice undergoes Bloch oscillations; in a highfinesse optical cavity the backaction of the atoms on the light leads to a  time-dependent modulation of the intracavity lattice depth at the Bloch frequency which can in turn transport the atoms up or down the lattice.”
(2) It is very helpful to connect cavity output spectra and forces. To my knowledge this was first discussed in: Gangl, M., EPJD, 8(1), 29-40. This should be mentioned.

We have added the following sentences on p3 (and four new references, including the one to Gangl’s work, in the bibliography): “A number of experiments have already demonstrated how the light transmitted by a cavity can be used to track the motion of atoms trapped inside \cite{Hood98,Hood00,Pinkse00}, and in particular, a theoretical analysis of the information stored in the frequency spectrum has been given in \cite{Gangl00}, showing that atomic motion introduces sidebands either side of the pump frequency. In our case, Bloch oscillations at angular frequency $\omega_{B}$ generates sidebands separated from the pump frequency by $\pm \omega_{B}$ (and harmonics thereof in the strong coupling regime).
Because the Bloch frequency is proportional to the applied force $\omega_{B}= F d/\hbar$, where $d=\lambda/2$ is the
lattice period, a detection of the spectrum of the transmitted light gives $F$ directly.”

(3) The authors claim there should be no backaction of the field on the oscillation frequencies. This is somehow in contradiction with the
eigenfrequency calculations (see Fig.4) in Horak, P.,Physical Review A, 61(3), 033609, 2000. Also the BEC will lead to an effective change of the cavity length and lattice constant, which should renormalize the wavelength and thus om_recoil.

Actually, we do not claim that oscillation frequencies in general will be unaffected by the backaction. Rather, we claim that one particular frequency, the Bloch frequency, will be unaffected. The Horak paper shows that Bloch energies (i.e. eigenmodes for a particle in a periodic potential…not to be confused with the Bloch oscillation frequency) will be changed by the backaction. This is because the presence of atoms in the cavity shifts the cavity resonance and hence the depth of the intracavity lattice: the Bloch energies are dependent on the potential depth. This change in potential depth due to backaction is fully included in our treatment. However, a change in the depth of a periodic potential does not change the Bloch frequency because this depends on the lattice’s spatial period, not the depth. Because this key aspect of our treatment can clearly lead to confusion we have modified the following sentence in the conclusions on p19, and added the reference to Horak’s paper: “Chief among these is that the backaction does not alter the frequency of the Bloch oscillations. By contrast, in the harmonic oscillator case there is the so-called optical spring effect which gives a dependence of oscillator frequency on field amplitude and detuning. To be clear, other motional frequencies are altered: because the intracavity lattice depth is  modulated by the backaction this will affect certain types of atomic motion, for example the oscillation frequency of an atom about the bottom of one of the potential minima \cite{Horak00}. Nevertheless, the Bloch oscillation frequency is robust against this depth modulation because it only depends on the lattice period, not its depth.”

Concerning the change in wavelength (and hence change in lattice constant) due to the refractive index of the gas, this effect is tiny. We consider it in section 6, on p17 and show that the correction to the refractive index is of order 10^-9 for our parameters. This is to be distinguished from the effective change in the cavity length which, due to the resonance effect, is substantial and is precisely what lies behind the amplitude modulation of the lattice.
(4) the cavity is an open quantum system. This has two consequences: (a) any measurement of the output field will generate backaction onto the condensate (Mekhov, J.Phys. B: Atomic, Molecular and Optical Physics, 45(10), 102001). This will generate extra noise, but might also help as it reduces atom number uncertainty in the cavity. (b) There will be heating limiting the available measurement time. In
particular for the blue detuned case not only the average momentum but also the momentum spread will grow. as the time scale can be faster than om_recoil, it could limit the measurement time to a fairly short period. Some rough estimate on the time scales should be added.

Yes we agree that there will be extra noise due to the open nature of the cavity, and we already refer to this on p3 of the Introduction: “The disadvantage of working in a cavity is that quantum measurement backaction, in the form of random fluctuations in the cavity field due to photons spontaneously leaking out of the cavity, heats up the cold atoms and limits the coherence time of the measurement \cite{Ven13}.” where the cited paper is one of ours [Prasanna Venkatesh, B.; O’Dell, D. H. J.; Bloch oscillations of cold atoms in a cavity: Effects of quantum noise. Phys. Rev. A 2013, 88, 013848] where we directly
calculated the heating effect for Bloch oscillating atoms in a cavity within the Bogoliubov approximation. Following the referee’s suggestion we have added the reference to Mekhov’s paper.
The coherence time for the many-particle Bloch oscillating state is in fact very hard to calculate (mainly because even the coherent part is time-dependent rather than stationary). It can, however, be estimated and so we have added in the following sentences on p3:
“\textcolor{red}{The coherence time is particularly hard to calculate in the Bloch oscillating case \cite{Ven13} due to the time dependence introduced by the Bloch oscillations, especially in the presence of many particles, but it can be roughly estimated to be $\tau=\tau_{\mathrm{sp}}/(1+C)$ \cite{Ven09} at cavity resonance, where$\tau_{\mathrm{sp}}^{-1}=2 \gamma \vert \alpha \vert^2 \Omega_{0}^2/\Delta_{a}^2$ is the spontaneous emission rate at an antinode. The factors $\vert \alpha \vert^2$ and
$\Delta_{a}$ are the mean number of cavity photons and the detuning of the laser from atomic resonance, respectively, and will be properly defined in the next  section. The numerical value of $\tau$ for the parameters considered in this paper will be given in Section \ref{sec:metrology}. Of course, Bose-Einstein condensates can be continuously measured and used for sensing
without a cavity, e.g.\ \cite{Ruostekoski98,Dalvit02,Saba05,Lee12,Java13}, but the cavity case is particularly interesting because it allows for a strong atom-light interaction even in the quantum regime.} “In addition, in Section 6 (paragraph 4, Pg 18) we have provided further details required to estimate the coherence time for the parameters used in the paper. We find that
coherence times of up to a couple of seconds (many  thousands of Bloch oscillation periods) are possible