Fast control of interactions in an ultracold two atom system: Managing correlations and irreversibility

We would like to thank the referee for their careful reading of our manuscript and their helpful suggestions. Below we address each of the comments individually.

1) As our protocol is based on a variational ansatz, the referee is correct in pointing out that it is not a theorem. However, we clearly show that this approximate approach is a good way to obtain a highly-effective STA for a nontrivial system. Indeed, when dealing with non-integrable systems with realistic interactions which, in particular, do not possess scale invariant solutions [see for example Deffner et al, PRX 4, 021013], approximative methods are the only techniques known. This has also been found in other approaches which deal with many-body systems [Sels and Polkovnikov, PNAS 114 (20) E3909]. Our work shows that the choice of an appropriate ansatz allows to create an efficient STA for interacting systems, which can be effective for short timescales. Due to the approximate nature of the ansatz our STA is not perfect and will, in particular, fail for very fast ramp times. In Fig. 2(a) in the manuscript we show the amount of irreversible work created during the STA and explain that it diverges for times $t_f \leq 1$, as at this point the STA approaches negative values and possesses a more oscillatory form rather than the smooth and slow increase shown in Fig. 1(b). To highlight this fact we have modified Fig. 1(b) to include the STA pulse for $t_f=1$, and include even more examples of STA ramps in the attached Figure R1. Irregardless of the final interaction $g_f$ the STA always gives good results for $t_f>1$, and we have modified the text in the manuscript to emphasize this point.

2) The referee is correct and our approach also works for finite initial g, as good functions for the ansatz are known. Actually, the STA works even better as the initial repulsive interaction between the particles introduces a degree of stiffness to the system which can allow for faster ramps of the interaction compared to starting from $g=0$. The attached Figure R2 shows the irreversible work for a ramp from $g_i=1$ to $g_f=20$, which confirms the improvement over the reference ramp. A sentence has been added to the conclusions of the manuscript to highlight this point.

3) The solutions to the Euler-Lagrange equations allow us to derive interaction ramps as a function of the ramp duration, $t_f$, which means that the shape of the ramp does change for different choices of $t_f$. In response to the first point raised by the referee, we have already modified Fig. 1(b) in the manuscript to highlight this. In general one finds that for larger $t_f$ the change of $g(t)$ is a continuous increase, initially slow and becoming faster towards the end of the ramp. For shorter ramp times the $g(t)$ starts to become oscillatory, initially increasing, going through two turning points and then finally ramping up quickly towards $t_f$ (see $t_f=1$ in Figure R1). For ramping times faster than $t_f=1$ the $g(t)$ does become negative and starts to increase in magnitude (see $t_f=0.5$ in Figure R1). It is on these timescales that the STA starts to fail, as too many dynamical excitations are created and cannot be compensated by our approximate protocol. We have expanded the text in the manuscript that discusses this behaviour. The referee correctly points out that fast protocols that require a lot of changes are difficult to experimentally implement perfectly and we therefore welcome the suggestion to check the robustness of our protocol. In Fig. R3a and R4a we show the interaction ramp for $g_f=5$ and $g_f=20$ respectively. We show the ramps following two paths, one which overshoots (red dashed line) and one which undershoots (blue dotted line) the calculated STA (black solid line). We construct these ramps by adding or subtracting the derivative of the STA (whose maximum is normalised to 1) given by $\tilde{g}(t)=g(t) \pm \gamma \frac{d g(t)}{dt} g_f$, which has the largest deviation when the interaction ramp changes quickly. We scale this error by a factor $\gamma=0.1$ (at most the “error” is a 10% deviation of the final interaction) and calculate the irreversible work (see Fig. R3b and R4b). We see that the irreversible work created does not change significantly, with a slightly larger deviation seen for $g_f=5$, although this is still a less than $10\%$ increase in irreversible work. As long as the interaction ramp approximately follows the path of the STA it will be successful, which highlights the robustness of our technique.

4) In the manuscript we calculate the continuous variable entanglement between the position coordinates of particle 1 ($x_1$) and particle 2 ($x_2$). This is of course different to quantifying the entanglement in the basis of left and right modes of a physically split system, but the continuous variable entanglement of 2 particles is no less valid and is well established [Sun, Zhou, and You, PRA 73, 012336; Wang, Law and Chu, PRA 72, 022346]. By tracing out one of the particles the eigenvalues of the reduced single particle density matrix (RSPDM) then describe the Schmidt rank of the state, that is the degree of mixedness of the reduced state. If the Schmidt rank is greater than one (there are more than one finite eigenvalue of the RSPDM) the state is entangled, and for bipartite pure states the von Neumann entropy is a well defined measure of entanglement, being zero if and only if the state is a product state [Paskauskas and You, PRA 64 042310; Ghirardi and Marinatto, PRA 70, 012109]. The description of the continuous variable entanglement has been expanded upon in the paper.

5) We choose the reference ramp that begins and ends smoothly in an effort to reduce non-equilibrium excitations during the beginning and end of the ramping procedure. The specific form of the reference is not unique and any similarly shaped pulse may be employed. However it shows clear advantages over the linear ramp (see attached figure R5) as it approaches adiabaticity at a faster rate and the irreversible work smoothly decays with increasing ramp time. In comparison, the linear ramp reaches adiabaticity at a much slower rate and can exhibit oscillations in the irreversible work due to resonantly driving the system. Therefore, choosing a reference ramp like the one we consider (or any other smoothly changing function) gives more consistent and better results but also offers a genuinely good comparison to the STA. A line has been added to the manuscript to emphasise this point.

The minor issues raised have also been addressed in the new version of the manuscript.

Attachment:

Figures.pdf

We’d like to thank the referee for reading our manuscript and his/her report. We are happy to see that they think that our work deserves to be published in Sci. Post and are grateful for their suggestions. Below we respond in detail to each point made in the report.

We fully agree with the referee that the extension of the presented STA protocol to larger systems is very interesting. In fact, it is a direction we are currently working on.
As the referee points out, this is firstly a problem of finding an appropriate ansatz and the one mentioned above by Girardeau is certainly interesting. Our first extension, however, is based on the work of Zinner et al. (Sci. Rep. 6, 28362(2016), PRA 95, 053632 (2017)), as it allows to treat few-body systems of bosons and fermions, as well as Bose-Fermi mixtures. As this work is quite involved, and as the two-particle case demonstrates the full breadth of our method, we have decided not to include any details about larger systems in the current manuscript.

Approaching the problem of creating a many-body STA through optimisation of the one-body reduced density matrix (OBRDM) is indeed a very interesting idea. We fully agree that direct manipulation of the OBRDM would allow for more intuitive control of the correlations in the system. Also this approach could potentially allow for easier scaling to larger systems as long as one can calculate the OBRDM efficiently (for instance in the TG regime).

For the final point the referee is correct, indeed we envisaged the work as a means to create stable entanglement between particles, and also to control it via careful application of the Feshbach resonance, which was inspired by proposals of collisional quantum gates [T. Calarco et al, Phys. Rev. A 61, 022304 (2000); O. Mandel et al, Nature 425, 937 (2003), and A. Negretti et al, Quantum Inf Process 10: 721 (2011)]. The successful suppression of oscillations in the von Neumann entropy due to collisions between the particles (as seen in Fig. 3a) points to the possibility to create entanglement in a highly precise manner, which is in contrast to the fluctuating correlations created by the reference pulse. We have amended the revised version of the manuscript to emphasise this point.