As a complement of the study of a quantum sensing device based on a BEC carrying OAM in a ring potential presented in this Chapter, in this section we discuss some possible strategies to prepare the initial state consisting of an imbalanced superposition of counter-propagating OAM l = 1 states. As we outlined in Sec. 3.2.1, there are two main possibilities to prepare such a state, namely:
i) Directly imprint to a BEC trapped in a ring potential the phase and density profiles associated with the state given by Eq. (3.11).
ii) Prepare the BEC in one of the OAM modes, e.g. in state |+1i state, then deform adiabatically the ring in order to induce a coupling with the|−1istate and finally return adiabatically the ring potential to its original shape.
Next, we discuss separately the implementation of these two strategies.
3.4.1 Direct imprinting of the phase and density profiles
The most direct strategy to generate an imbalanced superposition of the OAM l = 1 counter-rotating modes is to imprint the corresponding density and phase profiles onto the BEC by manipulating the trapping potential. The wave function of such a superposition, which is given by Eq. (3.11), can be written as Ψ(r, ϕ) =|Ψ(r, ϕ)|eiβ(ϕ). The modulus |Ψ(r, ϕ)|is given by Eq. (3.12), and the phase profile reads (we assume a relative phase between the counter-propagating modes α= 0 for simplicity)
β(ϕ) = arctan
Im[Ψ(r, ϕ)]
Re[Ψ(r, ϕ)]
= arctan √
p+1−√ p−1
√p+1+√
p−1 tanϕ
(3.34) This density and phase patterning could be done, for instance, by using highly pro-grammable digital micromirror devices [165].
3.4.2 Adiabatic deformation of the ring trap
The method described in the previous section might be difficult to implement exper-imentally due to the high degree of precision required in the phase and density imprint-ing. An alternative approach which does not demand such a fine control of the system is to adiabatically deform the ring trap to induce a coupling between the counter-rotating OAM modes. In order to do so one could, for instance, implement the following protocol 1. Load the OAMl = 1 state with clock-wise circulation, |+1i, in a ring potential of radiusR. This could be done by preparing the BEC in the ground state of the ring and then imprinting a 2π round phase with a Laguerre-Gaussian beam [120, 122].
2. Adiabatically deform the ring potential into an ellipse with the same area as the original ring,
such thatab=R2. Theaandbsemiaxes of the ellipse are varied in time according to the relations
where t(1)r is the total time of the adiabatic ramp and k is a factor that sets the maximum eccentricity of the ellipse.
3. Keep the elliptic potential with semiaxesa=R(1 +k) and b=R/(1 +k) during a hold timeth in order to populate the counter-propagating OAM mode|−1idue to the coupling induced by the breaking of the cylindrical symmetry of the potential [137].
4. Adiabatically deform the semiaxes of the ellipse into their original form according to the relations
where t(2)r is the total time of the second adiabatic ramp.
62 Chapter 3 – Quantum sensing using imbalanced counter-rotating BEC modes We have performed numerical simulations of this protocol with the parametersk= 0.2, ωt(1)r = ωt(2)r = 100 and ωth = 250 for different values of the ring radius R and g2d. For rings of small radius, R . 5σ, we observe that the final state is an imbalanced superposition of |1,±i OAM modes with very small populations of higher odd OAM modes. In Fig. 3.6 (a) we plot, for a ring ofR = 5σ, the time evolution of the populations of the OAM l = 1,3 states during the protocol for g2d = 0.5 and 1; and in Fig. 3.6 (b) we show several snapshots of the density distribution for g2d = 0.5. For rings of larger radius, the final state contains significant populations of higher odd OAM modes. An example of this is shown in Fig. 3.7, which contains the same information as Fig. 3.6 but for a ring of R = 15σ. As can be seen in Fig. 3.7 (a), in this case the final state contains a significant population of the|−3istate, and therefore, as shown Fig. 3.7 (b), its density profile has two minimal density lines. In both Fig. 3.6 (a) and Fig. 3.7 (a), it can be seen that the final population of the |−1i state is bigger for smaller values of g2d.
The reason why higher OAM states are populated in bigger rings is that the energy separation between OAM states decreases as the ring radius increases, as illustrated in table 3.1. As we discussed in Sec. 3.3.1, the only difference in energy between the OAM states comes from the centrifugal term of the kinetic energy. According to Eqs. (3.27), (3.28), this term is smaller for bigger values of R because the amplitude of the ground state wave function ψ0(r) is maximal around the radial position r = R. During the process of adiabatic deformation of the potential, the total energy of the BEC is slightly increased. When this slight increment is of the order of the energy separation between the OAM modes, states with higher values oflbecome significantly populated. Although a more detailed study of this effect is outside of the scope of this Chapter, the adiabatic trap deformation protocol could in principle be optimized to selectively populate a desired OAM state. Alternatively, one could design a faster scheme to obtain the desired state based on shortcuts to adiabaticity [203].
R/σ Ec/~ω 5 2.13×10−2 7.5 9.13×10−3 10 5.07×10−3 15 2.23×10−3 20 1.25×10−3
Table 3.1: Energy separation between the l = 1 and l = 0 OAM states for different values of the ring radius.
Figure 3.6: (a) Time evolution of the populations of the |±1i and |±3i OAM states during the trap deformation protocol described in the main text. The instants A, B, C, D correspond toωt= 0, ωt(1)r , ω(t(1)r +th), ω(t(1)r +th+t(2)r ), respectively. (b) Snapshots of the BEC density profile at the instants A, B, C, D for g2d = 0.5. The parameters of the simulation are R= 5σ, ωt(1)r =ωt(2)r = 100,ωth = 250.
64 Chapter 3 – Quantum sensing using imbalanced counter-rotating BEC modes
Figure 3.7: (a) Time evolution of the populations of the |±1i and |±3i OAM states during the trap deformation protocol described in the main text. The instants A, B, C, D correspond toωt= 0, ωt(1)r , ω(t(1)r +th), ω(t(1)r +th+t(2)r ), respectively. (b) Snapshots of the BEC density profile at the instants A, B, C, D for g2d= 0.5. The parameters of the simulation areR = 15σ, ωt(1)r =ωt(2)r = 100, ωth = 250.