Atomic interaction effects

The effect of atomic interactions on the phase evolution in an atom interferometer is commonly regarded as a major obstacle for achieving good sensitivities in interferometers based on Bose-Einstein condensates. In the following, we will estimate the mean field shifts based on the Thomas-Fermi density distributions described in the previous section. For a quantitative analysis of the relative phase evolution of the two states in the interferometer, we follow the

2.2 Bose-Einstein condensates for atom interferometry 39 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 Time of flight (ms) FWHM along x ( µ m) 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 800 Time of flight (ms) FWHM along y ( µ m) (a) (b)

Figure 2.1:Time-of-flight central cloud widths (a) perpendicular to and (b) along the weak trapping direction. The red lines represent a thermal cloud ofT₌1µK, the blue long-dashed line a condensate in the Thomas-Fermi limit and the black short-dashed line a condensate with no interactions. The cloud width are calculated for an atom numberN₌106_{and trapping frequenciesω}

x=ωz=209.2 Hz and

ωy=18.3 Hz.

approach in[86]and write the two-state wave function as

Φ(r,t) =ca(t)φa(r) +cb(t)φb(r). (2.16) The number of atoms in each state is given byN_a=|c_a(t)|2andN_b=|c_b(t)|2, and the overall wave function is normalised to the total atom number, �|Φ(r,t)|2d3r =Na+Nb=N. It follows thatφa(r)andφb(r)are normalised to one. The equations of motion forca(t)and

c_b(t)are obtained from the Gross-Pitaevskii equation (2.2): i˙c_a=ω_ac_{a+ (}U_aaN_a+U_abN_b)c_a≡α_ac_a

i˙c_b=ω_bc_b+ (UbbNb+UabNa)cb≡αbcb.

(2.17) Here,Ui j = g_ħhi j

�

|φi(r)|2|φj(r)|2d3r is related to the scattering length ai j between statesi and j viag_{i j} =4πħh2a_{i j}/m. The energies of the two states in the absence of interactions are

ħhω_a andħhω_b. The phase evolution of stateiis described byα_i, and the relative phase evolves

at a rate

α_≡α_a−α_b=ω_a−ω_b+ (Uaa−Uab)Na−(Ubb−Uab)Nb. (2.18) The interaction-induced reduction of interferometric sensitivity is due to uncertainties in the numbers N_a and N_b. We are particularly interested in the uncertainty in number and the dephasingduring the free evolution in the interferometer, i.e. after the first beam splitter. For a non-entangled ensemble of atoms, the limit of how well we can know the number after the first beam splitter is given by the quantum projection noise, which sets a lower bound for the uncertainty in relative atom number,σNa−Nb =

�_{N. Assuming a 50}

/50 split on the beam splitter, we setN_{a= (}N_±�N₎/2 andN_{b= (}N_∓�N₎/2. The relative phase evolves as

α=ωa−ωb+ (Uaa−Uab)

N_±�N

2 −(Ubb−Uab)

N_∓�N

2 . (2.19)

The important quantity is the uncertainty in the phase evolution due to relative number fluctuations. The uncertainty in the phase evolution rateαis given by

σα=

U_aa+Ubb−2Uab 2

�_N

40 Chapter 2. Bose-condensed sources for precision measurements

Figure 2.2: Calculated dephas- ing rateσα as a function of ex- pansion time. The maximum dephasing rate is 3.5 Hz for a trapped condensate. More than 99% of the atomic interaction en- ergy is converted to kinetic en- ergy during the first 10 ms of free expansion. The dephasing rates are obtained for condensate of N₌106_{atoms and trapping}

frequencies ωx=ωz=209.2 Hz andωy=18.3 Hz. 0 5 10 15 20 25 30 10−3 10−2 10−1 100 101 Time of flight (ms) Dephasing rate σ α / 2 π (Hz)

σ_α is the dephasing rate during the free evolution in the interferometer. We shall calculate σα for the time-of-flight Thomas-Fermi density profiles introduced in the previous section. For

simplicity, we will consider the case where the two states are spatially separated; this condition is fulfilled for interferometers with atomic clouds of sufficiently narrow spatial width (BECs) and a large enough momentum transfer during the beam splitting. For spatially separated states, we can neglect the interstate interaction termUabin equation (2.20). We shall assume a condensate withN =106 87Rb atoms and the trapping parameters of the QUIC trap described in section 2.4. The differences in the scattering lengths of different internal states in87Rb differ by�10%. For reasons of simplicity, we usea=100a0, independent of the internal atomic

state. From equation (2.8), we calculateU_aa=Ubband the dephasing rateσα. The results are

graphed in figure 2.2. The maximum dephasing rate isσ_α=2π_×3.5 Hz; it decreases by more than two orders of magnitude within the first 10 ms of free expansion. To achieve quantum projection noise limited operation with 106 _atoms,3 _{the (mean) dephasing rateσ}

α must be

kept below 2π_×5×10−4_{/T. Here,T is the free evolution time in the interferometer. For an}

interferometer withT=100 ms, this means a dephasing rate below 5 mHz which is achieved after 18 ms of free expansion in our setup.