Técnicas de Auditoría

To generate the state-selective potentials, we launch a microwave at fre- quencyω into the single CPW structure, which is blue detuned byδ= ∆2₁,_,−₋1₁ with respect to the transition |1i ↔ |F = 2, mF = −1i ≡ |3i (see Fig-

ure 5.2.1). The detuning δ is chosen such that the microwave primarily

couples |1_i to the auxiliary state _|3_i, with position-dependent Rabi fre- quency Ωπ(r) = −

p

3/4(µB/~)Bπ(r). All transitions other than |1i ↔ |3i

are much further off resonance and therefore have only minor effects (i.e.

∆2₁,_,−₋1₁ ∆2₁,i_,j

, [i, j] 6= [−1,−1]). The coupling results in a dressed state |1ithat is shifted in energy byVmw(r)with respect to|1i; the overall potential

seen by |1_i is thus V_|1i = VZ+Vmw. In contrast, state |2i and its potential

remain essentially unchanged, |2_{i ≈ |}2_i and V_{|2i ≈} V_|2i = VZ (see Figure

connecting to this state. In the experiments described in this chapter, we focus on the regime |Ωπ|2 |δ|2, where |1i contains only a small admixture

of state |3_i, which is important because _|3_i has opposite magnetic moment, and a large admixture would spoil the good coherence properties of our state pair. In this limit, Vmw(r)≈~|Ωπ(r)|2/4δ(r)and|1i ≈ |1i+ Ωπ(r)/2δ(r)|3i.

To demonstrate state-selective splitting, we prepare BECs in our exper- iment trap in an equal superposition of states |1_i and _|2_i. Here and in the following, the two-photon microwave intermediate state detuning ∆/2π = 280 kHzand Ω2P/2π = 1.47 kHz.

5.2.1

Adiabatic splitting

We prepare BECs containing N = 400 atoms in an equal superposition of states |1_i and _|2_i by applying a π/2 pulse of 170 µs duration on the two- photon transition. Right after this pulse, which is fast compared with the trap oscillation periods, the motional wave functions of|1i and |2i, ψ_|1i and

ψ_|2i, overlap completely. Then, within 150 ms, we smoothly ramp up the

microwave power in the CPW to a final value Pmw = 120 mW, at fixed

detuning δ(rm) = δm = 2π×150 kHz. This corresponds to a ramp of Vmw

that is adiabatic with respect to the dynamics of the internal state, ensuring population of only state |1_i, but not_|3_i, as well as adiabatic to the motion, enabling the BEC wave function ψ_|¯1i to follow the potential. At the end of the ramp, we switch off the combined static and microwave potential within 0.3 ms and image the atomic density distributions quasiin situ, using state-

selective absorption imaging [48]. Figure 5.2.3a shows images taken in this way. We observe that the BEC is state-selectively split alongxby a distance s = 9.4µm, which is 3.9 times the radius of each of the two trapped clouds [143].

The splitting is due to the strong near-field gradient in |Ωπ(r)| around r = rm (see Figure 5.2.2). By comparison, the spatial dependence of δ(r)

is weak. Although |Ωπ(r)| has gradients of similar magnitude along x and z (see Figure5.2.2b), the spatial splitting is nearly one-dimensional because f_⊥2 f_x2. Figure 5.2.3b shows the measured s as a function of Pmw/δ for

different values of δ. The data points lie on top of each other as expected

from the scaling Vmw ∼ |Ωπ|2/δ ∼ Pmw/δm in the regime |Ωπ|2 |δ2|. The

maximally applied Pmw = 120 mW corresponds to |Ωπ(rm)|= 2π×122 kHz,

which we measure independently by driving resonant Rabi oscillations with the microwave near-field. Note that for δ > 0, the repulsive microwave

potential pushes state |1_i into regions where _|Ωπ(r)| |Ωπ(rm)| so that

|Ωπ|2 |δ|2 is always satisfied.

Pmw

high resistivity silicon

microwave coplanar waveguide 1 µm 24 µm gold mirror Ig1=-Imw/2 Isig=Imw x y z 5 cm ~6 µm polymide Ig2=-Imw/2 25 MHz 0 MHz |ΩR(x,ym,z)| 2 π a) b) −40 −20 0 20 40 60 −60 0 10 20 30 40 50 60 Position, x [µm] Di stance from chi p, z [µm] Ideal CPW Mode −40 −20 0 20 40 60 −60 0 10 20 30 40 50 60 2 4 6 8 10 12 14 Position, x [µm] MHz + +

Sonnet current distribution c)

Figure 5.2.2: Visualization of the near-field gradients in the experiment

region. (a) Photograph of the atom chip and (b) close up of the experimen- tal region. The position rm = (xm, ym, zm) = (12,0,44)µm of the minimum

of the static trap VZ(r) is indicated by the black cross (r = 0 corresponds

to the top surface of the wire in the center of the CPW). An ideal CPW mode with microwave current amplitudes Isig = Imw on the signal wire and

Ig1 = Ig2 = −Imw/2 on each of both grounds is indicated. Equipotential

lines of |Ωπ|/2π are indicated for Imw = 76 mA (line spacing 70 kHz). The

asymmetry in|Ωπ|/2π with respect tox= 0is due to the spatial dependence

of the static magnetic field B0(r), that gives rise to VZ(r). (c) Comparison

between the calculated distribution of|Ωπ|/2πfor an ideal CPW mode (left)

and for Sonnet’s current distribution, see Section4.3. Both distributions are calibrated by the measured Rabi frequency |Ωπ(rm)|/2π = 122 kHz. Even

though both distributions differ significantly, the field gradients at the posi- tion of the trap are very similar.

F=1& F=1 F=2 F=2 a) c) 23 0 s 〉 〉 full simulation full simulation s xm Potential, V(x,y m ,zm )/h [kHz] Position, x [µm] 2 ℏ x z atoms/px b) δ π δ π δ π δ π δ π · · · · · · Splitting distance, s [µm] δ π δ adjusted δ π 2 1

Figure 5.2.3: State-selective splitting of a BEC. (a) Absorption images

of the adiabatically split BEC (Pmw = 120 mW, δm = 2π×150 kHz). By

imaging both hyperfine states simultaneously (top), only F = 1 (middle) or only F = 2 (bottom), the state-selectivity of the splitting is demonstrated. (b) Measured splitting distance as a function ofPmw/δmfor different values of

δmas indicated. The solid red line is the result of a static simulation assuming

an ideal CPW mode; the dashed line assumes a slightly asymmetric mode (Ig1 =−0.45×Imw, Ig2 =−0.55×Imw). The green line shows the splitting

resulting from the adjusted static current distribution, which reproduced the measured microwave field distribution best in Section 4.3. Notice that even though the measured field distribution shows a significant deviation from the ideal CPW mode for z _≥50µm (see Chapter 4), the field gradients along x

are of similar magnitude nearrm. (c) Simulated potentials along the splitting

direction (see the dashed line in Figure 5.2.2b), for an ideal CPW mode and

Imw = 76 mA, corresponding to the parameters of a). The potential minimum

of V_|1i is shifted by the microwave, whereas V_{|2i ≈} VZ. The full microwave

potential Vmw (dashed blue) and the approximation Vmw ≈ ~|Ω|2/4δ for |Ω|2 |δ_|2 (dash dotted green) are shown in comparison.

Figure 5.2.4: Measurement of the splitting s alongx as a function of Pmw

andδm with BECs. Fors >11µmthe atoms are lost because the trap opens.

The displacement along z is below the resolution of our imaging system. V_|2ithat takes the full 8-level system into account (see Section1.8), where the

microwave current amplitudeImw is calibrated using the measured|Ωπ(rm)|.

Figure 5.2.3c shows a slice through the simulated potentials along the split- ting direction, assuming an ideal CPW mode. In agreement with the experi- ment, the simulation shows that we can selectively displace the wave function of state |1i with the microwave potential gradient.

Sign of detuning δm By changing the sign of the detuning δm from blue

to red, the microwave potential can be changed from repulsive to attractive for |1_i. Measurements of s along x for various positive and negative values

ofδm are shown in Figure5.2.4. We observe that the trapped atoms are lost

for small negative values of δm and s >11µmbecause the trap opens.