1. Sobre el territorio
2.2 Ejes pedagógicos de la escuela
In a conductor of conductivityσ at temperature T, thermal agitation of the electrons leads to current noise. This effect is known as Johnson-Nyquist noise in electronics. The conductor may be a chip wire used for trapping, but thermal currents exist in any conductor, independent of whether an external current is applied or not. The currents are the source of a fluctuating magnetic field, which is many orders of magnitude stronger than the field due to black body radiation in the near field of the conductor [31]. At the relevant frequencies ω/2π <10 GHz corresponding to wavelengths >3 cm, the trap is in this near-field regime. If the magnetic trap is operated with very stable current sources, technical magnetic field noise can be made negligible. This leaves thermal magnetic near-field noise as the dominant effect limiting chip trap performance near conductors, as predicted in [30,31] and subsequently observed in [32, 88, 44].
Spectral density
At distance d from a conducting, non-magnetic layer of thickness t, see Fig. 1.8(a), the spectral density of the magnetic field fluctuations is [89, 31]
SBαβ(ω) =
µ20σkBT
16πd ·sαβ ·g(d, t, δ), (α, β =x, y, z), (1.34) wheresαβ = diag(12,1,12) is a tensor which is diagonal in the coordinate sys-
1.9 Atom-surface interactions a) b) d B0 x y z t 10−1 100 101 102 10−2 10−1 100 101 d[µm] τ [s ] |1,−1i → |1,0i |2,2i → |2,1i |2,1i → |2,0i
Figure 1.8: Trap loss due to spin flips caused by thermal magnetic near field noise. (a) Atoms near a conducting layer on top of a dielectric substrate. In addition to the geometry, the skin depthδ is an important length scale of the problem (see text). (b) Trap lifetime as a function ofd for the indicated transitions, t= 1 µm,δ max(d, t), andτbg = 5 s.
dimensionless functiong(d, t, δ) depends on the geometry and the skin depth δ=p2/σµ0ω, which carries the frequency dependence of SB(ω). The spec-
tral density is related to the mean square fluctuations of the magnetic field components hBα2(t)i= 1 π Z ∞ 0 SBαα(ω)dω. (1.35)
It is difficult to obtain exact expressions for g, even for simple geometries. In various limiting cases, analytical expressions or empirical interpolation formulae exist [89, 44]: g = 1 ford δ t, 3δ3/2d3 forδ min(d, t), t/d 1 + [4dt/(π2δ2)]2 fort min(δ, d), t/(t+d) forδ max(d, t), t t+d · w
w+ 2d forδ max(d, t) and finitew,
1 Atom chip theory
where the last formula is for a wire of thickness t and width w (for a more accurate treatment of this case, see [90]). Note that if δ is the largest length scale, it drops out of the problem and SB is a white noise spectrum at the
relevant frequencies. For gold conductors and ω/2π ∼ 1 MHz (a typical Larmor frequency), one obtains δ ∼ 75 µm, and we are well inside this regime for distances where surface effects matter. In this regime, the last two formulae in Eq. (1.36) give the following general scaling: for a metallic half space, g is a constant, for a thin layer, g ∝ t/d, and for a thin and narrow wire, g ∝ tw/d2. To reduce magnetic field noise, it is thus desirable to make the on-chip conductors as thin and narrow as possible.11 On chips with multiple metal layers where t δ for each layer, we roughly estimate SB by adding the spectral densities due to the individual layers.
Although it is a thermal effect, magnetic near-field noise near pure metal wires cannot be reduced by simply cooling the chip. Due to the temperature dependence ofσ(T),SB∝σ(T)T actually increases if T is decreased [90]. If
special alloys are used for the wires, a decrease of SB by cooling is possible
[90]. The use of superconducting wires would decrease SB as well, although
there is a theoretical controversy on how big this effect is [91, 92].
Trap loss due to spin flips
The fluctuating magnetic near-field couples to the magnetic momentµof the atoms in the trap. Components perpendicular to µ drive spin flips, which result in decoherence and loss from the magnetic trap since only low-field seeking states are trapped. The transition rate γs between states|ii and |fi
at frequency ωf i is given by Fermi’s golden rule
γs= 1 ~2 X α,β=x,y,z hi|µα|fihf|µβ|iiSBαβ(ωf i). (1.37)
Let us consider the spin-flip transition between adjacent magnetic sublevels
|ii=|F = 1, mF =−1iand |fi=|F = 1, mF = 0iat the Larmor frequency
ωf i = ωL. In most magnetic chip traps, the static field in the trap center
B0 is parallel to the surface, and we have chosen this direction as the z-
axis in Fig.1.8(a). In this case the evaluation of the matrix elements yields
|h1,0|µx|1,−1i| = |h1,0|µy|1,−1i| = µB/
√
8 and h1,0|µz|1,−1i = 0, using
µα = −µBgFFα and Eq. (A.7). For δ max(d, t) we obtain a lifetime due
11Interestingly, for dδ √dt, the scaling is S
B ∝δ4/td4, and a smaller t actually
1.9 Atom-surface interactions to spin flips of τs = 1 γs = 256π~ 2 3µ2 0µ2BσkBT · d(t+d) t for state |1,−1i. (1.38) The overall trap lifetime is τ = (τ−1
s +τ
−1 bg )
−1, where we take into account
a background lifetime τbg at large d where surface effects are negligible. In
Fig. 1.8(b), we plot τ for different states in comparison. Atoms in |2,2i are lost in a cascade process |2,2i → |2,1i → |2,0i [44], however the first spin flip already destroys coherent dynamics.
The spectral density SB(ω) decreases at high frequencies due to the skin
effect. Atω/2π= 6.8 GHz, δ = 0.9 µm, which has to be compared with the geometrical dimensions d and t, see Eq. (1.36). For typical chip geometries, the loss rate for hyperfine-changing transitions at GHz frequencies is thus smaller than that for transitions between adjacent Zeeman sublevels at MHz frequencies.
In addition to spin flips, magnetic near-field noise causes dephasing, heat- ing and motional decoherence. As discussed in the following paragraphs, the rates of these processes are all comparable to or smaller than γs.
Dephasing of spin superposition states
The component of the fluctuating magnetic field parallel to µ leads to de- coherence of spin superposition states α|0i + β|1i, without changing the populations (“pure dephasing”). The dephasing rate is given by [31]
γφ=
∆µ2
k
2~2 SBk(ω = 0), (1.39)
where ∆µk = h1|µk|1i − h0|µk|0i is the differential magnetic moment of the
states|1iand|0i,kdenotes the direction of the static trapping field, andSBk
is the component ofSBparallel to it. In Fig.1.8(a),µk =µz andSBk =SBzz.
In Eq. (1.39) we have made use of the fact that SB(ω) is flat for small ω so
that we can replace its low-frequency average by its value at ω= 0 [31]. For most states |0i and |1i, ∆µk ∼ µB, so that γφ is comparable in
magnitude to the spin flip rateγs. For a state pair with nearly equal magnetic
moments, as we use it in the experiments of chapter 4, ∆µk µB and
therefore γφ is negligible compared to γs.
Heating and decoherence of the center-of-mass motion
We now discuss how thermal magnetic near-field noise perturbs the center- of-mass motion of the atoms in the trap.
1 Atom chip theory
Spatially inhomogeneous fluctuations lead to heating and motional deco- herence. Consider a transition between motional states ψi(x) and ψf(x) in
the trap, assuming that the atom’s internal state does not change. Fermi’s golden rule yields a transition rate [31]
γh = µ2 k ~2 Z d3xd3x0Mf i∗(x)Mf i(x0)SBk(x,x0, ωf i), (1.40)
where Mf i(x) =ψf∗(x)ψi(x) is the wave function overlap, SB(x,x0, ω) is the
magnetic field correlation spectrum, and the transition frequency ωf i is of
order of the trap frequency ωt. For small deviations |x−r|, |x0 −r| lc
from the trap center r, the field correlation spectrum is related to the noise spectrum by [31] SBk(x,x0, ω)≈SBk(ω) 1− (x−x 0)2 l2 c , (1.41)
where SBk(ω) is evaluated at r, and lc is the coherence length of the field
fluctuations. It can be shown [31] that lc ≈ d. In order to estimate the
heating rate, we now consider the transition ψ0 → ψ1 between the ground
and first excited state along one dimension of a harmonic trap. The trap is tightly confining such that the size of ψ0 is alc. In this limit one obtains
γh ≈ µ2 k ~2SBk(ωt) a2 l2 c ≈γs a2 d2, (1.42)
where we have related γh to the spin flip rate γs, making use of the fact
that µk ∼ µB and SBk(ω) is flat in the relevant frequency range. Motional
decoherence occurs at a rate similar to γh [31]. For a d, heating and
motional decoherence due to near-field noise are thus negligible compared to spin flips. A similar reasoning applies to atoms split in a double-well potential, withagiven by the separation of the wells [93]. For ad, on the other hand, one obtainsγh ≈γs.
Spatially homogeneous fluctuations can also lead to heating and decoher- ence, since they add to the trapping fields and thus lead to a fluctuating trap position and curvature. The rate for the transitionψ0 →ψ1 is related to the
spectral density of position fluctuations Sr(ω) by [94]
γh =
mω3
t
2~ Sr(ωt), (1.43)
whereSr is normalized so that the equivalent of Eq. (1.35) holds. For trans-
1.9 Atom-surface interactions
can relate Sr(ω) to the transverse component SB⊥(ω) of the magnetic field
noise spectral density, Sr(ω) =SB⊥(ω)/B02. Using ωt =ω⊥, Eq. (1.10), and
ωL=µB|gF|B/~, we obtain γh ≈ µ2 B ~2 SB⊥(ω⊥) ω⊥ ωL ≈γs ω⊥ ωL . (1.44)
Since ω⊥ ωL to avoid Majorana spin flips, we again find γh γs. A
similar reasoning also applies to fluctuations of the trap curvature.
Summary
In summary, one finds that all loss, heating, and decoherence rates due to thermal magnetic near-field noise are of the order of the spin-flip rate γs or
smaller. In the following chapters, we will thus use γs as a measure for the
importance of surface effects above metal layers.
Finally, we note that the effect of technical magnetic field noise can be analyzed along the same lines by simply identifying SB(ω) with the corre-
sponding technical noise spectral density. Noise from technical current fluc- tuations in a wire, for example, hasSB= (µ0∆I)2/(4πd2∆ω), where we have
assumed that the spectrum is flat, ∆I is the RMS amplitude of the current fluctuations, and ∆ω/2π the bandwidth of the current source.