Data analysis

Single-plane tomography

...

Vertical magnetic field gradient

0

Combined magnetic field

209.3G 208 G

B_exp

33 G/cm

0

m

F

1

3 2

-9/2 -7/2 -5/2 -3/2

Activate spin-changing collision

time

F = 9/2

Imaging

Singles Down Singles Up

5

6 4

4

Figure 4.4:Experimental detection sequence for the spin sector.

• 6 : In-situimaging. Successive imaging, using the fast kinetics scheme, of the two spin states results in twoin-situoccupation profiles ofS_↑andS_↓, as shown in Fig.4.6.

Eq. (4.16) for averaged quantities in a two-dimensional lattice, we have

ndµ=sdT+a²dP−mdh, (4.17)

where the quantitiesn,sandmare the corresponding particles, entropy, and magnetisationper lattice siterespectively, anda²is the area of lattice unit cell.

Local density approximation Since Eq. (4.16) is constructed under the grand canonical ensemble, it describes a system in equilibrium with external reservoirs,i.e.a particle bath characterised by the chemical potentialµ. This motivates the use of local density approximation (LDA) in analysing our quantum gases sample. Under thermal equilibrium, the LDA asserts that an inhomogeneous system can be partitioned into homogeneous subsystems that are in equilibrium with each other. Obviously, this assumes that the Hubbard parameters are sufficiently stationary with respect to the spatial change of thermodynamical variables.

In optical lattices, the inhomogeneity arises mainly from the spatially-varying potential landscape. The inter-site tunnelling and interaction terms can be treated as constant throughout the whole lattice. As shown in Fig.4.5a, the LDA allows the partition of the whole inhomogeneous system (the trap shown as the grey solid line) into multiple, spatially-connected homogeneous subsystems (approximated by red solid steps). In thermal equilibrium, the subsystems have thesamethermodynamic potentials (µ₀,T,P,h). However, the potential landscapeV_trap(x,y)offsets part of the global chemical potential and thus leads to a local chemical potentialµ_localgiven by

µ(x,y)= µ₀−V_trap(x,y). (4.18)

For the region closer to the trap centre, the local chemical potential is higher, and thus the atom number increases accordingly. This leads to the coexistence of phases within the different position in the systems, as depicted for repulsive interaction in Fig.4.5bThe low-filling region corresponds to a low local chemical potential µ and hence realises the compressible metallic phase. Upon increasing density, the interaction contribution rises and is maximum at half-filling, thus leading to the interaction induced Mott-insulating phase.

Mott insulator μ 0~~ hole-doped Metal μ < 0

particle-doped Metal μ > 0

a b

position

potential Subsystem 1

(μ μ +V ,T,P,h)₀^~~ _{1 1}

Subsystem 2 (μ μ +V ,T,P,h)₀^~~ _{2 2}

Figure 4.5:Principle of local density approximation (LDA) . aSketch of trapping potential landscape.

The LDA approximates region with same trapping depths as a homogeneous system. The deeper the trap (closer to centre), the higher the local chemical potentialµand total densityn. Within a single experimental realisation, the entire system can be separated into coupled subsystems.bCoexistence of phases. The harmonic confinement provides a natural knob to adjust the chemical potentialµand hence supports the coexistence of phases within the optical trap.

0.0 0.5 1.0

Occupation

0.0 0.5

Occupation

−12 −8 −4 0 4

μ/t

μm

0.0 0.2 0.4 0.6 0.8

Occupation

S↑(μ) D(μ)

D S↑

−12 −8 −4 0 4

μ/t 0.0

0.1 0.2 0.3 0.4 0.5

Occupation

S^↑(μ) S^↓(μ)

a Density-resolved Spin-resolved

LDA

μ μ

r 0

Occupation

μ

r 0

Occupation

μm

b

( x,y) ( x,y) S^↑( x,y) S↑( x,y)

25

LDA

Figure 4.6:Experimental detection in density and spin sectors. aDensity sector measurement. In-situ images of singly-occupiedS_↑(x,y)and doubly-occupied sitesD(x,y), with a Gaussian spatial filter on the order of our imaging resolution to remove noise. By averaging within iso-potential region, we obtain the density profile as a function of chemical potentialµ.bSpin sector measurement. A spin-resolved measurement opens access to the spatial profile of the magnetisationm=n_↑−n_↓. The relative ratio betweenS_↑(µ)andS_↓(µ) allows the extraction of the effective Zeeman fieldhfrom a numerical fit with DQMC simulations. Thein-situ images and data shown are averaged within∼40 experimental realisations.

Equations of state In Fig.4.6, we depict the analysis that converts raw experimental images to the equations of state. For measurements in the density sector, we typically probe the density distribution ofS_↑(x,y)andD(x,y). S_↑(x,y)displays a hole near the centre as the filling increases beyond half-filling. And a Mott insulating phase is realised near the peak ofS_↑(x,y), forming a ring-like appearance. The doubly-occupied sitesD(x,y)rise monotonically with trap depth. We then map the spatial coordinates to chemical potential using Eq. (4.18). To this end, we obtain the density profile as a function ofµ,i.e. S_↑(µ)andD(µ). The sum of two measurements leads to the density equation of staten(µ),

n(µ)=2

S_↑(µ)+D(µ)

. (4.19)

The equation of state, in general, is a function of temperatureT and we implicitly assume in our notation for clarity. As for the spin sector, the main observable is the magnetisationm= n_↑−n_↓.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 RF width (MHz)

0 1 2 3 4

Zeeman field h/t

b

12 10 8 6 4 2 0 2 4

/t 0.0

0.1 0.2 0.3 0.4 0.5 0.6

magnetisation m

a

_{h/t = 0.21}

h/t = 0.26 h/t = 0.36 h/t = 0.44 h/t = 0.69 h/t = 1.01 h/t = 1.72

Figure 4.7:Determination of the effective Zeeman field h. aIn addition to temperature extraction, the numerical fit to the magnetisation profileh(µ)provides a method to calibrate the effective Zeeman fieldh, which is a global system parameter. Solid lines are results of numerical DQMC simulations. bTuning the spin-mixtures. By varying the frequency width of the spin-mixing pulses, a global population imbalance can be achieved, and thus a finiteh. The extractedhversus RF frequency width is plotted. The 50/50 balance point is calibrated at 2.9(4)MHz.

For a spin-balanced system,i.e. h=0, we havehmi=0. The effective Zeeman fieldhis varied by imbalancing population of the two spin-states. This can be achieved by varying the sweep width of the RF pulses used for creating a 50/50 mixtures of spin states, see Fig4.7. Our detection techniques give the spatial profile ofS_↑(µ)and S_↓(µ)using the same procedures. Again, the magnetisation can be obtained as a function of local chemical potentialµvia LDA. To this end, we arrive at the magnetisation, for a given Zeeman fieldh,

m(µ)=S_↑(µ) −S_↓(µ). (4.20)

Similarly, the magnetisation equation of statem(h)at fixed chemical potentialµcan be extracted from various realisations with different overall spin imbalance, and is given by,

m(h)=S_↑(h) −S_↓(h). (4.21)

The resulting density and magnetisation equation of state can be found in the next section, at which we derive the corresponding susceptibilities in each sector.

Temperature determination We then compare our experimental data for the equations of state n(µ) and m(µ) with numerical DQMC simulations, as discussed in Sec. 2.5. By performing a numerical chi-squared fit to the measured equations of state, we obtain the temperaturek_BT/tand global chemical potentialµ₀of the system. As shown in the lower part of Fig.4.6 a, for the density sector, a combined fit is performed for bothS_↑(µ)andD(µ).

As for the spin sectors, the numerical fit allows us to extract, in addition to the temperaturek_BT/t and the chemical potentialµ, the effective Zeeman field h. As shown in bottom part of Fig.4.6 b, we perform the numerical fit on S_↑(µ) and S_↓(µ)instead. By allowing two spin components to have different global chemical potential (µ_↑and µ_↑), the effective Zeeman field is obtained as h=(µ_↑−µ_↓)/2. The global spin imbalance is achieved by varying the frequency width of the RF mixing pulses, and the resultingh/tis shown in Fig.4.7.

Thermodynamical observables The thermodynamics of the system is fully characterised by the equation of state. Thus, thermodynamical quantities such as pressure, compressibility and entropy can be derived from it. At thermal equilibrium, the temperature throughout the optical lattice is constant,i.e. dT=0. Eq. (4.16) can be rewritten as

∂P

∂ µ = N

V =n. (4.22)

Integrating this gives the pressureP,

P(µ)=

∫ µ

−∞

n(µ⁰)dµ⁰. (4.23)

Next, from the first law of thermodynamics, the isothermal compressibilityκis the inverse of the bulk modulus, and is define as [52]

κ=−1 V

∂V

∂P

T

, (4.24)

hence κ is the change of volume with respect to a change in pressure, where dP > 0 (dP < 0) represents a compression (relaxation) applied to the system. The compressibility is sensitive to interaction effect and offers important characterisation of quantum phases. In order to experimentally measure this, we have to rewrite Eq. (4.24) into experimental observables. Using the differential chain rule, we obtain

κ_T =−1 V

∂V

∂P

T

=−1 V

∂V

∂ µ

T

∂ µ

∂P

T

=− 1 nV

∂V

∂ µ

T

. (4.25)

Here, we make use of Eq. (4.22). Furthermore, we can write the differential dn = d(N/V) =

−(N/V²)dV=−(n/V)dV, and differentiating with respect toµ, and obtain ∂V

∂ µ

T

=−V n

∂n

∂ µ

T

. (4.26)

Substituting Eq. (4.26) into Eq. (4.25), we finally arrive at the common form of compressibility in quantum gases experiments, which reads

κ_T = 1 n²

∂n

∂ µ

T

. (4.27)

We note that this particular form ofκ_T is not symmetric around half-fillingn= 1, there we focus on a slightly different form of the compressibility, namely by dropping the 1/n²and compute, and define

κ = ∂n

∂ µ

T

. (4.28)

Therefore, the compressibility can be deduced by differentiating the equation of staten(µ). In a similar fashion, the spin susceptibility χ_s characterises the response of magnetisation to an external field, and is defined as

χ_s = ∂M

∂H

T

∼ ∂m

∂h

T

. (4.29)

In later sections, we demonstrate that the spin susceptibility offers equivalent information as the compressibility in the density sector. In other words, the same physics in a metal/Mott-insulator

crossover can be studied using attractive interaction.⁴

Im Dokument Quantum simulation of strongly-correlated two-dimensional fermions in optical lattices (Seite 73-78)