Connection to experiment and conclusion

Chapter 4 Superlattice modulation spectroscopy of ultracold fermions in optical superlattices: Study of the excitation spectrum of the one-dimensional ionic Fermi-Hubbard model

4.6 Connection to experiment and conclusion

the presence of the trap and the local bond order order parameter, shown in Fig. 4.28(b), survives in the presence of a trap, acquiring only a slight asymmetry. Furthermore, the global bond order parameter is practically unchanged away from Ising criticality, see inset of Fig. 4.28(b). At the Ising critical point the system is much more sensitive to external influences due to the closing of the charge gap. At Ising criticality, see Fig. 4.28(a), the unit cell density is modified considerably by the presence of the trap compared to the homogeneous system, in contrast to within the bond order wave. Note that the unit cell density in the homogeneous system at Ising criticality is already asymmetric due to boundary effects which was discussed in Sec. 2.5.2 (cf. Fig. 2.7(a)). The local bond order parameter at Ising criticality acquires a larger asymmetry in the presence of the trap than within the bond order wave phase. In fact, at Ising criticality a homogeneous trapping potential as realized in Refs. [69–71] would be advantageous to unambiguously detect the scaling behavior we identified in this chapter.

The effect of a finite temperature present in experiment needs to be discussed in the Mott insulating phase and at the Ising critical point where the gap closes. It would be desirable to conduct experiments at temperatures below the exchange interaction of the order∼ J²/(∆ +U), the lowest temperature scale for spin and charge fluctuations. This is a similar scale as for the antiferromagnetic phase (∼J²/U) in the repulsive Fermi Hubbard model. Cold fermionic experiments are currently working on reaching these temperature scales and the experimental observation of relevant correlations was recently reported [17–

19].

Let us now comment on suitable parameters ranges of the ionic Hubbard model for the detection of the bond order wave. The ionic Hubbard model was experimentally realized for a wide range of parameters [36], for example,∆could be tuned between 0Jand 40J. In our case, it is important to choose parameters such that the bond order wave phase is robust and has a finite extension, preferably as large as possible, such that both phase transitions can be resolved inU. It has been shown that the extent of the bond order wave phase and the strength of the bond order parameter both increase with increasing∆[98].

Therefore, using a large value for∆would be beneficial. However, at the same time, the temperature should be below∼ J²/(∆ +U) in order to obtain robust results at the Ising critical point as explained above. As this temperature bound decreases with increasing∆andU, experimentally, one needs to find an acceptable compromise between these two requirements. Moreover, the strength of the applied superlattice modulation should be chosen a small fraction of the hopping amplitude (∼0.05%−0.5%) in order to obtain results within the linear response regime (cf. App. B)

C H A P T E R 5

Thermometry of ultracold fermions in optical lattices by modulation spectroscopy

In this chapter we present a scheme to directly measure the temperature of a fermionic gas confined to an optical lattice by means of modulation spectroscopy.

In fermionic lattice experiments one important macroscopic control parameter is the thermal energy which is intrinsically difficult to control due to the fact that the system is not coupled to a reservoir. A major challenge in the quantum simulation of the Fermi Hubbard model is a further reduction in temperature in order to realize interesting phases, for example long range antiferromagnetic order or unconventional superfluidity [37]. This is accompanied by the need of accurate temperature measurements, in particular when pursuing the goal to map out the equilibrium Fermi-Hubbard phase diagram as a function of temperature. However, experimentalists still lack reliable thermometry methods at low temperatures.

In the absence of an optical lattice potential, for the harmonically trapped gas in the weakly interacting regime, the temperature can be extracted from the integrated density profile imaged after a time-of-flight expansion which reflects the temperature-dependent momentum distribution of the particles trapped in the harmonic potential [53]. This is more difficult at strong interactions as the expansion is no longer ballistic but the temperature can be extracted from the ’tail’ of the distribution in many cases. However, this becomes inaccurate at very low temperatures of a few percent of the recoil energy. Typically, the temperature is measured before and after ramping into the optical lattice. Assuming entropy conservation during the loading process, one can determine the temperature in the lattice from the initial entropy. This is markedly limited by non-adiabatic heating processes caused by the ramping of the lattice or by light scattering. Furthermore, it cannot be used for in-lattice cooling schemes. Thus, the development of in-lattice thermometry techniques is of relevance. Different possible schemes to directly measure the temperature of fermionic particles in the optical lattice have been proposed and tested in experiments. All these methods have their limitations and most cannot be extended into the low-energy regime of interest.

Ref. [38] gives an overview on some of the thermometry schemes available for bosonic and fermionic particles. Here, we shortly summarize the most important thermometry setups for fermionic particles.

For instance, the temperature can be extracted from the double occupancy [156, 157] which is sensitive to thermal fluctuations for temperatures on the order or above the on-site interaction strength. The authors of Ref. [158] suggest a temperature measurement based on the fluctuation-dissipation theorem and spatially resolved density-density correlations. This requires in-situ resolution in the measurement which has only recently been achieved [159–162]. It was suggested to use Raman spectroscopy, transferring the atoms to a third hyperfine state, such that the Raman signal depends on the temperature-dependent Fermi factor [163], or to use off-resonant light diffraction from atoms in the lattice as a thermometer since the

Chapter 5 Thermometry of ultracold fermions in optical lattices by modulation spectroscopy

scattered intensity carries information on density-density fluctuations [164]. Later on, it was suggested to study the temperature-dependent fluctuations in the momentum distribution as a response to an artificial gauge field [165]. Moreover, the temperature of fermions in an anisotropic three-dimensional lattice has been extracted from a measurement of the nearest-neighbor spin correlator [166] and by spin-sensitive Bragg scattering of light [167].

Taking this background as a starting point we suggest lattice modulation spectroscopy as a possible thermometer for non-interacting fermionic particles confined to an optical lattice which is easy to implement in experiment and works particularly well for temperatures below the Néel temperature where antiferromagnetic ordering is expected to occur in contrast to the above methods which mostly fail in this temperature regime. Here, we present a thermometry setup based on the normal lattice modulation spectroscopy (cf. Sec. 2.6). In Sec. 5.1 we develop a multiple band tight-binding description for the considered setup, non-interacting fermionic particles in an optical lattice. This section is rather technical, no results will be given. The obtained tight-binding model will be used for all calculations in the subsequent sections. In Sec. 5.2 we explain in detail the detection scheme, a measurement of the number of excited atoms by means of the adiabatic band mapping technique and how this relates to the atom excitation rate within linear response theory. We then study the atom excitation rate as a response to a time-dependent modulation of the lattice amplitude for the one-dimensional homogeneous system in Sec. 5.3 and for the trapped system in one and two dimensions in Sec. 5.4. We show that the response reflects a clear signature of the Fermi distribution of the equilibrium system and we discuss the possibilities of thermometry considering realistic experimental parameters. In Sec. 5.5 we compare the results for the normal lattice modulation scheme to the superlattice modulation spectroscopy, another suitable setup for thermometry, which was investigated in the author’s master thesis [168, 169]. Note that in the figures in Secs. 5.1 and 5.2 we will directly show the comparison to the superlattice modulation although the results will not be discussed before Sec. 5.5. In Sec. 5.6 we summarize and comment on the applicability of our scheme to systems of interest. The results of this chapter are published in Ref. [169].

More recently, by means of in-situ measurements, it has been demonstrated that the temperature can be determined from the equation of state which relates the temperature to the in-situ density profile [17, 54, 170, 171] or by a reconstruction of the local entropy from occupation probabilities [162]. At lower temperatures on the order of or below the hopping parameter, the temperature can be extracted from the spin structure factor [18, 19] or the nearest-neighbor spin correlations [17, 19]. However, away from half-filling numerical simulations are challenging such that adequate thermometry methods remain an important open question in some parameter ranges.

5.1 Two-band tight-binding model

In this section we introduce a multiple band tight-binding description which is a convenient description to study the response of the system to lattice amplitude modulation. We consider non-interacting fermionic atoms confined to the three-dimensional periodic potentialV₀(~x) (2.2) in the absence of an additional external trapping potential. We assume the lattice to be sufficiently deep such that the lowest Bloch bands are well separated. We compute the eigenfunctions which are the Bloch functions along the different directionsxi = x, y, zcorresponding toi=1, 2, 3. We denote the Bloch functionsφ^k_α^xi

xi(xi) with corresponding eigenvaluesE_αxi(kxi) whereαxi denotes the band index and the quasimomentumk_xi lies within the first Brillouin zone ]−kL,kL]. For details see Sec. 2.2.2 where we outline how to obtain the Bloch spectrum from the single-particle Schrödinger equation. The full spectrum in three dimensions is given byE_α(~k) = E_αx(kx)+E_αy(ky)+E_αz(kz) as we consider non-interacting fermions for which the potential is separable. The indexα ≡ {αx, α_y, αz}labels the band. Note, that the Bloch functions

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5.1 Two-band tight-binding model

factorizeφ^~k_α(~x)=φ^k_α^x_x(x)φ^k_α^y_y(y)φ^k_α^z_z(z) in the tight-binding approximation, neglecting the coupling to other Bloch bands and the overlap of different sites. The unperturbed Hamiltonian (2.7) (forV_trap =0 and V(~x−~x⁰)=0) in Bloch basis is then given by

H₀ ≈ X

α,~k,σ

E_α(~k)−µ c^†

α~kσc

α~kσ, (5.1)

wherec^(†)

α~kσare the fermionic annihilation (creation) operators and where we include a chemical potential µwhich controls the particle number.

We apply normal lattice amplitude modulation along one directionx. We assume that initially only the lowest bandα=1≡ {αx =1, α_y=1, αz=1}is occupied and we consider excitations to a higher Bloch bandα⁰={α⁰_x >1, α⁰_y=1, α⁰_z=1}. We count the bands starting from one, that means the lowest band α=1 is called the first band. Referring to the bands of excitations we also start counting from one. This means that the first band of excitations corresponds to the second bandα⁰=2 and the second band of excitations corresponds to the third bandα⁰ = 3. The basic idea is that the frequency-resolved atom excitation rate to higher Bloch bands will reflect the Fermi factor as it will depend on the temperature-dependent filling of the lowest band. The atom excitation rate will be investigated for the homogeneous and the trapped system in Secs. 5.3 and 5.4. From a measurement of the atom excitation rate to higher Bloch bands, the temperature can be extracted. A suitable detection scheme will be discussed in Sec. 5.2.

In the mean time, we construct the perturbing term (2.39) (added to the Hamiltonian by the lattice modulation) in Bloch Basis representation. In order to construct the perturbing operator ˆO_N in Bloch basis representation, we need to determine the transition matrix elements from the initial state to the final state by the perturbing potentialδV(x)=sin²(kLx) (cf. Sec. 2.6 where we derive the time-dependence of the modulated potential). Thus, the transition matrix elements of interest are given by

M_{(α=1,~k)→(α}0,~k⁰) = 1 Ωx

Z x_max

x_min

dxφ^k_α^0x0^∗

x (x)δV(x)φ^k_α^x

x=1(x) Y

i,1

δ_α⁰

xi,α_xi=1δk⁰_xi,k_xi, (5.2) whereΩx=(L−1)ais the system size inx-direction withLthe number of lattice sites andathe lattice spacing, x_min = −(L/2−1)a and x_max = La/2. We used that the Bloch functions are orthonormal, i.e. (1/Ωx_i)Rx_i,max

xi,min dxiφ^k_α^0xi0^∗

xi (x_i)φ^k_α^xi_xi(xi) =δ_α⁰_xi,αxiδk⁰_xi,k_xi such that the contributions alongyandzdirection reduce toδ-functions. We insert the perturbing potentialδV(x)=sin²(kLx) and find that the perturbation, to a good approximation, only couples momenta~kand~k⁰ ≈~k. This is illustrated in Fig. 5.1(a) where the transition probability to the first band of excitationsα⁰_x=2 as a function ofkxandk⁰_xis shown. All other matrix elements for∆~k,0 are strongly suppressed. Quasimomentum is approximately conserved.

This is in accordance with the common lowest band tight-binding approximation which conserves quasimomentum as may be seen from a Fourier transform of the operator ˆON (2.40) in the lowest band tight-binding description, ˆON =2P

k_x,σcos(ak)c^†_k

xσc_k

xσ. We approximate M_{(α=1,~k)→(α}0,~k⁰) = M_α⁰_x(kx,k⁰_x)Y

i,1

δk⁰_xi,k_xi ≈ M_α⁰_x(kx)δ_~k,~k0, (5.3) such that in the multiple band tight-binding approximation the perturbing operator becomes

Oˆ_N = X

~k,σ

M_α0 x(kx)

c^†

α⁰~kσc

α=1~kσ+H.c.

. (5.4)

Chapter 5 Thermometry of ultracold fermions in optical lattices by modulation spectroscopy

The transition probability|M_α⁰

x(kx)|² for exciting atoms from the lowest band to the bandα⁰_x = 2 or α⁰_x =3 with zero momentum transfer∆kx =0 is shown in Fig. 5.1(b). Excitations to the first excited bandα⁰_x =2 (blue solid line) are prohibited at the centerk_x =0 and the borderk_x=k_Lof the Brillouin zone due to the symmetry of the Bloch functions. The maximum probability lies in between coinciding with the region of interest around the Fermi surface of a half-filled lowest band. This is advantageous as the temperature dependence of the Fermi surface will be captured by the response (atom excitation rate) of the system to lattice modulation spectroscopy. Additionally, the amplitude is sufficiently strong such that the atom excitation rate is strong enough to obtain a detectable signal at small perturbing amplitudes and reasonable perturbing times. We comment on the detectability in more detail, giving numbers for an experimental example (in the presence of an external trap) in Sec. 5.4.

Figure 5.1: (a) The transition matrix elements squared|Mα⁰_x=2(kx,k⁰_x)|² for exciting atoms at quasimomentum kxfrom the lowest bandα = 1 to the first excited band{α⁰_x = 2, αy = 1, αz = 1}for allk⁰_xby normal lattice amplitude modulation. The only non-zero elements correspond tok⁰_x=kx. Inset: The transition matrix elements squared|Mα⁰_x=3(kx,k⁰_x)|²for exciting atoms at quasimomentumkxfrom the lowest bandα=1 to the second excited band{α⁰_x=3, αy=1, αz =1}for allk⁰_xby superlattice modulation spectroscopy. In the case of the superlattice modulation, the only non-zero elements correspond to∆kx=kL. This case will be discussed in Sec. 5.5. (b) The transition probability|Mα⁰_x(kx)|²for exciting atoms to the first (blue solid line)α⁰_x=2 and the second excited (blue dashed line)α⁰_x=3 band at quasimomentumk⁰_x=kxby normal lattice modulation and for exciting atoms to the second excited band (orange dash-dotted line)α⁰_x=3 at quasimomentumk⁰_x=kx+kLby superlattice modulation.

In (a) and (b) we consider a lattice depthV0=7E_rinx-direction. Figures adapted from Ref. [169].

In contrast, excitations to the second excited bandα⁰=3 (blue dashed line) are non-zero for all quasimo-mentakxand the amplitude is about three times larger, also see Fig. 5.1(b). The transition probability

|M₃(kx)|²is maximum atk_x =0 andk_x =k_Lsuch that the response at the corresponding frequencies will be dominated by these contributions if considering the full range of excitations. Consequently, it is more favorable to use excitations toα⁰=2 for a temperature measurement at intermediate lattice heights.

Let us remark, that the increase of transition amplitude with increasingα⁰from 2 to 3 is in agreement with the harmonic oscillator approximation for deep lattices. In this case, the lattice wells almost decouple and each well can be approximated by a harmonic oscillator (cf. Sec. 2.2.2). The modulation spectroscopy then corresponds to a frequency modulation of the quantum harmonic oscillator, ˜ωHO =ωHO(1+ρ), whereρis a small parameter. This modulation couples to the second excited band, but transitions to the first excited band are prohibited by symmetry. We confirm that the transition probability to the first excited band|M₂(kx)|²decreases for increasing lattice depths in accordance with the harmonic oscillator approximation.

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