for the normal lattice modulation). At an amplitude of the modulation A = 0.05V0,x, considering a typical time of the modulationt=10ms, and at~ω =7Eras an example (indicated by a dashed line in Fig. 5.6(a)), we determine the mean number of excited atomshnαω0=3(t)iusing Eq. (5.16). We find the number of excited atoms to lie between 2 and 6 for temperatureskBT =0.5J−1.5J. This means that we excite a couple of hundreds of atoms when considering∼100 parallel one-dimensional tubes of approximately equal filling with an atom number difference of∼150 between the different temperatures with temperature difference∆kBT =0.5J which is detectable in experiments.
In two dimensions, the improved frequency resolution is a clear advantage compared to the normal lattice modulation. We find that a frequency resolution∆(~ω)≈ ±0.025Er(as compared to∆(~ω)≈ ±0.01Er
in the case of the normal lattice modulation) is already sufficient in experiment which corresponds to a perturbing timet≈9ms. For an amplitudeA=0.05V0,xat~ω=7Erwe excite mean atom numbers hnαω0=3(t)ibetween 300 and 400 for temperatureskBT =0.5J−1.5Jwith an atom number difference of approximately 30 atoms between the curves with temperature difference∆kBT =0.5Jwhich is easier to detect in experiment than the smaller atom number difference in the case of the normal lattice modulation (∼10 atoms).
Figure 5.6: The atom excitation rate by superlattice modulation along one direction to the second excited band α0x=3 in the presence of a harmonic trapping potential. The response is shown at temperatureskBT =0.5J, 1J and 1.5Jwith error bars corresponding to 5% uncertainty on the initial atom numberNfor the one-dimensional system (a) with initial atom numberN =60 and for the two-dimensional system (b) with initial atom number N=4000. The vertical lines indicate~ω=7Erfor which we determine the number of excited atoms (see main text). Figures adapted from Ref. [169].
In this sense, the superlattice modulation is advantageous compared to the normal lattice modulation as less frequency resolution is required to extract the temperature in experiment. However, the superlattice modulation of the ’normal’ equilibrium lattice is more challenging to implement experimentally compared to the standard amplitude modulation. For the superlattice modulation several laser beams have to be superimposed as explained in Sec. 2.6. Consequently, if the frequency resolution obtained by the normal lattice modulation is sufficient, the normal lattice modulation would be the preferred choice.
5.6 Conclusion
In this chapter we have investigated the temperature-dependent atom excitation rate to higher Bloch bands of non-interacting fermions in an optical lattice as a response to a time-dependent modulation of
Chapter 5 Thermometry of ultracold fermions in optical lattices by modulation spectroscopy
the lattice amplitude. We demonstrated that it shows clear signatures of the Fermi distribution of the equilibrium system, also in the presence of an external trapping potential. We explored the possibilities of thermometry for different dimensionalities of the equilibrium system. We find that quasimomentum is conserved by the normal lattice amplitude modulation (cf Fig. 5.1) and excitations to the first excited band are most suitable for thermometry. In the case of a superlattice modulation, a finite quasimomentumkLis transferred to the system and excitations to the second excited band are most suitable for thermometry.
We considered temperatures of a few percent of the hopping amplitude J and find that the response shows a clear signature of the temperature-dependent Fermi factor in the lowest band for one- and two-dimensional equilibrium systems for both modulation schemes. The Fermi dependence in the atom excitation rate is strongly broadened in energy due to the much larger bandwidth of the higher bands compared to the lowest band. This is beneficial as it strongly reduces the required experimental frequency resolution such that the Fermi dependence can be resolved within typical durations of the perturbation.
We estimate the number of excited atoms which we find to be sufficiently large to be measured in experiments.
We emphasize that the temperature dependence of the atom excitation rate becomes more pronounced for decreasing temperatures and the applicability of our scheme covers the regime of interest where antiferromagnetic ordering is expected to occur. In two dimensions, for which our thermometry scheme is well suited, spin ordering at half-filling in the homogeneous system is expected at entropies per particle belowsN ∼0.4kBat intermediate interactions (U/J)∼5−10 [176] and at smaller entropies for weaker interactions. We relate temperature to entropy for the square homogeneous lattice at half-filling of a typical system sizeL= 100 in each direction within the grand-canonical ensemble. An entropy sN ∼0.4kBcorresponds to a temperaturekBT ∼0.3Jwhere our thermometry scheme is well suited. Our scheme also works at lower temperatures corresponding to lower required entropies for spin ordering at weaker interactions. Consequently, our thermometry scheme is applicable in the regime of interest where spin ordering is expected to occur in the homogeneous system. Note that homogeneous trapping potentials were recently realized [69–71]. The assumption of a homogeneous system also give a first rough estimate for the harmonically trapped system considering the Mott insulating regime at intermediate filling. In this case the entropy is small and approximately constant across the central Mott plateau [177]. Furthermore, the density is constant in the bulk, resembling a homogeneous system although some additional entropy will be added to the liquid wings. Note that in a recent experiment in two dimensions temperatures kBT ∼0.25Jwere reached [19] and relevant correlations were observed [17–19].
Here, we investigated the applicability of our thermometry scheme for non-interacting particles but we believe it to be an important step in order to get also more complex systems under control by an adiabatic connection (entropy conserving) to the interacting system. Although for some strongly correlated atomic states, the assumption of an adiabatic process from the non-interacting atoms might not be suitable, there are several phases which could adiabatically be connected to the non-interacting atoms in the optical lattice that would be worth a thorough investigation. One example are weakly interacting Fermi liquids in the optical lattice where the influence of temperature and interactions on the properties of the quasiparticles, such as the effective mass or the life time of the quasiparticles, are not fully understood.
This would be worth a detailed experimental measurement in order to test theoretical predictions from the microscopic model. In the solid state context, a direct measurement of the properties of quasiparticles is possible by angle-resolved photoemission spectroscopy which gives access to the single-particle spectral function [178]. However, a detailed comparison with theory remains difficult in solid states systems due to the complexity of the underlying system. In cold atom systems, a measurement of the single-particle spectral function is possible by momentum-resolved Raman or radio-frequency spectroscopy [31]. The influence of the interactions can be directly probed due to the high tunability of interactions in cold atom systems and due to the absence of other influences such as impurities or phonon-scattering. Some
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5.6 Conclusion
properties of the Fermi liquid have been probed for the harmonically trapped gas, for example the quasiparticle dispersion and the effective mass have been studied in three- and two-dimensions [179, 180]. In the presence of an optical lattice potential, Raman spectroscopy has also been suggested to investigate the momentum-resolved quasiparticle peak [163]. However, further experimental investigation is required to connect better to the microscopic model and in particular to study the effect of temperature and interactions on the properties of the quasiparticle excitations.
C H A P T E R 6
Superlattice modulation spectroscopy of ultracold bosons in optical lattices
In this chapter we study the excitations in the one-dimensional Bose-Hubbard model (2.25) by means of superlattice modulation spectroscopy at zero temperature using t-DMRG. We study both, the Mott insulating phase as well as the superfluid phase, and we compare to analytical results. A detailed discussion of the ground state phase diagram is given in Sec. 2.4 and a possible implementation of the superlattice modulation is discussed in Sec. 2.6.
We begin with the derivation of a description of the Mott insulating regime at integer filling within perturbation theory at strong interactions in Sec. 6.1. This enables us to obtain an analytical description of the energy absorption rate in this regime. Sec. 6.1 is rather technical and can be skipped if one is only interested in the results which are discussed in the subsequent section. In Sec. 6.2 we study the energy absorption in the Mott insulator using t-DMRG and we compare to the perturbation theory result.
We discuss how, deep in the Mott insulator, a sharp spectral peak around~ω ∼ U enables a precise determination of the interaction parameter U in experimental setups. Moreover, we investigate the energy absorption rate close to the phase transition to the superfluid phase where the spectral response is broadened. In Sec. 6.3 we study the energy absorption rate using t-DMRG at weak interactions in the superfluid regime, both at commensurate and incommensurate fillings, where we observe excitations at low frequencies as well as a pronounced absorption peak at larger frequencies. Finally, we summarize and conclude in Sec. 6.4. The main results of this chapter are going to be published [181].
6.1 Perturbation theory at U J
In this section we determine the eigenstates and eigenenergies of the Bose-Hubbard model (2.25) to lowest order inJ/Uwithin perturbation theory in the strong coupling regime. We consider the interaction termHU as the unperturbed Hamiltonian and the kinetic termHkinas a small perturbation. The obtained spectrum gives an intuitive notion of the nature of excitations at strong interactions and enables us to analytically obtain an approximate expression for the energy absorption rate in the Mott insulating regime within linear response which will be discussed in Sec. 6.2 in comparison to t-DMRG calculations.
The Bose-Hubbard model is given by Eq. (2.25). Here, we use a slightly different definition of the
Chapter 6 Superlattice modulation spectroscopy of ultracold bosons in optical lattices
interaction term such that the Bose-Hubbard model becomes, H= Hkin+HU=−J
L−1
X
j=1
(b†jbj+1+H.c.)+ U 2
L
X
j=1
(nj−n)¯ 2, (6.1)
whereb(†)j are the bosonic annihilation (creation) operators at site jandnjis the particle number operator, Uis the on-site interaction strength andJis the tunneling matrix element. We replaced the usual notation P
jnj(nj −1) → P
j(nj −n)¯ 2 which amounts to a shift in energy. We work at fixed particle number N = L¯nwhereLis the number of lattice sites and we assume a Mott insulating ground state with ¯n bosons per site (commensurate filling). Locally, an excess particle (occupation ¯n+1 at site j) and a hole (occupation ¯n−1 at site j0) now both have an energyU/2 with respect to the initial occupation ¯n in contrast to the usual notation (2.25) where the energy of an excess particle (hole) with respect to the initial occupation depends on the filling. However, in both cases a particle-hole pair has energyUwith respect to the ground state.
Zero order The eigenstates of the unperturbed HamiltonianHU at commensurate filling and fixed particle number are given by Fock states with eigenenergies that are multiples of the on-site interaction strengthU. The ground state is given by
|0i=|n,¯ n, ...,¯ ni,¯ (6.2)
with eigenenergyE0 =h0|HU|0i=0. The first excited state in the same particle number sector is created by removing one particle from a site denotedm+dand putting it onto a different sitemwheredis the distance to the right from the site with occupation ¯n+1 to the site with occupation ¯n−1. This corresponds to the creation of a ’doublon-hole’ pair on top of the commensurately filled background which is sketched in the inset of Fig. 6.1(b). The excited state is given by
|m,di= 1
√n(¯¯ n+1)am+da†m|0i, (6.3) withm= 1, . . . ,Landd = 1, . . . ,L−1 such that the state is a L(L−1) degenerate eigenstate of HU with eigenenergyEm,d=hm,d|HU|m,di=U. Note that we have periodic boundary conditions inmbut open boundary conditions indbecaused=1 is not connected withd= L−1. The second excited state corresponds to the creation of two ’doublon-hole’ pairs with eigenenergy 2Uand the third excited state with eigenenergy 3Uis either given by three ’doublon-hole’ pairs or one site with occupation ¯n+2 and two sites with occupation ¯n−1.
First order Switching on the kinetic part in the Hamiltonian, all states and eigenenergies get modified.
The first order correction to the ground state energy vanishes ash0|Hkin|0i=0. The first order correction to the ground state is given by
|Ψ10i=X
m,d
hm,d|Hkin|0i
E0−Em,d |m,di= J U
pn(¯¯ n+1)X
m
(|m,1i+|m,L−1i), (6.4)
where we usedhm,d|Hkin|0i=−J√
¯
n(¯n+1) δd,1+δd,L−1
. Other excited states do not contribute to the sum as their overlap withHkin|0iis zero. The modified ground state is then given by|Ψ0i ≈ |0i+|Ψ10ito first order inJ/U.
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6.1 Perturbation theory atU J The corrections to the first excited state have to be determined within degenerate perturbation theory [182].
The lowest order correction is obtained by identifying the unperturbed degenerate eigenstates and to diagonalize the perturbation in this basis. This corresponds to determining the first-order energy shifts that lift the degeneracy while the lowest order states are not modified. To obtain higher order corrections one can use the formulas from non-degenerate perturbation theory but one has to exclude all contributions from the corresponding degenerate subspace. Here, we are only interested in the lowest order correction.
Essentially, we need to diagonalize the matrix element hm,d|Hkin|m0,d0i=−J
(1+n)¯ δm0,m+1δd0,d−1+δm0,m−1δd0,d+1+n¯ δd0,d−1+δd0,d+1δm0,m. (6.5) Note that this expression is invariant form →m+Ldue to periodic boundary conditions inmbut in dwe have open boundary conditions (see above). The diagonalization of the above expression can be achieved by a double Fourier transform in themanddcoordinates. The procedure is outlined in the following. The Fourier transform of themcoordinate is given by
|K,di= 1
√L
L
X
m=1
eiK xm|m,di, (6.6)
withxm=amwhereais the lattice spacing andK=2πq/(La) withq=1, ...,L. One obtains hK,d|Hkin|K0,d0i = 1
L
L
X
m,m0=1
e−i(K xm−K0xm0)hm,d|Hkin|m0,d0i
= −JδK,K0r(K)
eiθ(K)δd0,d−1+e−iθ(K)δd0,d+1
, (6.7)
wherer(K) = p
(¯n+1)2+n¯2+2¯n(¯n+1) cos(Ka) and θ(K) = (¯n+1) sin(Ka)/[¯n+(¯n+1) cos(Ka)].
Thed-sector cannot be diagonalized by the ’standard’ Fourier transform because of the open boundary conditions but a sine transform can be used,
|K,ki= r2
L XL−1
d=1
eidθ(K)sin(kxd)|K,di=
√ 2 L
XL−1
d=1 L
X
m=1
eidθ(K)sin(kxd)eiK xm|m,di, (6.8) wherexd=daandk=πp/(aL) withp=1, ...,L−1 and which is a good basis ashK,k|K0,k0i=δK,K0δk,k0. Finally, one can determine the matrix elements ofHkinin the new basis obtaining
hK,k|Hkin|K0,k0i = 2 L
L−1
X
d,d0=1
e−idθ(K)eid0θ(K0)sin(kxd) sin k0xd0hK,d|Hkin|K0,d0i
= −2Jr(K) cos(ka)δK,K0δk,k0, (6.9) which is diagonal and gives the first order correction to the energy forK =K0andk =k0. This correction completely lifts the degeneracy of the first band of excitations centered aroundUexcept for a translational invariance inKby 2π/a. One can identifyKwith the center of mass momentum of the ’doublon-hole’
pair created andkis related to its relative momentum. To first order, the lowest band of excitations is consequently given by
EK,k=U−2Jr(K) cos(ka). (6.10)
Chapter 6 Superlattice modulation spectroscopy of ultracold bosons in optical lattices
The lowest band of excitations for different center of mass momentaK is shown in Fig. 6.1(a). The maximum bandwidthW = 12J is obtained for zero center of mass momentum K = 0 whereas the minimum bandwidthW =4J is obtained forK =π. Excitations in the first case can be created by a perturbation which conserves quasimomentum such that a ’doublon-hole’ pair is created with zero center of mass momentum. This is the case for the normal lattice modulation which was considered in Ref. [39].
In order to create ’doublon-hole’ pairs withK=π, a finite momentum transferπof the perturbation is required. This is the case for the superlattice modulation such that excitations are created in a much narrower energy region compared to the normal lattice modulation.
Energy absorption Let us now consider the effect of the superlattice modulation spectroscopy. The energy absorption rate within linear response theory is given by Eq. (3.31). Our aim is to obtain a frequency-resolved excitation peak expected to occur at frequencies corresponding to the lowest band of excitationsEK,k (6.10) such that we need to considerEm=EK,kandE0=0 for the resonance condition δ(~ω−(Em−E0)) in Eq. (3.31). For the amplitude
D0 OˆS
mE
2 it is sufficient to only consider the lowest order, that is|0igiven by Eq. (6.2) and|mi=|K,kigiven by Eq. (6.8)1.
Figure 6.1: (a) The lowest band of excitations in the Bose-Hubbard model in lowest order inJ/Ufor different center of mass momentaK=0, π/aof the ’doublon-hole’ pair. In both cases, the energy band is centered around Ubut has different bandwidthW =12J, 4Jrespectively. (b) The energy absorption ratedE/dtas a response to the normal lattice modulation (orange dashed line) and to the superlattice modulation (blue solid line). The width of the excitation peak corresponds to the bandwidth of the lowest band of excitations forK=0, π/arespectively.
Inset: Sketch of the excited state|m,diin real space. One particle is added at positionmwhile one particle is removed at positionm+d, i.e. a ’doublon-hole’ pair is created on top of a commensurately filled background.
For the superlattice modulation spectroscopy the perturbing operator ˆOS is given by Eq. (2.41) and the relevant matrix element becomes
hK,k|OˆS|0i = p
2¯n(¯n+1) sin(ka)ηpδKa,π, (6.11) withηp =(1−(−1)p). The finite momentum transferπof the superlattice modulation is encoded in the Kronecker deltaδKa,π. In the case of the momentum conserving normal lattice modulation the matrix
1In principle, the matrix element to the considered order is given by|hΨ1|OˆS|Ψ0i|where|Ψ0i=|0i+|Ψ10i+O(J2/U2) and
|Ψ1i=|K,ki −(J/U)√
2¯n(¯n+1)ηpsin(ka)|0i+(J/U)P
α|αi+O(J2/U2) where|αiare states in addition to the Fock state|0i that are directly coupled via the kinetic term to the states|K,ki. However, we find that the transition matrix element squared simplifies|hΨ1|Oˆs|Ψ0i|2=|hK,k|OˆS|0i|2+O(J2/U2) such that it is sufficient to consider only the lowest order contribution.
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