Negative Temperature for Motional Degrees of Freedom in an Optical

So far, negative temperatures have only been created for spin degrees of freedom. In these systems, the discrete and finite spectrum of the Zeeman states naturally provides both lower and upper energy bounds. The upper energy bound is the key challenge for the realization of negative temperatures for motional degrees of freedom, i.e. where the occupation of the kinetic energy levels is described by an inverted thermal distribution.

Ultracold atoms in optical lattices have been theoretically suggested as a possible im-plementation scheme for negative temperatures for motional degrees of freedom [6, 7]. A great advantage of such systems lies in the very good isolation from the environment which is at positive temperature and would immediately destroy the negative temperature state upon thermal contact. As a key feature, optical lattices provide an elegant way of cre-ating an upper bound on kinetic energy: If the thermal energy of the atoms trapped in an optical lattice potential is much smaller than the band gap between the lowest and the first excited band, the atoms will be confined to the lowest band. The kinetic energy for ultracold atoms in an optical lattice is therefore effectively limited to the lowest band, providing both a lower and an upper limit. In the case of negative temperatures, the atoms will predominantly occupy the kinetic energy states close to the upper band edge. At this upper limit, the spectrum of the available states is identical but inverted compared to the lower band edge; only are the quasimomenta shifted by half the Brillouin zone. This strong symmetry between the lower and the upper band edge is another advantage of the Hubbard Hamiltonian: It directly follows that thermalization at the upper band edge is just as efficient as at the lower limit and a thermalized state with population inversion is actually realizable experimentally.

To create a negative temperature state, however, not only kinetic energy but all relevant degrees of freedom have to be limited from above. This prevents the realization of negative temperatures for motional degrees of freedom in solids: Here, the electrons also experience a band structure like in an optical lattice but couple to phonons that provide a spectrum with only a lower bound.

Negative temperature in the Bose-Hubbard Hamiltonian

Ultracold bosons in the tight-binding limit and confined to the lowest band are described by the Bose-Hubbard Hamiltonian (Section3.3.2),

H =−JX

hi,ji

ˆ

a^†_iˆaj+U 2

X

i

ˆ

ni(ˆni−1) +V X

i

r²_inˆi, (4.15)

containing interaction and potential energy in addition to kinetic energy. Thus, to realize a negative temperature state for bosons in this system, also interaction and potential energy need to be limited from above. Figure4.3illustrates the energy bounds of the three terms in the Bose-Hubbard model and shows how lower and upper bounds can be realized. As

4.3 Experimental Realizations of Negative Absolute Temperatures

already outlined, kinetic energyEkin is restricted to the lowest band of width 4dJ, where dis the dimensionality (Section3.2.3).

The energy bound of the on-site interaction term depends on the sign of the interaction U. Bose-Einstein condensates are initially prepared at repulsive interactions withU >0 in ultracold atom experiments to prevent collapse [27]. In principle, all atoms could occupy the same lattice site in an optical lattice. In the thermodynamic limit, this would lead to a diverging interaction energy per particle. For repulsive interactions, interaction energy therefore contains a lower bound at zero energy, in the case when all atoms occupy different lattice sites, but not an upper bound. The situation is inverted in the case of attractive interactions withU <0 where zero interaction energy indeed constitutes an upper bound.

The energy bound of the potential term analogously depends on the sign of the external potential V, where ultracold atoms in experiments are initially trapped in a trapping potential withV >0. In such a potential, a minimum in potential energy is reached when all atoms gather at the trap center, while potential energy can be increased to arbitrarily high values by atoms occupying remote sites. When the external confinement is instead converted to an anti-trapping potential withV <0, the trap center instead constitutes an upper limit for potential energy. As a conclusion, an upper bound for the total energy of the Bose-Hubbard model can be realized with attractive interactions and an anti-trapping potential.

E_q q

E_q q

r E E

0

-2dJ 2dJ

T, U, V > 0 T, U, V < 0

E_kin E_int E_pot

E_kin E_int E_pot

Figure 4.3: Energy bounds of the kinetic (Ekin), interaction (Eint) and potential energy term (Epot) of the two-dimensional Bose-Hubbard Hamiltonian. The insets illustrate the bounds either in quasimomentum or real space. In a sufficiently deep optical lattice, if the atoms are confined to the lowest band, kinetic energy is limited both from below and above. At positiveUandV, interaction and potential energy are limited from below. For attractive interactions (U <0) and an anti-trapping potential (V <0), they provide an upper bound such that all three terms are limited from above.

When realizing the Bose-Hubbard model with attractive interactions and negative tem-perature, an important question arises: How does the phase diagram in this regime look like, e.g. does the superfluid to Mott insulator phase transition happen at the same abso-lute values of|U|/J as for the repulsive Bose-Hubbard model at positive temperature (cf.

Fig.3.10)?

Any thermal state of a system with Hamiltonian ˆHand therefore also the phase diagram of this system is determined by the equilibrium density matrix [7]

ˆ ρ=e⁻

Hˆ

kBT. (4.16)

Apparently, if a system is prepared with an inverted Hamiltonian ˆH⁰=−Hˆ and at negative temperature T⁰ = −T, the resulting density matrix will be identical, ˆρ⁰ = ˆρ. In our experiment, the Bose-Hubbard Hamiltonian (Eq.4.15) consists of three terms, i.e. inverting the total Hamiltonian corresponds to inverting each of the three terms. In the experimental sequence (Section5.1), however, we only invert the interactionU → −U and the external potentialV → −V, as required for the upper energy bound, but the tunneling parameter J is unchanged. However, the tight-binding dispersion relation of the lowest band (Eq.

3.16)

E_kin=−2Jcos

π q klat

(4.17) shows that the sign ofJ can effectively be changed if the quasimomentum~qis shifted by half the width of the Brillouin zone,~q→~q+~klat [7]:

Ekin→ −2Jcos

πq+klat

klat

= 2Jcos

π q klat

(4.18) This is precisely what happens in the experiment (cf. Section5.1.3): The strongest occu-pation shifts from the ~q= 0 state to the edge of the Brillouin zone at~q=±~k_lat and, thus, the quasimomenta are shifted by~klat. Therefore, the phase diagram of the attractive Bose-Hubbard model at negative temperature is indeed the same as the phase diagram of the repulsive Bose-Hubbard model at positive temperature under the condition that all quasimomenta are shifted by half the Brillouin zone [7]. Therefore, also the superfluid to Mott insulator transition is expected at the same|U|/J values in both cases [154].

Negative temperature in the Fermi-Hubbard Hamiltonian

In contrast, fermionic atoms are described by the Fermi-Hubbard Hamiltonian, H =−J X

hi,ji,σ

ˆ

c^†_i,σˆcj,σ+UX

i

ˆ

ni,↓nˆi,↑+V X

i

r²_i(ˆni,↓+ ˆni,↑), (4.19)

where the indexσ∈ {↓,↑}denotes two different hyperfine states of the atoms, ˆc_i,σand ˆc^†_i,σ are the annihilation and creation operators for a fermion, respectively, and ˆni,σ = ˆc^†_i,σcˆi,σ

is the atom number operator.

The crucial difference compared to the Bose-Hubbard Hamiltonian lies in the Pauli exclusion principle that limits the occupation numbers to ˆn_i,σ ∈ {0,1}. The on-site in-teraction therefore, in addition to the lower bound of 0, also possesses an upper bound ofU/2 per particle. Negative temperatures are therefore possible for both attractive and repulsive interactions [155]. They have been realized, at least for local thermal equilibrium, in expansion experiments of fermionic band insulators in a homogeneous lattice [156] in