Quantum Phase Transition from a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms
(Experiments by M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch in Munich)
One of the most intriguing aspects of quantum mechanics is that even at absolute zero temperature quantum fluctuations prevail in a system, whereas all thermal fluctuations are frozen out. Although these quantum fluctuations are microscopic in nature, they are nevertheless able to induce a macroscopic phase transition in the ground state of a many body system, when the strength of two competing terms in the underlying Hamiltonian is varied across a critical value.
When atoms with repulsive interactions are transferred from a magnetically trapped Bose-Einstein condensate into a three-dimensional optical lattice potential, they will undergo such a quantum phase transition from a Superfluid to a Mott Insulator as the potential depth of the lattice is increased. For low potential depths, the atoms form a superfluid phase, where each atom is spread out over the entire lattice, with long-range phase coherence across the lattice. For high potential depths the repulsive interactions between the atoms cause a transition to a Mott insulator phase. In this phase the atoms are localized at the lattice sites with an exactly defined atom number, but no phase coherence throughout the lattice. The Mott insulator phase is characterized by a gap in the excitation spectrum, which is detected in the experiment. We also demonstrate that it is possible to reversibly change between the two ground states of the system.
Entering the Mott Insulator Phase
In the weakly interacting regime a Bose-Einstein condensate in a 3D optical lattice can be well described by a macroscopic wave function. In the ground state only one bloch state is occupied, which means that the phase of the macroscopic wave function is the same at each lattice site. This state can be clearly identified by detecting sharp peaks in the multiple matter wave interference pattern after a time of flight expansion (see also the section about the 3D lattice).
If we now increase the lattice depth to deeper and deeper values, we observe that the interference peaks dim out and a non coherent background gains strength. For a deep lattice the phase coherence is totally lost.
One might think that the system in the deep lattice is still superfluid but that the coherence is lost because the lattice sites are dephased, which means that one has a statistical mixture of Bloch states. But we were able to show that this is not the case. Instead a totally different regime is entered, the regime of a Mott Insulator.
The phase coherence can be restored very rapidly when the lattice potential is lowered again. A significant amount of coherence is restored already after about one tunneling time. This is shown in the graph below (filled circles), where the width of the central interference peak, which is a measure for coherence in the system, is plotted against the ramp down time.
The open circles show the same measurement for an ensemble, which was intentionally statistically dephased before reaching the Mott insulator phase. The coherence is not restored at all in this case, even for of times up to 400 ms. This is one of the evidences that the state in the deep lattice is not just a statistically dephased superfluid state, but really the Mott insulator.
The Bose-Hubbard Hamiltonian
The system of interacting bosonic particles in a lattice potential is very well described by a Bose-Hubbard Hamiltonian in second quantisation. It can be derived by expanding the field operator in the wannier basis of localized wave functions at each lattice site, and consists of three parts:
The first term is the kinetic energy term with the tunneling matrix element J. J is basically determined by the overlap between adjacent localized wave functions and decreases exponentially with the lattice depth.
The second term describes an energy offset in each lattice site for example due to an external confinement.
The third term is the potential energy term characterized by the onsite atom-atom interaction energy U. In our case U is very large, about h*1.6 kHz, due to the strong confinement at each lattice site in a 3D lattice. U tells you how much energy it costs to put a second atom into a lattice with already one atom present at this lattice site.
For a shallow lattice J is large compared to U and the Hamiltonian is dominated by the kinetic energy term with a superfluid ground state (left side of diagram below). For a deep lattice J becomes small, the Hamiltonian is dominated by the interaction energy term and the ground state is the state of a Mott insulator for commensurate filling (right side).
In the superfluid case, each atom is delocalized over the entire lattice. Each lattice site is populated with a superposition of different number states which form a coherent state with a well defined phase for each lattice site. In the Mott Insulator state the atoms are localized to lattice sites with a defined number of atoms at each site. Now only one number state is occupied, for example with exactly one atom per site. But with a minimized number uncertainty, the phase uncertainty is maximized and the interference pattern vanishes. The phase coherence is lost but replaced by atom number correlations.
Measuring the Gap in the Excitation Spectrum
The gas of atoms can move freely through the lattice in the superfluid regime. But when the Mott insulator regime is entered, the mobility of the atoms is blocked due to the repulsive interaction between the atoms. With an exact number of atoms at each lattice site, it costs the onsite interaction energy U for an atom to tunnel to an adjacent site. This energy U must be provided to create a particle-hole excitation. Since this excitation is the lowest possible excitation in the Mott insulator state, a gap in the excitation spectrum is formed.
This gap, which is an important characteristic of a Mott insulator, was measured by applying a potential gradient to the system. The tunneling of atoms to adjacent sites is blocked until the gradient becomes large enough to overcome the gap: A difference of the potential energy between neighbouring lattice sites of U can provide the energy needed for tunneling (see right picture above).
The following graphs show the response to an applied potential gradient, plotted against the strength of the gradient, for different lattice potential depths. The response is the perturbation of the system. It is measured as the width of the interference peaks after the lattice depth is reduced again and the coherence is restored.
For a shallow lattice with a depth of 10 recoil energies the system can be easily perturbed, even for small gradients. But at a depth of about 13 recoil energies two resonances start to appear, and for a lattice depth of 20 recoil energies the situation has dramatically changed: We observe two narrow resonances ontop of a otherwise flat perturbation probability. For small gradients, the system cannot be perturbed at all, but for an energy difference between neighbouring lattice sites of the onsite interaction energy U, the system is perturbed resonantly. This gap in the excitation spectrum directly proves that we have indeed entered the Mott insulator regime.
The position of the first resonance in the three-dimensional lattice (closed circles) is in good agreement with the calculated value for the onsite interaction energy U.