Superfluid-to-Mott-Insulator Transition

Take a Bose-Einstein condensate — the coherent matter wave assembled in the three articles this one builds on — and pour it into a crystal made entirely of light. Two counter-propagating laser beams interfere to a standing wave, and through the optical dipole force every atom feels a conservative egg-crate potential $V_0\sin^2(kx)$ whose depth follows the laser intensity. The result is a solid with no ions, no phonons, no defects and no disorder: a perfectly periodic lattice whose every parameter is set by a knob on the laser table — the cleanest realization physics has of a lattice model of interacting quantum matter, and why cold atoms in optical lattices became the premier platform for quantum simulation. When the wells are deep enough that only the lowest Bloch band is occupied, the whole many-body problem collapses onto a single Hamiltonian — the Bose-Hubbard model — governed by exactly two energies. One is the tunnelling amplitude $J$ that lets an atom hop to a neighbouring site; the other is the on-site repulsion $U$ paid whenever two atoms share a site. Their ratio $U/J$ is not fixed but tuned, exponentially, by lattice depth, because deepening the wells suppresses the wavefunction overlap that sets $J$ far faster than it raises $U$ . At zero temperature and integer filling the ground state changes character abruptly as $U/J$ crosses a critical value. Weakly interacting atoms spread coherently over the whole lattice — a superfluid with one macroscopic phase and large number fluctuations per site. Strongly interacting atoms lock to exactly $n$ per site — a Mott insulator with fixed number and, by the number-phase uncertainty relation, no phase at all. The crossing between them is a genuine quantum phase transition, driven by quantum rather than thermal fluctuations, and its reversible signatures make it the textbook example of strongly correlated matter engineered with light.

Key takeaways

Ultracold bosons in a periodic optical lattice are described by the Bose-Hubbard model, a contest between the tunnelling energy $J$ and the on-site repulsion $U$ .
At sufficiently small $U/J$ and at appropriate filling, the ground state is a superfluid: particles remain delocalized, phase coherence extends across the lattice, and sharp matter-wave interference peaks appear in time-of-flight.
At sufficiently large $U/J$ and integer filling, the ground state enters a Mott-insulating regime: the occupation is pinned close to an integer $n$ per site, particle-number fluctuations are strongly suppressed, the state is gapped and incompressible, and long-range phase coherence is lost.
The knob is exponential — deepening the lattice suppresses $J$ as $e^{-2\sqrt{V_0/E_R}}$ while $U$ grows only as $V_0^{3/4}$ — so a modest change in laser power sweeps $U/J$ across the critical point.
At unit filling in 3D the transition sits near $(U/J)_c\approx 35$ in mean field and $\approx 29.3$ from quantum Monte Carlo, corresponding to a depth $V_0\approx 13\,E_R$ — as first observed by Greiner and co-workers in 2002.
Key idea: the dimensionless ratio $U/J$ , tuned exponentially by lattice depth, drives a zero-temperature quantum phase transition between a phase-coherent, compressible superfluid and a number-locked, gapped, incompressible Mott insulator.

The core insight

The dimensionless ratio $U/J$ of on-site repulsion to tunnelling, tuned exponentially by lattice depth, drives a zero-temperature quantum phase transition between a phase-coherent, compressible superfluid and a number-locked, gapped, incompressible Mott insulator at integer filling.

Where this lives today

The clean bosonic transition was only the opening move. Its real value is as a calibrated benchmark for quantum simulation: because the Bose-Hubbard phase diagram is known with high accuracy from quantum Monte Carlo, an experiment that reproduces the measured critical depth and the wedding-cake shell structure provides strong evidence that the lattice and temperature are well controlled and that the imaging faithfully captures the underlying physics — a licence to then explore Hamiltonians theory cannot solve. The direct heir is the Fermi-Hubbard model, where fermionic atoms realize the physics believed to underlie high-temperature superconductivity and the Mott insulator becomes a stage for quantum antiferromagnetism. The enabling instrument is the quantum-gas microscope, which images single atoms on single sites with a high-resolution objective. There a unit-filling Mott insulator is a defect-free register of exactly one atom per site — a natural initializer for neutral-atom quantum computing and single-atom addressing — and its vanishing number fluctuations make it a resource for spin-squeezed, sub-shot-noise clocks and interferometers. Sweeping the depth across the critical point at a finite rate turns the same apparatus into a laboratory for non-equilibrium physics, testing the Kibble-Zurek theory of defect formation, while a quench and its collapse-and-revival dynamics probe interactions and thermalization in a closed quantum system. The optical lattice has become a programmable analog computer for strongly correlated matter, with the superfluid-Mott transition as its "hello, world".

The physical picture

Picture each lattice site as a tiny well and the atoms as balls that can quantum-tunnel to their neighbours. If tunnelling is cheap — a shallow lattice — an atom lowers its kinetic energy by spreading out, and the whole cloud becomes a single wave draped over the crystal, every site sharing one phase. This is the superfluid. If tunnelling is expensive compared with the repulsion cost $U$ of doubling up on a site, the atoms enter a Mott-insulating regime with integer average occupancy; at unit filling the occupation is close to one atom per site, with strongly suppressed number fluctuations. In the ideal deep-lattice limit the Mott state approaches exactly $n$ atoms per site, while moving an atom away from this configuration costs an energy of order $U$ . The deep reason the two phases look so different is that number and phase are conjugate variables, $\Delta N\,\Delta\phi \gtrsim 1$ . The superfluid buys a sharp global phase — and hence interference — at the price of huge on-site number fluctuations; the Mott insulator buys a razor-sharp number, $\Delta N = 0$ , at the price of a completely random phase. The transition is nothing more than the system switching which side of that uncertainty relation it wants to make small.

Figure

Two ways to satisfy the number–phase uncertainty ΔN·Δφ ≳ 1: at shallow depth the atoms form a phase-coherent state with appreciable number fluctuations, while at deep depth the occupation becomes nearly fixed at one atom per site, with strongly suppressed ΔN and an ill-defined phase

The reason a laser can drive this transition at all is that the knob is exponential. The hopping $J$ is the overlap integral of two neighbouring localized Wannier functions, and as the wells deepen the wavefunctions pull in and their tails stop touching, so $J$ collapses as $e^{-2\sqrt{V_0/E_R}}$ . The interaction $U$ , by contrast, is set by how tightly a single Wannier function is squeezed, and grows only as $V_0^{3/4}$ . A power divided by an exponential swings across orders of magnitude for a small change in the exponent, so a factor-of-two turn of the laser intensity carries $U/J$ cleanly from one phase, through the critical point, into the other. The whole story is read out in a single image. Switch the lattice off, let the cloud expand freely for a few milliseconds, and the density that lands on the camera maps the momentum distribution the atoms had inside the lattice. A phase-coherent superfluid interferes with itself and prints a sharp grid of Bragg peaks at the reciprocal-lattice vectors; a Mott insulator, with a random phase on every site, prints only the smooth broad envelope of one well. The reversible collapse and recovery of those peaks as the lattice is deepened and lowered again is the fingerprint of the transition, and the phase diagram below is the map the experiment is walking across.

Figure

Ground-state mean-field phase diagram of the Bose-Hubbard model: incompressible Mott lobes of fixed integer filling n = 1, 2, 3 sit inside a surrounding superfluid. Deepening the lattice moves the system leftward along a line of fixed density and, at unit filling, enters the

n=1

lobe through its tip at

zJ/U\simeq0.172

, corresponding to

U/(zJ)\simeq5.83

or, for the simple-cubic lattice with

z=6

U/J\simeq34.9

The core equations

Everything begins with the potential the light makes. Two counter-propagating beams of wavevector $k$ interfere to a standing wave, and the dipole force turns intensity into a conservative periodic potential; a photon absorbed and re-emitted transfers a recoil, whose energy $E_R$ is the natural unit of the whole problem. Retro-reflecting three orthogonal beams gives the simple cubic lattice below.

V_{\mathrm{lat}}(\mathbf r) = V_0\big[\sin^2(kx)+\sin^2(ky)+\sin^2(kz)\big],\qquad E_R=\frac{\hbar^2 k^2}{2m} \qquad (1)

where: $V_0$ — lattice depth, proportional to laser intensity through the dipole force; $k=2\pi/\lambda$ — laser wavevector, so the lattice spacing is $d=\lambda/2$ ; $m$ — atomic mass; $E_R$ — recoil energy, the energy scale set by one photon momentum. Depths are always quoted in units of $E_R$ , and "deep" means $V_0/E_R \gg 1$ . When the band gap to the second Bloch band exceeds every other energy in the problem, only the lowest-band Wannier states matter, and the field operator is expanded in them. The kinetic and interaction terms collapse to nearest-neighbour hopping and an on-site penalty, giving the single-band Bose-Hubbard Hamiltonian — the model this whole article lives inside.

\hat H = -J\sum_{\langle i,j\rangle}\hat a_i^{\dagger}\hat a_j + \frac{U}{2}\sum_i \hat n_i(\hat n_i-1) - \mu\sum_i \hat n_i \qquad (2)

where: $J$ — tunnelling amplitude between nearest-neighbour sites $\langle i,j\rangle$ ; $U$ — on-site repulsion for two atoms on one site; $\hat a_i^{\dagger},\hat a_i$ — bosonic creation and annihilation operators at site $i$ ; $\hat n_i=\hat a_i^{\dagger}\hat a_i$ — the number operator; $\mu$ — chemical potential fixing the mean filling. The first term favours delocalization, while the interaction favours integer occupancy at commensurate filling; their competition drives the transition. The hopping $J$ is not arbitrary: it is determined by the overlap of neighbouring localized Wannier functions through the single-particle Hamiltonian. This integral is what falls off so steeply as the wells deepen.

J = -\int \mathrm{d}^3r\; w^*(\mathbf r-\mathbf R_i)\left[-\frac{\hbar^2}{2m}\nabla^2 + V_{\mathrm{lat}}(\mathbf r)\right] w(\mathbf r-\mathbf R_j) \qquad (3)

where: $w(\mathbf r-\mathbf R_i)$ — the lowest-band Wannier function localized at site $\mathbf R_i$ ; the bracket is the single-particle Hamiltonian. Because $J$ measures the overlap of two neighbouring functions, squeezing them tighter kills it exponentially — the origin of the exponential knob. The interaction is a different overlap: the contact repulsion of two atoms averaged over the shape of a single Wannier function. It depends only on how tightly one site is squeezed, not on how far a wavefunction reaches, so it grows only mildly with depth.

U = \frac{4\pi\hbar^2 a_s}{m}\int |w(\mathbf r)|^4\,\mathrm{d}^3r \qquad (4)

where: $a_s$ — the s-wave scattering length setting the strength of the contact interaction; $g=4\pi\hbar^2 a_s/m$ — the coupling constant; $\int|w|^4$ — the inverse effective mode volume of a site. Positive $a_s$ (as in $^{87}$ Rb) means repulsion, the case that produces a Mott insulator. The two phases are distinguished operationally by which member of a conjugate pair is sharp. Number and phase fluctuations are complementary: suppressing on-site number fluctuations makes the phase less well defined, while a well-defined phase requires appreciable number fluctuations.

\Delta N_i\,\Delta\phi_i \gtrsim 1 \qquad (5)

where: $\Delta N_i$ — the on-site atom-number fluctuation; $\Delta\phi_i$ — the on-site phase uncertainty. The superfluid takes $\Delta\phi\to 0$ (coherence, interference) at the cost of large $\Delta N$ ; the Mott insulator takes $\Delta N\to 0$ (a Fock state) at the cost of a random phase. That choice is read out directly in a time-of-flight image. The measured density after free expansion is the momentum distribution, which factorizes into the single-site envelope and the coherence between sites — the quantity that is long-ranged in a superfluid and dies at one site in a Mott insulator.

n(\mathbf k) \propto |\tilde w(\mathbf k)|^2 \sum_{i,j} e^{i\mathbf k\cdot(\mathbf R_i-\mathbf R_j)}\langle \hat a_i^{\dagger}\hat a_j\rangle \qquad (6)

where: $n(\mathbf k)$ — density after expansion, with position mapping to initial momentum; $\tilde w(\mathbf k)$ — Fourier transform of the Wannier function, the smooth envelope; $\langle \hat a_i^{\dagger}\hat a_j\rangle$ — the one-body coherence. Long-ranged coherence builds sharp Bragg peaks; short-ranged coherence leaves only the envelope. Two thermodynamic quantities cleanly separate the phases and serve as order parameters of the transition. The Mott insulator is both gapped and incompressible; the superfluid is neither.

E_{\mathrm{gap}} \approx U \;\;(U/J\to\infty),\qquad \kappa = \frac{\partial n}{\partial \mu} \qquad (7)

where: $E_{\mathrm{gap}}$ — the energy to create a particle-hole excitation by moving one atom off its site; $\kappa$ — the compressibility. In a Mott lobe $\kappa=0$ exactly (adding atoms costs the gap, so $n$ is pinned to an integer over a finite range of $\mu$ ); in the superfluid $\kappa\gt 0$ and the spectrum is gapless.

Derivation AMott-lobe boundaries from second-order perturbation theory

The goal is to find, within a controlled approximation, exactly where the Mott insulator stops being stable. We start from the trivial limit $J=0$ , where the ground state is a perfect Fock state with $n$ atoms per site, and ask when it first pays to let one atom hop — making a mobile particle or hole. Treating hopping as a perturbation and demanding that both the particle and hole excitation energies stay positive traces out the boundary of the Mott phase in the plane of chemical potential versus hopping — the famous lobe — whose closing point is the quantum critical point at fixed density.

Step 1. At $J=0$ the sites decouple and the energy per site of the Fock state with occupancy $n$ is $E_0(n)=\tfrac{U}{2}n(n-1)-\mu n$ ; comparing $E_0(n)$ with $E_0(n\pm1)$ shows the $n$ -per-site state is the ground state for $(n-1)U \lt \mu \lt nU$ .

E_0(n) = \frac{U}{2}\,n(n-1) - \mu n

Why: with no hopping the problem is purely local, so the ground state is simply whichever integer occupancy minimizes the on-site energy at the given chemical potential — a staircase of Fock states.

Step 2. The energy costs to add one atom (particle) or remove one (hole), evaluated at $J=0$ , are both positive inside the Mott window; the smaller of these costs gives the particle or hole excitation gap.

\Delta E_{\mathrm{part}} = nU-\mu,\qquad \Delta E_{\mathrm{hole}} = \mu-(n-1)U

Why: an incompressible state is one that resists adding or removing particles; a finite positive cost for both is the definition of the gap that makes the Mott insulator insulating.

Step 3. Turn on hopping. A particle or hole is no longer stuck: it can lower its energy by delocalizing over the $z$ neighbours, and second-order perturbation theory gives that gain as proportional to $zJ^2/U$ near the tip. Adding it to the bare costs shifts the excitation energies downward.

E^{(2)} = -zJ^2\sum_{\alpha}\frac{|\langle\alpha|\hat T|n\rangle|^2}{E_\alpha - E_0}

Why: delocalization is a kinetic-energy reward, and it is what eventually overwhelms the interaction cost of a mobile defect — the mechanism by which hopping melts the Mott insulator.

Step 4. Requiring the corrected particle and hole energies to remain non-negative gives a quadratic in $\mu$ ; solving it yields the two branches that bound the lobe, meeting at the tip where the discriminant vanishes.

\left(\frac{\mu}{U}\right)_{\pm} = \tfrac12\big[(2n-1) - u\big] \pm \tfrac12\sqrt{u^2 - 2u(2n+1) + 1},\quad u=\frac{zJ}{U}

Why: the width of the lobe in $\mu$ is the gap; where the two branches collide the gap has closed, so the system can turn superfluid without any change of density.

Step 5. Setting the discriminant to zero locates the tip, the critical point reached at fixed integer filling. For $n=1$ this gives $(U/zJ)_c=3+2\sqrt2\approx 5.83$ ; on the 3D cubic lattice ( $z=6$ ) that is $(U/J)_c\approx 35$ , which quantum Monte Carlo corrects to $\approx 29.3$ .

\left(\frac{zJ}{U}\right)_c = \frac{1}{2n+1+2\sqrt{n(n+1)}}

Why: this is the number an experiment actually crosses — the fixed-density mean-field prediction against which the measured and Monte-Carlo transition points are compared. *The numerical values are: for $n=1$ , $3+2\sqrt{2}=5.828$ ; for $n=2$ , $5+2\sqrt{6}=9.899$ .*

Go deeper · Derivation B · 1 minWhy the lattice depth is an exponential knob on U/J

Derivation BWhy the lattice depth is an exponential knob on U/J

The goal is to show quantitatively that $J$ and $U$ respond to lattice depth in dramatically different ways — one exponential, one a slow power — so that $U/J$ becomes an enormously sensitive function of laser intensity. We approximate each Wannier function by the Gaussian ground state of the harmonic well at the bottom of a site, evaluate the two overlap integrals, and read off their depth scalings — which is why a factor-of-two turn of a knob spans the transition.

Step 1. Near a minimum the potential is harmonic with frequency $\omega_{\mathrm{ho}}=2\sqrt{V_0 E_R}/\hbar$ , so the Wannier function is roughly a Gaussian of width $\ell\propto (E_R/V_0)^{1/4}$ ; the tunnelling is set by this Gaussian's overlap with its neighbour a distance $d=\lambda/2$ away, which is exponentially small in $d/\ell$ .

\frac{J}{E_R} \approx \frac{4}{\sqrt{\pi}}\left(\frac{V_0}{E_R}\right)^{3/4} e^{-2\sqrt{V_0/E_R}}

Why: the exponent $2\sqrt{V_0/E_R}$ is essentially the WKB tunnelling action through the barrier between wells; deeper barriers suppress the overlap geometrically, which is the entire source of the transition's sensitivity.

Step 2. The interaction depends only on how tightly one site is squeezed, so it scales as the inverse mode volume $\propto \ell^{-3}\propto (V_0/E_R)^{3/4}$ times the coupling $g$ — a slow power of the depth with no exponential.

U \approx \sqrt{\frac{8}{\pi}}\,k\,a_s\,E_R\left(\frac{V_0}{E_R}\right)^{3/4}

Why: squeezing a site raises the local density and thus the repulsion, but only algebraically; unlike $J$ there is no barrier to tunnel through, so no exponential appears.

Step 3. Dividing the two, the common power cancels and only the exponential survives, so the ratio that controls the transition is essentially a pure exponential in $\sqrt{V_0/E_R}$ .

\frac{U}{J} \approx \sqrt{2}\,\pi\,k\,a_s\, e^{+2\sqrt{V_0/E_R}}

Why: this is the punchline — a linear change in $\sqrt{V_0/E_R}$ multiplies $U/J$ by a fixed factor, so sweeping the intensity moves the system across orders of magnitude in $U/J$ and cleanly through the critical point. *Note: these deep-lattice forms — $J\sim (V_0/E_R)^{3/4}e^{-2\sqrt{V_0/E_R}}$ and $U\sim (V_0/E_R)^{3/4}$ — slightly overestimate $J$ ; the exact lowest-band value from the Mathieu band structure is a little smaller, lowering the transition depth marginally.*

Go deeper · Derivation C · 1 minThe superfluid ground state and its interference peaks

Derivation CThe superfluid ground state and its interference peaks

The goal is to explain why the weakly interacting side of the transition is a phase-coherent state that prints sharp Bragg peaks in time-of-flight, and why those peaks vanish in the Mott insulator. For $J\gg U$ the atoms ignore their repulsion and condense into the single lowest Bloch state — uniform phase, zero quasimomentum. We write that state, take its coherent-state form, feed it through the momentum-distribution formula, and watch the lattice sum sharpen into a delta-comb.

Step 1. Minimizing the hopping term $-J\sum \hat a_i^{\dagger}\hat a_j$ favours the uniform, zero-quasimomentum single-particle orbital, and all $N$ bosons pile into it, giving a state that is an equal superposition over sites raised to the $N$ -th power.

|\Psi_{\mathrm{SF}}\rangle = \frac{1}{\sqrt{N!}}\Big(\frac{1}{\sqrt{M}}\sum_{i=1}^{M}\hat a_i^{\dagger}\Big)^{N}|0\rangle \;\xrightarrow{\ \text{coherent}\ }\; \prod_i e^{-|\alpha|^2/2}\,e^{\alpha\, \hat a_i^{\dagger}}|0\rangle

Why: for large $N$ and $M$ the fixed-number state is well approximated by a product of on-site coherent states sharing one phase — the mathematical statement of macroscopic phase coherence.

Step 2. Each site is then a coherent state with the same amplitude and phase, $\langle \hat a_i\rangle=\alpha=\sqrt{n}\,e^{i\phi}$ , so the one-body coherence stays finite at arbitrarily large separation — off-diagonal long-range order.

\langle \hat a_i^{\dagger}\hat a_j\rangle \to n \quad\text{as}\quad |\mathbf R_i-\mathbf R_j|\to\infty

Why: a common phase across the whole lattice is exactly what "superfluid" means here, and it is the ingredient the Mott insulator throws away.

Step 3. Put this coherence into the structure factor. Because every $\langle \hat a_i^{\dagger}\hat a_j\rangle$ contributes with a definite phase, the double lattice sum factorizes into the squared modulus of a single sum, which is a comb of sharp peaks at the reciprocal-lattice vectors $\mathbf G$ .

n(\mathbf k) = n\,|\tilde w(\mathbf k)|^2 \sum_{\mathbf G}\delta(\mathbf k-\mathbf G)

Why: constructive interference at $\mathbf k=\mathbf G$ is the matter-wave analogue of a diffraction grating; the peak width narrows as one over the system size, so a large coherent cloud gives needle-sharp peaks.

Step 4. In the opposite limit the Mott state has $\langle \hat a_i^{\dagger}\hat a_j\rangle = n\,\delta_{ij}$ , killing every cross term, so the sum collapses to a constant and the momentum distribution is just the smooth envelope $|\tilde w(\mathbf k)|^2$ — the peaks are gone.

n(\mathbf k) = n\,|\tilde w(\mathbf k)|^2 \quad(\text{Mott insulator})

Why: no inter-site phase relation means no interference, so the reversible appearance and disappearance of the Bragg peaks is a direct, model-independent readout of the transition.

Limiting cases

The Bose-Hubbard ground state at fixed density is controlled entirely by $U/J$ and by whether the filling is an integer:

Case	Result	Meaning
Deep superfluid, $U/J\to 0$	$\vert\Psi\rangle \propto \big(\sum_i \hat a_i^{\dagger}\big)^{N}\vert 0\rangle$	kinetic energy dominates; atoms coherently delocalize; large $\Delta N$ , sharp global phase, gapless and compressible
Deep Mott, $U/J\to\infty$ (integer $n$ )	$\vert\Psi\rangle = \prod_i \dfrac{(\hat a_i^{\dagger})^{n}}{\sqrt{n!}}\vert 0\rangle$	interaction dominates; a product of Fock states, $\Delta N=0$ , gap $\approx U$ , incompressible ( $\kappa=0$ )
Critical point (mean field)	$(U/J)_c = z\big(2n+1+2\sqrt{n(n+1)}\big)$	$z=2d$ is the coordination number; for $n=1,\,d=3$ this is $\approx 35$ , which quantum Monte Carlo lowers to $\approx 29.3$
Non-integer filling	superfluid for any $J\gt 0$	no Fock state fits; the extra atoms are always mobile, so the system stays compressible and coherent — Mott lobes exist only at integer $n$

Common trapThe loss of interference peaks is just heating or decoherence, not a real quantum phase transition

It is a genuine $T=0$ quantum phase transition, driven by $U/J$ , and the distinction is experimentally decisive rather than semantic. Heating would degrade coherence irreversibly, but the superfluid interference peaks come back sharply when the lattice is lowered again — the process is reversible, which no incoherent heating can be. Two further signatures nail it down: an excitation gap $\approx U$ opens in the Mott phase (seen by lattice-modulation or tilt spectroscopy), and the compressibility drops to zero (flat integer- $n$ plateaus versus $\mu$ , imaged directly as concentric Mott shells in a quantum-gas microscope). The transition is driven by quantum, not thermal, fluctuations — it survives at temperatures far below $U/k_B$ and disappears only when thermal energy smears the gap.

Worked examples

1 · Where does the transition sit for real rubidium? (confidence builder)

Take $^{87}$ Rb ( $a_s = 100.4\,a_0 = 5.31$ nm) in a simple cubic optical lattice made from $\lambda = 852$ nm light. Estimate $U/J$ at a depth $V_0 = 13\,E_R$ and decide which phase the unit-filled gas is in.

Step 1. Fix the energy scale. With $k=2\pi/\lambda = 7.37\times10^{6}\ \mathrm{m^{-1}}$ and $m = 1.44\times10^{-25}$ kg, the recoil energy is $E_R=\hbar^2k^2/2m = 2.10\times10^{-30}$ J, or $E_R/h \approx 3.2$ kHz. Every energy below is quoted as a multiple of $E_R$ .

Step 2. Hopping from the deep-lattice formula at $V_0/E_R = 13$ : with $2\sqrt{13}=7.21$ and $(13)^{3/4}=6.85$ , $J/E_R = (4/\sqrt\pi)(6.85)\,e^{-7.21} = 2.257\times6.85\times7.38\times10^{-4} = \mathbf{0.0114}$ , i.e. $J/h\approx 36$ Hz.

Step 3. Interaction from its formula: $k a_s = (7.37\times10^{6})(5.31\times10^{-9}) = 0.0392$ , so $U/E_R = \sqrt{8/\pi}\,(0.0392)(6.85) = 1.596\times0.0392\times6.85 = \mathbf{0.428}$ , i.e. $U/h\approx 1.35$ kHz.

Step 4. The ratio: $U/J = 0.428/0.0114 = \mathbf{37.5}$ . The mean-field critical value at unit filling is $(U/J)_c = 6\times5.83\approx 35$ , and the sharper quantum-Monte-Carlo value is $\approx 29.3$ . Since $37.5$ exceeds both, the gas at $V_0 = 13\,E_R$ has already crossed into the Mott insulator — the transition depth is a shade below $13\,E_R$ , exactly where Greiner and co-workers observed the peaks collapse in 2002.

2 · The Mott gap, its temperature, and the revival clock

For the same lattice ( $U/h\approx 1.35$ kHz), find the excitation gap as a temperature and the period of the collapse-and-revival of the matter-wave field.

Step 1. Deep in the Mott phase the gap to a particle-hole excitation is $E_{\mathrm{gap}}\approx U$ , so $E_{\mathrm{gap}}/h \approx 1.35$ kHz.

Step 2. As a temperature, $T_{\mathrm{gap}} = U/k_B = h(1.35\times10^{3})/k_B = (6.63\times10^{-34})(1.35\times10^{3})/(1.38\times10^{-23}) = \mathbf{65\ nK}$ . The sample must be held well below this for a clean Mott insulator — routine, since lattice gases reach a few nK.

Step 3. After a sudden quench to a deep lattice, each site's coherent state dephases and re-phases because its energy levels are spaced by $U$ ; the field fully revives after one period $T_{\mathrm{rev}} = h/U = 1/(1.35\times10^{3}\ \mathrm{Hz}) = \mathbf{0.74\ ms}$ . Measuring this period is a direct, calibration-free reading of the on-site interaction $U$ — the basis of the collapse-and-revival experiment.

3 · Why the second Mott shell is harder to reach

Compare the critical $U/J$ for the $n=1$ and $n=2$ Mott lobes on the 3D cubic lattice, and say what it means for the shell structure of a trapped cloud.

Step 1. The lobe-tip formula gives $(U/zJ)_c = 2n+1+2\sqrt{n(n+1)}$ . For $n=1$ : $3+2\sqrt2 = \mathbf{5.83}$ . For $n=2$ : $5+2\sqrt6 = \mathbf{9.90}$ .

Step 2. With $z=6$ , that is $(U/J)_c \approx \mathbf{35}$ for the first lobe and $\approx \mathbf{59}$ for the second — the $n=2$ Mott insulator needs a substantially deeper lattice (larger $U/J$ ) than the $n=1$ one.

Step 3. In a harmonic trap the local chemical potential is highest at the centre, so a real cloud is a stack of shells: an $n=2$ Mott core, ringed by a superfluid, ringed by an $n=1$ Mott shell, and so on outward. Because higher lobes require larger $U/J$ , deepening the lattice grows the shells from the outside in — the $n=1$ ring appears first, the $n=2$ core last. This concentric "wedding-cake" structure is exactly what quantum-gas microscopes image.

Physics of the portrait

The superfluid-to-Mott transition is where three strands of physics braid together: a lattice model born in the theory of dirty bosons, a proposal to build it from lasers and cold atoms, and an experiment that turned an abstract phase diagram into a photograph.

Origins · 1 minImmanuel Bloch and how this was discovered

Immanuel Bloch

born 1972 · optical-lattice quantum simulation and the observation of the superfluid–Mott transition

Immanuel Bloch is the experimentalist most associated with turning the Bose-Hubbard model from a theorist's toy into a laboratory instrument. Trained under Theodor Hänsch in Munich, he was a leading member of the team, with Markus Greiner, Olaf Mandel, Tilman Esslinger and Hänsch, behind the 2002 Nature paper that demonstrated the reversible superfluid-to-Mott transition in a $^{87}$ Rb condensate by ramping the depth of a three-dimensional optical lattice — watching the interference peaks collapse and, on lowering the lattice again, spring back. That reversibility provided strong evidence that the observed change was a quantum phase transition rather than an irreversible heating effect. In the same year his group saw the collapse and revival of the matter-wave field, reading the on-site interaction $U$ straight off the revival clock. The proposal his experiment realized had come from Dieter Jaksch, Christoph Bruder, Ignacio Cirac, Crispin Gardiner and Peter Zoller in 1998, who mapped cold lattice bosons onto the Bose-Hubbard model of Matthew Fisher and co-workers (1989). Bloch went on to pioneer quantum-gas microscopy with single-atom, single-site resolution, enabling direct imaging of individual atoms in optical lattices and detailed measurements of Mott-insulating states, and to pioneer optical lattices as programmable quantum simulators of Hubbard physics, artificial gauge fields and topological bands. As a director at the Max Planck Institute of Quantum Optics, he made "quantum simulation with light" a defining tool of modern AMO physics.

History

1989

— Fisher, Weichman, Grinstein and Fisher introduce the Bose-Hubbard model and predict the superfluid-to-Mott transition

1998

— Jaksch, Bruder, Cirac, Gardiner and Zoller propose realizing it with a Bose-Einstein condensate in an optical lattice

2002

— Greiner, Mandel, Esslinger, Hänsch and Bloch observe the reversible transition in rubidium-87, and the collapse and revival of the matter-wave field

2007

— Capogrosso-Sansone and co-workers fix the 3D unit-filling critical point at U/J ≈ 29.3 by quantum Monte Carlo

2010

— quantum-gas microscopes resolve single atoms on single sites, imaging the concentric Mott shells and their number statistics

Going further

Texts. Foot, Atomic Physics, ch. 10 (Bose-Einstein condensation and optical lattices) · Bloch, Dalibard and Zwerger, Rev. Mod. Phys. 80, 885 (2008), "Many-body physics with ultracold gases" · Pethick and Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed., ch. 14 · Sachdev, Quantum Phase Transitions, 2nd ed. (the Bose-Hubbard universality class).

Papers. Jaksch, Bruder, Cirac, Gardiner and Zoller, Phys. Rev. Lett. 81, 3108 (1998) (the proposal) · Greiner, Mandel, Esslinger, Hänsch and Bloch, Nature 415, 39 (2002) (the observation) · Fisher, Weichman, Grinstein and Fisher, Phys. Rev. B 40, 546 (1989) (the model and mean-field lobes) · Capogrosso-Sansone, Prokof'ev and Svistunov (2007) (the 3D quantum-Monte-Carlo critical point). Key numerical benchmarks: the mean-field lobe-tip is $(zJ/U)_c = 1/[2n+1+2\sqrt{n(n+1)}]$ , giving $0.172$ for $n=1$ and hence $(U/J)_c\approx 35$ at $z=6$ ; the quantum-Monte-Carlo value is $(U/J)_c\approx 29.3$ for the 3D cubic lattice at unit filling; the deep-lattice scalings are $J\sim (V_0/E_R)^{3/4}e^{-2\sqrt{V_0/E_R}}$ and $U\sim (V_0/E_R)^{3/4}$ ; and for $^{87}$ Rb at $\lambda=852$ nm, $V_0=13\,E_R$ : $E_R/h\approx 3.2$ kHz, $J/E_R\approx 0.011$ , $U/E_R\approx 0.43$ , $U/J\approx 37$ . The recoil energy is $E_R=\hbar^2k^2/2m$ and the lattice spacing is $d=\lambda/2$ .

Builds on: Bose-Einstein Condensation in Dilute Gases · Dipole Force and the Optical Dipole Potential · Evaporative Cooling. Leads to: Degenerate Fermi Gases and the BEC-BCS Crossover · Quantum Simulation with Optical Lattices.

Ask the tutor: Want to walk through how the exponential separation of $J$ and $U$ turns a modest change in laser power into a full quantum phase transition?

Is the Mott insulator here the same thing as a Mott insulator in a real solid?

It is the same underlying idea — interactions localizing particles that would otherwise move freely — stripped to its cleanest form. In a real solid a Mott insulator arises when strong Coulomb repulsion between electrons on the same orbital freezes them despite a partially filled band, tangled up with disorder, phonons, long-range forces and spin. The optical-lattice version keeps only the essential competition, $U$ versus $J$ , on a defect-free lattice with tunable short-range interactions and no phonons. That is exactly why it is valuable: it isolates interaction-driven localization so the phase diagram can be measured against exact theory, which no real crystal permits.

Why is the transition called "quantum" if it is driven by turning up a laser?

Because the control parameter is a coupling in the Hamiltonian, not the temperature. A quantum phase transition happens at $T=0$ and is driven by quantum fluctuations — here, the zero-point delocalization that tunnelling provides. Raising the laser intensity changes $U/J$ , a parameter in the ground-state energy, and the ground state itself changes character at the critical value even in the limit of zero temperature. Thermal transitions, by contrast, are driven by entropy and disappear at $T=0$ . In practice experiments run at a small but finite temperature, which broadens the sharp critical point into a crossover, but the transition is quantum in origin and survives as long as $k_BT$ stays well below the gap $U$ .

How is U/J actually measured, rather than just calculated?

Several independent ways, which is what makes the platform trustworthy. The depth $V_0/E_R$ is calibrated by pulsing the lattice and reading the Kapitza-Dirac or Bragg diffraction of the atoms, fixing $V_0$ absolutely; band-structure theory then converts $V_0/E_R$ into $J$ and $U$ with no free parameters. Independently, $U$ is read straight off the collapse-and-revival period $T_{\mathrm{rev}}=h/U$ after a quench, and the gap $\approx U$ from lattice-modulation spectroscopy, which resonantly creates particle-hole excitations when the drive matches $U$ . When these routes agree, the mapping onto the Bose-Hubbard model is confirmed.

Can you have a Mott insulator at non-integer filling, say 1.5 atoms per site on average?

No — that is the point of the lobe structure. A Mott insulator is a Fock state with a definite integer occupancy on every site, so it can only form when the average filling is itself an integer. At 1.5 atoms per site the extra half-atom's worth of population has nowhere to localize: there are always mobile atoms hopping on top of a filled background, and any non-zero tunnelling makes them condense into a superfluid. In the phase diagram this shows up as the superfluid filling the whole region between the Mott lobes; the lobes exist only along the integer-filling lines. In a harmonic trap, where the local filling varies smoothly with radius, this is why superfluid rings always separate the integer Mott shells.

What sets how deep the lattice must be — is it always around 13 recoils?

No, it depends on the atom and the laser wavelength through the combination $k a_s$ , which controls how large $U$ is relative to $E_R$ . The critical value of the dimensionless ratio $(U/J)_c\approx 29$ – $35$ (at unit filling, 3D) is universal, but the depth $V_0/E_R$ needed to reach it is not: a larger scattering length or a longer lattice wavelength raises $U$ and lets the transition happen in a shallower lattice, while a smaller $k a_s$ pushes it deeper. For $^{87}$ Rb at $852$ nm it lands near $13\,E_R$ ; change the species, the wavelength, or use a Feshbach resonance to tune $a_s$ , and the critical depth moves. What is fixed is $(U/J)_c$ , not $V_0$ .

Why does time-of-flight imaging show momentum, and what exactly are the sharp peaks?

When trap and lattice switch off, the atoms fly ballistically and, after a long enough expansion, each lands at a position proportional to the momentum it had at release — so the camera image is a picture of the in-trap momentum distribution. For a phase-coherent superfluid the localized wavepackets interfere like light through a diffraction grating, and the constructive directions are the reciprocal-lattice vectors: a sharp grid of Bragg peaks under a smooth single-site envelope. In the Mott insulator the sites have random relative phases, the interference washes out, and only the envelope remains. The peaks are matter-wave diffraction orders, and their sharpness measures the range of phase coherence.