Controlling coherence via tuning of the population imbalance in a bipartite optical lattice

Abstract

The control of transport properties is a key tool at the basis of many technologically relevant effects in condensed matter. The clean and precisely controlled environment of ultracold atoms in optical lattices allows one to prepare simplified but instructive models, which can help to better understand the underlying physical mechanisms. Here we show that by tuning a structural deformation of the unit cell in a bipartite optical lattice, one can induce a phase transition from a superfluid into various Mott insulating phases forming a shell structure in the superimposed harmonic trap. The Mott shells are identified via characteristic features in the visibility of Bragg maxima in momentum spectra. The experimental findings are explained by Gutzwiller mean-field and quantum Monte Carlo calculations. Our system bears similarities with the loss of coherence in cuprate superconductors, known to be associated with the doping-induced buckling of the oxygen octahedra surrounding the copper sites.

ARTICLE

Received 8 Apr 2014 | Accepted 3 Nov 2014 | Published 11 Dec 2014

The control of transport properties is a key tool at the basis of many technologically relevant effects in condensed matter. The clean and precisely controlled environment of ultracold atoms in optical lattices allows one to prepare simplied but instructive models, which can help to better understand the underlying physical mechanisms. Here we show that by tuning a structural deformation of the unit cell in a bipartite optical lattice, one can induce a phase transition from a superuid into various Mott insulating phases forming a shell structure in the superimposed harmonic trap. The Mott shells are identied via characteristic features in the visibility of Bragg maxima in momentum spectra. The experimental ndings are explained by Gutzwiller mean-eld and quantum Monte Carlo calculations. Our system bears similarities with the loss of coherence in cuprate superconductors, known to be associated with the doping-induced buckling of the oxygen octahedra surrounding the copper sites.

DOI: 10.1038/ncomms6735 OPEN

Controlling coherence via tuning of the population imbalance in a bipartite optical lattice

M. Di Liberto1, T. Comparin1,2, T. Kock3, M.lschlager3, A. Hemmerich3 & C. Morais Smith1

1 Institute for Theoretical Physics, Centre for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584CE Utrecht, The Netherlands.

2 Laboratoire de Physique Statistique,cole Normale Suprieure, UPMC, Universit Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France. 3 Institut fr Laser-Physik, Fachbereich Physik, Universitat Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. Correspondence and requests for materials should be addressed to A.H. (email: mailto:[email protected]

Web End [email protected] ).

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Rapid and precise control of transport properties are at the heart of many intriguing and technologically relevant effects in condensed matter. Small changes in some

external parameters, for example, an electric or a magnetic eld, may be used to signicantly alter the mobility of electrons. Prominent examples are eld effect transistors1 and systems showing colossal magneto-resistance2. Often, the control is achieved via structural changes of the unit cell, leading to an opening of a band gap. In iron-based superconductors, the variation of pressure is a well-known technique to control their transport properties3. In certain high-Tc superconductors, pulses of infrared radiation, which excite a mechanical vibration of the unit cell, can for short periods of time switch these systems into the superconducting state at temperatures at which they are actually insulators4. In La-based high-Tc cuprates, the drastic reduction of Tc at the doping value of x 1/8, known as the 1/8

mystery, is connected to a structural transition that changes the lattice unit cell5.

Ultracold atoms in optical lattices provide a particularly clean and well-controlled experimental platform for exploring many-body lattice physics6. Schemes for efcient manipulation of transport properties can be readily implemented and studied with great precision. In conventional optical lattices, tuning between a superuid and a Mott insulating phase has been achieved by varying the overall lattice depth V0, with the consequence of changing the height of the tunnelling barriers and the on-site contact interaction energy7. The equivalent is not easily possible in condensed-matter systems, since the lattice depth is practically xed.

In this work, we present an ultracold atom paradigm, where tuning the system between a superuid and a Mott insulator becomes possible via controlled distortion of the unit cell. This distortion acts to adjust the relative depth DV between two classes of sites (denoted by A and B) forming the unit cell and allows us to drive a superuid-to-Mott insulator transition

without altering the average lattice depth. We can access a rich variety of Mott insulating states with different integer populations of the A and B sites, which give rise to a shell structure in the nite harmonic trap potential, leading to characteristic features in the visibility of Bragg maxima in momentum spectra. We compare our observations with quantum Monte Carlo (QMC) and Gutzwiller mean-eld calculations, thus obtaining a detailed quantitative understanding of the system. In the following, we rst describe our experimental set-up; then, we theoretically investigate the behaviour of the visibility for two different cases: rst, for xed barrier height V0, by varying DV (bipartite lattice), and second, for DV 0 (monopartite lattice),

by tuning the lattice depth V0. Although monopartite lattices have been previously studied in great detail, and QMC calculations have provided a good tting of the visibility curve measured experimentally8, here we show more accurate data and argue that the main features of the curve can be understood in terms of a precise determination of the onset of new Mott lobes in the phase diagram.

ResultsDescription of the experimental set-up. We prepare an optical lattice of 87Rb atoms using an interferometric lattice set-up912. A two-dimensional (2D) optical potential is produced, comprising deep and shallow wells (A and B in Fig. 1a) arranged as the black and white elds of a chequerboard. In the xy-plane, the optical potential is given by V (x, y) V0

[cos2(kx) cos2(ky) 2cos(y) cos(kx) cos(ky)], with the tunable

well depth parameter V0 and the lattice distortion angle y. An additional lattice potential Vz (z) Vz,0 cos2(kz) is applied along

the z-direction. To study an effectively 2D scenario, Vz,0 is

adjusted to 29Erec, such that the motion in the z-direction is frozen out. Here, k 2p/l, Erec :2k2/2m, m denotes the

atomic mass and l 1,064 nm is the wavelength of the lattice

Phase angle ([afii9843])

/2

0.5 0.54 0.58

y

Energy ( rec)

6

8

10

12

Phase angle ([afii9843])

B

A

Population (a.u)

V V0 V+

A

B

x

0 1 Length ( /2)

0

0 0.5 1[afii9797]V/V0

2nd band

Energy ( rec)

7.2

7.6

8.0

0.48 0.49 0.5 0.51 0.52

Momentum (hk)

2

0

2

Population (a.u.)

1st band

2 0 2 2 0 2

max.

0

Momentum (hk)

Momentum (hk)

Figure 1 | Lattice potential. (a) Sketch of the lattice geometry within the xy-plane. l 1,064 nm denotes the wavelength of the laser light. (b) The potential

along the dashed trajectory in a is plotted for y 0.51p and V0 6Erec (thick grey line) with the rst and second bands represented, respectively, by the red

and blue horizontal bars. (c) The rst two bands are plotted versus y for V0 6Erec. (d) The red and blue squares show the relative number of atoms

(normalized to the total particle number and plotted versus DV/V0) associated with the Bragg peaks enclosed by red and blue circles in e, respectively. The lled (open) squares are recorded for Vz,0 0 (Vz,0 22Erec). The error bars indicate the statistical errors for ve measurements. The solid lines are

determined by a full-band calculation (neglecting interaction) with no adjustable parameters. (e) Momentum spectra (V0 6Erec, Vz,0 0) are shown with DV 0 (left) and DV/V0 0.5 (right) with the respective rst Brillouin zones imprinted as dashed rectangles.

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Phase angle ( )

beams. Apart from the lattice, the atoms experience a nearly isotropic harmonic trap potential. Adjustment of y permits controlled tuning of the effective well depths of the deep and shallow wells V V0(1cos(y))2 and their difference

DV V V 4V0cos(y) (see Fig. 1b). The effective mean

well depth

V0 V V =2 V01 cos2y is only weakly

dependent on y. For example, within the interval 0.46oy/ po0.54 one has cos2(y)o0.015 and hence

V0 V0. Tuning of y

signicantly affects the effective bandwidth, as shown in Fig. 1c. At y p/2, the A- and B wells become equal, which

facilitates tunnelling as compared with values yap/2, where the broad lowest band of the y p/2-lattice splits into two

more narrow bands.

We record momentum spectra, which comprise pronounced Bragg maxima with a visibility V (specied in the Methods

section) depending on the parameters V0 and DV. The distribution of Bragg peaks reects the shape of the underlying rst Brillouin zone, which changes size and orientation as DV is detuned from zero. This is illustrated in Fig. 1d,e. In Fig. 1e two spectra recorded for DV 0 (left) and DV/V0 0.5 (right)

are shown. For DV 0 (the special case of a monopartite square

lattice), the increased size of the rst Brillouin zone gives rise to destructive interference, such that the (1, 1):k-Bragg peaks indicated by the red circle vanish. As DV is detuned from zero, a corresponding imbalance of the A- and B populations yields a retrieval of the (1, 1):k-Bragg peaks. This is shown in Fig. 1d for the case of approximately vanishing interaction energy per particle UE0 (Vz,0 0) by the lled red squares

and for UE0.3Erec (Vz,0 22Erec) by the open red squares.

It is seen that the interaction energy signicantly suppresses the formation of a population imbalance and corresponding (1, 1):k-Bragg peaks.

Model. For low temperatures and for large lattice depths V0, the

system is described by the inhomogeneous BoseHubbard model13,14

H J X

hi;ji

ay

1.0

14

0.49 0.5 0.51 0.52 0.53 0.54

V/V0

0.8

12

0.6

10

V 0 (E rec)

Visibility

0.4

8

6

0.2

4

0

0.0 0.25 0.5

Figure 2 | Visibility measurements in the bipartite lattice. The visibility (parametrized by the colour code shown on the right edge) is plotted as a function of the well depth parameter V0 (measured in units of the recoil energy Erec) and the potential energy offset difference DV between shallow and deep wells in the bipartite lattice. The dashed line corresponds to the theoretical calculation of the points where the fraction of particlesnB P

iABni/N of the B sublattice vanishes (nBo5.5 10 3). The grid with

the experimental points is shown in Supplementary Fig. 9.

i aj h:c:

X

i

~mini

U

2

Xini ni 1 ;

1

where J is the coefcient describing hopping between nearest-neighbour sites, U accounts for the on-site repulsion and ~mi is a local chemical potential, which depends on the frequency o of the trap and on the sublattice: ~mi mA;B mo2r2i=2. The ratio U/J is

a monotonously increasing function of V0/Erec.

Bipartite lattice DVa0. The visibility measured for xed V0 as a function of DV (see Fig. 2) exhibits a region of rapid decrease. When the lattice barrier is large, for example, V0 12Erec, a

modest detuning DVB0.25V0 is able to completely destroy phase coherence with the consequence of a vanishing visibility. At smaller barrier heights, for example, V0 6Erec, superuidity

remains robust up to signicantly larger values of DV. To explain this behaviour, we performed a mean-eld calculation using the Gutzwiller technique15 for the BoseHubbard model given by equation (1). The values of J and Dm mA mB have been estimated from the exact band

structure and U has been calculated within the harmonic approximation. The total number of particles has been xed to N 2 103 and the trap frequency takes into account

the waist of the laser beam (see Methods and Supplementary Note 1). We performed large-scale Gutzwiller calculations in presence of a trap, thus going beyond local density approximation16,17 (see Methods).

In Fig. 3a, we show the evolution of the fraction of particles in the B sites (which we assumed to be the shallow wells). As DV increases, the number of bosons in the B sites decreases because of the excess potential energy required for their population. Within the tight-binding description, this is captured by the increased chemical potential difference between A- and B sites as DV grows. Our calculations predict a critical value DVc for which the population of the B sublattice vanishes. As shown in Fig. 3a, DVc becomes smaller as V0 increases. This corresponds to the observation in the phase diagram shown in Supplementary Fig. 5 and discussed in Supplementary Notes 2 and 3 that the area covered by the Mott insulating regions with vanishing B populations (lling gB 0) increases as the hopping amplitude is reduced.

The critical values DVc for different values of V0 are also shown in Fig. 2 as a dashed white line on top of the experimental data for the visibility. This line consistently lies on experimental points corresponding to constant visibility (V 0:5),

where phase coherence is rapidly lost, and suggests the onset of a new regime.

In Fig. 3b it is shown that, in addition to the population of the B sites, also the condensate fraction at the A sites approaches zero beyond the critical value DVc (see the inset in Fig. 3a for the total condensed fraction); in this regime, the density prole displays only sharp concentric Mott shells of the form (gA, gB) (g, 0)

where the integer lling g of the Mott regions can reach g 4 (see

Supplementary Fig. 6). This can be understood by considering that in the new regime where B sites are empty, the particles populating A sites can only delocalize (and thus establish phase coherence) by hopping through the intermediate B sites. Since these are second-order processes, they are highly suppressed when Dm is large enough, and the system has to become an imbalanced Mott insulator.

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[afii9835]/[afii9843]

0.50 0.51 0.52 0.53

0.4

0.5

0.3

0.2

0.1

0.0

0.5

0.4

0.3

0.2

0.1

0.00.0 0.1 0.2 0.3 0.4 V/V0

0.8

12 11 10

98760.5 0.51 0.52 0.53

0.8

Total condensate fraction [afii9845]

0.6

Condensate fraction [afii9845] AParticle fraction n B

V 0(E rec)

Visibility

0.6

0.4

0.4

0.2

0.2

0

Phase angle [afii9835] ([afii9843])

0. 0.49 0.5 0.51 0.52 0.53

[afii9835]/[afii9843]

V0 =8 Erec

V0 =9 Erec

V0 =10 Erec

V0 =11 Erec

V0 =12 Erec

Figure 4 | Comparison of the measured visibility with the theory at large imbalance. The data shown are for V0 10.8 Erec (squares) and

V0 11.44 Erec (circles). The red dashed (dash-dotted) line is obtained by

tting the last four data points with equation (3) using the average lling g as a tting parameter. We obtain respectively g 2:75 0:23 and

g 3:77 0:31. The data for V0 10.8 Erec are shifted along the vertical

axis by 0.1. The error bars represent the statistical variance of typically 45 independent measurements.

Figure 3 | Gutzwiller results in the trap. (a) Particle number fraction on the B sites (nB). The inset shows the total condensed fraction r P

iri/N.

(b) Condensate fraction on the A sites (rA P

iAAri/N, where ri jci j 2,

with ci the mean-eld order parameter) as a function of DV for increasing values of V0 and xed total number of particles N 2 103, calculated with

the Gutzwiller ansatz. The key shows the colour code for both, the curves in a and b.

In the new Mott insulating regime, particlehole pairs are responsible for a non-vanishing visibility, as in the conventional case in absence of imbalance18. By performing the perturbation theory on top of the ideal Mott insulating state |MIi PiAA|giiPjAB|0ij, the ground state can therefore be

written as (see Supplementary Note 4).

j cGi 1

J2 2D2

jMIi

J D

X

hi;ji

ay

i aj jMIi

description of the visibility data for large y by means of equation (3) is only possible in a window V0E111 Erec, where sufcient data points are available in the low-visibility tail with values of the visibility large enough to be measured with sufcient precision to allow tting.

Monopartite lattice DV 0. Adjustment of DV 0 produces the

special case of a conventional monopartite square lattice, extensively studied in the literature during the past decade7,1820. Experiments in three-dimensional cubic lattices have suggested that the formation of Mott shells within the external trap could be associated with the appearance of kinks in the visibility18,19, whereas experiments in 2D triangular lattices have rather detected an instantaneous decrease21. Arguable attempts were made to interprete small irregularities in the observed visibility in this respect. On the theoretical front, a QMC study of the one-dimensional trapped BoseHubbard model22 has shown the appearance of kinks in V as a function of U/J. Unfortunately, this

study, employing a trap curvature proportional to J rather than V0, appears to have limited relevance for experiments. More realistic QMC simulations of 2D and three-dimensional conned systems have been able to quantitatively describe the momentum distribution23 and the experimental visibility8,24, however, with no indications for distinct features associated with Mott shells. To clarify this long-standing discussion, we have recorded the visibility of Fig. 2 along the DV 0 trajectory versus V0 with

increased resolution in Fig. 5. Guided by an inhomogeneous mean-eld calculation indicating that the local lling g is lower than 4, we computed the critical J/U values for the tips of Mott lobes with g 1,2 and 3, making use of the worm algorithm as

implemented in the ALPS libraries2527. Superimposed upon the experimental data, we mark in Fig. 5 with (blue) dashed lines the values of V0/Erec corresponding to the values of J/U at the tip of the Mott lobes obtained by QMC. As V0 is increased in Fig. 5, four different regimes are crossed. For small values of V0 (regimeI), most of the system is in a superuid phase. Increasing V0 yields only little loss of coherence due to increasing depletion, and hence the visibility remains nearly constant. When the rst Mott ring with g 1 particle per site is formed, the system enters

regime II, where the visibility decreases slowly but notably as the g 1-Mott shell grows. When the second Mott insulating ring

with g 2 arises (regime III), a sharp drop of the visibility occurs

2J2 UD

X

hi;jiA

ay

i aj jMIi

J2 UD

X

hhi;jiiA

i aj jMIi;

where D U(g 1) Dm. The rst term is simply the unperturbed

term with a wavefunction renormalization, whereas the linear term in J describes particlehole pairs with the particle sitting on the A site and the hole in the neighbour B site, or vice versa. The last two terms are second-order processes that involve intermediate B sites and describe particlehole pairs within the A sublattice only. This ground state leads to the visibility

V c1J=D c2J2=UD c3J2=D2; 3 where c1 2 g 1

1 r1

, c2 4 g 1

2r1 r2 3

,

c3 4 g 1

2 r1 3

1 r1

, with r1 cos 2

p p

ay

0:266 and r2 cos 8

p p 0:858. By using the average

lling g in the trap as a tting parameter, we found that the theoretical visibility curve compares reasonably well with the experimental data both in magnitude and scaling behaviour, with an average lling of the order g 3 (see Fig. 4). A perturbative

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a

1

I II III IV

g=1 g=2 g=3

0.8

0.6

Visibility

0.4

0.2

0

b

0.2

Derivative

0

0.2

0.4

6 7 8 9 10 11 12 13 14 15 16 17 18 19

V0 (Erec)

Figure 5 | Visibility measurement in the monopartite lattice. (a) Visibility of 87Rb, plotted as a function of the well depth V0, for DV 0 and

Vz,0 29 Erec. Vertical dashed lines: values of V0/Erec corresponding to the

tips of the Mott lobes with different lling g, as computed through QMC (see Supplementary Note 5). Grey solid lines in regions IIII are a guide to the eyes, whereas the red line in region IV displays a t to the function A(U/zJ)a with A 4.00.7 and a 1.000.06, showing good

agreement with the theoretical prediction18. The error bars representthe statistical variance of typically 45 independent measurements.(b) Numerical derivative of the visibility data; vertical lines as in a. The error bars are derived from the ones in a.

indicating a signicantly increased growth of the Mott insulating part of the system with V0. Finally, when the third Mott ring with g 3 forms or closes in the centre of the trap, only a small

superuid fraction remains in the system, such that the visibility cannot further rapidly decrease with V0 (regime IV), that is, a quasi-plateau arises in Fig. 5. The red solid line shows that for large V0 the visibility acquires a (U/J) 1 dependence, in agreement with a result obtained by the rst-order perturbation theory in J/U (ref. 18).

DiscussionSeveral conclusions can be drawn from our experimental and theoretical investigations: for monopartite lattices, the visibility comprises characteristic signatures, which can be connected to the position of the tips of the Mott insulator lobes in a m/U versus

J/U phase diagram calculated by QMC. Mean-eld calculations are insufcient, even when the inhomogeneity due to the trap is taken into account. Deforming the unit cell of a bipartite lattice is a means to efciently tune a transition from a superuid to a Mott insulating state. The visibility displays distinct regions with explicitly different slopes, as a function of the detuning between the A and B sublattices. A pronounced loss of coherence occurs at the critical value of the detuning DVc, at which the population of the shallow wells vanish. Our work may shed some light also on the behaviour of condensed-matter systems, where loss of phase coherence occurs due to a structural modication of the lattice. For example, in La2 xBaxCuO4 high-Tc cuprate, superconduc

tivity is weakened at the structural transition from a low-temperature orthorhombic into a low-temperature tetragonal phase28. The same occurs for La2 x yNdySrxCuO4 (ref. 5). This

structural transition corresponds to a buckling of the oxygen octahedra surrounding the copper sites, which changes the nature of the copperoxygen lattice unit cell28. The critical buckling angle yc 3.6 deg for the destruction of superconductivity29 bears

similarities with the critical deformation angle yc (or equivalently DVc) found here (see Supplementary Note 6 for a more detailed discussion). Most of the present theoretical studies of high-Tc superconductivity concentrate only on the copper lattice. We hope that our results will inspire further investigations of the specic role played by the oxygen lattice, and its importance in preserving A phase coherence

Methods

Experimental details. Our experimental procedure begins with the production of a nearly pure BoseEinstein condensate of typically 5 104 rubidium atoms (87Rb)

in the F 2, mF 2 state conned in a nearly isotropic magnetic trap with about

30 Hz trap frequency. The adjusted values of the lattice depth V0 are determined with a precision of about 2% by carefully measuring the resonance frequencies with respect to excitations into the third band along the x- and y-directions. The adjustment of y is achieved with a precision exceeding p/300 by an active stabilization with about 10 kHz bandwidth. In a typical experimental run, the lattice potentials V(x, y) and Vz(z) are increased to the desired values by an exponential ramp of 160 ms duration. After holding the atoms in the lattice for 20 ms, momentum spectra are obtained by rapidly (o1 ms) extinguishing the lattice and trap potentials, permitting a free expansion of the atomic sample during 30 ms, and subsequently recording an absorption image. The magnetic trap and the nite Gaussian prole of the lattice beams (beam radius 100 mm) give rise to a com

bined trap potential. For Vz,0 29Erec and V0 18Erec this yields trap frequencies

of 73 Hz in the xy-plane and 65 Hz along the z-direction. The observed momentum spectra comprise pronounced Bragg maxima with a visibility depending on the parameters V0 and DV. These spectra are analysed by counting the atoms (nd,0) in a disk with a 5-pixel radius around some higher-order Bragg peak and within a disk of the same radius, but rotated with respect to the origin by 45 (nd,45). The

visibility is obtained as V nd;0 nd;45=nd;0 nd;45 (ref. 18).

Gutzwiller scheme. The Gutzwiller ansatz approximation used in this work is an extension of the well-known procedure employed for the BoseHubbard model in conventional monopartite lattices16,17 that takes into account the different local energies for the sites of type A and B. The wavefunction is assumed to be a product of single-site wavefunctions |fi Pi|fii. On each site the ansatz reads

j fii X

1 f in jni: 4

We have included states up to n 7 and considered real Gutzwiller coefcients for

an extended 69 69 lattice, which is allowed because of the U(1) symmetry and the

fact that the ground state cannot have nodes, according to Feynmans no-node theorem.

As shown in Supplementary Note 2, the mean-eld Hamiltonian can be written as a sum of site-decoupled local Hamiltonians represented in the local Fock basis, HMF P

iHi. Each local Hamiltonian needs, as an input, the order parameters of the neighbour sites (cB for the local Hamiltonian on sites of type A and vice versa).

One can thus use the following iterative procedure to determine the ground state at a given value of J/U and ~mi=U: start with a random guess of the order parameters cA,B, diagonalize the local Hamiltonians Hi, take the eigenvectors of the lowest energy state (that is, the Gutzwiller coefcients f in), calculate the new order

parameters ci hayii P

n

p n 1f inf in1 and repeat the procedure until convergence. In this way, we have obtained Fig. 3 and Supplementary Fig. 6 for the density proles. By collecting the points where the fraction nB of particles on the B sites vanishes, as a function of DV/V0, for several values of V0, we nd the white line plotted in Fig. 2.

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Acknowledgements

This work was partially supported by the Netherlands Organization for Scientic Research (NWO), by the German Research Foundation DFG-(He2334/14-1, SFB 925) and the Hamburg centre of ultrafast imaging (CUI). A.H. and C.M.S. acknowledge support by NSF-PHYS-1066293 and the hospitality of the Aspen Center for Physics. We are grateful to Peter Barmettler, Matthias Troyer and Juan Carrasquilla for helpful discussions.

Author contributions

M.D.L. and T.C. carried out the calculations; T.K. and M.. carried out the measurements; A.H. and C.M.S. devised the experimental and theoretical parts of the research, respectively. C.M.S. and A.H. wrote the manuscript with input from all authors.

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How to cite this article: Di Liberto, M. et al. Controlling coherence via tuning of the population imbalance in a bipartite optical lattice. Nat. Commun. 5:5735 doi: 10.1038/ ncomms6735 (2014).

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