Probing Matter-Field and Atom-Number Correlations in Optical Lattices by Global Nondestructive Addressing
Wojciech Kozlowski, Santiago F. Caballero-Benitez, Igor B. Mekhov
PProbing matter-field and atom-number correlations in optical lattices byglobal nondestructive addressing
W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov
Department of Physics, Clarendon Laboratory, University of Oxford,Parks Road, Oxford OX1 3PU, United Kingdom
We show that light scattering from an ultracold gas reveals not only density correlations, butalso matter-field interference at its shortest possible distance in an optical lattice, which defineskey properties such as tunnelling and matter-field phase gradients. This signal can be enhancedby concentrating probe light between lattice sites rather than at density maxima. As address-ing between two single sites is challenging, we focus on global nondestructive scattering, allowingprobing order parameters, matter-field quadratures and their squeezing. The scattering angulardistribution displays peaks even if classical diffraction is forbidden and we derive generalized Braggconditions. Light scattering distinguishes all phases in the Mott insulator - superfluid - Bose glassphase transition.
I. INTRODUCTION
The modern field of ultracold gases is success-ful due to its interdisciplinarity [1, 2]. Originallycondensed matter effects are now mimicked in con-trolled atomic systems finding applications in areassuch as quantum information processing (QIP). Areally new challenge is to identify novel phenomenawhich were unreasonable to consider in condensedmatter, but will become feasible in new systems.One such direction is merging quantum optics andmany-body physics [3, 4]. The former describesdelicate effects such as quantum measurement andstate engineering, but for systems without strongmany-body correlations (e.g. atomic ensembles).In the latter, decoherence destroys these effectsin conventional condensed matter. Due to recentexperimental progress, e.g. Bose-Einstein conden-sates (BEC) in cavities [5–7], quantum optics ofquantum gases can close this gap.Here we develop a method to measure proper-ties of ultracold gases in optical lattices (OL) bylight scattering. Recent quantum non-demolition(QND) schemes [8–10] probe density fluctuations,thus inevitably destroy information about phase,i.e. the conjugate variable, and as a consequencedestroy matter-field coherence. In contrast, we fo-cus on probing the atom interference between lat-tice sites. Our scheme is nondestructive in con-trast to totally destructive methods such as time-of-flight. It enables in-situ probing of the matter-field coherence at its shortest possible distancein an OL, i.e. the lattice period, which defineskey processes such as tunnelling, currents, phasegradients, etc. This is achieved by concentrat-ing light between the sites. By contrast, stan-dard destructive time-of-flight measurements dealwith far-field interference and a relatively near-field one was used in Ref. [11]. Such a counter-intuitive configuration may affect works on quan-tum gases trapped in quantum potentials [3, 12–17] and quantum measurement-induced prepara-tion of many-body atomic states [18–21]. Withinthe mean-field (MF) treatment, this enables prob- ing the order parameter, matter-field quadraturesand squeezing. This can have an impact on atom-wave metrology and information processing in ar-eas where quantum optics already made progress,e.g., quantum imaging with pixellized sources ofnon-classical light [22, 23], as an OL is a natu-ral source of multimode nonclassical matter waves.The scattering angular distribution is nontrivial,even when classical diffraction is forbidden. We de-rive generalized Bragg conditions and give param-eters for the only two relevant experiments to date[11, 24]. The method works beyond MF, which wesupport by distinguishing all phases in the Mottinsulator (MI) - superfluid (SF) - Bose glass (BG)phase transition in a 1D disordered OL. We under-line that transitions in 1D are much more visiblewhen changing an atomic density rather than forfixed-density scattering. It was only recently thatan experiment distinguished a MI from a BG [25].
II. THE MODEL
The theory is based on the Bose-Hubbard (BH)model generalized for quantum light [26]. We con-sider off-resonant probe and detected light at an-gles θ , with N atoms at M sites, K of which areilluminated (Fig. 1). Light modes can be in freespace or cavities. The Hamiltonian isˆ H = ˆ H BH + (cid:88) l (cid:126) ω l a † l a l + (cid:126) (cid:88) l,m U lm a † l a m ˆ F lm , ˆ H BH = − J cl M (cid:88) (cid:104) i,j (cid:105) b † i b j + U M (cid:88) i ˆ n i (ˆ n i − − µ M (cid:88) i ˆ n i , (cid:104) i, j (cid:105) gives summation over nearest sites, a l ( l =0,1) are the annihilation operators of the lightmodes with frequency ω l , atom-light coupling con-stant g l ( U lm = g l g m / ∆ a ), light-atom detuning∆ a = ω − ω a . b i (ˆ n i ) is the atom annihilation(number) operator with hopping amplitude J cl ,interaction strength U , and chemical potential µ . a r X i v : . [ qu a n t - ph ] J un The atomic operator is ˆ F lm = ˆ D lm + ˆ B lm ,ˆ D lm = K (cid:88) i =1 J lmi,i ˆ n i , ˆ B lm = K (cid:88) (cid:104) i,j (cid:105) J lmi,j b † i b j , (1)comes from overlaps of light mode functions u l ( r )and density operator ˆ n ( r ) = ˆΨ † ( r ) ˆΨ( r ), afterthe matter-field operator is expressed via Wan-nier functions: ˆΨ( r ) = (cid:80) i b i w ( r − r i ). ˆ D lm sumsthe density contributions ˆ n i , while ˆ B lm sums thematter-field interference terms. J lmi,j are the con-volutions of Wannier and light mode functions andare given by J lmi,j = (cid:90) w ( r − r i ) u ∗ l ( r ) u m ( r ) w ( r − r j )d r . (2)This equation encapsulates the simplicity and flex-ibility of the measurement scheme that we areproposing. The operators given by Eq. (1) areentirely determined by the values of these coeffi-cients and despite its simplicity, this is sufficientto give rise to a host of interesting phenomena viameasurement back-action such as the generationof multipartite entangled spatial modes in an op-tical lattice [20, 27], the appearance of long-rangecorrelated tunnelling capable of entangling distantlattice sites, and in the case of fermions, the break-up and protection of strongly interacting pairs [21].Additionally, these coefficients are easy to manip-ulate experimentally by adjusting the optical ge-ometry via the light mode functions u l ( r ). FIG. 1: (Color online) Setup. Atoms in an opticallattice are illuminated by a probe beam. The lightscatters in free space or into a cavity and is measuredby a detector.
Here, we will use the fact that the light is sensi-tive to the atomic quantum state due to the cou-pling of the optical and matter fields via operatorsaccording to Eq. (1) in order to develop a methodto probe the properties of an ultracold gas. There-fore, we neglect the measurement back-action andwe will only consider expectation values of lightobservables. Since the scheme is nondestructive(in some cases, it even satisfies the stricter re-quirements for a QND measurement [3, 26]) andthe measurement only weakly perturbs the sys-tem, many consecutive measurements can be car-ried out with the same atoms without preparing a new sample. Again, we will show how the ex-treme flexibility of the the measurement operatorˆ F allows us to probe a variety of different atomicproperties in-situ ranging from density correlationsto matter-field interference.For a coherent beam probe which has a negligi-ble effect on atomic dynamics, the stationary lightamplitude a is given by ˆ F [26] (we drop the sub-scripts in ˆ F , ˆ D , and ˆ B ). In a cavity with the decayrate κ and probe-cavity detuning ∆ p , a = U a ∆ p + iκ ˆ F = C ˆ F . (3)In free space, the electric field operator in the far-field point r isˆ E = ω a d A E π (cid:126) (cid:15) c ∆ a r ˆ F = C E ˆ F , (4)where d A is the dipole moment and E is probeelectric field [28].The light quadrature operators ˆ X φ = ( a e − iφ + a † e iφ ) / F quadratures, ˆ X Fβ ,ˆ X φ = | C | ˆ X Fβ = | C | ( ˆ F e − iβ + ˆ F † e iβ ) / , (5)where β = φ − φ C , C = | C | exp( iφ C ), and φ isthe local oscillator phase . The means of ampli-tude and quadrature, (cid:104) a (cid:105) and (cid:104) ˆ X φ (cid:105) , only dependon atomic mean values. In contrast, the means oflight intensity (cid:104) a † a (cid:105) = | C | (cid:104) ˆ F † ˆ F (cid:105) and quadraturevariance(∆ X φ ) = (cid:104) ˆ X φ (cid:105)−(cid:104) ˆ X φ (cid:105) = 1 / | C | (∆ X Fβ ) (6)reflect atomic correlations and fluctuations, whichis our main focus. Alternatively, one can measurethe light intensity, where the “quantum addition”to light due atom quantum fluctuations (classicaldiffraction signal is subtracted), R = (cid:104) a † a (cid:105)−|(cid:104) a (cid:105)| ,behaves similarly to (∆ X Fβ ) . III. GLOBAL NONDESTRUCTIVEMEASUREMENTA. On-site density measurements
Typically, the dominant term in ˆ F is the density-term ˆ D , rather than inter-site matter-field interfer-ence ˆ B [26, 29–32], because the Wannier functions’overlap is small. Our aim is to enhance the ˆ B -termin light scattering by suppressing the density sig-nal. To clarify typical light scattering, we startwith a simpler case when scattering is faster thantunneling and ˆ F = ˆ D . This corresponds to a QNDscheme [8–10, 26]. The density-related measure-ment destroys some matter-phase coherence in theconjugate variable [27, 33, 34] b † i b i +1 , but this termis neglected. FIG. 2: (Color online) Light intensity scattered into astanding wave mode from a SF in a 3D lattice (unitsof
R/N K ). Arrows denote incoming travelling waveprobes. The Bragg condition, ∆ k = G , is not fulfilled,so there is no classical diffraction, but intensity stillshows multiple peaks, whose heights are tunable bysimple phase shifts of the optical beams: (a) ϕ = 0;(b) ϕ = π/
2. Interestingly, there is also a significantuniform background level of scattering which does notoccur in its classical counterpart.
In a deep lattice,ˆ D = K (cid:88) i u ∗ ( r i ) u ( r i )ˆ n i , (7)which for travelling [ u l ( r ) = exp( i k l r + iϕ l )] orstanding [ u l ( r ) = cos( k l r + ϕ l )] waves is just adensity Fourier transform at one or several wavevectors ± ( k ± k ). The quadrature for two trav-elling waves is reduced toˆ X Fβ = K (cid:88) i ˆ n i cos[( k − k ) · r i − β ] . (8)Note that different light quadratures are differentlycoupled to the atom distribution, hence varying lo-cal oscillator phase and detection angle, one scansthe coupling from maximal to zero. An identicalexpression exists for ˆ D for a standing wave, where β is replaced by ϕ l , and scanning is achieved byvarying the position of the wave with respect toatoms. Thus, variance (∆ X Fβ ) and quantum ad-dition R , have a non-trivial angular dependence,showing more peaks than classical diffraction andthe peaks can be tuned by the light-atom coupling.Fig. 2 shows the angular dependence of R forstanding and travelling waves in a 3D OL. Theisotropic background gives the density fluctuations[ R = K ( (cid:104) ˆ n (cid:105) − (cid:104) ˆ n (cid:105) ) / K sites, N K /
2, in the deep SF. There exist peaksat angles different than the classical Bragg onesand thus, can be observed without being maskedby classical diffraction. Interestingly, even if 3Ddiffraction [11] is forbidden (Fig. 2), the peaksare still present. As (∆ X Fβ ) and R are squared variables, the generalized Bragg conditions for thepeaks are 2∆ k = G for quadratures of travellingwaves, where ∆ k = k − k and G is the recipro-cal lattice vector, and 2 k = G for standing wave a and travelling a , which is clearly different fromthe classical Bragg condition ∆ k = G . The peakheight is tunable by the local oscillator phase orstanding wave shift as seen in Fig. 2b.We estimate the mean photon number per sec-ond integrated over the solid angle for the onlytwo experiments so far on light diffraction fromtruly ultracold bosons where the measurement ob-ject was light n Φ = (cid:18) Ω ∆ a (cid:19) Γ K (cid:104) ˆ n (cid:105) − (cid:104) ˆ n (cid:105) ) , (9)where Ω = d A E / (cid:126) and Γ is the atomic relaxationrate. The background signal should reach n Φ ≈ s − in Ref. [24] (150 atoms in 2D), and n Φ ≈ s − in Ref. [11] (10 atoms in 3D). B. Matter-field interference measurements
We now focus on enhancing the interferenceterm ˆ B in the operator ˆ F . For clarity we willconsider a 1D lattice, but the results can be ap-plied and generalised to higher dimensions. Cen-tral to engineering the ˆ F operator are the coeffi-cients J i,j given by Eq. (2). The operators ˆ B andˆ D depend on the values of J i,i +1 and J i,i respec-tively. These coefficients are determined by theconvolution of the light mode product, u ∗ ( r ) u ( r )with the relevant Wannier function overlap shownin Fig. 3a. For the ˆ B operator we calculatethe convolution with the nearest neighbour over-lap, W ( r ) ≡ w ( r − d / w ( r + d / D operator we calculate the convolution with thesquare of the Wannier function at a single site, W ( r ) ≡ w ( r ). Therefore, in order to enhancethe ˆ B term we need to maximise the overlap be-tween the light modes and the nearest neighbourWannier overlap, W ( r ). This can be achieved byconcentrating the light between the sites ratherthan at atom positions. Ideally, one could measurebetween two sites similarly to single-site address-ing [35, 36], which would measure a single term (cid:104) b † i b i +1 + b i b † i +1 (cid:105) , e.g., by superposing a deeper OLshifted by d/ J i,j coefficients we per-form numerical calculations using realistic Wan-nier functions [37]. However, it is possible to gainsome analytic insight into the behaviour of thesevalues by looking at the Fourier transforms of theWannier function overlaps, F [ W , ]( k ) , shown inFig 3b. This is because the light mode product, - � � � � / � - ��������� � � �� � � ( � ) - � - � � � � �� / π ���� | ℱ [ � � ] | �� | ℱ [ � � ] | ( � ) � ξ π ξ + π � π φ � - ������������ � � � � + � - ����������� � � � � ( � ) � ��� � ��� + � � π / � π � π / � � π φ � - ��������� � � � � + � - ������� � � � � ( � ) � ��� � ��� � ��� � ��� FIG. 3: (Color online) The Wannier function products: (a) W ( x ) (solid line, right axis), W ( x ) (dashed line,left axis) and their (b) Fourier transforms F [ W , ]. The Density J i,i and matter-interference J i,i +1 coefficients(1) in diffraction maximum (c) and minimum (d) as are shown as functions of standing wave shifts ϕ or, if onewere to measure the quadrature variance (∆ X Fβ ) , the local oscillator phase β . The black points indicate thepositions, where light measures matter interference ˆ B (cid:54) = 0, and the density-term is suppressed, ˆ D = 0. Thetrapping potential depth is approximately 5 recoil energies. u ∗ ( r ) u ( r ), can be in general decomposed into asum of oscillating exponentials of the form e i k · r making the integral in Eq. (2) a sum of Fouriertransforms of W , ( r ). We consider both the de-tected and probe beam to be standing waves whichgives the following expressions for the ˆ D and ˆ B op-erators ˆ D = 12 [ F [ W ]( k − ) (cid:88) m ˆ n m cos( k − x m + ϕ − )+ F [ W ]( k + ) (cid:88) m ˆ n m cos( k + x m + ϕ + )] , ˆ B = 12 [ F [ W ]( k − ) (cid:88) m ˆ B m cos( k − x m + k − d ϕ − )+ F [ W ]( k + ) (cid:88) m ˆ B m cos( k + x m + k + d ϕ + )] , (10)where k ± = k x ± k x , k (0 , x = k , sin( θ , ),ˆ B m = b † m b m +1 + b m b † m +1 , and ϕ ± = ϕ ± ϕ . Thekey result is that the ˆ B operator is phase shiftedby k ± d/ D operator since itdepends on the amplitude of light in between thelattice sites and not at the positions of the atoms,allowing to decouple them at specific angles.Firstly, we will use this result to show howone can probe (cid:104) ˆ B (cid:105) which in MF gives informa-tion about the matter-field amplitude, Φ = (cid:104) b (cid:105) .The simplest case is to find a diffraction maxi-mum where J i,i +1 = J B . This can be achievedby crossing the light modes such that θ = − θ and k x = k x = π/d and choosing the light modephases such that ϕ + = 0. Fig. 3c shows the value of the J i,j coefficients under these circum-stances. In order to make the ˆ B contribution tolight scattering dominant we need to set ˆ D = 0which from Eq. (10) we see is possible if ϕ = − ϕ = arccos[ −F [ W ](2 π/d ) / F [ W ](0)] /
2. Thisarrangement of light modes maximizes the interfer-ence signal, ˆ B , by suppressing the density signal,ˆ D , via interference compensating for the spreadingof the Wannier functions. Hence, by measuring thelight quadrature we probe the kinetic energy and,in MF, the matter-field amplitude (order parame-ter) Φ: (cid:104) ˆ X Fβ =0 (cid:105) = | Φ | F [ W ](2 π/d )( K − X bα = ( be − iα + b † e iα ) /
2, which in MF can beprobed by measuring the variance of ˆ B . Acrossthe phase transition, the matter field changes itsstate from Fock (in MI) to coherent (deep SF)through an amplitude-squeezed state as shown inFig. 4(a,b). We consider an arrangement wherethe beams are arranged such that k x = 0 and k x = π/d which gives the following expressionsfor the density and interference termsˆ D = F [ W ]( π/d ) (cid:88) m ( − m ˆ n m cos( ϕ ) cos( ϕ )ˆ B = −F [ W ]( π/d ) (cid:88) m ( − m ˆ B m cos( ϕ ) sin( ϕ ) . (11)The corresponding J i,j coefficients are shown inFig. 3d for ϕ = 0. It is clear that for ϕ = ± π/ D = 0, which is intuitive as this places the lat-tice sites at the nodes of the mode u ( r ). Thisis a diffraction minimum as the light amplitude isalso zero, (cid:104) ˆ B (cid:105) = 0, because contributions from al-ternating inter-site regions interfere destructively.However, the intensity (cid:104) a † a (cid:105) = | C | (cid:104) ˆ B (cid:105) is propor-tional to the variance of ˆ B and is non-zero. Assum-ing Φ is real in MF: (cid:104) a † a (cid:105) = 2 | C | ( K − F [ W ]( πd ) × [( (cid:104) b (cid:105) − Φ ) + ( n − Φ )(1 + n − Φ )] (12)and it is shown as a function of U/ ( zJ cl ) in Fig. 4.Thus, since measurement in the diffraction maxi-mum yields Φ we can deduce (cid:104) b (cid:105) − Φ from theintensity. This quantity is of great interest as itgives us access to the quadrature variances of thematter-field(∆ X b ,π/ ) = 1 / n − Φ ) ± ( (cid:104) b (cid:105)− Φ )] / , (13)where n = (cid:104) ˆ n (cid:105) is the mean on-site atomic density. � � � � � /( �� �� ) ��������������� 〈 � � † � � 〉 / � ���� � � � � � /( �� �� ) ���������������� ( Δ � � � π / �� ) � ( � ) � � � - � � � - ��� Δ � π / �� Δ � �� ��� ( � ) FIG. 4: (Color online) Photon number scattered ina diffraction minimum, given by Eq. (12), where˜ C = 2 | C | ( K − F [ W ]( π/d ). More light is scat-tered from a MI than a SF due to the large uncer-tainty in phase in the insulator. (a) The variances ofquadratures ∆ X b (solid) and ∆ X bπ/ (dashed) of thematter field across the phase transition. Level 1/4 isthe minimal (Heisenberg) uncertainty. There are threeimportant points along the phase transition: the coher-ent state (SF) at A, the amplitude-squeezed state atB, and the Fock state (MI) at C. (b) The uncertaintiesplotted in phase space. Alternatively, one can use the arrangement for adiffraction minimum described above, but use trav-elling instead of standing waves for the probe anddetected beams and measure the light quadraturevariance. In this case ˆ X Fβ = ˆ D cos( β ) + ˆ B sin( β )and by varying the local oscillator phase, one canchoose which conjugate operator to measure. For β = π/
2, (∆ X Fπ/ ) looks identical to Eq. (12).Probing ˆ B gives us access to kinetic energy fluc-tuations with 4-point correlations ( b † i b j combinedin pairs). Measuring the photon number variance,which is standard in quantum optics, will lead upto 8-point correlations similar to 4-point densitycorrelations [26]. These are of significant interest,because it has been shown that there are quantumentangled states that manifest themselves only inhigh-order correlations [38]. Surprisingly, inter-site terms scatter more lightfrom a MI than a SF Eq. (12), as shown in Fig.(4), although the mean inter-site density (cid:104) ˆ n ( r ) (cid:105) istiny in a MI. This reflects a fundamental effect ofthe boson interference in Fock states. It indeedhappens between two sites, but as the phase isuncertain, it results in the large variance of ˆ n ( r )captured by light as shown in Eq. (12). The inter-ference between two macroscopic BECs has beenobserved and studied theoretically [39]. When twoBECs in Fock states interfere a phase difference isestablished between them and an interference pat-tern is observed which disappears when the resultsare averaged over a large number of experimentalrealizations. This reflects the large shot-to-shotphase fluctuations corresponding to a large inter-site variance of ˆ n ( r ). By contrast, our method en-ables the observation of such phase uncertainty ina Fock state directly between lattice sites on themicroscopic scale in-situ. C. Mapping the quantum phase diagram
We have shown how in MF, we can track theorder parameter, Φ, by probing the matter-fieldinterference using the coupling of light to the ˆ B operator. In this case, it is very easy to follow theSF-MI quantum phase transition since we have di-rect access to the order parameter which goes tozero in the insulating phase. In fact, if we’re onlyinterested in the critical point, we only need ac-cess to any quantity that yields information aboutdensity fluctuations which also go to zero in theMI phase and this can be obtained by measuring (cid:104) ˆ D † ˆ D (cid:105) . However, there are many situations wherethe MF approximation is not a valid descriptionof the physics. A prominent example is the BHMin 1D [40–44]. Observing the transition in 1D bylight at fixed density was considered to be difficult[10] or even impossible [45]. By contrast, here wepropose to vary the density or chemical potential,which sharply identifies the transition. We per-form these calculations numerically by calculatingthe ground state using DMRG methods [46] fromwhich we can compute all the necessary atomicobservables. Experiments typically use an addi-tional harmonic confining potential on top of theOL to keep the atoms in place which means thatthe chemical potential will vary in space. How-ever, with careful consideration of the full ( µ/ J cl , U/ J cl ) phase diagrams in Fig. 5(d,e) our analysiscan still be applied to the system [47].The 1D phase transition is best understood interms of two-point correlations as a function oftheir separation [48]. In the MI phase, the two-point correlations (cid:104) b † i b j (cid:105) and (cid:104) δ ˆ n i δ ˆ n j (cid:105) ( δ ˆ n i = ˆ n i −(cid:104) ˆ n i (cid:105) ) decay exponentially with | i − j | . On theother hand the SF will exhibit long-range orderwhich in dimensions higher than one, manifests it-self with an infinite correlation length. However, FIG. 5: (Color online) (a) The angular dependence of scattered light R for SF (thin black, left scale, U/ J cl = 0)and MI (thick green, right scale, U/ J cl = 10). The two phases differ in both their value of R max as well as W R showing that density correlations in the two phases differ in magnitude as well as extent. Light scatteringmaximum R max is shown in (b, d) and the width W R in (c, e). It is very clear that varying chemical potential µ or density (cid:104) n (cid:105) sharply identifies the SF-MI transition in both quantities. (b) and (c) are cross-sections of thephase diagrams (d) and (e) at U/ J cl = 2 (thick blue), 3 (thin purple), and 4 (dashed blue). Insets show densitydependencies for the U/ (2 J cl ) = 3 line. K = M = N = 25. in 1D only pseudo long-range order happens andboth the matter-field and density fluctuation cor-relations decay algebraically [48].The method we propose gives us direct accessto the structure factor, which is a function of thetwo-point correlation (cid:104) δ ˆ n i δ ˆ n j (cid:105) , by measuring thelight intensity. For two travelling waves maximallycoupled to the density (atoms are at light intensitymaxima so ˆ F = ˆ D ), the quantum addition is givenby R = (cid:88) i,j exp[ i ( k − k )( r i − r j )] (cid:104) δ ˆ n i δ ˆ n j (cid:105) , (14)The angular dependence of R for a MI and aSF is shown in Fig. 5a, and there are two vari-ables distinguishing the states. Firstly, maximal R , R max ∝ (cid:80) i (cid:104) δ ˆ n i (cid:105) , probes the fluctuations andcompressibility κ (cid:48) ( (cid:104) δ ˆ n i (cid:105) ∝ κ (cid:48) (cid:104) ˆ n i (cid:105) ). The MI is in-compressible and thus will have very small on-sitefluctuations and it will scatter little light leadingto a small R max . The deeper the system is in theMI phase (i.e. that larger the U/ J cl ratio is), thesmaller these values will be until ultimately it willscatter no light at all in the U → ∞ limit. In Fig. 5a this can be seen in the value of the peak in R .The value R max in the SF phase ( U/ J cl = 0) islarger than its value in the MI phase ( U/ J cl = 10)by a factor of ∼
25. Figs. 5(b,d) show how the valueof R max changes across the phase transition. Wesee that the transition shows up very sharply as µ is varied.Secondly, being a Fourier transform, the width W R of the dip in R is a direct measure of the cor-relation length l , W R ∝ /l . The MI being aninsulating phase is characterised by exponentiallydecaying correlations and as such it will have avery large W R . However, the SF in 1D exhibitspseudo long-range order which manifests itself inalgebraically decaying two-point correlations [48]which significantly reduces the dip in the R . Thiscan be seen in Fig. 5a and we can also see thatthis identifies the phase transition very sharply as µ is varied in Figs. 5(c,e). One possible concernwith experimentally measuring W R is that it mightbe obstructed by the classical diffraction maximawhich appear at angles corresponding to the min-ima in R . However, the width of such a peak ismuch smaller as its width is proportional to 1 /M .It is also possible to analyse the phase transitionquantitatively using our method. Unlike in higherdimensions where an order parameter can be easilydefined within the MF approximation there is nosuch quantity in 1D. However, a valid descriptionof the relevant 1D low energy physics is providedby Luttinger liquid theory [48]. In this model cor-relations in the SF phase as well as the SF densityitself are characterised by the Tomonaga-Luttingerparameter, K b . This parameter also identifies thephase transition in the thermodynamic limit at K b = 1 /
2. This quantity can be extracted fromvarious correlation functions and in our case it canbe extracted directly from R [41]. By extractingthis parameter from R for various lattice lengthsfrom numerical DMRG calculations it was evenpossible to give a theoretical estimate of the criti-cal point for commensurate filling, N = M , in thethermodynamic limit to occur at U/ J cl ≈ . R in a lab which can then be used to ex-perimentally determine the location of the criticalpoint in 1D.So far both variables we considered, R max and W R , provide similar information. Next, we presenta case where it is very different. BG is a local-ized insulating phase with exponentially decayingcorrelations but large compressibility and on-sitefluctuations in a disordered OL. Therefore, mea-suring both R max and W R will distinguish all thephases. In a BG we have finite compressibility, butexponentially decaying correlations. This gives alarge R max and a large W R . A MI will also haveexponentially decaying correlations since it is aninsulator, but it will be incompressible. Thus, itwill scatter light with a small R max and large W R .Finally, a SF will have long range correlations andlarge compressibility which results in a large R max and a small W R . FIG. 6: (Color online) The MI-SF-BG phase diagramsfor light scattering maximum R max /N K (a) and width W R (b). Measurement of both quantities distinguishall three phases. Transition lines are shifted due tofinite size effects [49], but it is possible to apply wellknown numerical methods to extract these transitionlines from such experimental data extracted from R [41]. K = M = N = 35. We confirm this in Fig. 6 for simulations withthe ratio of superlattice- to trapping lattice-period r ≈ .
77 for various disorder strengths V [49].Here, we only consider calculations for a fixed den-sity, because the usual interpretation of the phase diagram in the ( µ/ J cl , U/ J cl ) plane for a fixedratio V /U becomes complicated due to the pres-ence of multiple compressible and incompressiblephases between successive MI lobes [49]. This way,we have limited our parameter space to the threephases we are interested in: SF, MI, and BG. FromFig. 6 we see that all three phases can indeed bedistinguished. In the 1D BHM there is no sharpMI-SF phase transition in 1D at a fixed density[40–44] just like in Figs. 5(d,e) if we follow thetransition through the tip of the lobe which corre-sponds to a line of unit density. However, despitethe lack of an easily distinguishable critical point itis possible to quantitatively extract the location ofthe transition lines by extracting the Tomonaga-Luttinger parameter from the scattered light, R ,in the same way it was done for an unperturbedBHM [41].Only recently [25] BG was studied by combinedmeasurements of coherence, transport, and exci-tation spectra, all of which are destructive tech-niques. Our method is simpler as it only requiresmeasurement of the quantity R and additionally,it is nondestructive. IV. CONCLUSIONS
In summary, we proposed a nondestructivemethod to probe quantum gases in an OL. Firstly,we showed that the density-term in scatteringhas an angular distribution richer than classicaldiffraction, derived generalized Bragg conditions,and estimated parameters for the only two rele-vant experiments to date [11, 24]. Secondly, weproposed how to measure the matter-field interfer-ence by concentrating light between the sites. Thiscorresponds to interference at the shortest possibledistance in an OL. By contrast, standard destruc-tive time-of-flight measurements deal with far-fieldinterference and a relatively near-field one was usedin Ref. [11]. This defines most processes in OLs.E.g. matter-field phase changes may happen notonly due to external gradients, but also due tointriguing effects such quantum jumps leading tophase flips at neighbouring sites and sudden can-cellation of tunneling [50], which should be acces-sible by our method. In MF, one can measure thematter-field amplitude (order parameter), quadra-tures and squeezing. This can link atom opticsto areas where quantum optics has already madeprogress, e.g., quantum imaging [22, 23], using anOL as an array of multimode nonclassical matter-field sources with a high degree of entanglement forQIP. Thirdly, we demonstrated how the methodaccesses effects beyond MF and distinguishes allthe phases in the MI-SF-BG transition, which iscurrently a challenge [25]. Based on off-resonantscattering, and thus being insensitive to a detailedatomic level structure, the method can be extendedto molecules [51], spins, and fermions [31].
Acknowledgments
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