Quantum Measurement-induced Dynamics of Many-Body Ultracold Bosonic andFermionic Systems in Optical Lattices

Gabriel Mazzucchi,∗ Wojciech Kozlowski,† Santiago F. Caballero-Benitez, Thomas J. Elliott, and Igor B. MekhovDepartment of Physics, Clarendon Laboratory, University of Oxford,

Parks Road, Oxford OX1 3PU, United Kingdom(Dated: March 31, 2015)

Trapping ultracold atoms in optical lattices enabled numerous breakthroughs uniting several dis-ciplines. Although the light is a key ingredient in such systems, its quantum properties are typicallyneglected, reducing the role of light to a classical tool for atom manipulation. Here we show howelevating light to the quantum level leads to novel phenomena, inaccessible in setups based onclassical optics. Interfacing a many-body atomic system with quantum light opens it to the en-vironment in an essentially nonlocal way, where spatial coupling can be carefully designed. Thecompetition between typical processes in strongly correlated systems (local tunnelling and interac-tion) with global measurement backaction leads to novel multimode dynamics and the appearanceof long-range correlated tunnelling capable of entangling distant lattices sites, even when tunnellingbetween neighbouring sites is suppressed by the quantum Zeno effect. We demonstrate both thebreak-up and protection of strongly interacting fermion pairs by different measurements.

Ultracold gases trapped in optical lattices representa successful interdisciplinary field: Atomic systems al-low quantum simulations of phenomena predicted in con-densed matter and particle physics, and find applicationsin quantum information processing [1]. Although lightplays a key role in such systems, its quantum propertieshave been neglected in most works, reducing it to a clas-sical tool for creating intriguing atomic states. Here, weintroduce the quantumness of light into strongly corre-lated atomic systems and demonstrate phenomena thatcannot be realised using classical optical setups. The rec-ognized advantage of ultracold gases in contrast to con-densed matter materials is their isolation from the envi-ronment. However, this creates a challenge to introduceany controlled dissipation (e.g. atom losses and collisionsare difficult to manipulate). Here we show that couplingatoms to quantized light, which can be continuously mea-sured, introduces a controllable decoherence channel into

Figure 1. Setup. Atoms in an optical lattice are probed bya coherent light beam, and the light scattered at a particularangle is enhanced and collected by a leaky cavity.

∗ gabriel.mazzucchi@physics.ox.ac.uk† wojciech.kozlowski@physics.ox.ac.uk

many-body dynamics. Moreover, global light scatter-ing from multiple lattice sites creates nontrival, spatiallynonlocal coupling to the environment, which is impossi-ble to obtain with local interactions [2]. Such a quantumoptical approach can broaden the field even further, al-lowing quantum simulation of models unobtainable usingclassical light, and the design of novel systems beyondcondensed matter analogues.

Collapse of the atomic wave function, when only thescattered light is measured, reflects the measurementbackaction. This is one of the most fundamental manifes-tations of quantum mechanics [3], here due to the light-matter entanglement. We demonstrate how this backac-tion and the spatially nonlocal quantum Zeno effect, be-ing introduced in a strongly correlated system, competewith many-body dynamics defined by standard local pro-cesses (tunnelling and on-site interaction). Highlightingsuch competition takes one beyond recent quantum non-demolition (QND) approaches [4–10], where either many-body dynamics or measurement backaction did not playany role. For bosons, it leads to the emergence of dynam-ical macroscopic superpositions (multimode Schrödingercat states). For strongly interacting fermions, we demon-strate the measurement-induced break-up and protectionof fermion pairs. Even if standard tunnelling betweenneighbouring sites is suppressed by the quantum Zenoeffect, long-range correlated tunnelling appears, leadingto dynamical generation entanglement between distantsites. The generated spatial modes of matter fields canbe considered as designed systems and reservoirs. We canmake analogy with famous cavity QED experiments [3],where the quantum light states in a cavity were probedand projected by measuring atoms. Here, light and mat-ter are reversed and we probe matter-fields with light.A striking advancement is that the number of quantummatter waves (sites with trapped atoms) can be easilyscaled from few to thousands by changing the numberof illuminated sites, while scaling cavities is an extremechallenge [3]. This work, together with recent experi-

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ments where our predictions can be tested (Bose-EinsteinCondensates trapped inside a cavity [11–13]), will helpto close the gap between quantum optics and quantumgases, by joining quantized light and strongly correlatedmany-body systems, thus approaching the fully quantumregime of light-matter interaction [5, 14, 15].

We consider off-resonant light scattering from N atomstrapped in an optical lattice with period d and L sites [14](see Methods and Fig. 1). The light scattered at a par-ticular angle can be selected and enhanced by a cav-ity [16] with decay rate κ. Similar to classical optics,the light amplitude is given by a sum of scatterings fromall atoms with coefficients dependent on their positions:a = C(D + B), where a is the photon annihilation oper-ator, C is the Rayleigh scattering coefficient and

D =

L∑j=1

Jjj nj , B =

L∑〈i,j〉

Jijb†i bj , (1)

where bj and nj = b†jbj are the atomic annihilationand number operators at site j (〈i, j〉 sums over neigh-bouring sites), and Jij are given in Methods. In equa-tion (1) D describes scattering from the on-site densi-ties, while B that from the small inter-site densities [17].For well-localised atoms, the second term is usuallyneglected, and a = CD with Jjj = u∗out(rj)uin(rj),where u(r) are the mode functions of input and scat-tered light (e.g., uin,out(r) = exp(ikin,out·r) for travellingand uin,out(r) = cos(kin,out·r) for standing waves withwave vectors kin,out). (A peculiar case of light scat-tering from inter-site regions [17], where a = CB, willbe underlined here later). For spin- 1

2 fermions we usetwo light polarizations ax,y that couple differently to twospin densities n↑j , n↓j allowing measurement of their lin-ear combinations, e.g., ax = CDx = C

∑Lj=1 Jjj ρj and

ay = CDy = C∑Lj=1 Jjjmj , where ρj = n↑j + n↓j and

mj = n↑j − n↓j are the mean density and magnetisa-tion. This property has recently been used to investigatespin-spin correlations in Fermi gases [18, 19].

We focus on a single run of a continuous measurementexperiment using the quantum trajectories technique [20](see Methods). The evolution is determined by a stochas-tic process described by quantum jumps (the jump op-erator c =

√2κa is applied to the state when a pho-

todetection occurs) and non-Hermitian evolution withthe Hamiltonian Heff = H0 − i~c†c/2 between jumps,where H0 is the usual (Bose-)Hubbard Hamiltonian. Im-portantly, the measurement introduces a new energy andtime scale γ = |C|2κ, which competes with the twoother standard scales responsible for unitary dynamicsof closed systems (tunnelling J and on-site interactionU). If atoms scatter light independently in uncontrolleddirections (as in spontaneous emission [21] or local ad-dressing), independent jump operators cj would be ap-plied to each site, projecting the atomic state to a fullymixed state [21]. In contrast, here we consider global

coherent scattering, where the single global jump oper-ator c is given by the sum over all sites, and the localcoefficients Jjj (1) responsible for the atom-environmentcoupling (via the light mode a) can be engineered byoptical geometry. Thus, atoms are coupled to the envi-ronment globally, and atoms that scatter light with thesame phase are indistinguishable to light scattering (i.e.there is no “which-path information”). As a striking con-sequence, the quantum superpositions are strongly pre-served in the final projected states, and the systems splitsinto several spatial modes [22], where all atoms belong-ing to the same mode are indistinguishable, while beingdistinguishable from atoms belonging to different modes.

We engineer the atom-environment coupling coeffi-cients Jjj using standing or traveling waves at differentangles to the lattice. If both probe and scattered lightare standing waves crossed at such angles to the latticethat projection kin·r is equal to kout·r and shifted suchthat all even sites are positioned at the nodes (do notscatter light), one gets Jjj = 1 for odd and Jjj = 0for even sites. Thus we measure the number of atomsat odd sites only (the jump operator is proportional toNodd), introducing two modes, which scatter light differ-ently: odd and even sites. The coefficients Jjj = (−1)j

are designed by crossing light waves at 90◦ such thatatoms at neighboring sites scatter light with π phasedifference [4, 10, 23–25], giving D = Neven − Nodd, in-troducing the same modes, but with different coherencebetween them. Moreover, using travelling waves crossedat the angle such that each R-th site is indistinguish-able ((kin−kout)·rj = 2πj/R), introduces R modes withmacroscopic atom numbers Nl: D =

∑Rl=1 Nle

i2πl/R.Here two (odd- and even-site modes) appear for R = 2.Therefore, we reduce the jump (measurement) operatorfrom being a sum of numerous microscopic contributionsfrom individual sites to the sum of smaller number ofmacroscopically occupied modes with a very nontrivialspatial overlap between them. In the following, we willshow how such globally designed measurement backac-tion introduces spatially long-range interactions of thesemodes, and demonstrate novel effects resulting from thecompetition of mode dynamics with standard local pro-cesses in a many-body system.

We start with non-interacting bosons (U/J = 0), anddemonstrate the competition between global measure-ment and local tunnelling. Because of such competi-tion, the weak measurement (γ � J) is unable to freezethe atom numbers as expected for quantum Zeno dy-namics [26–28]. In contrast, atom-number measurementleads to giant oscillations of particle number between themodes. In Figs. 2a-c we show the atom number distribu-tions in one of the modes for R = 2 (Nodd) and R = 3(N1). Without measurement, these distributions wouldspread strongly and may show only oscillation amplitudesproportional to the initially created imbalance (thus, tinyoscillations for initially tiny imbalance). In contrast, herewe observe (i) full exchange of atoms between the modesindependent of the initial state, (ii) the distributions con-

3

Figure 2. Large oscillations between the measurement-induced spatial modes resulting from the competitionbetween tunnelling and weak-measurement backac-tion. The plots show single quantum trajectories. a-d, Atomnumber distributions p(Nl) in one of the modes, which showvarious number of well-squeezed components, reflecting thecreation of macroscopic superposition states depending on themeasurement configuration (U/J = 0, γ/J = 0.01, L = N ,initial states: superfluid for bosons, Fermi sea for fermions).a, Measurement of the atom number at odd sites Nodd cre-ates one strongly oscillating component in p(Nodd) (N = 100

bosons). b, Measurement of (Nodd−Neven)2 introduces R = 2

modes and preserves the superposition of positive and nega-tive atom number differences in p(Nodd) (N = 100 bosons). c,Measurement for R = 3 modes (see text) preserves 3 compo-nents in p(N1) (N = 108 bosons). d, Measurement of Nodd forfermions leads to oscillations in p(Nodd), though not as wellsqueezed as for bosons because of Pauli blocking (L = 8 sites,N↑ = N↓ = 4 fermions). e, On-site atom numbers 〈ni〉 formeasurement of the matter-field coherence between the sitesB. Atoms, all initially at the edge site, are quickly spreadacross the whole lattice leading to the large uncertainty inthe atom number, while the matter-phase related variable isdefined (projected) by the measurement (bosons, N = L = 6,U/J = 0, γ/J = 0.1). Simulations for 1D lattice.

sist of a small number of well-defined components, and(iii) these components are squeezed even by weak mea-surement. The multi-component structure is explainedby the degeneracy of spatially multimode states, as themeasurement of photon number (intensity) a†a is not sen-sitive to the light phase. Therefore, all permutations of

mode occupations that scatter light with the same inten-sity are indistinguishable. For R = 2, the state is 2-folddegenerate, for R > 2 it is 2R-fold degenerate. Fig. 2breflects the superposition of two states with positive andnegative Nodd − Neven, while Fig. 2c reflects the super-position in the three-mode case. As an important conse-quence, one gets a dynamical method to prepare multi-component macroscopic superpositions (Schrödinger cator NOON states), which are useful in quantum metrologyand information. This method is deterministic withoutneed for external quantum control [29, 30], as the con-tinuously monitored intensity directly shows, when thenumber splitting in the components reaches its maximum(corresponding to the maximal macroscopicity [31]), andthe lattice depth can be ramped up to freeze further dy-namics. Such states are indeed fragile to decoherencesuch as photon losses, but setups can be modified to makethem more robust [32]. In addition, the state in Fig. 2aconsists of only one component (here Nodd is measureddirectly) and is therefore insensitive to photon losses. Re-ducing the number of modes and components can enablestudy the transition between quantumness and classical-ity, where nontrivial superpositions cannot exist [33].

We found that for many atoms, the R-mode prob-lem can be treated analytically reducing it to R sites,if the initial state is superfluid. Giant oscillations areexplained by the action of an effective quasi-periodicdriving force due to the measurement backaction. Inparticular, for R = 2 (effective double-well [34, 35]),we showed that the system leaves its stationary state(Nodd = Neven) with the increment Nγ/2. Note thatdynamics described here will be visible already in a sin-gle experimental run. In contrast, averaging over manyruns, which corresponds to the master equation solution,masks such effects completely. This happens because theoscillation phase changes from realization to realizationas is known from works involving single and multiplemeasurements [36].

In contrast to bosons, dynamics for two modes of non-interacting fermions does not show well-defined oscilla-tions (Fig. 2d) due to Pauli exclusion. However, whilethe initial ground state is a product of ↑ and ↓ wave func-tions (Slater determinants), the measurement introducesan effective interaction between two spin components andthe state gets entangled by measurement.

Carefully choosing geometry, one can suppress the on-site contribution to light scattering and effectively con-centrate light between the sites, thus in-situ measuringthe matter-field interference b†i bi+1 [17]. In this case,a = CB, and the coefficients Jij (1) can be engineered[17]. For Jij = 1, the jump operator is proportional tothe kinetic energy EK and tends to freeze the systemin eigenstates of the non-interacting Hamiltonian. Beinginsensitive to the sign of EK , the measurement projectsto a superposition of states with opposite kinetic ener-gies: a superposition of matter waves propagating withdifferent momenta. The measurement freezes dynamicsfor any γ/J , since the jump operator and Heff have the

4

γ/J = 10γ/J = 1γ/J = 0.25γ/J = 0.1no measurement

γ/J = 10γ/J = 1γ/J = 0.1γ/J = 0.01no measurement

γ/J = 10γ/J = 1γ/J = 0.1γ/J = 0.01no measurement

0 2 4 6 8 10U/J

0.0

0.5

1.0

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2.0⟨σ

D2⟩ tr

aj(a)

0

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⟨σD

x2⟩ tr

aj

(b)

0 2 4 6 8 10U/J

0

1

2

3

⟨σD

y2⟩ tr

aj

(c)

Figure 3. Atom number fluctuations demonstratingthe competition of global measurement with localinteraction and tunnelling. Number variances are aver-aged over many trajectories. a, Bose-Hubbard model withrepulsive interaction. The fluctuations of the atom num-ber at odd sites Nodd in the ground state without a mea-surement (dashed line) decrease as U/J increases, reflect-ing the transition between superfluid and Mott insulatorphases. For the weak measurement, 〈σ2

D〉traj is squeezed be-low the ground state value, but then increases and reachesits maximum as the atom repulsion prevents oscillations andmakes the squeezing less effective. In the strong interactinglimit, the Mott insulator state is destroyed and the fluctua-tions are larger than in the ground state. (100 trajectories,N = L = 6.) b,c, Fermionic Hubbard model with attrac-tive interaction; fluctuations of the total atom number at oddsites Dx = N↑odd + N↓odd (b) and of the magnetisation atodd sites Dy = Modd = N↑odd − N↓odd (c). Without mea-surement, the interaction favours formation of doubly occu-pied sites so the density fluctuations in the ground state areincreasing while the magnetisation ones are decreasing. Themeasurement creates singly occupied sites decreasing the den-sity fluctuations and increasing the magnetisation ones, whichmanifests the break-up of fermion pairs by measurement. Themeasurement-based protection of fermion pairs is shown inthe next figure. (100 trajectories, L = 8, N↑ = N↓ = 4.)Simulations for 1D lattice.

same eigenstates. As a result of the detection, the atomsquickly spread across the lattice, and the density distri-bution becomes uncertain (Fig. 2e), clearly illustratingthe quantum uncertainty relation between the number-and phase-related variables (ni and b

†i bi+1). Engineering

Jij , can lead to the measurement-based preparation ofpeculiar multicomponent momentum (or Bloch) states.

Including the interactions (U/J 6= 0), dynamicschanges as the measurement competes with both tun-nelling and interaction. To highlight this, we focus onthe variance of the measured variable averaged over manytrajectories 〈σ2

D〉traj, when it reaches its steady state. InFig. 3, we show results for the measurement of Nodd.

For bosons (Fig. 3a), without measurement the num-ber fluctuations σ2

D monotonically decrease for increasingU , reflecting the superfluid - Mott insulator phase transi-tion. However, the measurement changes this behaviourand 〈σ2

D〉traj varies non-monotonically. For weak interac-

tion, the fluctuations are strongly squeezed below thoseof the ground state; they then quickly increase, reachtheir maximum, and subsequently decrease as the inter-action becomes stronger. We explain this effect by look-ing at single trajectories: For small U/J , as describedabove, the measurement strongly squeezes the occupa-tion of one mode (Fig. 2a). However, the local atomrepulsion prevents the formation of states with high pop-ulation in one of two modes (Fig. 4a). Because of the lo-cal interaction, states with different imbalances oscillatewith different frequencies and, hence, the measurementdoes not decrease the fluctuations of Nodd as efficientlyas in the non-interacting case. For large U , both globalmeasurement and local interaction tend to squeeze thefluctuations, but as the measurement destroys the Mottinsulator, the fluctuations are larger than in the groundstate.

For fermions (Figs. 3b,c), the ground state of the at-tractive Hubbard model in the strong interacting regimecontains mainly doubly occupied sites (pairs). There-fore, without measurement, the fluctuations of total pop-ulation Dx = N↑odd + N↓odd (σ2

Dx) increase with U/J

as empty or doubly occupied sites become more proba-ble. We can probe either the total population Dx, orboth total population and magnetisation Dy = Modd =

N↑odd − N↓odd. In the first case (Figs. 3b,c), the mea-surement quickly squeezes σ2

Dx, but destroys the pairs as

it does not distinguish between singly or doubly occu-pied sites. This manifests the break-up of local fermionpairs by the global measurement and the correspondingtrajectory is shown in Figs. 4c: while initially dynamicscontains only even values of Nodd (i.e. pairs), the mea-surement also creates unpaired (odd) fermion numbers.In contrast, measuring both density and magnetisationreduces their fluctuations (i. e. unpaired fermions) andincreases the lifetime of doubly occupied sites. This man-ifests the measurement-induced protection of fermionpairs and the trajectory is shown in Fig. 4d: The numberdistribution contains only even values of Nodd indicatingthat fermions tunnel only in pairs.

When the measurement is strong (γ � J), we willshow the following nontrivial spatial effects. First, themeasurement can freeze the mode atom numbers, thusdecorrelating the numbers in different modes, while pro-tecting typical dynamics within each mode. Second, evenif the modes are number-decorrelated and the standardtunnelling between them is suppressed by the quantumZeno effect (i.e. close to full projection), they can stillbe entangled by another process. We show that higher-order long-range correlated multi-tunnelling events ap-pear: e.g., atoms can tunnel only in pairs (second ordertunnelling) and, importantly, this pair is delocalized andcorrelated in space. Although for clarity we demonstratesuch phenomena in Fig. 5 for small systems, they alsohold for macroscopic modes.

First (Fig. 5a), we freeze the atom number Nillum inthe central illuminated region by detecting light in thediffraction maximum (a = CNillum) [10, 32]. The lattice

5

Figure 4. Conditional dynamics of the atom-numberdistributions at odd sites illustrating competition ofthe global measurement with local interaction andtunnelling. Single quantum trajectories, initial states arethe ground states. a, Weakly interacting bosons: the on-site repulsion prevents the formation of well-defined oscilla-tions in the population of the mode. As states with differ-ent imbalance evolve with different frequencies, the squeez-ing due to the measurement is not as efficient as one ob-served in the non-interacting case (N = L = 6, U/J = 1,γ/J = 0.1). b, Strongly interacting bosons: oscillationsare completely suppressed and the number of atoms in themode is rather well-defined, although is less squeezed as inthe Mott insulator. (N = L = 6, U/J = 10, γ/J = 0.1).c, Attractive fermionic Hubbard model in the strong in-teraction limit. Measuring only the total population atodd sites Dx = N↑odd + N↓odd quickly creates singly occu-pied sites, demonstrating measurement-induced break-up offermion pairs (L = 8, N↑ = N↓ = 4, U/J = 10, γ/J = 0.1).d, The same as c, but with added measurement of the mag-netisation at odd sites Dy = Modd = N↑odd − N↓odd. Thisprotects the doubly occupied sites, thus, demonstrating pro-tection of fermion pairs by measurement. The distribution ofNodd vanishes for odd numbers, implying that the fermionstunnel only in pairs. Simulations for 1D lattice.

is thus divided into two modes: non-illuminated zones1 and 3 and the illuminated one 2 (Fig. 5a.1). Typi-cal dynamics occurs within each zone, but the standardtunnelling between the zones is suppressed (Fig. 5a.2).Importantly, atoms can still penetrate, if the processdoes not change Nillum: An atom from 1 can tunnelto 2, if simultaneously one atom tunnels from 2 to 3.Thus, effective long-range tunnelling between two spa-

tially disconnected zones 1 and 3 happens due to two-step processes 1 → 2 → 3 or 3 → 2 → 1. This is sup-ported by the negative (anti-)correlations 〈δN1δN3〉 =〈N1N3〉 − 〈N1〉〈N3〉 showing that an atom disappearingin 1 appears in 3, while there are no correlations be-tween illuminated and non-illuminated regions, 〈(δN1 +δN3)δN2〉 = 0 (Fig. 5a.4). Surprisingly, the correlatedtunnelling still builds entanglement between illuminatedand non-illuminated regions (Fig. 5a.3) underlying theintermediate (virtual) step in the two-tunnelling event.This effect may lead to creation of intriguing multipar-tite entangled states with no simple classical correlations[37].

To make correlated tunnelling visible even in the meanatom number, we consider illumination of all even sites(Fig. 5b) thus freezing both Neven and Nodd. Fig. 5b.2shows that while Neven is fixed, there is a slow atomexchange between odd sites, although they are discon-nected. Fig. 5b.1 clarifies the nontrivial character of cor-related tunnelling. The correlations can appear betweenall possible tunnellings leading to the fixed atom num-bers of modes. This again suggests that global coher-ent addressing favours correlated tunnelling, in contrastto local uncorrelated addressing of individual sites thatshould decrease the probability of individual tunnellingevents being correlated. Scheme in Fig. 5b.1 can help todesign a nonlocal reservoir for the tunnelling (or decay)of atoms from one region to another. For example, ifthe atoms are placed only at odd sites, their standardtunnelling is strongly suppressed as there is no secondtunnelling event. If, however, one adds some atoms toeven sites (even if they are far from the initial atoms),the slow correlated tunnelling ("decay") becomes allowedand its rate can be tuned by the number of added atoms.This resembles the repulsively bound pairs created bylocal interactions [38]. In contrast, here the atoms arelong-range correlated due to the global measurement.

The negative number correlations are typical for sys-tems with constraints (superselection rules) such as fixedatom number. The constraints implied by our globalmeasurement are more general. For example, in Fig. 5cwe show the generation of positive number correlations(shown in Fig. 5c.4) by freezing the atom number differ-ence between the sites (Nodd − Neven, by measuring atthe diffraction minimum). Thus, atoms can only enter orleave this region in pairs, which again is possible due tocorrelated tunnelling (Figs. 5c1-2) and manifests positivecorrelations. Note that, using more modes, the design ofhigher-order multi-tunnelling events is possible.

In summary, we introduced global quantum measure-ment in strongly correlated many-body systems anddemonstrated the competition between the global back-action and standard local processes. It leads to spatiallymultimode dynamics producing macroscopic superposi-tions useful for quantum information and metrology. Weshowed the possibility of measurement-induced break-up and protection of strongly interacting fermion pairs.For strong measurement, when the quantum Zeno ef-

6

⟨δN1δN3⟩⟨δNIδNNI ⟩

⟨(δn1+δn5 )δn3⟩⟨δNIδNNI ⟩

⟨δn1δn4⟩⟨δNIδNNI ⟩

(a.1) (a.2)

0 5 10 15Jt

-1

0(a.4)

0

1

2

3(a.3)

Ent

angl

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tC

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(b.1) (b.2)

0 10 20 30 40 50Jt-1

0(b.4)

0

1

2(b.3)

Ent

angl

emen

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(c.1) (c.2)

0 50 100 150 200 250Jt

-2

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Ent

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Figure 5. Long-range correlated tunnelling and entanglement, dynamically induced by strong global mea-surement. a,b,c show different measurement geometries, implying different constraints. Panels (1): schematics of long-rangetunnellings, when standard short-range ones are Zeno-suppressed. Panels (2): evolution of on-site densities; atoms effectivelytunnel between disconnected regions due to correlations. Panels (3): entanglement entropy growth between illuminated andnon-illuminated regions. Panels (4): correlations between different modes (orange) and within the same mode (green); atomnumber NI (NNI) in illuminated (non-illuminated) mode. a, Atom number in the central region is frozen: system is dividedinto three regions and correlated tunnelling occurs between non-illuminated zones (a.1). Standard dynamics happens withineach region, but not between them (a.2). Entanglement build up (a.3). Negative correlations between non-illuminated regions(green) and zero correlations between two modes (orange) (a.4). Initial state: |1, 1, 1, 1, 1, 1, 1〉, γ/J = 100. b, Even sites areilluminated, freezing Neven and Nodd. Long-range tunnelling is represented by any pair of blue and red arrows (b.1). Cor-related tunnelling occurs between non-neighbouring sites without changing mode populations (b.2). Entanglement build up(b.3). Negative correlations between edge sites (green) and zero correlations between two modes (orange) (b.4). Initial state:|0, 1, 2, 1, 0〉, γ/J = 100. c, Atom number difference between two central sites is frozen. Correlated tunnelling leads to exchangeof long-range atom pairs between illuminated and non-illuminated regions (c.1,2). Entanglement build up (c.3). In contrastto previous examples, sites in the same zones (illuminated/ non-illuminated) are positively correlated (green), while atoms indifferent zones are negatively correlated (orange) (c.4). Initial state: |0, 3, 1, 0〉, γ/J = 500. 1D lattice, U/J = 0.

fect suppresses usual short-range tunnelling, the atomscan still effectively tunnel even to distant sites due tothe long-range correlated tunnelling induced by globalmeasurement. Moreover, such a nonlocal high-order tun-nelling creates entanglement between zones disconnectedby the measurement, which is not visible in density-density correlations. Quantum optical engineering of

nonlocal coupling to environment, combined with quan-tum measurement, can allow the design of nontrivialsystem-bath interactions, enabling new links to quan-tum simulations [39] and thermodynamics [40]. Ourpredictions can be tested using both macroscopic mea-surements [11–13, 41] as well as novel methods based onsingle-site resolution [42–45]. Based on off-resonant scat-

7

tering and thus being non-sensitive to a detailed levelstructure, our approach can be applied to other arrays ofnatural or artificial quantum objects: molecules [46], ions[47], atoms in multiple cavities [48], semiconductor [49]or superconducting [50] qubits.

ACKNOWLEDGMENTS

The work was supported by the EPSRC (DTA andEP/I004394/1).

METHODS

We describe atomic dynamics by the (Bose-) HubbardHamiltonian. For the bosonic case,

H0 = −~J∑〈i,j〉

b†jbi + ~U

2

∑i

ni (ni − 1) , (2)

while for the fermionic case

H0 = −~J∑σ=↑,↓

∑〈i,j〉

f†j,σfi,σ − ~U∑i

ni,↑ni,↓, (3)

where b and fσ are respectively the bosonic and fermionicannihilation operators, n is the atom number operator,U and J are the on-site interaction and tunnelling co-efficients. The Hamiltonian of the light-matter systemis [14]

H = H0 +∑l

~ωla†l al + ~∑l,m

Ulma†l amFlm (4)

where al are the photon annihilation operators for thelight modes with frequencies ωl, Ulm = glgm/∆a, gl arethe atom-light coupling constants, and ∆a = ωp − ωa isthe probe-atom detuning. The operator Flm = Dlm +Blm couples atomic operators to the light fields:

Dlm =∑i

J lmii ni, Blm =∑〈i,j〉

J lmij b†i bj , (5)

J lmij =

∫w(r− ri)u

∗l (r)um(r)w(r− rj) dr. (6)

Flm originations from the overlaps between the lightmode functions ul(r) and density operator n(r) =

Ψ†(r)Ψ(r), after the matter-field operator is expressedvia Wannier functions: Ψ(r) =

∑i biw(r−ri). Dlm sums

the density contributions ni, while Blm sums the matter-field interference terms. The light-atom coupling via op-erators assures the dependence of light on the atomicquantum state. These equations can be extended tofermionic atoms introducing an additional index for thepolarisation of light modes.

From (3) we can compute the Heisenberg equation forlight operators in the stationary limit [14]. Specifically,we consider two light modes: a coherent probe beam a0

and the scattered light a1, which is enhanced by a cavitywith the decay rate κ. This allows us to express a1 as

a1 =iU10a0

i∆p − κF10 ≡ CF10 (7)

where ∆p = ω0−ω1 is the probe-cavity detuning, and C isthe cavity analogue of the Rayleigh scattering coefficientin free space [17]. In the main text, we drop the subscriptin a1 and superscripts in J lmij .

Quantum trajectories describe the evolution of a quan-tum system conditioned on the result of a measure-ment [20]. It is possible to represent dynamics of asystem with two different processes: non-Hermitian dy-namics and quantum jumps. The first one describesevolution of the system between two consecutive mea-surement events, while the second one models the pho-todetections. These can be simulated introducing thejump operator c =

√2κa1 and effective Hamiltonian

Heff = H0 − i~c†c/2. Starting from the time t0, a ran-dom number r between 0 and 1 is generated with uniformprobability, and we solve the effective Schrödinger equa-tion

d

dt|ψ(t)〉 = − i

~Heff |ψ(t)〉 (8)

until the time moment tj such that 〈ψ(tj)|ψ(tj)〉 = r.At this time moment, a photon is detected and the jumpoperator is applied to the state of the system, which issubsequently normalised:

|ψ(tj)〉 →c|ψ(tj)〉√

〈ψ(tj)|c†c|ψ(tj)〉. (9)

Finally, a new random number is generated and the pro-cedure is iterated setting tj as a new starting time. Thistechnique allows to simulate a single quantum trajectory,thus modelling result of a single experimental run.

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