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

Trapping ultracold atoms in optical lattices enabled numerous breakthroughs uniting several disciplines. Coupling these systems to quantized light leads to a plethora of new phenomena and has opened up a new field of study. Here we introduce a physically novel source of competition in a many-body strongly correlated system: We prove that quantum backaction of global measurement is able to efficiently compete with intrinsic short-range dynamics of an atomic system. The competition becomes possible due to the ability to change the spatial profile of a global measurement at a microscopic scale comparable to the lattice period without the need of single site addressing. In coherence with a general physical concept, where new competitions typically lead to new phenomena, we demonstrate novel nontrivial dynamical effects such as large-scale multimode oscillations, long-range entanglement and correlated tunneling, as well as selective suppression and enhancement of dynamical processes beyond the projective limit of the quantum Zeno effect. We demonstrate both the break-up and protection of strongly interacting fermion pairs by measurement. Such a quantum optical approach introduces into many-body physics novel processes, objects, and methods of quantum engineering, including the design of many-body entangled environments for open systems.

Trapping ultracold atoms in optical lattices enabled numerous breakthroughs uniting several disciplines. Although the light is a key ingredient in such systems, its quantum properties are typically neglected, reducing the role of light to a classical tool for atom manipulation. Here we show how elevating light to the quantum level leads to novel phenomena, inaccessible in setups based on classical optics. Interfacing a many-body atomic system with quantum light opens it to the environment in an essentially nonlocal way, where spatial coupling can be carefully designed. The competition between typical processes in strongly correlated systems (local tunnelling and interaction) with global measurement backaction leads to novel multimode dynamics and the appearance of long-range correlated tunnelling capable of entangling distant lattices sites, even when tunnelling between neighbouring sites is suppressed by the quantum Zeno effect. We demonstrate both the break-up and protection of strongly interacting fermion pairs by different measurements.
Ultracold gases trapped in optical lattices represent a successful interdisciplinary field: Atomic systems allow quantum simulations of phenomena predicted in condensed matter and particle physics, and find applications in quantum information processing [1]. Although light plays a key role in such systems, its quantum properties have been neglected in most works, reducing it to a classical tool for creating intriguing atomic states. Here, we introduce the quantumness of light into strongly correlated atomic systems and demonstrate phenomena that cannot be realised using classical optical setups. The recognized advantage of ultracold gases in contrast to condensed matter materials is their isolation from the environment. However, this creates a challenge to introduce any controlled dissipation (e.g. atom losses and collisions are difficult to manipulate). Here we show that coupling atoms to quantized light, which can be continuously measured, introduces a controllable decoherence channel into Atoms in an optical lattice are probed by a coherent light beam, and the light scattered at a particular angle 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 scattering from multiple lattice sites creates nontrival, spatially nonlocal coupling to the environment, which is impossible to obtain with local interactions [2]. Such a quantum optical approach can broaden the field even further, allowing quantum simulation of models unobtainable using classical light, and the design of novel systems beyond condensed matter analogues.
Collapse of the atomic wave function, when only the scattered light is measured, reflects the measurement backaction. This is one of the most fundamental manifestations of quantum mechanics [3], here due to the lightmatter entanglement. We demonstrate how this backaction and the spatially nonlocal quantum Zeno effect, being introduced in a strongly correlated system, compete with many-body dynamics defined by standard local processes (tunnelling and on-site interaction). Highlighting such competition takes one beyond recent quantum nondemolition (QND) approaches [4][5][6][7][8][9][10], where either manybody dynamics or measurement backaction did not play any role. For bosons, it leads to the emergence of dynamical macroscopic superpositions (multimode Schrödinger cat states). For strongly interacting fermions, we demonstrate the measurement-induced break-up and protection of fermion pairs. Even if standard tunnelling between neighbouring sites is suppressed by the quantum Zeno effect, long-range correlated tunnelling appears, leading to dynamical generation entanglement between distant sites. The generated spatial modes of matter fields can be considered as designed systems and reservoirs. We can make analogy with famous cavity QED experiments [3], where the quantum light states in a cavity were probed and projected by measuring atoms. Here, light and matter are reversed and we probe matter-fields with light. A striking advancement is that the number of quantum matter waves (sites with trapped atoms) can be easily scaled from few to thousands by changing the number of illuminated sites, while scaling cavities is an extreme challenge [3]. This work, together with recent experi-ments where our predictions can be tested (Bose-Einstein Condensates trapped inside a cavity [11][12][13]), will help to close the gap between quantum optics and quantum gases, by joining quantized light and strongly correlated many-body systems, thus approaching the fully quantum regime of light-matter interaction [5,14,15].
We consider off-resonant light scattering from N atoms trapped in an optical lattice with period d and L sites [14] (see Methods and Fig. 1). The light scattered at a particular angle can be selected and enhanced by a cavity [16] with decay rate κ. Similar to classical optics, the light amplitude is given by a sum of scatterings from all atoms with coefficients dependent on their positions: a = C(D +B), where a is the photon annihilation operator, C is the Rayleigh scattering coefficient and where b j andn j = b † j b j are the atomic annihilation and number operators at site j ( i, j sums over neighbouring sites), and J ij are given in Methods. In equation (1)D describes scattering from the on-site densities, whileB that from the small inter-site densities [17]. For well-localised atoms, the second term is usually neglected, and a = CD with J jj = u * out (r j )u in (r j ), where u(r) are the mode functions of input and scattered light (e.g., u in,out (r) = exp(ik in,out ·r) for travelling and u in,out (r) = cos(k in,out ·r) for standing waves with wave vectors k in,out ). (A peculiar case of light scattering from inter-site regions [17], where a = CB, will be underlined here later). For spin-1 2 fermions we use two light polarizations a x,y that couple differently to two spin densitiesn ↑j ,n ↓j allowing measurement of their linear combinations, e.g., a x = CD x = C L j=1 J jjρj and a y = CD y = C L j=1 J jjmj , whereρ j =n ↑j +n ↓j and m j =n ↑j −n ↓j are the mean density and magnetisation. This property has recently been used to investigate spin-spin correlations in Fermi gases [18,19].
We focus on a single run of a continuous measurement experiment using the quantum trajectories technique [20] (see Methods). The evolution is determined by a stochastic process described by quantum jumps (the jump operator c = √ 2κa is applied to the state when a photodetection occurs) and non-Hermitian evolution with the HamiltonianĤ eff =Ĥ 0 − i c † c/2 between jumps, where H 0 is the usual (Bose-)Hubbard Hamiltonian. Importantly, the measurement introduces a new energy and time scale γ = |C| 2 κ, which competes with the two other standard scales responsible for unitary dynamics of closed systems (tunnelling J and on-site interaction U ). If atoms scatter light independently in uncontrolled directions (as in spontaneous emission [21] or local addressing), independent jump operators c j would be applied to each site, projecting the atomic state to a fully mixed state [21]. In contrast, here we consider global coherent scattering, where the single global jump operator c is given by the sum over all sites, and the local coefficients J jj (1) responsible for the atom-environment coupling (via the light mode a) can be engineered by optical geometry. Thus, atoms are coupled to the environment globally, and atoms that scatter light with the same phase are indistinguishable to light scattering (i.e. there is no "which-path information"). As a striking consequence, the quantum superpositions are strongly preserved in the final projected states, and the systems splits into several spatial modes [22], where all atoms belonging to the same mode are indistinguishable, while being distinguishable from atoms belonging to different modes.
We engineer the atom-environment coupling coefficients J jj using standing or traveling waves at different angles to the lattice. If both probe and scattered light are standing waves crossed at such angles to the lattice that projection k in ·r is equal to k out ·r and shifted such that all even sites are positioned at the nodes (do not scatter light), one gets J jj = 1 for odd and J jj = 0 for even sites. Thus we measure the number of atoms at odd sites only (the jump operator is proportional tô N odd ), introducing two modes, which scatter light differently: odd and even sites. The coefficients J jj = (−1) j are designed by crossing light waves at 90 • such that atoms at neighboring sites scatter light with π phase difference [4, 10, 23-25], givingD =N even −N odd , introducing the same modes, but with different coherence between them. Moreover, using travelling waves crossed at the angle such that each R-th site is indistinguishable ((k in − k out )·r j = 2πj/R), introduces R modes with macroscopic atom numbersN l :D = R l=1N l e i2πl/R . Here two (odd-and even-site modes) appear for R = 2. Therefore, we reduce the jump (measurement) operator from being a sum of numerous microscopic contributions from individual sites to the sum of smaller number of macroscopically occupied modes with a very nontrivial spatial overlap between them. In the following, we will show how such globally designed measurement backaction introduces spatially long-range interactions of these modes, and demonstrate novel effects resulting from the competition of mode dynamics with standard local processes in a many-body system.
We start with non-interacting bosons (U/J = 0), and demonstrate the competition between global measurement and local tunnelling. Because of such competition, the weak measurement (γ J) is unable to freeze the atom numbers as expected for quantum Zeno dynamics [26][27][28]. In contrast, atom-number measurement leads to giant oscillations of particle number between the modes. In Figs. 2a-c we show the atom number distributions in one of the modes for R = 2 (N odd ) and R = 3 (N 1 ). Without measurement, these distributions would spread strongly and may show only oscillation amplitudes proportional to the initially created imbalance (thus, tiny oscillations for initially tiny imbalance). In contrast, here we observe (i) full exchange of atoms between the modes independent of the initial state, (ii) the distributions con- sist of a small number of well-defined components, and (iii) these components are squeezed even by weak measurement. The multi-component structure is explained by the degeneracy of spatially multimode states, as the measurement of photon number (intensity) a † a is not sensitive to the light phase. Therefore, all permutations of mode occupations that scatter light with the same intensity are indistinguishable. For R = 2, the state is 2-fold degenerate, for R > 2 it is 2R-fold degenerate. Fig. 2b reflects the superposition of two states with positive and negative N odd − N even , while Fig. 2c reflects the superposition in the three-mode case. As an important consequence, one gets a dynamical method to prepare multicomponent macroscopic superpositions (Schrödinger cat or NOON states), which are useful in quantum metrology and information. This method is deterministic without need for external quantum control [29,30], as the continuously monitored intensity directly shows, when the number splitting in the components reaches its maximum (corresponding to the maximal macroscopicity [31]), and the lattice depth can be ramped up to freeze further dynamics. Such states are indeed fragile to decoherence such as photon losses, but setups can be modified to make them more robust [32]. In addition, the state in Fig. 2a consists of only one component (here N odd is measured directly) and is therefore insensitive to photon losses. Reducing the number of modes and components can enable study the transition between quantumness and classicality, where nontrivial superpositions cannot exist [33].
We found that for many atoms, the R-mode problem can be treated analytically reducing it to R sites, if the initial state is superfluid. Giant oscillations are explained by the action of an effective quasi-periodic driving force due to the measurement backaction. In particular, for R = 2 (effective double-well [34,35]), we showed that the system leaves its stationary state (N odd = N even ) with the increment N γ/2. Note that dynamics described here will be visible already in a single experimental run. In contrast, averaging over many runs, which corresponds to the master equation solution, masks such effects completely. This happens because the oscillation phase changes from realization to realization as is known from works involving single and multiple measurements [36].
In contrast to bosons, dynamics for two modes of noninteracting fermions does not show well-defined oscillations (Fig. 2d) due to Pauli exclusion. However, while the initial ground state is a product of ↑ and ↓ wave functions (Slater determinants), the measurement introduces an effective interaction between two spin components and the state gets entangled by measurement.
Carefully choosing geometry, one can suppress the onsite contribution to light scattering and effectively concentrate light between the sites, thus in-situ measuring the matter-field interference b † i b i+1 [17]. In this case, a = CB, and the coefficients J ij (1) can be engineered [17]. For J ij = 1, the jump operator is proportional to the kinetic energy E K and tends to freeze the system in eigenstates of the non-interacting Hamiltonian. Being insensitive to the sign of E K , the measurement projects to a superposition of states with opposite kinetic energies: a superposition of matter waves propagating with different momenta. The measurement freezes dynamics for any γ/J, since the jump operator andĤ eff have the same eigenstates. As a result of the detection, the atoms quickly spread across the lattice, and the density distribution becomes uncertain (Fig. 2e), clearly illustrating the quantum uncertainty relation between the numberand phase-related variables (n i and b † i b i+1 ). Engineering J ij , can lead to the measurement-based preparation of peculiar multicomponent momentum (or Bloch) states.
Including the interactions (U/J = 0), dynamics changes as the measurement competes with both tunnelling and interaction. To highlight this, we focus on the variance of the measured variable averaged over many trajectories σ 2 D traj , when it reaches its steady state. In Fig. 3, we show results for the measurement ofN odd .
For bosons (Fig. 3a), without measurement the number fluctuations σ 2 D monotonically decrease for increasing U , reflecting the superfluid -Mott insulator phase transition. However, the measurement changes this behaviour and σ 2 D traj varies non-monotonically. For weak interac-tion, the fluctuations are strongly squeezed below those of the ground state; they then quickly increase, reach their maximum, and subsequently decrease as the interaction becomes stronger. We explain this effect by looking at single trajectories: For small U/J, as described above, the measurement strongly squeezes the occupation of one mode (Fig. 2a). However, the local atom repulsion prevents the formation of states with high population in one of two modes (Fig. 4a). Because of the local interaction, states with different imbalances oscillate with different frequencies and, hence, the measurement does not decrease the fluctuations ofN odd as efficiently as in the non-interacting case. For large U , both global measurement and local interaction tend to squeeze the fluctuations, but as the measurement destroys the Mott insulator, the fluctuations are larger than in the ground state.
For fermions (Figs. 3b,c), the ground state of the attractive Hubbard model in the strong interacting regime contains mainly doubly occupied sites (pairs). Therefore, without measurement, the fluctuations of total pop-ulationD x =N ↑odd +N ↓odd (σ 2 Dx ) increase with U/J as empty or doubly occupied sites become more probable. We can probe either the total populationD x , or both total population and magnetisationD y =M odd = N ↑odd −N ↓odd . In the first case (Figs. 3b,c), the measurement quickly squeezes σ 2 Dx , but destroys the pairs as it does not distinguish between singly or doubly occupied sites. This manifests the break-up of local fermion pairs by the global measurement and the corresponding trajectory is shown in Figs. 4c: while initially dynamics contains only even values ofN odd (i.e. pairs), the measurement also creates unpaired (odd) fermion numbers. In contrast, measuring both density and magnetisation reduces their fluctuations (i. e. unpaired fermions) and increases the lifetime of doubly occupied sites. This manifests the measurement-induced protection of fermion pairs and the trajectory is shown in Fig. 4d: The number distribution contains only even values of N odd indicating that fermions tunnel only in pairs.
When the measurement is strong (γ J), we will show the following nontrivial spatial effects. First, the measurement can freeze the mode atom numbers, thus decorrelating the numbers in different modes, while protecting typical dynamics within each mode. Second, even if the modes are number-decorrelated and the standard tunnelling between them is suppressed by the quantum Zeno effect (i.e. close to full projection), they can still be entangled by another process. We show that higherorder long-range correlated multi-tunnelling events appear: e.g., atoms can tunnel only in pairs (second order tunnelling) and, importantly, this pair is delocalized and correlated in space. Although for clarity we demonstrate such phenomena in Fig. 5 for small systems, they also hold for macroscopic modes.
First (Fig. 5a), we freeze the atom numberN illum in the central illuminated region by detecting light in the diffraction maximum (a = CN illum ) [10,32]. The lattice c, Attractive fermionic Hubbard model in the strong interaction limit. Measuring only the total population at odd sitesDx =N ↑odd +N ↓odd quickly creates singly occupied sites, demonstrating measurement-induced break-up of fermion pairs (L = 8, N ↑ = N ↓ = 4, U/J = 10, γ/J = 0.1). d, The same as c, but with added measurement of the magnetisation at odd sitesDy =M odd =N ↑odd −N ↓odd . This protects the doubly occupied sites, thus, demonstrating protection of fermion pairs by measurement. The distribution of N odd vanishes for odd numbers, implying that the fermions tunnel only in pairs. Simulations for 1D lattice.
is thus divided into two modes: non-illuminated zones 1 and 3 and the illuminated one 2 (Fig. 5a.1). Typical dynamics occurs within each zone, but the standard tunnelling between the zones is suppressed (Fig. 5a.2). Importantly, atoms can still penetrate, if the process does not change N illum : An atom from 1 can tunnel to 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 twostep processes 1 → 2 → 3 or 3 → 2 → 1. This is supported by the negative (anti-)correlations δN 1 δN 3 =  N 1 N 3 − N 1 N 3 showing that an atom disappearing in 1 appears in 3, while there are no correlations between illuminated and non-illuminated regions, (δN 1 + δN 3 )δN 2 = 0 (Fig. 5a.4). Surprisingly, the correlated tunnelling still builds entanglement between illuminated and non-illuminated regions (Fig. 5a.3) underlying the intermediate (virtual) step in the two-tunnelling event. This effect may lead to creation of intriguing multipartite entangled states with no simple classical correlations [37].
To make correlated tunnelling visible even in the mean atom number, we consider illumination of all even sites (Fig. 5b) thus freezing both N even and N odd . Fig. 5b.2 shows that while N even is fixed, there is a slow atom exchange between odd sites, although they are disconnected. Fig. 5b.1 clarifies the nontrivial character of correlated tunnelling. The correlations can appear between all possible tunnellings leading to the fixed atom numbers of modes. This again suggests that global coherent addressing favours correlated tunnelling, in contrast to local uncorrelated addressing of individual sites that should decrease the probability of individual tunnelling events being correlated. Scheme in Fig. 5b.1 can help to design a nonlocal reservoir for the tunnelling (or decay) of atoms from one region to another. For example, if the atoms are placed only at odd sites, their standard tunnelling is strongly suppressed as there is no second tunnelling event. If, however, one adds some atoms to even sites (even if they are far from the initial atoms), the slow correlated tunnelling ("decay") becomes allowed and its rate can be tuned by the number of added atoms. This resembles the repulsively bound pairs created by local interactions [38]. In contrast, here the atoms are long-range correlated due to the global measurement.
The negative number correlations are typical for systems with constraints (superselection rules) such as fixed atom number. The constraints implied by our global measurement are more general. For example, in Fig. 5c we show the generation of positive number correlations (shown in Fig. 5c.4) by freezing the atom number difference between the sites (N odd − N even , by measuring at the diffraction minimum). Thus, atoms can only enter or leave this region in pairs, which again is possible due to correlated tunnelling (Figs. 5c1-2) and manifests positive correlations. Note that, using more modes, the design of higher-order multi-tunnelling events is possible.
In summary, we introduced global quantum measurement in strongly correlated many-body systems and demonstrated the competition between the global backaction and standard local processes. It leads to spatially multimode dynamics producing macroscopic superpositions useful for quantum information and metrology. We showed the possibility of measurement-induced breakup and protection of strongly interacting fermion pairs. For strong measurement, when the quantum Zeno ef- Long-range correlated tunnelling and entanglement, dynamically induced by strong global measurement. a,b,c show different measurement geometries, implying different constraints. Panels (1): schematics of long-range tunnellings, when standard short-range ones are Zeno-suppressed. Panels (2): evolution of on-site densities; atoms effectively tunnel between disconnected regions due to correlations. Panels (3): entanglement entropy growth between illuminated and non-illuminated regions. Panels (4): correlations between different modes (orange) and within the same mode (green); atom number NI (NNI ) in illuminated (non-illuminated) mode. a, Atom number in the central region is frozen: system is divided into three regions and correlated tunnelling occurs between non-illuminated zones (a.1). Standard dynamics happens within each 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 are illuminated, freezing Neven and N odd . Long-range tunnelling is represented by any pair of blue and red arrows (b.1). Correlated 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 exchange of long-range atom pairs between illuminated and non-illuminated regions (c.1,2). Entanglement build up (c.3). In contrast to previous examples, sites in the same zones (illuminated/ non-illuminated) are positively correlated (green), while atoms in different 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 atoms can still effectively tunnel even to distant sites due to the long-range correlated tunnelling induced by global measurement. Moreover, such a nonlocal high-order tunnelling creates entanglement between zones disconnected by the measurement, which is not visible in densitydensity correlations. Quantum optical engineering of nonlocal coupling to environment, combined with quantum measurement, can allow the design of nontrivial system-bath interactions, enabling new links to quantum simulations [39] and thermodynamics [40]. Our predictions can be tested using both macroscopic measurements [11][12][13]41] as well as novel methods based on single-site resolution [42][43][44][45]. Based on off-resonant scat-tering and thus being non-sensitive to a detailed level structure, our approach can be applied to other arrays of natural 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 and EP/I004394/1).

METHODS
We describe atomic dynamics by the (Bose-) Hubbard Hamiltonian. For the bosonic case, while for the fermionic casê where b and f σ are respectively the bosonic and fermionic annihilation operators,n is the atom number operator, U and J are the on-site interaction and tunnelling coefficients. The Hamiltonian of the light-matter system is [14]Ĥ where a l are the photon annihilation operators for the light modes with frequencies ω l , U lm = g l g m /∆ a , g l are the atom-light coupling constants, and ∆ a = ω p − ω a is the probe-atom detuning. The operatorF lm =D lm + B lm couples atomic operators to the light fields: J lm ij = w(r − r i )u * l (r)u m (r)w(r − r j ) dr.
F lm originations from the overlaps between the light mode functions u l (r) and density operatorn(r) =Ψ † (r)Ψ(r), after the matter-field operator is expressed via Wannier functions:Ψ(r) = i b i w(r−r i ).D lm sums the density contributionsn i , whileB lm sums the matterfield interference terms. The light-atom coupling via operators assures the dependence of light on the atomic quantum state. These equations can be extended to fermionic atoms introducing an additional index for the polarisation of light modes. From (3) we can compute the Heisenberg equation for light operators in the stationary limit [14]. Specifically, we consider two light modes: a coherent probe beam a 0 and the scattered light a 1 , which is enhanced by a cavity with the decay rate κ. This allows us to express a 1 as a 1 = iU 10 a 0 i∆ p − κF 10 ≡ CF 10 where ∆ p = ω 0 −ω 1 is the probe-cavity detuning, and C is the cavity analogue of the Rayleigh scattering coefficient in free space [17]. In the main text, we drop the subscript in a 1 and superscripts in J lm ij . Quantum trajectories describe the evolution of a quantum system conditioned on the result of a measurement [20]. It is possible to represent dynamics of a system with two different processes: non-Hermitian dynamics and quantum jumps. The first one describes evolution of the system between two consecutive measurement events, while the second one models the photodetections. These can be simulated introducing the jump operator c = √ 2κa 1 and effective Hamiltonian H eff =Ĥ 0 − i c † c/2. Starting from the time t 0 , a random number r between 0 and 1 is generated with uniform probability, and we solve the effective Schrödinger equation until the time moment t j such that ψ(t j )|ψ(t j ) = r. At this time moment, a photon is detected and the jump operator is applied to the state of the system, which is subsequently normalised: .
Finally, a new random number is generated and the procedure is iterated setting t j as a new starting time. This technique allows to simulate a single quantum trajectory, thus modelling result of a single experimental run.