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 an unusual additional 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 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 breakup 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.

Figure 1

Setup. 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. The photons escaping the cavity are detected, perturbing the atomic evolution via measurement backaction. This process competes with the usual (Bose-)Hubbard dynamics and leads to the phenomena described in the text.

Figure 2

Large oscillations between the measurement-induced spatial modes resulting from the competition between tunneling and weak-measurement backaction. The plots show single quantum trajectories. [(a)–(d)] Atom number distributions $p\left(N_l\right)$ in one of the modes, which show various number of well-squeezed components, reflecting the creation of macroscopic superposition states depending on the measurement configuration ($U/J=0,\gamma /J=0.01,L=N$, initial states: superfluid for bosons, Fermi sea for fermions). (a) Measurement of the atom number at odd sites ${\widehat{N}}_{\text{odd}}$ creates one strongly oscillating component in $p\left(N_{\text{odd}}\right)$ ($N=100$ bosons, $J_{jj}=1$ if $j$ is odd and 0 otherwise). (b) Measurement of ${({\widehat{N}}_{\text{odd}}-{\widehat{N}}_{\text{even}})}^2$ introduces $R=2$ modes and preserves the superposition of positive and negative atom number differences in $p\left(N_{\text{odd}}\right) [N=100$ bosons, $J_{jj}={(-1)}^{j+1}]$. (c) Measurement for $R=3$ modes (see text) preserves three components in $p\left(N_1\right)$ ($N=108$ bosons, $J_{jj}=e^{ij2\pi /3})$. (d) Measurement of ${\widehat{N}}_{\text{odd}}$ for fermions leads to oscillations in $p\left(N_{\text{odd}}\right)$, though not as clearly defined as for bosons because of Pauli blocking ($L=8$ sites, $N_{\uparrow }=N_{\downarrow }=4$ fermions, $J_{jj}=1$ if $j$ is odd and 0 otherwise). Simulations for 1D lattice.

Figure 3

Evolution of the on-site atomic density, while measuring the matter-field coherence between the sites $\widehat{B}$. The plots show single quantum trajectories. (a) Atoms, all initially at the edge site, are quickly spread across the whole lattice, leading to the large uncertainty in the atom number, while the matter-phase-related variable is defined (projected) by the measurement (bosons, $N=L=6,U/J=0,\gamma /J=0.1,J_{jj}=1$). (b) Atomic density spread and revival due to coherent tunneling in the absence of measurement (bosons, $N=L=6,U/J=0,\gamma /J=0,J_{jj}=1$). Simulations for 1D lattice.

Figure 4

Atom number fluctuations demonstrating the competition of global measurement with local interaction and tunneling. Number variances are averaged over many trajectories. Error bars are too small to be shown $(\sim 1$%), which emphasizes the fundamental nature of the squeezing. (a) Bose-Hubbard model with repulsive interaction. The fluctuations of the atom number at odd sites ${\widehat{N}}_{\text{odd}}$ in the ground state without a measurement (thin solid line) decrease as $U/J$ increases, reflecting the transition between the superfluid and Mott insulator phases. For the weak measurement, ${\langle {\sigma }_D^2\rangle }_{\mathrm{traj}}$ is squeezed below the ground-state value but then increases and reaches its maximum as the atom repulsion prevents oscillations and makes the squeezing less effective. In the strong interacting limit, the Mott insulator state is destroyed and the fluctuations are larger than in the ground state (100 trajectories, $N=L=6,J_{jj}=1$ if $j$ is odd and 0 otherwise). [(b) and (c)] Fermionic Hubbard model with attractive interaction; fluctuations of the total atom number at odd sites ${\widehat{D}}_x={\widehat{N}}_{\uparrow \text{odd}}+{\widehat{N}}_{\downarrow \text{odd}}$ (b) and of the magnetization at odd sites ${\widehat{D}}_y={\widehat{M}}_{\mathrm{odd}}={\widehat{N}}_{\uparrow \text{odd}}-{\widehat{N}}_{\downarrow \text{odd}}$ (c). Without measurement, the interaction favors formation of doubly occupied sites so the density fluctuations in the ground state are increasing while the magnetization ones are decreasing. The measurement creates singly occupied sites decreasing the density fluctuations and increasing the magnetization ones, which manifests the breakup of fermion pairs by measurement. The measurement-based protection of fermion pairs is shown in the next figure (100 trajectories, $L=8,N_{\uparrow }=N_{\downarrow }=4,J_{jj}=1$ if $j$ is odd and 0 otherwise.) Simulations for 1D lattice.

Figure 5

Conditional dynamics of the atom-number distributions at odd sites illustrating competition of the global measurement with local interaction and tunneling (single quantum trajectories, initial states are the ground states). (a) Weakly interacting bosons: The on-site repulsion prevents the formation of well-defined oscillations in the population of the mode. As states with different imbalance evolve with different frequencies, the squeezing due to the measurement is not as efficient as one observed in the noninteracting case ($N=L=6,U/J=1,\gamma /J=0.1,J_{jj}=1$ if $j$ is odd and 0 otherwise). (b) Strongly interacting bosons: Oscillations are completely suppressed and the number of atoms in the mode is rather well-defined, although less squeezed than in the Mott insulator ($N=L=6,U/J=10,\gamma /J=0.1,J_{jj}=1$ if $j$ is odd and 0 otherwise). (c) Attractive fermionic Hubbard model in the strong interaction limit. Measuring only the total population at odd sites ${\widehat{D}}_x={\widehat{N}}_{\uparrow \text{odd}}+{\widehat{N}}_{\downarrow \text{odd}}$ quickly creates singly occupied sites, demonstrating measurement-induced breakup of fermion pairs ($L=8,N_{\uparrow }=N_{\downarrow }=4,U/J=10,\gamma /J=0.1,J_{jj}=1$ if $j$ is odd and 0 otherwise). (d) The same as in (c) but with added measurement of the magnetization at odd sites ${\widehat{D}}_y={\widehat{M}}_{\mathrm{odd}}={\widehat{N}}_{\uparrow \text{odd}}-{\widehat{N}}_{\downarrow \text{odd}}$. This protects the doubly occupied sites, thus demonstrating protection of fermion pairs by measurement. The distribution of $N_{\mathrm{odd}}$ vanishes for odd numbers, implying that the fermions tunnel only in pairs. Simulations for 1D lattice.

Figure 6

Long-range correlated tunneling and entanglement, dynamically induced by strong global measurement in a single quantum trajectory. Panels (a), (b), and (c) show different measurement geometries, implying different constraints. Panels (1): Schematic representation of the long-range tunneling processes, when standard tunneling between different zones is 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 nonilluminated regions. Panels (4): Correlations between different modes (solid orange line) and within the same mode (dashed green line); atom number $N_I \left(N_{NI}\right)$ in illuminated (nonilluminated) mode. (a) Atom number in the central region is frozen: The system is divided into three regions and correlated tunneling occurs between nonilluminated zones (a.1). Standard dynamics happens within each region but not between them (a.2). Entanglement build up (a.3). Negative correlations between nonilluminated regions (dashed green line) and zero correlations between the $N_I$ and $N_{NI}$ modes (solid orange line) (a.4). Initial state: $|1,1,1,1,1,1,1\rangle ,\gamma /J=100,J_{jj}=[0,0,1,1,1,0,0]$. (b) Even sites are illuminated, freezing $N_{\text{even}}$ and $N_{\text{odd}}$. Long-range tunneling is represented by any pair of one blue and one red arrow (b.1). Correlated tunneling occurs between non-neighboring sites without changing mode populations (b.2). Entanglement build up (b.3). Negative correlations between edge sites (dashed green line) and zero correlations between the modes defined by $N_{\text{even}}$ and $N_{\text{odd}}$ (solid orange line) (b.4). Initial state: $|0,1,2,1,0\rangle ,\gamma /J=100,J_{jj}=[0,1,0,1,0]$. (c) Atom number difference between two central sites is frozen. Correlated tunneling leads to exchange of long-range atom pairs between illuminated and nonilluminated regions (c.1,2). Entanglement builds up (c.3). In contrast to previous examples, sites in the same zones (illuminated or nonilluminated) are positively correlated (dashed green line), while atoms in different zones are negatively correlated (solid orange line) (c.4). Initial state: $|0,2,2,0\rangle ,\gamma /J=100,J_{jj}=[0,-1,1,0]$. 1D lattice, $U/J=0$, the trajectory is smooth as no photons are detected.