Continuous measurement of an atomic current

We are interested in dynamics of quantum many-body systems under continuous observation, and its physical realizations involving cold atoms in lattices. In the present work we focus on continuous measurement of atomic currents in lattice models, including the Hubbard model. We describe a Cavity QED setup, where measurement of a homodyne current provides a faithful representation of the atomic current as a function of time. We employ the quantum optical description in terms of a diffusive stochastic Schrödinger equation to follow the time evolution of the atomic system conditional to observing a given homodyne current trajectory, thus accounting for the competition between the Hamiltonian evolution and measurement back action. As an illustration, we discuss minimal models of atomic dynamics and continuous current measurement on rings with synthetic gauge fields, involving both real space and synthetic dimension lattices (represented by internal atomic states). Finally, by “not reading” the current measurements the time evolution of the atomic system is governed by a master equation, where—depending on the microscopic details of our CQED setups—we effectively engineer a current coupling of our system to a quantum reservoir. This provides interesting scenarios of dissipative dynamics generating “dark” pure quantum many-body states.

Figure 1

Schematic of the current operators ${\mathcal{J}}_{i,i+1}$ and measurement setup in a cold atom context. (a) A 2D lattice system with finite flux $\theta$ resulting in a persistent current on each plaquette, and (b) a 1D setup. The current operator ${\mathcal{J}}_{i,i+1}$ involves the particles moving from site $i$ to site $i+1$. (c) Measurement schematic: Transitions between sites are made via Raman transitions driven by external lasers (black solid), and those to be measured are additionally coupled to a cavity field (red dashed). The cavity output is mixed with a local oscillator resulting in a homodyne current ${\mathcal{I}}^h\left(t\right)$ proportional to the local current. Coupling to all transitions results in a measurement of global current ${\widehat{\mathcal{J}}}_{\mathrm{tot}}$.

Figure 2

Current operators, for a three-site (state) system with enclosed flux $\theta$ (atomic version of a SQUID) realized equivalently with (a) external degrees of freedom on an optical lattice, and (b) internal atomic degrees of freedom, in a trapped gas.

Figure 3

Schematic of the atom-cavity coupling for (a) the asymmetric coupling and (b) the symmetric coupling. In both cases, the Raman transition mitigates hopping between the system levels ${\widehat{a}}_{1/2}$ where one or both legs of the Raman transition are coupled via the bare coupling $\tilde{g}$ to a cavity mode $\widehat{d}$. Together with standard laser fields with Rabi frequencies ${\mathrm{\Omega }}_2$ and ${\mathrm{\Omega }}_{L/R}$ the effective couplings are given, for large detunings, by $g=\tilde{g}{\mathrm{\Omega }}_2/\mathrm{\Delta }$ and $g_{L/R}=\tilde{g}{\mathrm{\Omega }}_{L/R}/\mathrm{\Delta }$ in the asymmetric and symmetric systems, respectively. The cavity is driven with an input beam ${\widehat{d}}_{\mathrm{in}}$, which is assumed to be the vacuum field, and the output ${\widehat{d}}_{\mathrm{out}}$ is measured via homodyne measurement.

Figure 4

Coupling transitions to different cavities can result in measurement of (a) noncommuting operators ${\mathcal{J}}_{1,2}$, ${\mathcal{J}}_{2,3}$ or (b) commuting operators ${\mathcal{J}}_{4,1}$, ${\mathcal{J}}_{2,3}$. Evolution is given by the SSE with multiple couplings and jump operators, as given in Eq. (35). Realizations of such setups could be made in both external (lattice) systems or made up from internal states.

Figure 5

(a) Landscape of the potential term in the phase representation of the BH Hamiltonian (see derivation in Appendix pp2) with $\theta =\pi$ as function of ${\varphi }_1-{\varphi }_2$ and ${\varphi }_2-{\varphi }_3$. The minima are at ${\varphi }_1-{\varphi }_2={\varphi }_2-{\varphi }_3=2\pi /3,4\pi /3$, and define the two states $|\pm \rangle$ which are characterized by opposite current. (b) The double-well structure taken from a one-dimensional (1D) cut through the potential landscape, indicated in panel (a) with a black dashed line. The black-dotted line is for matching parameters to panel (a) and the red-solid line is for $\theta =1.02\pi$. In the latter case the double well is offset, into a tilted well.

Figure 6

Homodyne current $\sim \langle {\widehat{\sigma }}_z\rangle$ in a single trajectory of the SSE in Eq. (47), with an initial state satisfying $\langle {\widehat{\sigma }}_z\rangle =+1$. The parameter $\omega$ is the coupling strength between the two states. The ratio $\gamma /\omega$ is $5,2,1,0.5$ from top left to bottom right. For small $\gamma /\omega$ the evolution is dominated by Rabi oscillations, where the measurement is not strong enough to project the state into one of the two eigenstates. For large $\gamma /\omega$ the measurement acts to project the system onto an eigenstate of ${\widehat{\sigma }}_z$ until tunneling events, due to finite $\omega$, occur.

Figure 7

Continuous homodyne measurement of the global current in a system of three sites, with three particles, at $\theta =\pi /3$. (a) Ten different realizations of the SSE at $U=0,\gamma =J$. The measurement is QND and each evolution ends up in one of the eigenstates of the current operator. (b)–(d) The interplay between the measurement (with strength $\gamma$, which tends to gradually project the system to eigenstates of the global current operator) and interaction (with strength $U$) with respect to the tunneling rate ($J$); for a weak measurement the state is not completely pinned onto a current eigenstate, as the finite interaction $\propto U$ causes transitions between eigenstates with fixed global current. (b) $U=2J,\gamma =0.1J$. However, for a strong measurement, the state is projected to a current eigenstate, with the interaction causing transitions between these well defined states, (c) $U=J/2,\gamma =5J$ and (d) $U=2J,\gamma =5J$.

Figure 8

Continuous homodyne measurement of the local current. Plotted are four trajectory solutions to the SSE for (a) the symmetric measurement scheme and (b) the asymmetric measurement scheme. Parameters $N=3,L=3,\theta =\pi /3,j=1,\gamma =J,U=0$ are identical for both schemes. In the symmetric case, the measurement is always competing with the Hamiltonian, leading to no well defined current throughout the evolution, whereas in the case of the asymmetric measurement, the system is driven into a dark state (on a time scale determined by $\gamma /J$).

Figure 9

Energy spectrum of the Hamiltonian ${\widehat{H}}_J$ as a function of $\theta$ for (a) three levels, and $N=3$ and 4, four levels and $N=4$. Sets of degenerate states where dark states appear are highlighted in red. For $\theta$ at odd multiples of $\pi /4$ there is a set of dark states, while at even multiples of $\theta =\pi /4$ there is a single dark state.

Figure 10

Purity ($\mathrm{Tr}\left[{\rho }^2\right]$) of the density matrix, evolved with respect to the master equation. In each evolution a random initial (pure) density matrix is taken. Red dashed: A three-level system at $\theta =\pi /3$, a QND measurement of the global current ${\widehat{\mathcal{J}}}_{\mathrm{tot}}$. Note that the evolution in the case of a QND measurement is initial-state dependent (see text for discussion). Black solid: A three-level system at $\theta =\pi /3$, a measurement of ${\widehat{\mathcal{J}}}_{1,2}$, a non-QND measurement. A dark state exists, and the density matrix evolves into a pure state. Blue dotted: A three-level system at $\theta =\pi /2$, a measurement of ${\widehat{\mathcal{J}}}_{1,2}$, a non-QND measurement. No dark state exists, and the system evolves into a totally mixed state for all initial states.

Figure 11

Energy spectrum of the Hamiltonian ${\widehat{H}}_J$ as a function of $\theta$ for a $L=3$ level system with a different total particle number: (a) $N=1$, (b) $N=2$, (c) $N=3$, (d) $N=4$. At the symmetry points, $\theta =\pi /3,2\pi /3,\pi$ degeneracies emerge; in particular there is one set of $N+1$ degenerate states (highlighted in red). This set of degenerate states will be shown in the text to host a dark state to the local current measurement operator.

Figure 12

a) Three-level Raman system with coupling to the cavity $\tilde{g}$, and laser driven transition $\mathrm{\Omega }$ and detuning to the excited state $\mathrm{\Delta }$. We consider spontaneous emission from the excited state to the ground states with equal rate $\gamma$. b) Effective two-level system after elimination of the excited state, resulting in an effective coupling $g$ between the two states, and effective dissipation coupling the two levels ${\gamma }_s$ and dephasing ${\gamma }_{s^{\prime }}$ and ${\gamma }_{s^{\prime \prime }}$.

Figure 13

Evolution of $\mathrm{Tr}\left[{\widehat{\rho }}_c{\mathcal{J}}_{\mathrm{tot}}\right]$ with respect to the stochastic master equation with one measurement channel and the dissipation channels corresponding to the processes of Fig. 12. Each trajectory is made with a random initial (pure) state. In these trajectories $\theta =\pi$, and all dissipation channels have equal strength ${\gamma }_s={\gamma }_{s^{\prime }}$. Different panels indicate different levels of spontaneous emission: Upper Panel (red) ${\gamma }_s/{\gamma }_m=0.05$; Middle panel (green) ${\gamma }_s/{\gamma }_m=0.1$; Lower panel (blue) ${\gamma }_s/{\gamma }_m=0.5$. Black in each panel indicates reference for ${\gamma }_s/{\gamma }_m=0$.

Figure 14

Evolution of $\mathrm{Tr}\left[{\widehat{\rho }}_c{\mathcal{J}}_{1,2}\right]$ with respect to the stochastic master equation with one measurement channel and the dissipation channels corresponding to the processes of Fig. 12. In these trajectories $\theta =\pi$, and all dissipation channels have equal strength ${\gamma }_s={\gamma }_{s^{\prime }}$. Different panels indicate different levels of spontaneous emission: Upper Panel (red) ${\gamma }_s/{\gamma }_m=0.05$; Middle panel (green) ${\gamma }_s/{\gamma }_m=0.1$; Lower panel (blue) ${\gamma }_s/{\gamma }_m=0.5$. Black in each panel indicates reference for ${\gamma }_s/{\gamma }_m=0$.