Physical currents for stochastic Einstein-Podolsky-Rosen quantum trajectories

Abstract

Theories of the measured homodyne current generated by a stochastic Schrödinger equation (SSE) can be tested in a simulation of the Einstein-Podolsky-Rosen (EPR) correlations for a two-mode squeezed state. We carry out such a simulation, and determine the correct stochastic term for the measured current in the broad-band limit. Stratonovich rather than Ito stochastic noise agrees with experiment. We show that this is relevant to measurement noise and errors in quantum technologies. By analyzing the SSE trajectories as measurement settings are changed, we propose a modern version of Schrodinger's gedanken experiment, where one measures position and momenta simultaneously, ``one by direct, the other by indirect measurement''.

Figures (4)

• Figure 1: The averaged unfiltered SSE homodyne current correlation $\left\langle j_{1}(\tau)j_{2}(\tau)\right\rangle$ vs the dimensionless time $\tau=\gamma\tau$ for a two-mode damped squeezed state, comparing different theories. The blue dashed line is the infinite bandwidth quantum solution, $\langle\hat{J}_{1}(\tau)\hat{J}_{2}(\tau)\rangle=-e^{-\tau}\sinh(2r)$, in agreement with the Stratonovich result (where $j_{k}\equiv j_{k}^{S}$), but far from the two Ito results (where $j_{k}\equiv j_{k}^{I}$), whether one evaluates the noise and the wave-function either at the same time or not. Here, $r=0.5$, the two lines are $\pm\sigma$ sampling error bars for a sample size of $2\times10^{5}$, and the time step-size is $0.05$.
• Figure 2: Graph of correlations when one phase angle is changed dynamically, so that at first the two quadratures have complementary phases and their quantum noise is not correlated. At a time $\tau=0.5$, one phase is changed so the quadratures become anti-correlated.
• Figure 3: The two-mode CIM success probability inferred from the filtered homodyne current outputs (black line) and from the wave-function (red dashed line). Both results are derived from solving the same homodyne SSE. Left graph, $\kappa=50$, right graph, $\kappa=10$, using $10^{4}$trajectories and a time-step of $\Delta\tau=0.003$.
• Figure 4: The averaged filtered homodyne current correlation $\langle J_{1}(\tau)J_{2}(\tau)\rangle$ vs the dimensionless time $\tau=\gamma t$ for a two-mode damped squeezed state with different detection bandwidth $\kappa$. The left and right plots correspond to detection bandwidths $\kappa$ of $10$ and $50$ respectively. The blue dashed line is the infinite bandwidth analytical solution, $\langle J_{1}(\tau)J_{2}(\tau)\rangle=-e^{-\tau}\sinh(2r)$. The noise in the current and the wave-function are evaluated at the same time. Here, $r=0.5$ is the squeezing parameter, the two lines are sampling error bars for a sample size of $2\times10^{5}$, and the time step-size is $0.001$.