Experimental observation of conditional past-future correlations

Conditional past-future correlations measure the lack of statistical independence between past and future system measurement outcomes when conditioned to a given state at a present time. Quantum non-Markovian memory effects are present whenever this correlation is not null. Conditional past-future correlations can also be used to detect initial system-environment correlated states. In this work, we provide theoretical and experimental evidence for these general properties. We build an optical setup that implements the dynamics of a qubit interacting with a dephasing spin bath. Both finite and infinite bath-size limits are observed. Confirmation of theoretical predictions for conditional past-future correlations is obtained. This work provides experimental support for quantum memory indicators based solely on outcomes of explicit system measurement processes.

Figure 1

The experimental setup for measuring the CPF correlation consists of five modules: (i) the single-photon source; (ii) preparation of initial system-bath states; (iii) the initial correlated states preparation (ICSP); (iv) dephasing dynamics between measurements; and (v) detecting devices for conditional probabilities. Projective measurements are performed by PBS ${\mathrm{\Omega }}_x, {\mathrm{\Omega }}_y$, and ${\mathrm{\Omega }}_z$. Key elements also include fiber coupler (FC) and interference filter (IF).

Figure 2

CPF correlations for the spin-bath model (3) with different sizes, $N=2,4,8$, and $N=\text{infinite}$. (a)–(d) are the analytical results. (e)–(h) are the respective experimental results obtained from the proposed optical platform with a precision of $15\times 15$ pixels in one period. The axes are the time intervals $gt$ and $g\tau$ between successive measurements.

Figure 3

System coherence behaviors $c\left(\tau \right)$ and $c^{yx}(t,\tau )$ between the first ($\left\{{\mathrm{\Omega }}_x\right\}$ and $\left\{{\mathrm{\Omega }}_y\right\}$) and last ($\left\{{\mathrm{\Omega }}_y\right\}$ and $\left\{{\mathrm{\Omega }}_z\right\}$) two measurements. (a) Finite spin bath with $N=8$. (b) Infinite bath size, $N=\infty$. Solid lines represent the theoretical predictions, and symbols represent experimental results.

Figure 4

CPF correlation for an initial system-environment correlated state induced by the total unitary dynamics during a time interval $g{t}_0$. (a) Finite bath, $N=8$. (b) Infinite bath size. In both cases, the symbols with error bars represent experimental results, and the solid lines are given by Eq. (5).