Bounds on adaptive quantum metrology under Markovian noise

We analyze the problem of estimating a scalar parameter $g$ that controls the Hamiltonian of a quantum system subject to Markovian noise. Specifically, we place bounds on the growth rate of the quantum Fisher information with respect to $g$, in terms of the Lindblad operators and the $g$ derivative of the Hamiltonian $H$. Our new bounds are not only more generally applicable than those in the literature—for example, they apply to systems with time-dependent Hamiltonians and/or Lindblad operators, and to infinite-dimensional systems such as oscillators—but are also tighter in the settings where previous bounds do apply. We derive our bounds directly from the master equation describing the system, without needing to discretize its time evolution. We also use our results to investigate how sensitive a single detection system can be to signals with different time dependences. We demonstrate that the sensitivity bandwidth is related to the quantum fluctuations of $\partial H/\partial g$, illustrating how “nonclassical” states can enhance the range of signals that a system is sensitive to, even when they cannot increase its peak sensitivity.

Figure 1

Top-left panel: Plots of $\mathcal{F}\left(t\right)$ bounds in the HLS case from Sec. 2, compared to those from previous paper. The solid curves correspond to the bound on $\mathcal{F}\left(t\right)$ from Eq. (31); the red curves take $c_1=1$, $c_2=1$ (all quantities are taken to be dimensionless), and the blue curves take $c_1=\sqrt{5}$, $c_2=1$. The gray-dotted line shows the $\mathcal{F}\left(t\right)\le 4c_{2}t$ bound from [8, 9] [Eq. (26)], while the red and blue-dotted curves correspond to the quadratic bounds from Eq. (33), for the appropriate $c_1$. Top-right panel: As per the top-left panel, but plotting $\dot{\mathcal{F}}\left(t\right)$ bounds. This illustrates how the large-$t$ scaling of our bounds is only affected by $c_2$, and not $c_1$. Bottom-left panel: Solid curves as in top row plots; dashed curves plot $\mathcal{F}\left(t\right)$ for different detection schemes with a damped harmonic oscillator probe, as considered in Sec. 2e (taking $\gamma =1$ and $n_T=0$). The dashed-red curve corresponds to preparing an oscillator in its ground state, and allowing it to evolve unimpeded under the influence of the linear forcing. This attains our $\mathcal{F}\left(t\right)$ bound (solid-red curve) for $c_1=c_2=1$ at $t\le t_c=2(c_2/c_1^2)ln2\simeq 1.4$, but then asymptotes to a constant value of $\mathcal{F}\left(t\right)$. The blue-dashed curve corresponds to preparing an oscillator in the $N=2$ Fock state, and allowing it to evolve. This has faster $\mathcal{F}\left(t\right)$ growth rate at small times, but that growth slows down for $t\gtrsim t_c=\frac{2ln2}{5}\simeq 0.3$. As well as representing the Eq. (31) bound with $c_1=1$, the solid-red curve also corresponds to $\mathcal{F}\left(t\right)$ for a harmonic oscillator prepared in its ground state, but coupled to a continuum of ancillae for $t\ge t_c=2ln2$, to maintain the system at $\dot{\mathcal{F}}=4c_2$. Bottom-right panel: As per the bottom-left panel, but plotting $\dot{\mathcal{F}}\left(t\right)$.

Figure 2

Plots of $\dot{\mathcal{F}}$ (at large times) for a damped harmonic oscillator, as a function of the detuning $\delta \omega$ of the signal frequency from the oscillator's resonant frequency. The oscillator is assumed to be governed by the master equation in Eq. (39) (with $n_T=0$). Left panel: The blue curve corresponds to an oscillator that is critically coupled to ancillary degrees of freedom (e.g., a cavity mode coupled to a waveguide output), and initialized in its ground state. As discussed in Sec. 2e, this setup attains the $\dot{\mathcal{F}}\le 4{\left|\epsilon \right|}^2/\gamma$ bound at large enough times, for a resonant forcing. The pink curve corresponds to an oscillator critically coupled to a source of squeezed vacuum, with squeezing parameter $G_s=4$. The dashed pink curve corresponds to the oscillator being overcoupled to the same squeezed state source by a factor $\sim 4G_s$, which decreases on-resonance sensitivity by a constant factor, but results in a sensitivity bandwidth $\sim G_s$ times larger than for preparation in the ground state (Sec. 3b). This is illustrated by the dashed-blue curve, which corresponds to preparation in the ground state, overcoupled such that the on-resonance sensitivity matches that of the dashed-pink curve. Right panel: Plot of the time-averaged rate of accumulation of (classical) Fisher information for a scheme in which the oscillator is prepared in a Fock state, and then measured in the Fock basis after evolving for a time $t_1$, as described in Sec. 3c. The blue curve corresponds to preparation in the ground state, followed by measurement at the time, which optimises on-resonance sensitivity ($t_1\simeq 2.5/\gamma$), while the pink curve corresponds to preparation in the $N=4$ state, followed by measurement at the time, which optimises on-resonance sensitivity ($t_1\simeq 0.12/\gamma$).