Energetic Cost of Measurements Using Quantum, Coherent, and Thermal Light

Quantum measurements are basic operations that play a critical role in the study and application of quantum information. We study how the use of quantum, coherent, and classical thermal states of light in a circuit quantum electrodynamics setup impacts the performance of quantum measurements, by comparing their respective measurement backaction and measurement signal to noise ratio per photon. In the strong dispersive limit, we find that thermal light is capable of performing quantum measurements with comparable efficiency to coherent light, both being outperformed by single-photon light. We then analyze the thermodynamic cost of each measurement scheme. We show that single-photon light shows an advantage in terms of energy cost per information gain, reaching the fundamental thermodynamic cost.

Figure 1

Schematic of the setup. (a) Upper two panels: frequency spectra of the coherent and thermal light. The dashed lines show the frequencies $f_g^{(c)}$ and $f_e^{(c)}$, corresponding to cavity resonances with qubit in $|g⟩$ and $|e⟩$ states. Lower panel: illustration of the effective single-photon light that utilizes a rotation in the $\{e,f\}$ manifold plus two sideband pumps. The green circle arrow represents a rotation between the qubit $|e⟩$ and $|f⟩$ states with a rotation angle $\theta$. The two purple arrows represent the two sideband pumps at frequencies $f_{ef}^{(q)}+\mathrm{\Delta }f$ and $f_e^{(c)}+\mathrm{\Delta }f$ where $f_{ef}^{(q)}$ is the frequency of the $|e⟩\leftrightarrow |f⟩$ transition and we use a detuning $\mathrm{\Delta }f=0.5\mathrm{GHz}$ in the experiment. These two sideband pumps are equivalent to a $\pi$ rotation of the $|f⟩\otimes |0⟩\leftrightarrow |e⟩\otimes |1⟩$ transition (the dashed purple circle arrow). (b) Schematic of the cavity-transmon system. The output from the cavity is further demodulated and analyzed to obtain the measurement signal (see [14] for details of the signal processing).

Figure 2

Measurement backaction of different light sources. (a) Ramsey experiment sequence for the coherent and thermal light. The orange bar represents the pulse for the measurement light. The projective qubit measurement at the end of the sequence is realized by a high-power readout [28]. “$Q$” denotes qubit and “$C$” denotes cavity. (b) Ramsey experiment sequence for the single-photon light. The purple bars represent the sideband pumps used to introduce a $\pi$ rotation of the $|f⟩\otimes |0⟩\leftrightarrow |e⟩\otimes |1⟩$ transition. ${\theta }_{ef}$ represents a rotation in the transmon $\{e,f\}$ manifold with a rotation angle $\theta$. (c) Oscillation of qubit population as a function of the phase on the second $\pi /2$ pulse for coherent light probing. The amplitude of the oscillation is proportional to the qubit coherence. Similar measurements are performed for thermal and single-photon light. Different colors represent different emitted photon number (red: $n^{(\mathrm{emit})}=0$; blue: $n^{(\mathrm{emit})}=1.0$; green: $n^{(\mathrm{emit})}=2.8$; black: $n^{(\mathrm{emit})}=6.5$). (d) The qubit coherence as a function of the emitted photon number $n^{(\mathrm{emit})}$ with the maximal qubit coherence normalized to 1. The error bars indicate two standard deviations from five measurement repetitions. Solid lines are the theoretical prediction [14]. The red line corresponds to the form $e^{-n^{(\mathrm{emit})}/2}$ and the yellow line corresponds to the form $\sqrt{1-n^{(\mathrm{emit})}}$. (e) The measurement SNR for coherent and thermal light as a function of the emitted photon number $n^{(\mathrm{emit})}$. The error bars indicate two standard deviations from five measurement repetitions. The black solid line is a fit using a theoretical model for coherent light [14]. The inset is a typical histogram of the measurement signal using coherent light for qubit at $|g⟩$ (blue dots) and $|e⟩$ (red dots) states respectively [14]. The solid lines are Gaussian fits.

Figure 3

Fundamental cost of measurement: theoretical insights. (a) The mutual information $I$ as a function of the cavity photon number $n^{(\text{cav})}$ for different light sources. (b) The entropy of the three light sources. (c) The ultimate measurement efficiency is given by comparing the information gain ($I$) to the entropy generated in the probe ($S$) with the limit set at ($I=S$), which is only achieved with pure measurement resources such as the single-photon light. Note that $I$ and $S$ obtained here are calculated after the qubit-cavity interaction with different light sources and the cavity measurement by the Maxwell’’s demon in the Fock state basis (see [14] for details of the calculation). The qubit is in state $(|g⟩+|e⟩)/\sqrt{2}$ before the readout process.