Quantum-To-Classical Transition in Cavity Quantum Electrodynamics

The quantum properties of electromagnetic, mechanical or other harmonic oscillators can be revealed by investigating their strong coherent coupling to a single quantum two level system in an approach known as cavity quantum electrodynamics (QED). At temperatures much lower than the characteristic energy level spacing the observation of vacuum Rabi oscillations or mode splittings with one or a few quanta asserts the quantum nature of the oscillator. Here, we study how the classical response of a cavity QED system emerges from the quantum one when its thermal occupation—or effective temperature—is raised gradually over 5 orders of magnitude. In this way we explore in detail the continuous quantum-to-classical crossover and demonstrate how to extract effective cavity field temperatures from both spectroscopic and time-resolved vacuum Rabi measurements.

Figure 1

(a) Coplanar microwave resonator with two qubit slots and flux bias lines (top) and a single embedded qubit (violet) shown on an enlarged scale (bottom). (b) Circuit diagram of setup. Input (left) is at room temperature (RT) with microwave sources for measurement (${\nu }_{\mathrm{probe}}$), qubit spectroscopy and qubit drive (${\nu }_{\mathrm{spec}}$), variable attenuation ($D$) quasi thermal field source ($S_{\mathrm{white}}$) at ${\nu }_r$ and an arbitrary waveform generator for fast flux biasing ($I_{\mathrm{dc}}$) through the on-chip flux line ($B$). At 20 mK the qubit is coupled via a capacitance $C_g$ to the transmission line resonator between the capacitances $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$. The transmitted microwave tone is amplified, down-converted with a local oscillator (LO) and digitized (ADC).

Figure 2

(a) Energy level diagram of dipole coupled dressed states $|n,\pm ⟩$ for excitation numbers $n=0$, 1 and 2 (black lines), uncoupled qubit and cavity energies (grey dashed lines) and allowed transitions at frequencies ${\nu }_{g0,1\pm }$ and ${\nu }_{1\pm ,2\pm }$ (blue arrows). (b) Dressed-state diagram for large excitation numbers $n>280$ and allowed transitions at the resonator frequency ${\nu }_r$ (red arrows). (c) Measured cavity transmission $A^2/A_{\max }^2$ for intracavity thermal photon numbers $0.05(\mathrm{\text{blue}})\lesssim n_{\mathrm{th}}\lesssim 400(\mathrm{\text{red}})$ and a fit to a Lorentzian line (black). Measured data sets are normalized to the transmission amplitude $A_{\max }$ obtained when the qubit is maximally detuned. The data are offset and colored for better visibility, see inset in Fig. 4 for color code.

Figure 3

(a) Measured (dots) and calculated (lines) cavity transmission [same data as in Fig. 2c] shown for applied $S_n=-221.5\mathrm{dBm}/\mathrm{Hz}$ (bottom) to $-190\mathrm{dBm}/\mathrm{Hz}$ (top) in steps of $4.5\mathrm{dBm}/\mathrm{Hz}$. Extracted thermal photon numbers $n_{\mathrm{th}}$ and cavity temperatures $T_c$ are indicated. (b) Measured qubit excited state population $P_e$ as a function of the resonant cavity interaction time $\tau$ (dots) and master equation simulation (lines) for $S_n=-214\mathrm{dBm}/\mathrm{Hz}$ to $-202\mathrm{dBm}/\mathrm{Hz}$ in steps of $3\mathrm{dBm}/\mathrm{Hz}$. (c), Similar measurement as in (b) with qubit prepared in the excited state for $S_n=-214\mathrm{dBm}/\mathrm{Hz}$. Inset: Vacuum Rabi pulse sequence.

Figure 4

Temperature $T_c$ and photon number $n_{\mathrm{th}}$ extracted from measured Rabi spectra (full dots) and Rabi oscillations (crosses) versus noise power spectral density $S_n$ or equivalently attenuation $D$. Theory (black line). Inset: extrapolation to higher $T_c$ provides temperature scale (color coded) for the measurements shown in Figs. 2c, 3.