Statistical properties of quantum jumps between macroscopic states of light: reading an operational coherence record

Abstract

We propose an experimental apparatus to reveal the quantum coherence manifested in downward quantum jumps of amplitude bistability. The underlying coherent superposition of macroscopic quantum states is translated into the statistical properties of the integrated charge deposited in the detector circuit of a mode-matched heterodyne/homodyne detection scheme. At first, the dynamical evolution of a signal transmitted from an auxiliary cavity is employed to pinpoint a macroscopic switching event in a bistable main cavity subject to direct photodetection. Once the decision is made on the occurrence of a downward switch, the main cavity mode is let to freely decay to the vacuum, monitored to the production of an integrated charge. In the long-time limit, the charge distribution over an identical collection of pure states generated during the jumps converges to the Q function (heterodyne detection) or marginals of the Wigner function (homodyne detection) dictated by the phase of the local oscillator. When fluctuations over the ensemble step in, we connect the statistical properties of several switching events and the ensuing production of current records, to the cavity field correlations associated with the breakdown of photon blockade.

Figures (4)

• Figure 1: Schematic illustration of the proposed experimental setup. The main cavity mode is coherently driven with drive amplitude $\varepsilon$ and strongly coupled to a two-state atom with strength $g$. Photon loss at rate $2\kappa$ establishes the only channel for leaking information to the environment, where an avalanche photodiode (APD) detector is placed. The auxiliary cavity mode is driven with amplitude $\varepsilon^{\prime}$ and is weakly coupled to the main cavity mode with strength $g^{\prime}$. The rapid decay from the auxiliary cavity determines the dominant rate $\kappa^{\prime}$ allowing for adiabatic elimination of its resonating mode field. The main cavity-atom system is operated under conditions of high-amplitude bistability with a typical (schematic) realization of the conditioned output flux rate pictured in inset (i). For the control switch at the position ${\rm P_A}$, the record obtained under direct photodetection consists of the alternation between time periods of closely spaced photoelectron "clicks", corresponding to the bright (B) state, and periods of several cavity lifetimes with no "clicks", corresponding to the dim (D) state. The latter two states, together with the unstable (U) state are also identified from the semiclassical hysteresis curve illustrated below the main trajectory of (i). Inset (ii) depicts the signal transmitted from the auxiliary cavity during a downward quantum jump, focused about the green-shaded area of the trajectory (i). It shows the dip dictating when the control switch is to be moved to position ${\rm P_B}$, indicated by the arrow, actuating balanced and mode-matched homodyne/heterodyne detection. In the balanced detection scheme (BHD) two fields are superposed past the second beam splitter, the cavity output field and the local-oscillator field, with a $\pi$ phase difference between the superpositions. The integrated charge deposited in the detector circuit as a result of their detection CarmichaelBook2 is $Q=Q_x+iQ_y$ for heterodyne and $Q_{\theta}$ for homodyne detection. Inset (iii) depicts a steady-state distribution $P(Q^{*},Q)$ (top) and $P(Q_{\theta})$ (bottom) over an ensemble average of realizations with a coherent-state superposition as a common start for the cavity state. The coherent offset (applied at the first beam splitter) is set to zero for heterodyne and to $-(\alpha_1+\alpha_2)/2$ for homodyne detection, where $\alpha_1$ and $\alpha_2$ are the conditioned coherent state amplitudes attributed to the bright and unstable states, respectively.
• Figure 2: Individual realizations, jump detection and initial state preparation.(a) Sample trajectory interval of the main cavity photon number $\langle a^{\dagger}a(t)\rangle=\langle a^{\dagger}a(t)\rangle_{\rm REC}$ against five cavity lifetimes centered about a downward switch from the bright (B) to the dim (D) state. The trajectory includes a short interval shown by the yellow shaded area of duration $\kappa\Delta t_{\rm loc}\approx 0.106$ during which coherent localization occurs from the bright state to an unstable state (U) while the two states coexist with varying weights. Inset (i) depicts the quadrature phase amplitude $x=\langle \mathcal{A}_0(t) \rangle_{\rm REC}$ of the main cavity field after a direct detection unraveling of ME \ref{['eq:MEJC']}, superimposed onto the scaled amplitude $(\kappa^{\prime}/g^{\prime})x^{\prime}$ of the auxiliary cavity produced by the solution of a Schrödinger equation with Hamiltonian \ref{['eq:HnonH']} and external drive $\varepsilon^{\prime}=0$. Similarly in inset (ii) for $y=\langle \mathcal{A}_{\pi/2}(t) \rangle_{\rm REC}$ and $(\kappa^{\prime}/g^{\prime})y^{\prime}$. (b) Scaled outgoing photon flux and transmission from the auxiliary cavity, as defined in Eq. \ref{['eq:TF']}, against the same dimensionless time interval focused upon in (a). The external field amplitude is set to $-\tilde{\alpha}_1/2$, where $\tilde{\alpha}_1\approx 2.082 -4.875i$ solves Eq. \ref{['eq:neocl']}. The purple line in frames (a, b) indicates the position of the dip in flux and transmission, occurring at $\kappa t_{\rm dip}=6.590$ [at the center of the yellow-shaded area in (a)]. The horizontal line marks the position of $|\tilde{\alpha}_1|^2/4 \approx 7.025$. Inset (i) shows an unlabelled contour plot of the conditioned quasiprobability distribution of the main cavity field $\mathcal{Q}(x+iy; t_{\rm dip})$. The peak locations, corresponding to the conditioned coherent state amplitudes are read as $\alpha_1 \approx 1.95-5.45i$ and $\alpha_2 \approx -1.40+0.85i$. Insets (ii), (iii) depict barplots of the real and imaginary parts, respectively, of the conditioned cavity density matrix $[\rho_{c}(t_{\rm dip})]_{mn}$. (c) (i) Conditioned quasiprobability distribution $\mathcal{Q}(x+iy; t_{1})$, at $\kappa t_1=6.650$, a time indicated by the dashed line in frame (d, i); (ii) Imaginary part of $[\rho_{c}(t_{1})]_{mn}$, and (iii) $\mathcal{Q}(x+iy; t_{2})$ at $\kappa t_2=6.695$. (d) Closer focus on the coherent localization within a cavity lifetime. Frame (i) depicts a close-up on the trajectory of (a). The inset on the left depicts the $Q$ function of the steady-state cavity field $\mathcal{Q}_{\rm ss}(x+iy)$ with photon average $\langle a^{\dagger}a \rangle_{\rm ss}\approx 14.65$ [ensemble average after the solution of ME \ref{['eq:MEJC']}]. In $\mathcal{Q}_{\rm ss}(x+iy)$, the D state is practically the vacuum while the B state peaks at $\approx 2.05-4.85i$. Frames (ii) and (iii) depict the scaled time derivative of the $x$ and $y$-quadratures of the auxiliary cavity field. The two vertical lines mark the position of the minimum and maximum, respectively. These two times, along with $\kappa t_{\rm dip}=6.590$, are all indicated in a cluster of closely-spaced lines in frame (i). The operating parameters read: $g/\kappa=60$, $\varepsilon/\kappa=13.5i$, $\Delta\omega/\kappa=-8$, and $\kappa^{\prime}/\kappa=\kappa^{\prime}/g^{\prime}=100$.
• Figure 3: Statistical properties of the integrated charge.(a) Two-dimensional histogram of the complex integrated charge $Q_x+iQ_y$ normalized to a probability density function $P(Q^{*},Q)$, collected from mode-matched heterodyne detection on the main cavity output at ten cavity lifetimes after the control switch has been moved to position $P_B$. For the $10^5$ realizations recorded, the initial cavity state is taken as \ref{['eq:psi01']} with $c_1=c_2=1/\sqrt{2}$, while the conditioned amplitudes are assumed identical for every jump across different realizations, equal to $\alpha_1=1.95-5.45i$ (bright state) and $\alpha_2=-1.40+0.85i$ (unstable state), corresponding to the peak positions of the $Q$ function displayed in inset (i) of Fig. \ref{['fig:FIG2']}(b). The bars are superimposed on the surface plot of the $Q$ function $\mathcal{Q}(Q_x + iQ_y)=[1/(2\pi)]\{\exp[-(Q_x-1.95)^2-(Q_y-5.45)^2] + \exp[-(Q_x+1.40)^2-(Q_y+0.85)^2]\}$. (b) Same principle as in (a) but instead with the unstable state coefficient $|c_2|^2$ assuming random values uniformly distributed in $(0,1)$, while $|c_1|^2=1-|c_2|^2$, and with the unstable state amplitude taking a fixed modulus $|\alpha_2|$ but a uniformly distributed phase in $(0, 2\pi)$. A surface density projection of the histogram is given in (i) and the bar graph is depicted in (ii). (c) Histograms normalized to a probability density function of $10^5$ homodyne current records $P(Q_{\theta})$ at $t\to \infty$ ($\eta \to 1$) for the initial state \ref{['eq:cat']} with $\phi_0=0$ and $A=9$. The LO phase is set to $\theta=0$ (i) and $\theta=\pi/2$ (ii). Solid black lines plot the analytical expression for $P(Q_{\theta})$ in Eq. \ref{['eq:PQan']}. (d) Same as in (c) for $\theta=\pi/2$ but with the amplitude $A$ exhibiting fluctuations across different trajectories, which are Gaussian-distributed with mean $\bar{A}=9$ and standard deviation $\sigma=0.1\sqrt{\bar{A}}$ (main frame) and $\sigma=0.5\sqrt{\bar{A}}$ (inset). B and U stand for the bright and unstable states, respectively.
• Figure 4: Fluctuating vs. metastable states. (a) Sample trajectory interval of the main cavity photon number $\langle a^{\dagger}a(t)\rangle=\langle a^{\dagger}a(t)\rangle_{\rm REC}$ against nine cavity lifetimes, including a downward quantum jump from the bright state which is then followed by large-photon fluctuations. Inset (i) depicts a schematic contour plot of the steady-state main cavity $Q$ function $\mathcal{Q}_{\rm ss}(x+iy)$, while inset (ii) shows the conditioned $Q$ function of the main cavity field $\mathcal{Q}(x+iy; t_{\rm dip})$ at the time $t_{\rm dip}$ indicated by the vertical purple line. (b) Scaled outgoing photon flux and transmission from the auxiliary cavity, as defined in Eq. \ref{['eq:TF']}, against the same dimensionless time interval focused upon in (a). The external field amplitude is set to $-\tilde{\alpha}_1/2$, where $\tilde{\alpha}_1\approx 2.448 -5.445i$ solves Eq. \ref{['eq:neocl']}. The purple line in frames (a, b) indicates the position of the dip in flux and transmission, occurring at $\kappa t_{\rm dip}=2.668$. The horizontal line marks the position of $|\tilde{\alpha}_1|^2/4 \approx 8.957$. Inset (i) focuses on the asymmetry of the dip over $0.2$ cavity lifetimes, and inset (ii) depicts barplots of the conditioned main cavity density matrix $[\rho_{c}(t_{\rm dip})]_{mn}$ (real and imaginary parts indicated accordingly). (c)$y$-quadrature of the main cavity field against the same time interval as in (a,b) and scaled quadrature $(\kappa^{\prime}/g^{\prime}) y^{\prime}$ of the auxiliary cavity field, plotted by curves (i) and (ii), respectively. (d) Three sample realizations against the scaled time $\eta \equiv 1-e^{-2\kappa t}$ of the integrated charge $Q_{\pi/2}$ produced as homodyne current records solving Eq. \ref{['eq:SDE2']}. The coherent offset is set to the value $(\alpha_1 + \alpha_2)/2$, with $\alpha_1 \approx 2.95-5.35i$ (bright state) and $\alpha_2 \approx -2.05-0.20i$ corresponding to the peak locations of the conditioned $Q$ function in [(a), inset (ii)]. The LO phase is set to $\theta=\pi/2$ relative to the axis joining $\alpha_1$ and $\alpha_2$. The initial state is set to that of Eq. \ref{['eq:cat']} with $A(\alpha_1,\alpha_2)=(1/2)|\alpha_1-\alpha_2|\approx 3.59$. (e) Histogram of $Q_{\pi/2}$ from an ensemble of $10^5$ realizations solving Eq. \ref{['eq:SDE2']} in the limit $\eta \to 1$ under the same measurement procedure followed in (d). The analytical solution of a Fokker--Planck equation [Eq. \ref{['eq:PQan']}] for $\theta=\pi/2$ and $A=3.59$ is plotted in thick black line. B, D, U denote the bright, dim and unstable states, respectively. The steady-state photon number is $\langle a^{\dagger}a \rangle_{\rm ss}\approx 22.83$ [ensemble average after the solution of ME \ref{['eq:MEJC']}]. In $\mathcal{Q}_{\rm ss}(x+iy)$, the D state is practically the vacuum while the B state peaks at $\approx 2.47-5.45i$. The operating parameters read: $g/\kappa=60$, $\varepsilon/\kappa=14.4i$, $\Delta\omega/\kappa=-7.2$, and $\kappa^{\prime}/\kappa=\kappa^{\prime}/g^{\prime}=100$.