Thermodynamics of quadrature trajectories in open quantum systems

We apply a large-deviation method to study the diffusive trajectories of the quadratures of light emitted from open quantum systems. We formulate the study of quadrature trajectories in terms of characteristic operators and show that, in the long-time limit, the statistics of such trajectories obey a large-deviation principle. We take our motivation from homodyne detection schemes which allow the statistics of light quadratures to be measured. We illustrate our approach with four examples of increasing complexity: a driven two-level system, a “blinking” three-level system, a pair of weakly coupled two-level driven systems, and the micromaser. We discuss how quadrature operators can serve as alternative order parameters for the classification of dynamical phases, which is particularly useful in cases where the statistics of quantum jumps cannot distinguish such phases. The formalism we introduce also allows us to analyze the properties of the light emitted in quantum-jump trajectories which deviate far from the typical dynamics.

Figure 1

A schematic diagram of the general Markovian system plus environment models we study. The interaction between the system and the environment is of form $c{b}^{†}+b{c}^{†}$, where the operators $b^{†}$ ($b$) are creation (annihilation) operators of an harmonic oscillator bath and the operators $c$ and $c^{†}$ are system operators which are related to the Lindblad projectors of the system.

Figure 2

Shown (left) is a setup for homodyne detection and (right) a typical photocurrent, $I_{\mathrm{homo}}$, plotted against time in such a measurement. A simple homodyne scheme consists of an input beam and an output beam which is split between a detector and a strong coherent field which acts as a local oscillator. If we denote $b_{\mathrm{Out}}$ as the output lowering operator and $\mathcal{E}$ as the local oscillator strength, the total transmitted field is $b_{\mathrm{Out}}^h=\mathcal{E}+b_{\mathrm{Out}}$. Looking at the photocurrent $I_{\mathrm{p}}=d{b}_{\mathrm{Out}}^{†h}{b}_{\mathrm{Out}}^h/dt$ one may define the homodyne current $I_{\mathrm{homo}}=\frac{I_p}{\mathcal{E}}-\mathcal{E}$ in the limit of infinite local oscillator strength. By expanding $I_p$ one finds that the homodyne current is a direct measure of the quantity $b+b^{†}$, which is directly related to the $X$ quadrature. Reference [2] contains further details about the various homodyne detection schemes which allow $X$-quadrature measurements. In this work we use the input and output fields to form the reservoir we depict in Fig. 1.

Figure 3

The quadrature operators define axes of an optical phase space. The generic quadrature operator $X^{\alpha }$ may be viewed as a rotation of the X-quadrature axis as shown above.

Figure 4

Schematic diagram of a laser-driven two-level system coupled to a vacuum reservoir.

Figure 5

A plot of the dimensionless $Y$-quadrature activity $y_s{\Omega }^{-1/2}$ and minus the dimensionless quantum-jump activity $-k_{s^{\prime }}{\Omega }^{-1}$ for the two-level system when biased by the appropriate fields $s$ and $s^{\prime }$. Note that the order parameters coincide for the physical dynamics at $s=s^{\prime }=0$.

Figure 6

Plots of the LD functions and associated activities and variances for the laser-driven two-level system. Shown in (a), the LD function for the $X$-quadrature statistics is completely symmetric about $s=0$. It is also worth noting that as $\left|s\right|$ becomes larger, the dynamical variance tends to 1/4, a consequence of the quadratic form of ${\theta }_X$ at large $\left|s\right|$. In (b) we show that the $Y$-quadrature LD function is now asymmetric close to $s=0$, while at large $\left|s\right|$ it adopts the same quadratic form as ${\theta }_X\left(s\right)$. Both LD functions exhibit a positive activity for $s<0$ and a negative activity for $s>0$.

Figure 7

Phase-space portraits displaying the marginal probability distributions $e^{-\varphi \left(x^{\alpha }\right)}$ for the two-level system at various photon biases $s^{\prime }$. We demonstrate that, as we make the system more photon active, the plot moves away from the origin in the negative $y$ direction. In (a), $s^{\prime }=-5$ and the center of the plot is (0, $-$1.534); in (b), $s^{\prime }=0$ and the plot center is (0, $-$0.667); in (c), $s^{\prime }=+5$ and the plot center is at (0, $-$0.289).

Figure 8

Reconstructed Wigner functions for the same system parameters as in Fig. 7. $\alpha$ is incremented by 0.01 rad over the range $[0,\pi ]$ in the reconstruction. The values $s^{\prime }=-5$, 0 and $+5$ are plotted in (a)–(c), respectively. Focus is on the central regions where the Wigner function is large: The regions far from the central maxima which take negative values are necessarily sensitive to the choice of $\alpha$ increment used. We have reproduced the Wigner functions with a broad range of $\alpha$ increments and see no changes to the central regions when choosing increments between 0.01 rad and 0.001 rad.

Figure 9

A three-level system coupled to a vacuum reservoir and driven by two resonant lasers with Rabi frequencies ${\Omega }_1$ and ${\Omega }_2$. When ${\Omega }_1\gg {\Omega }_2$ the photon emission trajectories are intermittent; the $|1\rangle \to |0\rangle$ transition is the active or light line transition while $|2\rangle$ is an inactive level.

Figure 10

(a) The LD function, activities, and dynamical variances for the $X$-quadrature statistics of the three-level system. This is completely symmetric about $s=0$, but it is not the same as the two-level LD function except in the large $\left|s\right|$ limit. This can be seen in the dynamical variance, which tends to 1/4 in this limit, a consequence of the quadratic form of ${\theta }_X$ at large $\left|s\right|$. (b) The corresponding plots for $Y$ quadratures for the three-level system are shown. The activity exhibits a marked crossover around $s=0$, which appears as a rounded step. At the crossover, the corresponding dynamical variance is shown to have a sharp peak.

Figure 11

Phase-space portraits of the marginal probability distributions $e^{-\varphi \left(x^{\alpha }\right)}$ at various photon biases $s^{\prime }$ for the three-level system. As we make the system more photon active the plot moves away from the origin in the negative $y$ direction. In (a), $s^{\prime }=-5$ and the center of the plot is (0, $-$1.534); in (b), $s^{\prime }=0$ and the plot center is (0, $-$0.4), and in (c), $s^{\prime }=+5$ and the plot center is at (0,0).

Figure 12

(a) A contour plot of the, $s^{\prime }=0$, two-level system density plot plus the three-level system (axes scaling for this part is ${\Omega }_1^{-1/2}$), $s^{\prime }=+5$, (effectively a one-level system due to inactivity) density plot with its center shifted by 0.1 along the $y{\Omega }_1^{-1/2}$ direction. (b) The contour plot of the three-level $s^{\prime }=0$ marginals. It is very similar to (a) indicating the three-level system's physical dynamics may be considered as being primarily composed of a two-level system plus an inactive two-level system. However, the shifts introduced in (a) and slight differences between (a) and (b) are an indication of some interference between these two pictures.

Figure 13

Plots of the dimensionless quantum-jump activity ${\Omega }_1^{-1}{k}_{s^{\prime }}$, for different quadrature biases $s$ in the three-level system. The jump activity grows as we bias the $X$ quadrature regardless of sign of $s$. However the photon count exhibits a dynamical crossover in the rare $Y$ trajectories about $s=0$, and shows that the sign of $s$ determines the behavior of $k_{s^{\prime }}$. Furthermore the photon variance in these quadratures also indicates the inactive-active phase crossover, with a peak at $s=0$ correlating with the dynamical phase transition at this point.

Figure 14

Plots of the dimensionless activity ${\Omega }_1^{-1}{y}_{s=0}$ as a function of photon bias $s^{\prime }$ in the three-level system. $y_{s=0}$ exhibits a crossover at $s^{\prime }=0$, confirming that while $k_{s^{\prime }}$ varies with $s$ in a highly correlated way, $y_s$ is similarly correlated with $s^{\prime }$.

Figure 15

A schematic diagram of two weakly coupled two-level systems driven by lasers of different polarisation. The coupling between the ground states is neglected in our case.

Figure 16

Plots of the $X$-quadrature LD function as well as the activity and dynamical variance, for the four-level system depicted in Fig. 15. Note that the dynamical variance has been rescaled to be a factor of 5 smaller.

Figure 17

Phase-space portraits for the two coupled two-level systems. The $X$-quadrature bias in (a)–(c) is $s^{\prime \prime }=-0.1$, 0, and $+0.1$, respectively. In (a) and (c) the probability distributions are more concentrated about $x>0$ and $x<0$, respectively, whereas at $s^{\prime \prime }=0$ they are even functions of $x_s$. These plots are indicative of the crossover in the $X$ activity at $s=0$ and highlight the less dramatic phase changes we may see using this order parameter as opposed to the jump activity.

Figure 18

Quadrature activity in the micromaser from mean-field theory. We find multiple first-order transition lines in the activity either side of $s=0$. Also we note some bending of these transition lines occurs as they approach $s=0$. The form of the mean-field theory is very similar to one obtained for the “atom” counting case of Ref. [5], highlighting the connection between transitions in quadrature and jump activities.

Figure 19

Quadrature activity phase diagrams in quantum-jump biased systems with $s^{\prime }=-0.005$, 0 and $+0.005$ in plots (a)–(c), respectively. In all cases the activity shows first-order transition lines as we change $\varphi$. Comparing (b) with the mean-field theory, agreement exists up to $\varphi \approx 0.7$, however, there is significant bending of these lines as we approach the normal system dynamics. The degree of this bending is linked with the jump activity: In the active phase (a), the transition lines bend less approaching $s=0$, while in the inactive jump phase (c), the bending is more prominent.

Figure 20

Cavity occupation number for double counting cases with jump bias $s^{\prime }=-0.005$, 0, and $+0.005$ in plots (a)–(c), respectively. In the jump-active case (a), the reduced bending of the transition lines compared with (b) correlates with an increased cavity occupation which remains larger than the typical value away from $s=0$. Conversely the increased bending in (c) compared with in (b) is associated with larger regions away from $s=0$ attaining typical ($s=0$) cavity occupation numbers rapidly as we increase $\varphi$.