Phase-diffusion pattern in quantum-nondemolition systems

We quantitatively analyze the dynamics of the quantum phase distribution associated with the reduced density matrix of a system, as the system evolves under the influence of its environment with an energy-preserving quantum nondemolition (QND) type of coupling. We take the system to be either an oscillator (harmonic or anharmonic) or a two-level atom (or equivalently, a spin-1/2 system), and model the environment as a bath of harmonic oscillators, initially in a general squeezed thermal state. The impact of the different environmental parameters is explicitly brought out as the system starts out in various initial states. The results are applicable to a variety of physical systems now studied experimentally with QND measurements.

Figure 1

Quantum phase distribution $\mathcal{P}\left(\theta \right)$ given by Eq. (25), for a harmonic oscillator initially in a coherent state, as a function of $\theta$ (in radians), for different environmental conditions and evolution times. The parameters have been taken as $\omega =1.0$, ${\omega }_c=100$, $\mid \alpha {\mid }^2=5$, $a=0.0$, ${\gamma }_0=0.0025$. The small-dashed and the large-dashed curves are for temperatures $T$ (in units with $\hslash \equiv k_B\equiv 1$) $=0$ and 300, respectively, with an environmental squeezing parameter $r=2$ and at an evolution time $t=0.1$. The dot-dashed and the double dot-dashed curves are at evolution times $t=0.1$ and 0.2, respectively, for $T=300$ and $r=1$. The continuous curve represents unitary evolution $({\gamma }_0=0)$.

Figure 2

Quantum phase distribution $\mathcal{P}\left(\theta \right)$ given by Eq. (29), for a harmonic oscillator initially in a squeezed coherent state, as a function of $\theta$ (in radians), for different environmental conditions and evolution times. The parameters have been taken as $\omega =1.0$, $\mid \alpha {\mid }^2=5$, $a=0.0$, ${\gamma }_0=0.0025$; $r_1=0.5$, and $\psi =\pi ∕4$ [$r_1$ and $\psi$ are the system squeezing parameters (26)]. The small-dashed and the large-dashed curves are for temperatures $T$ (in units with $\hslash \equiv k_B\equiv 1$) $=0$ and 300, respectively, at an environmental squeezing parameter (15) $r=2$ and evolution time $t=0.1$. The dot-dashed and the double dot-dashed curves are at evolution times $t=0.1$ and 0.2, respectively, with $T=300$ and $r=1$. The continuous curve represents unitary evolution $({\gamma }_0=0)$.

Figure 3

Quantum phase distribution $\mathcal{P}\left(\theta \right)$ given by Eq. (39), for an anharmonic oscillator initially in a Kerr state, as a function of $\theta$ (in radians), for different environmental conditions at a fixed time of evolution. The parameters have been taken as ${\gamma }_0=0.0025$, $\mid \alpha {\mid }^2=5$, $\omega =1.0$, $\chi =\lambda =0.02$, and evolution time $t=0.1$. The small-dashed and the large-dashed curves are for the bath squeezing parameter $r=0$ and 2, respectively, at $T$ (in units with $\hslash \equiv k_B\equiv 1$) $=50$. The dotted curve is for $T=0$ and $r=2$. The continuous curve represents unitary evolution $({\gamma }_0=0)$.

Figure 4

Time evolution of the quantum phase distribution $\mathcal{P}\left(\theta \right)$ given by Eq. (39), for an anharmonic oscillator initially in a Kerr state, as a function of $\theta$ (in radians), for different evolution times under fixed environmental conditions. The parameters have been taken as ${\gamma }_0=0.0025$, $\mid \alpha {\mid }^2=5$, $\omega =1.0$, $\chi =\lambda =0.02$ (as in Fig. 3), $T=0$, and $r=2$. The dotted curve is for an evolution time $t=0.1$, the small-dashed curve is for $t=0.5$, and the large-dashed curve is for $t=1.0$.

Figure 5

Quantum phase distribution $\mathcal{P}\left(\theta \right)$ given by Eq. (46), for an anharmonic oscillator initially in a squeezed Kerr state, as a function of $\theta$ (in radians), for different environmental conditions. The parameters have been taken as $t=0.1$, $\chi =\lambda =0.02$, ${\gamma }_0=0.025$, $r_1=0.4$, and $\varphi =0$ ($r_1$, $\varphi$ are the system squeezing parameters). The dot-dashed and the dotted curves are for $T$ (in units with $\hslash \equiv k_B\equiv$ 1) $=0$ and 100, respectively, with the environmental squeezing parameter $r=1$. The large-dashed curve is at $T=100$ and $r=0$. The continuous curve represents the unitary evolution $({\gamma }_0=0)$.

Figure 6

Quantum phase distribution $\mathcal{P}\left(\varphi \right)$ given by Eq. (62), for a two-level atom initially in an atomic coherent state, as a function of $\varphi$ (in radians), for different environmental conditions and evolution times. The parameters have been taken as $\alpha =\beta =\pi ∕4$ [Eq. (58)], and ${\gamma }_0=0.025$. The continuous curve and the dotted curve are for the bath squeezing parameter $r=0$ and 2, respectively, at a temperature $T=0$ and an evolution time $t=0.1$. The small-dashed and the large-dashed curves correspond to evolution times $t=0.1$ and 0.02, respectively, at $T$ (in units with $\hslash \equiv k_B\equiv 1$) $=300$ and $r=2$.

Figure 7

Quantum phase distribution $\mathcal{P}\left(\varphi \right)$ given by Eq. (72), at the south pole of the phase space for a two-level atom initially in an atomic squeezed state, as a function of $\varphi$ (in radians), for different environmental conditions. The system squeezing parameter (64) $\Theta$ is taken as $=-0.5494$, the bath squeezing parameter $r=1$, and ${\gamma }_0=0.025$. The continuous curve corresponds to a temperature $T=0$ and an evolution time $t=0.1$, while the small-dashed and large-dashed curves correspond to evolution times $t=0.1$ and 0.05, respectively, at $T$ (in units with $\hslash \equiv k_B\equiv$ 1) $=300$.

Figure 8

Quantum phase distribution $\mathcal{P}\left(\varphi \right)$ given by Eq. (73), at the north pole of the phase space for a two-level atom initially in an atomic squeezed state, as a function of $\varphi$ (in radians), for different environmental conditions. The system squeezing parameter (64) $\Theta$ has been taken as $=-0.5494$, the bath squeezing parameter $r=1$, and ${\gamma }_0=0.025$. The continuous curve corresponds to a temperature $T=0$ and an evolution time $t=0.1$, while the small-dashed and large-dashed curves correspond to evolution times $t=0.1$ and 0.05, respectively, at $T$ (in units with $\hslash \equiv k_B\equiv 1$) $=300$.