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This article is cited in 3 scientific papers (total in 3 papers)
Quantum integrability and quantum chaos in the micromaser
R. K. Bullougha, N. M. Bogolyubovb, R. R. Puric a University of Manchester, Department of Mathematics
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
c Bhabha Atomic Research Centre
Abstract:
The time-dependent quantum Hamiltonians
$$
\widehat H(t)=\begin{cases}
\widehat H,\quad &t_i<t<t_i+t_\mathrm{int},
\\ \omega_0\widehat N,\quad &t_i+t_\mathrm{int}<t<t_{i+1},
\end{cases}
$$
describe a maser with $N$ two-level atoms coupled to a single mode of a quantized field inside the maser cavity; here, $t_i$, $i=1,2,\dots,N_a$, are discrete times, $N_a$ is large ($\sim 10^5$), $\widehat{N}$ is the number operator in the Heisenberg–Weyl (HW) algebra, and $\omega_0$ is the cavity mode frequency. The $N$ atoms form an $(N+1)$-dimensional representation of the $su(2)$ Lie algebra, the single mode forming a representation of the HW algebra. We suppose that $N$ atoms in the excited state enter the cavity at each $t_i$ and leave at $t_i+t_\mathrm{int}$. With all damping and finite-temperature effects neglected, this model for $N=1$ describes the one-atom micromaser currently in operation with $^{85}$Rb atoms making microwave transitions between two high Rydberg states. We show that $\widehat{H}$ is completely integrable in the quantum sense for any $N=1,2,\dots$ and derive a second-order nonlinear ordinary differential equation (ODE) that determines the evolution of the inversion operator $S^Z(t)$ in the $su(2)$ Lie algebra. For $N=1$ and under the nonlinear condition $[S^Z(t)]^2= (1/)4\hat{I}$, this ODE linearizes to the operator form of the harmonic oscillator equation, which we solve. For $N=1$, the motion in the extended Hilbert space $\mathcal H$ can be a limit-cycle motion combining the motion of the atom under this nonlinear condition with the tending of the photon number $n$ to $n_0$ determined by $\sqrt{n_0+1}gt_\mathrm{int}=r\pi$ $($where $r$ is an integer and $g$ is the atom-field coupling constant$)$. The motion is steady for each value of $t_i$; at each $t_i$, the atom-field state is $|e\rangle|n_0\rangle$, where $|e\rangle$ is the excited state of the two-level atom and $\widehat{N}|n_0\rangle=n_0|n_0\rangle$. Using a suitable loop algebra, we derive a Lax pair formulation of the operator equations of motion during the times $t_\mathrm{int}$ for any $N$. For $N=2$ and $N=3$, the nonlinear operator equations linearize under appropriate additional nonlinear conditions; we obtain operator solutions for $N=2$ and $N=3$. We then give the $N=2$ masing solution. Having investigated the semiclassical limits of the nonlinear operator equations of motion, we conclude that “quantum chaos” cannot be created in an $N$-atom micromaser for any value of $N$. One difficulty is the proper form of the semiclassical limits for the $N$-atom operator problems. Because these $c$-number semiclassical forms have an unstable singular point, “quantum chaos” might be created by driving the real quantum system with an additional external microwave field coupled to the maser cavity.
Citation:
R. K. Bullough, N. M. Bogolyubov, R. R. Puri, “Quantum integrability and quantum chaos in the micromaser”, TMF, 122:2 (2000), 182–204; Theoret. and Math. Phys., 122:2 (2000), 151–169
Linking options:
https://www.mathnet.ru/eng/tmf562https://doi.org/10.4213/tmf562 https://www.mathnet.ru/eng/tmf/v122/i2/p182
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Abstract page: | 343 | Full-text PDF : | 170 | References: | 51 | First page: | 1 |
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