Dissipative quantum state preparation and metastability in two-photon micromasers

We study the preparation of coherent quantum states in a two-photon micromaser for applications in quantum metrology. While this setting can be in principle realized in a host of physical systems, we consider atoms interacting with the field of a cavity. We focus on the case of the interaction described by the Jaynes-Cummings Hamiltonian, which cannot be achieved by the conventional approach with three-level atoms coupled to the cavity field at two-photon resonance. We find that additional levels are required in order to cancel Stark shifts emerging in the leading order. Once this is accomplished, the dynamics of the cavity features a degenerate stationary state manifold of pure states. We derive the analytic form of these states and show that they include Schrödinger cat states with a tunable mean photon number. We also confirm these states can be useful in phase estimation protocols with their quantum Fisher information exceeding the standard limit. To account for realistic imperfections, we consider single-photon losses from the cavity, finite lifetime of atom levels, and higher order corrections in the far-detuned limit, which result in metastability of formerly stationary cavity states and long-time dynamics with a unique mixed stationary state. Despite being mixed, this stationary state can still feature quantum Fisher information above the standard limit. Our work delivers a comprehensive overview of the two-photon micromaser dynamics with particular focus on application in phase estimation and, while we consider the setup with atoms coupled to a cavity, the results can be directly translated to optomechanical systems.

Figure 1

(a) Atomic level structure: the transitions $|j-1\rangle \leftrightarrow |j\rangle$, $j=1,..,4$ are coupled to the cavity field with the strengths $g_j$ and detunings ${\mathrm{\Delta }}_j$. The transition $|3\rangle \leftrightarrow |\mathrm{a}\rangle$ is driven by a classical field with Rabi frequency $G$ and detuning $\delta$ (see Sec. 2a). (b) Micromaser: atoms are passing through a lossy cavity one at a time, interacting with a single-mode quantized cavity field of frequency $\omega$ (orange) and a classical Rabi field $G$ of frequency ${\omega }_{\mathrm{cl}}$ (green). (c) Effective dynamics: at the two-photon resonance ${\mathrm{\Delta }}_2=-{\mathrm{\Delta }}_3$, the $(5+1)$-level model reduces to an effective two-photon Jaynes-Cummings interaction with the coupling strength $\lambda$ between the cavity field (depicted as a quantum harmonic oscillator) and the effective two-level atom with ground and exited states $|1\rangle$ and $|3\rangle$ (see Sec. 2b). (d) Micromaser dynamics in weak-coupling regime: the Wigner function (15) for the cavity state is shown. The initial coherent state $|\alpha \rangle$ with $\alpha =0.6$ evolves first into a DFS spanned by the odd and even cat states (time $t_1$), which would be stationary if not for single-photon losses from the cavity that renders it metastable. After the first metastable regime, the macroscopic coherence dephases (time $t_2$), leading to metastable mixture of coherent states. This mixture then finally relaxes into a unique stationary state (time $t=\infty$) via mixing dynamics. In the second metastable regime ($t\ge t_2$), the system state features a single reflection symmetry, while the final parity-symmetric stationary state features two reflection symmetries (see Sec. 2c). The parameters are as in Fig. 6; see Sec. 4b for discussion.

Figure 2

Pure stationary states of cavity dynamics: (a) Wigner function [Eq. (15)] for even cavity stationary states corresponding to the parameters in panel (c) [and indicated in Fig. 8]. The two reflection symmetries (along diagonal gray lines) are due to the stationary states being parity symmetric and real valued (after adding the phase $\pi /4$) (see Sec. 2c). (b) The photon-number distribution of the states (blue bars, only even photon numbers) is compared to that of the coherent states with the same average photon number $\langle n\rangle$ (red dashed lines). Blue dashed lines show ${cot}_{2n}^2(\varphi /2)/10$, which diverges as $4/{sin}_{2n}^2\left(\varphi \right)$ [gray dashed lines] for soft walls concurring with the boundary condition for stationary states (see Sec. 3e). (c) Properties of stationary states (i)–(ix): the parameters $(K,c_{\mathrm{e}})$ [which determine $\varphi$ by Eq. (35), where the hard wall is at $m=20$; for $\varphi$ see also the last panel in Fig. 8, while $c_g=\sqrt{1-c_e^2}$, the mean photon number $\langle n\rangle$, the variance $\mathrm{\Delta }{n}^2$, the enhancement (79) in phase estimation, the maximal rate related to even soft wall ${max}_{0\le 2n\le m}1/{sin}_{2n}\left(\varphi \right)$, and the estimated number of atoms $k_{\mathrm{ss}}$ for which the stationary states are reached, as characterized by the fidelity $F[{\rho }_{\text{ss}};\rho \left(k\right)]=\mathrm{Tr}\sqrt{\sqrt{{\rho }_{\text{ss}}}\rho \left(k\right)\sqrt{{\rho }_{\text{ss}}}}\ge 0.99$, for the cavity initially in the vacuum state $|0\rangle$.

Figure 3

Steady states in the presence of hard walls. The photon-number distribution $P\left(n\right)$ and the Wigner function [Eq. (15)] for (a) the equal mixture of the odd pure stationary states obtained from the initial superposition of odd Fock states $(|1\rangle +|15\rangle )/\sqrt{2}$ for $c_{\mathrm{e}}=0.3$ and the hard wall (dashed gray) at $m=11$ with $K=1$ ($\varphi \approx 0.252$) and (b) the approximately pure stationary state obtained from the initial vacuum state $|0\rangle$ for $c_{\mathrm{e}}=0.4$ and the hard wall at $m=12$ with $K=8$ ($\varphi \approx 0.593$).

Figure 4

Soft walls. (a) The function ${sin}_n^{-2}\left(\varphi \right)$ for rational ${\varphi }_1/\pi =5/7$ (blue dots), ${\varphi }_2/\pi =6/7$ (red circles), and irrational ${\varphi }_3/\pi =7/\sqrt{210}$ (gray diamonds), with the hard walls (gray lines) at $m_1=13$ and $m_2=839$. For ${\varphi }_1$, the walls remain finite, in contrast to ${\varphi }_2$, where ${sin}_n^{-2}\left(\varphi \right)$ diverges as $n^{-2}$ [cf. Eq. (41)] and ${\varphi }_3$, where soft walls appear due to recurrence of the irrational rotation. (b) The orbits for both ${\varphi }_1$ and ${\varphi }_2$ are approximately periodic (with periods 14 and 7), while for ${\varphi }_3$ the orbit is dense.

Figure 5

Dynamics of $(5+1)$-level micromaser vs effective 2-photon micromaser. The fidelities $F[{\rho }_{\text{ss}};\rho \left(t\right)]=\mathrm{Tr}\sqrt{\sqrt{{\rho }_{\text{ss}}}\rho \left(t\right)\sqrt{{\rho }_{\text{ss}}}}$ of the stationary state ${\rho }_{\mathrm{ss}}$ in the two-photon micromaser with respect to its evolving state $\rho \left(t\right)$ (blue solid line), Eq. (11), and to the evolving state $\rho \left(t\right)$ of $(5+1)$-micromaser, Eq. (B9), for increasing values of detuning (orange, green, red solid lines), while keeping the integrated coupling $\varphi$ constant. Excellent agreement is observed during the metastable regime, whose length increases with the square of the detuning and coupling strength ratio, and is followed by the long-time dynamics well approximated by the effective dynamics in the DFS (black dotted lines), Eq. (45). These results are observed for different atom states, coupling strengths, and initial cavity states: $c_{\mathrm{e}}=0.3$, $\varphi =1.0$, $|{\psi }_{\mathrm{in}}\rangle =|0\rangle$ (the vacuum), (b) $c_{\mathrm{e}}=0.2$, $\varphi =0.3$, $|{\psi }_{\mathrm{in}}\rangle =|1\rangle$ (the single-photon state), and (c) $c_{\mathrm{e}}=0.1$, $\varphi =0.1$, $|{\psi }_{\mathrm{in}}\rangle =|\alpha \rangle$, $\alpha =1$ (a coherent state). The coupling strengths and the detunings in the $(5+1)$-level model are chosen uniformly as $g_1=g_2=g_3=g_4=g$ and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=-{\mathrm{\Delta }}_3=-{\mathrm{\Delta }}_4=\mathrm{\Delta }$, together with $G=2g$ and $\delta =2\mathrm{\Delta }$, and thus satisfy Eq. (5).

Figure 6

Dynamics of micromaser with single-photon losses. The fidelity [cf. Fig. 5] between the cavity state $\rho \left(t\right)$ and the stationary state (57) is compared for the dynamics of two-photon micromaser with single-photon losses (blue solid line), (52), and the effective dynamics in the DFS (black dashed line), (55). The effective dynamics approximates well the long-time dynamics of the cavity for the initial states $|0\rangle$ [(a), (c)] and $|\alpha \rangle$, $\alpha =0.6$ [(b), (d)], both in the weak-coupling limit [$c_{\mathrm{e}}=0.1$, $\varphi =0.1$ in panels (a) and (b)], where additional metastable regime (second plateau) is observed (b), and at the finite coupling [$c_{\mathrm{e}}=0.2$, $\varphi =1.0$ in panels (c) and (d)]. The loss rate was chosen as $\kappa /\nu ={10}^{-6}$, and the vertical lines indicate the timescales of the dynamics determined by the eigenvalues of Eq. (52), ${(-\mathrm{Re}{\lambda }_k)}^{-1}$ for $k=5,4,2$ (black, purple, red), which are ordered in decreasing real value [see also Eqs. (58) and (59) and cf. Appendix pp6].

Figure 7

Effective long-time dynamics due to single-photon losses. The DFS of the odd and even states (22) (the Bloch sphere in light gray) is shown under the effective dynamics in Eq. (55), for times $t={(-{\lambda }_4)}^{-1},{(-{\lambda }_3)}^{-1}$, and ${(-{\lambda }_2)}^{-1}$ (gray, purple, red) [see Eq. (58) and vertical lines in Fig. 6]. Because of the weak parity symmetry, the stationary state (black dot), Eq. (57), is found on the vertical axis (black line) representing mixtures of even and odd states, while when the initial state is odd or even, its dynamics remains confined to the vertical axis at all times (cf. purple dashed trajectory). As the effective dynamics is also real, the coherence eigenmodes correspond to the axis between the states in Eq. (62) (dashed gray) and the axis perpendicular to it that also crosses the equator. The trajectories for two initial states are also shown: $|{\mathrm{\Psi }}_{+}\rangle$ (dashed purple) and $cos(\pi /6)|{\mathrm{\Psi }}_{+}\rangle +e^{i\pi /4}sin(\pi /6)|{\mathrm{\Psi }}_{-}\rangle$ (dashed black). In panel (a), due to separation of the characteristic timescales of the dynamics as given by Eq. (59), classical metastable manifold emerges [blue; the image of DFS under the dynamics of at $t={(-{\lambda }_2)}^{-1}/100$], well approximated by mixtures of the states $|{\mathrm{\Psi }}_1\rangle$ and $|{\mathrm{\Psi }}_2\rangle$ in Eq. (62) (dashed gray axis). Here an initial state first relaxes onto the manifold (black arrow along black dashed trajectory) and only at later times relaxes toward the stationary state (blue arrow) [see also Fig. 6]. Parameters: (a) as in Figs. 6 and 6 leading to ${\eta }_{\text{loss}}\approx 1.00$, ${\langle n\rangle }_{+}\approx 1.92$ and ${\langle n\rangle }_{-}\approx 2.07$; (b) as in Figs. 6 and 6 leading to ${\eta }_{\text{loss}}\approx 0.99$, ${\langle n\rangle }_{+}\approx 0.11$, and ${\langle n\rangle }_{-}\approx 1.01$.

Figure 8

Phase estimation with dissipatively generated cavity states. The four panels show the ratio of the QFI to the performace of the corresponding coherent state, $F_Q\left(\rho \right)/4\langle n\rangle$ for the cavity initially in the vacuum $|0\rangle$ after the passage of $k=100$, ${10}^3$, ${10}^4$ atoms and for the stationary state [Eq. (22)]. The enhancement is shown as a function of the atom state [Eq. (8)] and integrated coupling $\varphi$. We sample the $\varphi$ axis for ${\varphi }_{20,K}$, Eq. (35), with odd $K=1,3,...,43$, which gives the hard wall at $m=20$ and allows convergence to stationary state also for $c_{\mathrm{e}}>1/\sqrt{2}$ (note that a larger $m$ would generally allow higher $\langle n\rangle$ and could also enable a higher enhancement in precision). The purple shading shows regions with reduced purity $\mathrm{Tr}\left({\rho }^2\right)<0.99$, whereas the green shading excludes low average photon number, $\langle n\rangle <1$. The red dots in the steady-state panel mark the stationary states (i)–(ix) analyzed in Fig. 2. The states (iii)–(ix) correspond to the states at the local maxima of the precision enhancement, while (i) and (ii) correspond to the standard and squeezed Schrödinger cat states. A complex phase of $c_{\mathrm{e}}$ does not change the results, but the stationary states are not periodic in $\varphi$, and thus here we show only a part of the parameter space.

Figure 9

Effect of single-photon losses on phase estimation precision. (a) The enhancement (79) in the phase estimation is shown as a function of the integrated coupling $\varphi$ [$c_{\mathrm{e}}=0.65$ corresponding to dashed red line in Fig. 8]. The enhancement in the stationary state of lossy dynamics (black) [Eq. (85)] is shown against the enhancement in the even (blue) and odd (green) states that are stationary for lossless cavity. For the majority of parameter space, we observe the enhancement in phase estimation, i.e., $F_Q\left(\rho \right)/4\langle n\rangle >1$ (values above the horizontal dashed gray line). Here the lossy stationary state is given by perturbative Eq. (57). (b) Average photon number in even and odd stationary states. We observe the correlation of high photon number to when the QFI of a lossy stationary state differs from Eq. (85) in panel (a), as it determines the size of the correction from the single-photon losses (together with the relaxation timescales in the lossless case [cf. Fig. 8].

Figure 10

Effects of the nonmonochromaticity of atomic beam. Dynamics of the purity (a), $\mathrm{Tr}\left({\rho }^2\right),$ and the QFI (b) [Eq. (77), normalized by the maximum value $F_Q\left(\rho \right)/4\langle n\rangle =7.39$ in dynamics with the monochromatic beam], with the number of atoms $k$ passing the cavity, is shown for different widths $\sigma$ of the integrated coupling distribution, which for simplicity is assumed normal, $g\left(\varphi \right)=exp[-{(\varphi -\langle \varphi \rangle )}^2/2{\sigma }^2]/\sqrt{2\pi {\sigma }^2}$. The initial state is the vacuum $|0\rangle$, the atom state is $c_{\mathrm{e}}=0.65$, and the coupling $\overline{\varphi }={\varphi }_{20,5}\approx 0.73707$ [equal to the parameters of the stationary state (iii) in Figs. 2 and 8]. Dynamics was averaged over 100 random trajectories [cf. Eq. (G66)]. Note the control of the order of $0.1\%$ in the velocity spread is required in order to achieve $\mathrm{Tr}\left({\rho }^2\right)>0.9$ and $>90\%$ of the QFI that was obtained with a monochromatic beam.

Figure 11

Effective dynamics of realistic micromaser with hard walls. First wall at even $m_1$ leads to hard walls of both parities (cf. Table 1) and multiple even and odd stationary states [(a), (c)], while first wall at odd $m_1$ leads to only odd hard walls and multiple odd stationary states [(b), (d)]. [(a), (b)] Single-photon losses and higher order corrections to far-detuned limit induce local transitions between states of opposite parity: from ${\rho }_k^{+}$ only to ${\rho }_{k-1}^{-}$ and ${\rho }_k^{-}$ (solid arrows), from ${\rho }_k^{-}$ only to ${\rho }_k^{+}$ or ${\rho }_{k+1}^{+}$ (dashed arrows) for $m_1$ even, from the unique even state $|{\mathrm{\Psi }}_{+}\rangle \langle {\mathrm{\Psi }}_{+}|$ to odd states ${\rho }_k^{-}$ (solid arrows), and from the odd states to the even state (dashed arrows) for $m_2$ odd, $k=0,1,...$. [(c), (d)] Atom decay and nonmonochromaticity of atom beam lead to local transitions only between the states of the same parity: from ${\rho }_k^{\pm }$ only to ${\rho }_{k-1}^{\pm }$ and ${\rho }_{k+1}^{\pm }$. All effective dynamics feature detailed balance.