Tuning photon statistics with coherent fields

Photon correlations, as measured by Glauber's $n\mathrm{th}$-order coherence functions $g^{\left(n\right)}$, are highly sought to be minimized and/or maximized. In systems that are coherently driven, so-called blockades can give rise to strong correlations according to two scenarios based on level repulsion (conventional blockade) or interferences (unconventional blockade). Here, we show how these two approaches relate to the admixing of a coherent state with a quantum state such as a squeezed state for the simplest and most recurrent case. The emission from a variety of systems, such as resonance fluorescence, the Jaynes-Cummings model, or microcavity polaritons, as a few examples of a large family of quantum optical sources, are shown to be particular cases of such admixtures, that can further be doctored up externally by adding an amplitude- and phase-controlled coherent field with the effect of tuning the photon statistics from exactly zero to infinity. We show how such an understanding also allows to classify photon statistics throughout platforms according to conventional and unconventional features, with the effect of optimizing the correlations and with possible spectroscopic applications. In particular, we show how configurations that can realize simultaneously conventional and unconventional antibunching bring the best of both worlds: huge antibunching (unconventional) with large populations and being robust to dephasing (conventional).

Figure 1

Schemes considered in this text. (a) A quantum field $s$ can be seen as composed of a classical component (its mean field) $\langle s\rangle$ and quantum fluctuations $\epsilon$. (b) By using a beam splitter, a classical source $a$ can be used to remove the classical component from a quantum source $d$ resulting in a more quantum output $s$. The excess of classical signal is redirected toward the other branch of the interferometer in $o$. (c) A particular case of great popularity in the literature admixes a coherent state $|\alpha \rangle$ to a squeezed state $|\xi \rangle$, producing a displaced squeezed thermal state in a density matrix form ${\rho }_s$ as the other output is discarded (traced over).

Figure 2

(a) Admixing a squeezed state with statistics $g_{\xi }^{\left(n\right)}$ (dashed-dotted lines, $n=2,3$) with a coherent state with $g_{\alpha }^{\left(n\right)}=1$ (dashed) yields a displaced thermal state whose statistics $g_s^{\left(2\right)}$ ranges between antibunching and superbunching, here shown as a function of squeezing $r$. (b) Decomposition of $g_s^{\left(2\right)}$ in terms of the $\mathcal{I}$ coefficients, with ${\mathcal{I}}_1=0$ for a squeezed coherent states admixture. (c) Map of the $g_s^{\left(2\right)}$ realized for arbitrary admixtures of coherent $|\alpha \rangle$ and $r$ squeezed states. The cases in (a) and (b) correspond to the purple-dashed cut shown at $\left|\alpha \right|=0.3$ and $\theta =2\varphi$. (d) Same as (c) but for $g_s^{\left(3\right)}$. In both cases, the black dashed line that optimizes two-photon antibunching shows the mismatch to different photon orders.

Figure 3

$N$-photon statistics of a two-level system driven coherently at rate ${\mathrm{\Omega }}_{\sigma }$ homodyned (laser corrected) with a field of phase $\varphi$ and amplitude a fraction $\mathcal{F}$ of ${\mathrm{\Omega }}_{\sigma }$, in the limit of vanishing driving [(a), Eq. (41)] and for small but finite driving [(b), Eq. (42)]. (c), (d) Maps of two- and three-photon statistics as a function of the homodyning field. (e) Shows how correlations are weakened with increasing driving (solid lines) and how antibunching is realized at a given photon number at a time, as opposed to bunching. Limits are given in Table 1. (f) Shows the dependence on the phase of the homodyning field, for both vanishing and finite drivings for the case of bunching ($\mathcal{F}=2$) and two and three photon antibunching ($\mathcal{F}=4,6$). The only parameter has been taken as ${\gamma }_{\sigma }=1$.

Figure 4

(a) Two-photon statistics of a two-level system in the Heitler limit with (dashed dark) and without (solid light blue) filtering as a function of the homodyning amplitude $\mathcal{F}$. Conventional antibunching gets spoiled by filtering but perfect unconventional antibunching is obtained with homodyning. (b) Evolution of the $\mathcal{I}$ parameters as a function of the filter's width $\mathrm{\Gamma }$ showing how filtering disrupts the perfect cancellation of $g_{s,\mathrm{\Gamma }}^{\left(2\right)}$. Homodyning restores the condition ${\mathcal{I}}_2=-2{\mathcal{I}}_0$. (c) Evolution of the $\mathcal{I}$ parameters as a function of pumping strength ${\mathrm{\Omega }}_{\sigma }$. Without filtering, the cancellation is still perfect at all pumpings but involves ${\mathcal{I}}_1$ and thus cannot be fully restored with homodyning.

Figure 5

Landscape of two-photon correlations in the Jaynes-Cummings system, for cavity driving only (left, with $\tilde{\chi }=0$) and mixed driving (right, with $\tilde{\chi }=0.5$ and $\varphi =\pi /2$), as a function of where the system is driven (${\omega }_{\mathrm{L}}$) and where it is emitting (${\omega }_a$). The top row shows the exact results, Eq. (55) left and Eq. (56) right, and the bottom row its classification in terms of conventional (C) and unconventional (U) features of antibunching (A) and bunching (B), namely, Eq. (57) with $N=1$ for CA (solid blue) and $N=2$ for CB (solid red), and Eq. (60) for UA (dashed blue). For the left case with cavity pumping only, the UA simplifes to Eq. (62). Parameters: $g=1, {\gamma }_a=0.1$, and ${\gamma }_{\sigma }=0.01$.

Figure 6

Conventional and unconventional features at the $N=1,2,3$, and 4-photon level. Upper row shows the numerically exact landscapes of $N$-photon observables, from the population normalized to the relative laser intensity ${\tilde{\mathrm{\Omega }}}_a^2={\mathrm{\Omega }}_a^2/{\gamma }_a^2$ ($N=1$, left) until four-photon correlations ($N=4$ right). Middle row shows the theoretical lines that reproduce these structures and their classifications as conventional (C, solid) and unconventional (U, dashed) bunching (red) and antibunching (blue), respectively. Bottom row shows the cuts along the horizontal dashed line in the top row. The inset in $g_a^{\left(4\right)}$ magnifies a forking of antibunching. Parameters are the same as in Fig. 5.

Figure 7

Decomposition of the Jaynes-Cummings statistics into its $\mathcal{I}$ coefficients (in log scales, separating the positive and negative components) for the two cuts shown in the middle panel. The upper case captures one of the exactly-zero antibunching, on the UA line. Parameters are the same as in Fig. 5.

Figure 8

Intersection of the UA and CA lines in the Jaynes-Cummings model. (a) Representation to show simultaneously the lines' shape in the correlation landscape and their magnitude, with two of them intersecting with the effect of the UA line dragging down the CA line. (b), (c) Correlation landscapes with (b) no intersection, when $\chi =0.87$ and $\varphi =\pi$ and (c) [case also shown in (a)] with intersection, when $\chi =0.5$ and $\varphi =\pi /2$. (d), (e) Magnitude of the UA and CA along their respective lines, as a function of ${\omega }_{\mathrm{L}}$ as the scanning parameter. Note that, as a consequence, the case ${\omega }_{\mathrm{L}}=0$ corresponds to infinite cavity detuning, indicated by a gap opening which is otherwise continuous. In the intersecting case, note the highly populated, strong, and all-order antibunching. Parameters are the same as in Fig. 5.

Figure 9

Decomposition of $g_a^{\left(2\right)}$ in terms of the $\mathcal{I}$ coefficients (Supplemental Material [63]) for microcavity polaritons, along the two cuts shown in the central panel, which intersect the points where $g_a^{\left(2\right)}$ becomes exactly zero to leading order for the upper cut and corresponds to the CA-UA intersection point for the lower cut. Parameters: $g=1, {\gamma }_a=0.1, {\gamma }_b=0.01, U=1, \chi =0$ ($\varphi =0$). The cuts are at ${\omega }_a/g\approx 8.63$ where UA is exactly zero and ${\omega }_a/g\approx -0.44$ where UA and CA intersect.

Figure 10

Conventional and unconventional statistics in a microcavity polariton system, with strong (top row, $U/{\gamma }_a=0.2$) and weak (bottom row, $U/{\gamma }_a=0.03$) interactions. (a), (d) Conventional (solid) and unconventional (dashed) antibunching as a 3D representation of the landscapes of correlations shown in (b)–(e). Note how the CA line gets pulled down by the UA one if they intersect. (c)–(f) $g_a^{\left(n\right)}$ for $n=2,3,4$ along the upper (UP) and lower (LP) polariton lines (i.e., CA). The upper polariton line gets a much better antibunching thanks to its proximity, or even intersection, with the unconventional antibunching. The inset in (f) shows for comparison the much smaller antibunching of the upper line (the one so far reported experimentally).

Figure 11

Polariton antibunching as a function of polariton interactions. (a) Antibunching on the two polariton branches, showing the much greater UP antibunching due to the proximity to the UA line, even at very small interactions [zoom-in (b)]. An optimum is obtained for a finite value of interactions, in contrast to antibunching on the LP line which is optimum in the Jaynes-Cummings limit $U\to \infty$. The solid lines in (a) show the optimum value of $g_a^{\left(2\right)}$ on the LP (black) and the UP (red). On the UP, where it is significantly stronger, this is due to the intersection, or proximity, between UA and CA, until interactions get very large, $U/\gamma \gtrsim 10$, where another local minimum overtakes and the $\mathrm{UA}+\mathrm{CA}$ carries on as the dashed line. (c) Details of the proximity (top) and intersection (bottom) effect for the small and high interactions, respectively. The vertical line shows the minimum antibunching on the UP line, which is CA. (d) Location on the UP line where to drive the system depending on its interaction $U/{\gamma }_a$ to optimize its two-photon antibunching.

Figure 12

Effect of dephasing on photon correlations $g_a^{\left(2\right)}$. In (b), the correlation landscape scanning in frequencies, with two cuts shown in (a) and (c) at ${\omega }_a/g=-3$ and 8, respectively, along with their dephasing-free counterparts (light blue). Some of the antibunching and bunching peaks, related to unconventional features, are strongly suppressed whereas conventional lines are left almost untouched. Additionally, new dephasing-induced lines emerge, labeled I and II in (a)–(c). (d) Shows the robustness of the antibunching against dephasing: (i) for the $\mathrm{UA}+\mathrm{CA}$ intersection [marked with a gray circle in (b)], (ii) for CA, and (iii) for UA at the position shown in (c). Parameters: $g=1, {\gamma }_a=0.1, {\gamma }_b=0.01, U=0.5, \chi =0$ (cavity driven), and ${\gamma }_{\varphi }=0.1{\gamma }_b$.