Measurement-based generation of shaped single photons and coherent state superpositions in optical cavities

We propose related schemes to generate arbitrarily shaped single photons, i.e., photons with an arbitrary temporal profile, and coherent state superpositions using simple optical elements. The first system consists of two coupled cavities, a memory cavity and a shutter cavity, containing a second-order optical nonlinearity and electro-optic modulator (EOM), respectively. Photodetection events of the shutter cavity output herald preparation of a single photon in the memory cavity, which may be stored by immediately changing the optical length of the shutter cavity with the EOM after detection. On-demand readout of the photon, with arbitrary shaping, can be achieved through modulation of the EOM. The second scheme consists of a memory cavity with two outputs, which are interfered, phase shifted, and measured. States that closely approximate a coherent state superposition can be produced through postselection for sequences of detection events, with more photon detection events leading to a larger superposition. We furthermore demonstrate that no-knowledge feedback can be easily implemented in this system and used to preserve the superposition state, as well as provide an extra control mechanism for state generation.

Figure 1

(a) The scheme consists of two coupled cavities, memory cavity (MC) and shutter cavity (SC). The former supports two modes, ${\widehat{a}}_h$ (heralding) and ${\widehat{a}}_e$ (emission), with frequencies ${\omega }_h$ and ${\omega }_e$, respectively. Pairs of photons ${\widehat{a}}_h^{†}{\widehat{a}}_e^{†}$ are created by pumping of an optical nonlinearity ${\chi }^{\left(2\right)}$ with a laser. SC supports a single mode $\widehat{b}$ of frequency ${\omega }_{sc}\left(t\right)$, which may be tuned by the EOM. The output channels of the two cavities may be monitored in a number of ways, and based on this we vary ${\omega }_{sc}\left(t\right)$ and the pumping $\mathrm{\Lambda }\left(t\right)$ of ${\chi }^{\left(2\right)}$.

Figure 2

The scheme for shaped single photon generation consists of two cavities coupled by a semitransparent mirror. MC contains a ${\chi }^{(2,n)}$ optical nonlinearity, pumped by a coherent field $\mathrm{\Lambda }\left(t\right)$. SC contains an EOM, which allows us to control its resonant frequency. MC is perfect while SC allows leakage at a rate ${\kappa }_{sc}$, with the output field continuously monitored by a photodetector.

Figure 3

Simulations of Eq. (14) during readout stage, with $\mathrm{\Delta }\left(t\right)$ chosen according to Eq. (17) in order to generate (a) Gaussian and (b) rising exponential pulse shapes. The dashed line shows the desired temporal profile ${\left|\xi \left(t\right)\right|}^2$, while the solid region is the emission probability density Eq. (18). We begin with a single photon in the ${\widehat{a}}_e$ mode, and choose parameters $g=100\gamma$ and ${\kappa }_{sc}=10g$ (with $\mathrm{\Lambda }=0$ during readout).

Figure 4

(a) The system for generation of CSS consists of a single cavity, MC, containing a ${\chi }^{(2,d)}$ optical nonlinearity, pumped by a coherent field of constant amplitude $\mathrm{\Lambda }$. Polarization-sensitive semitransparent mirrors control which mode is emitted from each mirror, and a wave plate $\lambda$ matches the polarisations so that the two fields are indistinguishable. They are interfered through a 50:50 beam splitter, then have a relative phase of $\frac{\pi }{2}$ introduced leading to two Hermitian decoherence channels, which are measured by detection blocks (DBs). During the no-knowledge feedback stage, a coherent field with frequency ${\omega }_e$ and complex amplitude $\beta \left(t\right)$ proportional to the measurement signal is introduced into the cavity. (b) A schematic of each DB, which allows us to choose between photodetection and homodyne measurement. During state generation (1.) the decoherence channel is directed along the path indicated by the solid arrows to the photodetector, where the grey block denotes a mirror of controllable reflectivity, and a simple switch in the bottom left controls which signal is received by the controller. For storage via no-knowledge feedback, the mirror is made transmissive and the channel follows the dashed arrows towards a homodyne measurement (2.), where it is interfered with a local oscillator ($LO$) and the two beam splitter outputs are subtracted. The corresponding photocurrent is used to modulate $\beta \left(t\right)$ incident on the MC, which has the effect of removing the decoherence channel from the system dynamics [32].

Figure 5

We plot in (a) the fidelity $\mathcal{F}$ of the state generated by $n$ consecutive detections from the ${\widehat{L}}_{+}$ channel with the normalized version of the CSS in Eq. (27). The solid (black) line shows the ideal fidelity given by (28), if the jumps occurred instantaneously with negligible evolution in-between. We see that this rapidly approaches a value of around 0.97. The effect of the no-jump evolution is shown by the vertical series, which simulate 1000 trajectories for a time $t\gamma =5$ and plot the average fidelity of the postselected states with Eq. (27). The error is taken to be the variance, and indicated by the range of the brackets. The numbers accompanying each series give the percentage of trajectories that survived postselection, which we interpret as the probability of the state being generated. The dashed (brown) series denotes postselection only for the correct jump sequence, while the solid (orange) series also selects for a maximum time between jumps of $t\gamma =\frac{1}{2}$. In (b) we show a histogram of the fidelities of the ideal state Eq. (27) with the generated CSS states for $n=6$, where we postselect only on jump sequence. We see that this is concentrated at the ideal value of 0.97.

Figure 6

We show Wigner functions and Fock basis expansions for (a) a superposition of the Fock states $|1\rangle$ and $|4\rangle$ and (b) a CSS, which approximates a four-component coherent state superposition, both generated through sequences of detections from the scheme in Fig. 4. In (c) we shown an actual Schrödinger's cat coherent states superposition (the Fock basis expansion is shown up to $n=14$), where $\alpha =2.825e^{i\frac{\pi }{4}}$. The CSS approximates this state with a fidelity of 0.98. In the top plots color denotes the height of the Wigner function. For the bottom plots color and the series labels denote the complex phase of the Fock basis coefficients.