Reduced decoherence using squeezing, amplification, and antisqueezing

Loss and decoherence are major problems in the transmission of nonclassical states of light over large distances. It was recently shown that the effects of decoherence can be reduced by applying a probabilistic noiseless attenuator before transmitting a quantum state through a lossy channel, followed by probabilistic noiseless amplification [M. Micuda, I. Straka, M. Mikova, M. Dusek, N. J. Cerf, J. Fiurasek, and M. Jezek, Phys. Rev. Lett. 109, 180503 (2012)]. Here we show that similar results can be obtained for certain kinds of macroscopic quantum states by squeezing the signal before transmission, followed by deterministic amplification and antisqueezing to restore the original amplitude of the state. This approach can greatly reduce the effects of decoherence in the transmission of non-Gaussian states, such as Schrödinger cat states, without any reduction in the data transmission rate.

Figure 1

(a) Reduction in the decoherence of a quantum signal by applying a noiseless attenuation factor $\nu$ before transmission through a lossy channel, followed by noiseless amplification with gain $g$ [23]. The probabilistic nature of the noiseless attenuation and amplification results in an exponential decrease in the data transmission rate in this approach. (b) Reduction in the decoherence of a Schrödinger cat state by applying an appropriate squeezing operation $\widehat{S}$ before transmission, followed by a deterministic amplifier (OPA) and antisqueezing ${\widehat{S}}^{†}$. This approach has the advantage that all of the operations are deterministic and the data transmission rate is not reduced as a result.

Figure 2

Phase-space diagram illustrating the squeezing of a Schrödinger cat state before transmission through a lossy channel to minimize decoherence as shown in Fig. 1. The real and imaginary axes labeled Re and Im correspond to either the Wigner distribution or the $Q$ function, while the solid and dashed lines represent the 1-$\sigma$ contours of the relevant Gaussian distributions. The initial cat state is assumed to have components that differ by a phase shift of $\pi$ as indicated by the blue (solid) lines. The squeezing parameters are chosen to reduce the amplitudes along the real axis as indicated by the red (dashed) lines. This process decreases the overall amplitude of the signal and thus the number of photons left in the environment due to loss and amplification.

Figure 3

Measurement of the amount of decoherence using a Schrödinger cat interferometer [3, 4, 5]. The source box on the left produces a Schrödinger cat state as illustrated in Fig. 2. Here $|{\alpha }_0\rangle$ is an initial coherent state and $K$ is a Kerr medium located in one path of a single-photon interferometer. The Kerr medium produces a phase shift of $2\varphi$ if single photon ${\gamma }_1$ passes through it. A constant phase shift of $-\varphi$ is applied in both paths (not shown), which gives a net phase shift of $\pm \varphi$ depending on the path taken by ${\gamma }_1$. A variable phase shift $\theta$ is also applied to ${\gamma }_1$ in one path of the single-photon interferometer, with postselection on the detection of ${\gamma }_1$ in the detector shown. After passing through a squeezer $\widehat{S},$ lossy transmission channel, OPA, and an antisqueezer ${\widehat{S}}^{†},$ the visibility of the quantum interference between the two components of the cat state is measured using the apparatus shown in the analyzer box on the right. Here a second photon ${\gamma }_2$ passes through a single-photon interferometer with a Kerr medium in one of its paths, which produces another phase shift of $\pm \varphi .$ The events are postselected based on single-photon detection in the detectors shown, along with a net phase shift of zero as measured by the homodyne detector (H.D.). This results in quantum interference between the two components of the original cat state, as described in more detail in Fig. 4.

Figure 4

Phase-space representation of the state of the system as it progresses through the apparatus shown in Fig. 3 for the case of $\varphi =\pi /2.$ The horizontal axes correspond to the real part of the Wigner distribution or $Q$ function as in Fig. 2, while the vertical axes correspond to the imaginary part. (a) The initial coherent state $|{\alpha }_0\rangle$ (solid circle) along with the idler mode of the OPA (dotted circle) which is initially in its vacuum state. (b) Cat state created by the first single-photon interferometer, where the Kerr medium and a constant phase shift apply a net phase shift of $\pm \pi /2$ depending on the path taken by the single photon. (c) Reduced overall amplitude of the cat state components due to squeezing along the real axis. The squeezing also increases the amplitude along the imaginary axis. (d) Compensation for the effects of loss using a deterministic amplifier, which also displaces the idler modes. This results in a state where the signal and idler modes are entangled. (e) Restoration of the original amplitude of the cat state using antisqueezing. (f) A second phase shift of $\pm \pi /2$ is produced in the analyzer of Fig. 3, depending on the path taken by the second single photon. Postselecting on a net phase shift of zero results in an overlap between the two original components of the cat state. This gives quantum interference, the visibility of which provides a measure of the amount of decoherence, as described in more detail in Ref. [5].

Figure 5

Plot of the visibility of the quantum interference as a function of the squeezing parameter $r$. The solid red curve corresponds to a relatively small gain of $g=1.001$ while the dotted blue line corresponds to a loss factor of $t=0.999.$ The black dashed curve shows the combined effects of both loss and gain. The amplitude of the coherent state was $|{\alpha }_0|=100$ and the parameters in the interferometer of Fig. 3 were chosen to be $\varphi =\pi /2$ and $\theta =0.$ These results show that squeezing and antisqueezing can produce a large improvement in the visibility even when the loss and gain are relatively small.

Figure 6

Plot of the average number of photons left in the environment as a function of the squeezing parameter $r$ under the same conditions used in Fig. 5. The solid red line corresponds to the number of idler photons emitted by the OPA, while the blue dots correspond to the number of photons left in the environment by a beam splitter used to model loss. The optimal value of the squeezing parameter $r$ minimizes the number of photons left in the environment.

Figure 7

Maximum achievable visibility as a function of the initial coherent-state amplitude $|{\alpha }_0|$ for several values of the gain $g.$ (a) The optimal value of the squeezing parameter $r$ as a function of $|{\alpha }_0|$. (b) The corresponding maximum visibility as a function of $|{\alpha }_0|$. It can be seen that the maximum visibility decreases more rapidly as a function of $|{\alpha }_0|$ for larger values of the gain.

Figure 8

Optimal squeezing parameter $r$ and the corresponding maximum visibility as a function of the gain for several values of $|{\alpha }_0|$.

Figure 9

Plots of the visibility for several different values of the squeezing parameter $r$. (a) The visibility plotted as a function of the gain with $|{\alpha }_0|$ fixed at a value of 10. (b) The visibility as a function of $|{\alpha }_0|$ with the gain fixed at a value of 1.1.

Figure 10

Contour plots of the visibility as a function of the amplitude gain and the squeezing parameter $r.$ (a) $|{\alpha }_0|=10$. (b) $|{\alpha }_0|=100$.

Figure 11

Contour plots of the visibility as a function of $|{\alpha }_0|$ and the squeezing parameter $r$. (a) Gain $g=1.01.$ (b) Gain $g=1.1.$