Quantum limits on phase-preserving linear amplifiers

The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.

Figure 1

Input signal $E_{\mathrm{in}}\left(t\right)=\cos \omega t$ (blue) and output signal $E_{\mathrm{out}}\left(t\right)=g\cos \omega t$ (red) for a single-mode, phase-preserving linear amplifier with amplitude gain $g=4$. Input and output have the same phase. The input complex amplitude, $\langle a_{\mathrm{in}}\rangle =\langle x_1\rangle /\sqrt{2}=1$, is amplified to become the output complex amplitude, $\langle a_{\mathrm{out}}\rangle =g$. The (phase-insensitive) noise on the signal is represented by smearing out the mean signal into a band with vertical height equal to the uncertainty in the field, ${\langle \Delta E^2\rangle }^{1/2}={\langle \left|\Delta a\right|}^2{\rangle }^{1/2}/\sqrt{2}$; for the ideal linear amplification with coherent-state input shown here, the height of the input band is $\frac{1}{2}$, and the height of the output band is $\sqrt{(g^2-\frac{1}{2})/2}$. The phase-space diagrams depict the same input and output as do the temporal plots. The mean complex amplitude is represented by an arrow, and the noise by a circle whose diameter is equal to the height of the band in the temporal plot. The temporal plots can be obtained by rotating the phase-space stick and ball about the origin and projecting onto the real axis.

Figure 2

Ball-and-stick phase-space depictions of input and output states for an ideal linear amplifier whose input is a coherent state $|\beta \rangle$ with $\left|\beta \right|=1$ and which has amplitude gain $g=4$, giving the output state a mean that lies on a circle of radius $g\left|\alpha \right|=4$ centered at the origin. The three cases correspond to normal ordering (Glauber-Sudarshan $P$ function), symmetric ordering (Wigner $W$ function), and antinormal ordering (Husimi $Q$ distribution). The input (blue) and output (red) noise are characterized by the variance ${\Sigma }^2$ of $\alpha$; the radius of the noise circle is chosen to be $\Sigma /2\sqrt{2}$. For normal ordering, ${\Sigma }_P^2=\langle \Delta a^{†}\Delta a\rangle$; for symmetric ordering, ${\Sigma }_W^2={\langle \left|\Delta a\right|}^2\rangle =\frac{1}{2}\langle \Delta a\Delta a^{†}+\Delta a^{†}\Delta a\rangle$ (${\Sigma }_W/2\sqrt{2}=\frac{1}{2}{\langle \Delta E^2\rangle }^{1/2}$); and for antinormal ordering, ${\Sigma }_Q^2=\langle \Delta a\Delta a^{†}\rangle$. At the input, ${\Sigma }_P^2=0$, ${\Sigma }_W^2=\frac{1}{2}$, and ${\Sigma }_Q^2=1$; at the output, ${\Sigma }_P^2=g^2-1$, ${\Sigma }_W^2=g^2-\frac{1}{2}$, and ${\Sigma }_Q^2=g^2$. For each ordering, blue is used for the input state and for the amplification of the input mean and noise by a factor of $g$; red depicts the total noise at the output. For symmetric ordering, the output noise consists of the amplified input noise, with variance $\frac{1}{2}{g}^2$, and added noise, with variance $\frac{1}{2}(g^2-1)$, which degrades the output signal-to-noise ratio relative to the perfect amplification of Eq. (2.8). For normal ordering, the input is represented by a (noiseless) point, which is amplified to another point; all of the output noise, which has variance $g^2-1$, appears to be added noise. The flip side is antinormal ordering, where the input has an extra half-quantum of noise relative to symmetric ordering; the amplified input noise, which has variance $g^2$, accounts for all of the output noise, and there appears to be no added noise. Even for the modest gain used here, it is difficult to distinguish the sizes of the output noise circles for the three orderings, because the output noise dwarfs the half-quantum difference between the orderings.

Figure 3

(a) Ideal phase-preserving linear amplifier realized as a parametric amplifier. The primary mode $a$ interacts with an ancillary mode $b$, which begins in the vacuum state $|0\rangle$, via a two-mode squeezing interaction. The output state $\mathcal{E}\left(\rho \right)$ of the primary mode is amplified with an amplitude gain $g=\cosh r$ and with the minimum amount of added noise permitted by quantum mechanics. (b) Parametric amplifier with an arbitrary initial state $\sigma$ for the ancillary mode. The main result of this paper is that any phase-preserving linear amplifier, no matter its physical realization, is equivalent to a parametric amplifier with some physical initial state $\sigma$ for the ancillary mode.

Figure 4

(a) Quantum circuit for measurement-based amplification of primary mode $a=(x_1+i{x}_2)/\sqrt{2}$. (b) Circuit for the equivalent coherent scheme. Solid circles represent control on the first quadrature of a mode; open circles represent control on the second quadrature. In both cases, the initial controlled operation is a sensing step in which the two quadratures, $x_1$ and $x_2$, of mode $a$ are written onto ancillary modes $b=(y_1+i{y}_2)/\sqrt{2}$ and $c=(z_1+i{z}_2)/\sqrt{2}$; $x_1$ is recorded in $y_1$, and $x_2$ is recorded in $z_2$. This sensing step is followed by a feedback operation from $b$ and $c$ to amplify the complex amplitude of $a$. In (a) the feedback operation is classical and based on the results, ${\beta }_1$ and ${\gamma }_2$, of measuring $y_1$ and $z_2$ (measurements are represented by rounded boxes); discarding the measurement outcomes yields a linear-amplification process on the primary mode. In (b) the feedback operation is coherent; the final measurement is irrelevant to the amplification and can be omitted. The amplified output state (2.57) of mode $a$ is obtained by tracing out modes $b$ and $c$. The equivalence of the two circuits, demonstrated in the text, is obvious from the circuit diagrams as an example of the principle of deferred measurement [20, 52]. The single-mode squeezers applied initially to the ancillary modes prepare the input state $|\varphi \rangle$ of Eq. (2.48). When the squeeze parameter $r$ is chosen to squeeze by a factor of two ($e^{2r}=2$), the model is based on the Arthurs-Kelly simultaneous measurement of $x_1$ and $x_2$ [50], which is described by coherent-state projectors, but the result is not quite an ideal linear amplifier. When the squeeze parameter is specified by $e^{2r}=2(g-1)/(g+1)$, the result is an ideal linear amplifier, based on a slightly different model of simultaneous measurement of $x_1$ and $x_2$; this realization of an ideal linear amplifier is different from a parametric amplifier.

Figure 5

Output $P$ functions for five amplifiers. For all five, the input is a coherent state $|\beta \rangle$, with $\beta =1$, and the amplitude gain is $g=4$; these correspond to the conditions in Figs. 1 and 2. The output $P$ function is given by the added-noise function as in Eq. (5.8). In (a) the ancillary-mode state $\sigma$ is the vacuum state, which gives an ideal linear amplifier. The inset shows the $P$ function of the input coherent state as a tiny black dot and the output $P$ function, $P_{\mathrm{out}}\left(\alpha \right)=e^{-{\left|\alpha \right|}^2/(g^2-1)}/\pi (g^2-1)$, as colored contours; this inset is the full contour plot of the $P$-function perspective that is given as a stick figure in Fig. 2. The main plot in (a) displays the same output $P$ function in a three-dimensional plot. In (b)–(e), the ancillary-mode “state” is $\sigma =(\frac{1}{2}-\lambda )|0\rangle \langle 0|+\lambda |1\rangle \langle 1|+\frac{1}{2}|2\rangle \langle 2|$ [Eq. (5.7)]: (b) $\lambda =0.5$, (c) $\lambda =0$, (d) $\lambda =-0.5$, and (e) $\lambda =-1.0$; the output $P$ functions, given by Eqs. (5.8) and (5.9), are displayed as three-dimensional plots. Plots (b) and (c) are physical amplifiers, whereas (d) and (e) are unphysical, this despite the similarity, at least by eye, of (d) to the Gaussian of the ideal linear amplifier in (a). All four values of $\lambda$ are such that the second-moment quantum limit is satisfied, which requires $\lambda \ge -1$, and the output $P$ function is everywhere non-negative, which requires $-1.37=-(1+\sqrt{3})/2\le \lambda \le 1/2$.

Figure 6

Output $P$ function (5.8), which is a displacement of the added-noise function (5.9), for amplifiers with ancillary-mode ``state'' $\sigma$ of Eq. (5.7). The parameter $\lambda$ varies from $0.5$ to $-1.5$ in steps of size $0.025$. As in Fig. 4, the input state is a coherent state $|\beta \rangle$ with $\beta =1$, and the amplitude gain is $g=4$. The $P$ function is plotted along the real $\alpha$ axis, which is sufficient because of the rotational symmetry of the noise. The plotted $P$ functions are not normalized to unity; instead, they are scaled to have maximum value equal to 1. The amplifiers in the range $0<\lambda \le 0.5$, all of which are physical, are plotted in green and blue; $\lambda =0$, the boundary between physical and unphysical, is plotted as a black dashed line. When $\lambda =0.5$, there is no vacuum contribution to $\sigma$, so the $P$ function has a deep hole in the middle. As $\lambda$ decreases from $0.5$, the vacuum contribution increases, and the hole disappears, to be replaced by a distribution with broad shoulders, still faintly evident at $\lambda =0$. The transition from distributions that have a minimum in the middle (blue) to those with a maximum in the middle (green) occurs at $\lambda =0.22$. The values in the range $-1<\lambda <0$, in which the second-moment constraint is satisfied ($A_1\ge 1/2$), are plotted in purple; $\lambda =-1$ is plotted as a black dashed line. The values $\lambda <-1$, plotted in red, violate the second-moment constraint, and for $\lambda <-(1+\sqrt{3})/2=-1.37$, the output $P$ function takes on negative values. The Gaussian output $P$ function of an ideal linear amplifier is plotted as a solid line; it reveals the non-Gaussian character of all the added-noise functions associated with the ancillary-mode state $\sigma$ of Eq. (5.7).