Quantum noise at the nonlinear mapping of phase space (Kouznetsov)

Quadrature components of a single-mode field are interpreted as coordinates of the phase space. It is assumed that a quantum amplifier transforms the state of a single mode and the initial state of the field in this mode is a coherent squeezed one. The transfer function relating the average values of the field in the initial and final states determines mapping of the phase state. In the case of amplification, the field uncertainty in the final state is usually greater than in the initial state. This increase is interpreted as quantum noise of the amplifier. The lower bounds of this noise are estimated in terms of the derivatives of the transfer function. A degenerate parametric amplifier with pumping depletion is considered to be an illustration. Transformation of an initially orthogonal rectangular net in the phase space and deformation of the body of uncertainty given by the Wigner function are constructed for such an amplifier.

Introduction
A classical field in a certain mode can be amplified without introducing additional noise, whereas a quantum field cannot, even if prepared in a coherent state. Field amplification in a quantum mode produces quantum noise, i.e., increases the total uncertainty of quadrature field components.

We call an amplifier linear, if operators of the amplified field may be represented in the form of a linear combination of operators of the initial field with C-numerical coefficients. The minimum noise of linear amplifiers is known [1-4]. But what is the minimum noise of nonlinear amplifiers? This question has as yet only been partially studied [5,6].

For a phase-invariant amplifier, we have the lower bound of quantum noise [6] (phase invariance means that the amplification coefficient does not depend on the phase of the initial field). These estimates are applicable only for a field prepared initially in a coherent state. In this case, the lower bounds of noise of linear amplifiers [1-4] are applicable to any (not necessarily phase-invariant) linear amplifier and any initial state of the field. Needless to say, any partial estimate of the lower bound of quantum noise may be useful for properties of nonlinear amplifiers. But at the same time, it is desirable to have a stronger and more general estimate, where possible.

We showed that a phase-invariant nonlinear amplifier may produce a lower noise than an ideal linear amplifier with the same amplification coefficient [6]. In this paper, we generalize the lower bounds of noise in two ways. First, we consider an initial state with arbitrary squeezing. Second, we remove the requirement of phase invariance. Thus, our lower estimates are also applicable to parametric quantum amplifiers. These two generalizations are related to each other.

The noise in our mode may be defined as $$~D=D_1+D_2-1/2~$$, where $$~D_1~$$ and $$~D_2~$$ are the dispersions of the quadrature components. We use varaible $$~i~$$ to numerate components, so, $$~i~$$ may have values 1 or 2. It should not be confused with imaginary unity $$~{\rm i}=\sqrt{-1}~$$, which is not variable at all; $$~{\rm i}^{2}=-1 ~$$. When a parametric quantum amplifier is used, any of the dispersions $$~D_{1}~$$ or $$~D_{2}~$$ (but not both of them) may be as small as desired.

In this case, the initial state of the field becomes squeezed. Thus, we should consider squeezing in order to obtain useful lower estimates for the noise of an amplifier of general type.

A parametric amplifier amplifies quadrature field components with different coefficients. Therefore, transformation of the average values of these components should be described by two functions or a single complex-valued transfer function. We will determine it in Section I. This function maps initial values of quadrature components onto their output mean values. The quadrature components of a single fixed mode correspond to the coordinates $$~(x, p)~$$ of the phase space. Therefore, the transfer function determines mapping of the phase space onto itself. This is a nonlinear mapping in the general case. As already noted, our object is to bind possible bound of quantum noise from below. In Section II, we derive the lower estimates for the noise $$~D~$$ of an arbitrary amplifier and for the dispersions $$~D_{1}~$$ and $$~D_2~$$ of the quadrature components.

In Section III, we illustrate our results for the case of a parametric quantum amplifier with nonlinearity resulting from the depletion of pumping. We represent the transfer function as a distortion of the initially uniform rectangular net of values of initial quadrature components. To represent the growth of the uncertainty of the components in graphic form, we also construct the distribution of the Wigner function as the quasiprobability of the distribution of the quadrature compo- nents and demonstrate how the body of uncertainty is distorted when the phase space is subjected to nonlinear transformations.

Amplification, coherent states and squeezing
Quantum mechanics of amplifiers suggests that an amplifier converts an input field a into an output field $$~A~$$ through a unitary transformation: $$~A=U^{\dagger}+aU~$$; we are using lower-case letters for the input field and uppercase letters for the output field. For simplicity, we restrict ourselves to the case of a single-mode amplifier. In terms of the operators of the input field $$~a~$$ and its Hermite conjugation $$~a^{\dagger} ~$$, the input quadrature components may be represented as $$(1) a_{1}=\frac{a+a^{\dagger}}{2},~ a_{1}=\frac{a-a^{\dagger}}{2\!~ {\rm i}} $$ These components are coordinates of the phase space and do not commute. Let us denote the mathematical expectations of the input and output fields as $$~\langle a_{i}\rangle~$$ and $$~\langle A_{i}\rangle~$$, respectively. Then, $$~U~$$ determines phase space mapping from $$~\langle a_1 \rangle + i\langle a_2 \rangle ~$$ to $$~\langle a_1 \rangle ~$$, $$~\langle a_2 \rangle ~$$. The transition from $$~\langle a_1 \rangle + i~\langle a_2 \rangle ~$$ to $$~\langle a_1 \rangle + i~\langle a_2 \rangle ~$$ determines the transfer function of the amplifier. The gain factor $$~G_i~$$ is the ratio of the output and the input mathematical expectations, $$~G_i = \langle A_i\rangle /\langle a_i\rangle~$$. In the general case, this factor is a function of both $$~\langle a_1 \rangle ~$$ and $$~\langle a_2 \rangle ~$$. We determine the noise $$~D~$$ of the amplified state as $$ (2)D=\langle A^{\dagger} A \rangle - \langle A^{\dagger} \rangle \langle A\rangle=D_1+D_2-\frac{1}{2} $$ where $$ (3)D_i= \langle A_i^2\rangle - \langle A_i\rangle^2 $$ The object of this work is to obtain the lower estimates for $$~D_1~$$, $$~D_2~$$, and $$~D~$$. These estimates are known in certain particular cases. For a linear quantum amplifier, $$~G_i~$$ is constant [1-4]. In this case, the uncertainties of the amplified quadrature components satisfy the inequality $$ (4) D_1 D_2 \ge \frac{2 G_1 G_2-1}{16} $$ from which it follows that $$ (5) D\ge G_1 G_2-1 $$ For $$~G_1 = G_2~$$, the amplifier is phase-invariant, and $$~D \ge G_2-1 ~$$. Both of these lower bounds (4) and (5) are true for any type of linear amplifier, regardless of the initial state of the field over which the mean values are taken. In the general (nonlinear) case, lower estimates require some knowledge of the initial state, and the minimum noise is no longer determined by the value of the gain factor for a given initial state. In the previous work [6], lower estimates of the noise of a phase-invariant amplifier are presented under the assumption that the field is initially prepared in a coherent state. Here, we allow for squeezing of the initial state $$~| \rangle~$$ $$( 6) $$ where $$~T_{\alpha}~$$ a is the displacement operator $$( 7) T_{\alpha}=\exp(\alpha a^{\dagger}-\alpha^{*}a) $$ and $$~S_z~$$ is the squeezing operator $$( 8) S_z=\exp\!\left( \frac{z}{2}(a^{\dagger})^2 - \frac{z}{2}a^2 \right) $$ Here, $$~\alpha~$$ and $$~z~$$ are C-numerical complex-valued parameters, $$~\alpha~$$ determines the amplitude and the phase of the initial coherent state, and $$~z~$$ determines the direction and degree of squeezing; we are using Greek letters to denote coherent states, and Latin letters to denote $$~n~$$-photon states. The state $$~|m,0\rangle~$$ represents $$~m~$$ photons and a certain specially prepared state of the amplifier uncorrelated to them. Although this notation could suggest that the amplifier is prepared in the ground state, we do not make such an assumption. In what follows, an expression $$~\langle Q \rangle ~$$ for any operator $$~Q~$$ determines the mathematical expectation of the operator $$~Q~$$ over the state $$~|a, 0\rangle _z~$$. We rewrite the state $$~|a,0\rangle_z~$$ in a a little bit different way. Using the identity $$( 9) S_z^\dagger a S_z = a\!~ \cosh|z| + \alpha^* \frac{z}{|z|} \sinh(z) $$ we can change the order of the translation and the squeezing operators in formula (6). We have $$(10) T_{\alpha} S_{z}|0,0\rangle= S_{z} T_{\gamma}|0,0\rangle $$ where $$(11) \gamma=\alpha \cosh|z|-\alpha^{*}\frac{z}{|z|} \sinh|z| $$
 * \rangle = |\alpha,p\rangle_{z}=T_\alpha S_z |0,0\rangle
 * \alpha,0\rangle_{z}=

In such a way, the amplification appears as operator, and the amplified field, generally, has no need fo be a coherent state. Even worse, the amplifier with amplification coefficient larger than unity cannot provide the coherent stte at the outout; the output state is entangled with state of an amplifier. This entanglement appears as increase of the uncertainty of the filed operator; this increase of uncertainty can be interpreted as quantum noise.

Lower bounds of noise and dispersions
For the lower bounds of the noise of nonlinear amplifiers to be obtained, two basic formulas are required. First, it follows from formulas (1) and (17) of [6] that $$(12) \frac{1}{\sqrt{m!\!~}}~ \frac{\partial ^{m}}{\partial \beta^m}~\! \langle \beta,0 \beta,0 \rangle = \langle 0,0 m,0 \rangle $$ for any operator $$~Q~$$ independent of B; here, $$~|\beta,0\rangle~$$ represents a coherent state of the field. Define the new operator $$(13) I_{0}= \sum_{m=0}^{\infty} S_z\!~ T_{\beta} \!~ |m,0\rangle\!~ \langle m,0 | T_\beta \!~ S_z $$ It is a projector; $$~I^2_{}0 = I_0~$$ and inequality $$~\langle {\rm any} | {\rm any} \rangle \ge \langle {\rm any} |\!~I_{0}\!~ |{\rm any}\rangle~$$ is valid for any state. In particular, $$(14) \langle A^{\dagger} A \rangle \ge \langle A^{\dagger} I_{0}A \rangle $$
 * ~Q~|
 * ~T_{\beta}^{\dagger}\!~Q\!~ T_{\beta}~|

From formulas (12)—(14), follows

Theorem 1. $$~ D \ge E(\alpha,z)~$$ and $$~D \ge F(\alpha,z)~$$, where $$(15) E(\alpha,z)=\sum_{n=1}^{\infty} \frac{1}{n!} \left| \left(				\cosh\!|z| \frac{\partial}{\partial \alpha}+	\frac{z}{|z|}\sinh\!|z|\frac{\partial}{\partial \alpha^{*}} \right)^{n} \langle A^{\dagger} \rangle \right|^2 $$ and $$(16) F(\alpha,z)=\sum_{n=1}^{\infty} \frac{1}{n!} \left| \left(				\cosh\!|z| \frac{\partial}{\partial \alpha}+	\frac{z}{|z|}\sinh\!|z|\frac{\partial}{\partial \alpha^{*}} \right)^{n} \langle A \rangle \right|^2-1 $$

Proof. Use identity (14) and definition (2) of the noise $$~D~$$ gives $$(17) D\ge \langle A^\dagger I_0 A \rangle - \langle A^\dagger \rangle \langle A \rangle = \sum_{n=1}^{\infty} \left| \langle 0,0| ~ T_{\beta}^{\dagger}~ S_{z}^{\dagger} A S_{z}~ T_{\beta}~ ~ \rangle \begin{array}{cc} \\ \\ \end{array}\!\!\!\!\! \right|^{2} $$ Use (12) for $$~Q = S+zA+Sz~$$ gives $$(18)~ D\ge \sum_{n=1}^{\infty} \frac{1}{n!} \left| \frac{\partial^{n}}{\partial \beta^{n}} \langle 0,0|\!~ T_{\beta}^{\dagger}\!~ S_{z}^{\dagger} A^{\dagger} S_{z}\!~ T_{\beta}\!~ \right|^{2} = \sum_{n=1}^{\infty} \frac{1}{n!} \left| \left(\!				\cosh\!|z| \frac{\partial}{\partial \alpha}+	\frac{z}{|z|}\sinh\!|z|\frac{\partial}{\partial \alpha^{*}} \!\right)^{\!n} \langle A^{\dagger}\rangle \right|^{2}=E(\alpha,z). $$ However, we may utilize the conjugation of formula (14). In this case, $$~A~$$ and $$~A^{\dagger}~$$ change places. Let us apply the relation $$~AA^{\dagger}=A^{\dagger}A - 1~$$ to definition (2). Then, using (12) for $$~Q = S^{\dagger}_z A S_z~$$, we get the lower estimate for $$~F(a,z)~$$. (end of proof)
 * 0,0
 * 0,0\rangle

Let us apply this theorem to special cases. By setting the squeezing parameter $$~z=0~$$ and assuming that $$~G = G_1 = G_2~$$ depends only on $$a^*a$$, we obtain conditions for a nonlinear phase-invariant amplifier. In this case, the output mean value of the field is $$~\langle A \rangle = G a~$$, and Theorem 1 reproduces the lower estimate of the noise of such an amplifier [6]. In a still more special case, we set $$G = \rm const$$ and obtain the lower bound [1-4] for a linear phase-invariant amplifier. Let us turn now to the dispersions $$~D_1~$$ and $$D_2$$ of the quadrature components. For a linear parametric amplifier, the noise in one component may be as small as desired. For a nonlinear parametric amplifier, we have

Theorem 2. $$~D_i~ \ge E_i(\alpha, z)~$$, where $$(19) E_{i}= \sum_{n=1}^{\infty}\frac{1}{n!} \left| \left( \frac{   1}{2}\left(\cosh|z|+\frac{z^{*}}{|z|}\sinh|z|\right)\frac{\partial}{\partial \alpha_{1}} -\frac{\rm i}{2}\left(\cosh|z|-\frac{z^{*}}{|z|}\sinh|z|\right)\frac{\partial}{\partial \alpha_{2}} \right)^{\!n} \langle A_{i}\rangle \right|^{2} $$ The Proof. Similar to the proof of Theorem 1, we start with the definition $$D_i = (A_i^2) - (A_i)^2 $$ and have $$(20) D_{i}\ge \langle A_{i} I_0 A_{i} \rangle - \langle A_{i} \rangle ^{2} = \sum_{n=1}^{\infty} \left| \langle 0,0| T_{\beta}^{\dagger}S_{z}^{\dagger}A_{i}S_{z}T_{\beta} \right|^2 $$ Taking (12) into account, we obtain $$(21) D_i\ge \sum_{n=1}^\infty \frac{1}{n!} \left| \left( \cosh|z| \frac{\partial} {\partial \alpha} + \frac{z}{|z|}\sinh|z|\frac{\partial}{\partial \alpha^*} \right)^{\!n} \langle A_i\rangle \right|^2 ~=~ $$ $$ =~\sum_{n=1}^\infty \frac{1}{n!} \left| \left(\frac{1}{2}\left( \cosh|z|+\frac{z}{|z|}\sinh|z|  \right)\frac{\partial}{\partial \alpha_1}  +		  \frac{1}{2}\left(  \cosh|z|+\frac{z}{|z|}\sinh|z|  \right)\frac{\partial}{\partial \alpha_2}	\right)^{\!n} \langle A_i\rangle \right|^2 $$ Note that only the first term ($$~m=1~$$) in expression (19) for $$~E_i(a, z)~$$ survives for a linear parametric amplifier. In a proper choice of the squeezing parameter $$~z~$$, any of the dispersions $$~D_1~$$ or $$~D_2~$$ may be made as small as desired. By choosing $$~z~$$, which makes $$~D_1~$$ small, we make $$~D_2~$$ large at the same time, and vice versa. Therefore, we cannot make both $$~D_1~$$ and $$~D_2~$$ as small as desired while retaining equation (4). To extend this equation to the case of nonlinear parametric amplifiers, we could multiply the estimates $$~D_1 \ge E_1(\alpha,z)~$$ and $$~D_2 \ge E_2(\alpha,z)~$$ of Theorem 2 and obtain $$~D_1 D_2 \ge E_1(\alpha,z) E_2(\alpha,z)~$$, but this would not be the best possible estimate. It does not reproduce formula (4) in the linear case [7].
 * 0,0 \rangle

Example
All real amplifiers are saturable. Saturation manifests itself in the fact that the gain factors $$~G_i~$$ become functions of input amplitudes $$~\alpha_1~$$ and $$~a_2~$$, and the map ping of the phase space becomes nonlinear. As an example of such nonlinear mapping, we consider a parametric amplifier with depleted pumping [8] $$(22) H=\frac{1}{2\rm i}\Big( a^{2}b^{\dagger} - (a^{\dagger})^{2} b\Big) $$ Here, $$~b~$$ is the operator of field in the pumping mode with the same commutation relations as in mode $$~a~$$; $$~a~$$ and $$~a^{\dagger}~$$ commute with $$~b~$$ and $$~b^{\dagger}~$$. At positive values of $$~t~$$, the operator $$~U = \exp(-iHt)~$$ is unitary; it determines the transformation. Assume, the initial state is coherent in each mode: $$~\alpha = \langle a \rangle~$$, $$~\beta = \langle b\rangle~$$. The transfer function maps $$~\alpha_1~$$ and $$~\alpha_{2}~$$ into $$~\langle A_1 \rangle ~$$ and $$~\langle A_2 \rangle ~$$.

For numerical calculations illustrating mapping of the phase space, we diagonalized the Hamiltonian on the sub- space of states with a number of photons no greater than 40. In doing so, we considered states $$~|k - 2m, m\rangle~$$ with $$~0 \le k \le 40~$$ and $$~0 \le m \le k/2~$$, ($$~k~$$ and $$~m~$$ are integers), in the Fock representation. Here, $$~a^\dagger a | k-2m,m\rangle = (k-2m)|k-2m, m\rangle~$$, and $$~b^\dagger b | k-2m,m\rangle =    m |k-2m, m\rangle~$$. Note that transformation $$~U~$$ does not mix states with different values of $$~k~$$, so that, in fact, we did not need to diagonalize matrices wider than $$~21 \times 21~$$. The MAPLE software was used for diagonalization, and eigenvalues and eigenfunctions were cal- culated with 20 significant digits. We chose $$~\beta = 2~$$ to show how nonlinear parametric amplification deforms the initial phase space. Let us have a uniform orthogonal net for $$~t=0~$$ (Fig. 1a). The quality of representation does not allows to see a deformation of this net in the right upper corner, it is of order of thickness of line. This indicates that our subspace is sufficiently representative.

Figures 1b and 1c show the deformation of this net at $$~t=0.3~$$ and $$~t=0.5~$$. These mappings are symmetric with respect to $$~\alpha_1\rightarrow -\alpha_{1}~$$ and $$~\alpha_2\rightarrow -\alpha_{2}~$$, so, only the first quadrant of the coordinate plane is presented in the figures. For $$~t = 0.3~$$, the mapping is almost linear for $$~|\alpha| < 2~$$. Coordinate squares are converted into prolate rectangles typical of linear squeezing. For larger values of the initial field, these rectangles are deformed. For $$~t=0.5~$$, this mapping becomes ambiguous. The uncertainty of the output field can be characterized in more detail by the Wigner function $$(23) W( \alpha_{1}', \alpha_{1}')= \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{k=0}^{\infty}\!~ c^{*}_{n,m}\!~ c^{*}_{k,m}\!~ \int_{-\infty}^{\infty} \psi_n\!\left(\alpha'_{1} \sqrt{2}-\frac{u}{2}\right)~\! \psi_k\!\left(\alpha'_{1} \sqrt{2}+\frac{u}{2}\right)~\! \exp\!\left( {\rm i}\sqrt{2}\!~ \alpha'_{2} u \right) \!~{\rm d}u $$ where $$~c_{n,m}~$$ are coefficients of expansion of a transformed state in states with $$~n~$$ photons of field and $$~m~$$ photons of pump, $$~\psi_{n}(x) = \mathcal{H}_n(x)\exp\left(-{x^2}/{2}\right)/\sqrt{{2^n n! \sqrt{\pi}}\!~}~$$, and $$~{\mathcal H}_{n}(x)~$$ are Hermitian polynomials [9]. The initial state for $$~\alpha=1~$$ is shown in Fig.2a by concentric circles centered at the point $$~\alpha'_1 = \alpha_1~$$, $$~\alpha'_2 = \alpha_2~$$, (we construct the Wigner function in the coordinates $$~\alpha'_1~$$, $$~\alpha'_2~$$, which correspond to $$~x/\sqrt{2}~$$, $$~p/\sqrt{2}~$$. For $$~t=0.3~$$, these circles are deformed. They look like ellipses, and this deformation corresponds to linear compression of the net in Fig.1b. For large $$~t~$$, the right part of the body of uncertainty becomes wider than the left one (Fig. 2c). Note than $$~A_1 = G_1a_1~$$ $$~A_2 = G_2a_2~$$, and for a linear parametric amplifier, the Wigner function $$~W~$$ of an amplified state may be expressed in terms of the Wigner function $$~w~$$ of the initial state: $$(24)~ W(\alpha'_1, \alpha'_2) = w(~\langle A_1 \rangle,\langle A_2 \rangle) $$ for $$~\langle A_1 \rangle ~$$ and $$~\langle A_2 \rangle ~$$, which is determined by the transfer

function for the input values $$~\alpha'_1~$$ and $$~\alpha'_2~$$. One needs to compare Figs 1c and 2c to realize this. The net in Fig.1c is condensed to the right, whereas the body of uncertainty becomes wider. Only the mean position of the body of uncertainty follows the mapping of the transfer function of the initial position. The nonlinear behavior of the body of uncertainty demonstrates how the second and higher derivatives of the transfer function increase the noise of an amplifier.

Consider an amplifier with the transfer function shown in Fig. 1. Figure 3 presents $$~\langle A_1 \rangle ~$$ as a function of $$~\alpha_1~$$ for $$~\alpha_2 = 0~$$. Only the first quadrature component of the field is amplified in such an amplifier. Let us examine the dispersion of this component, divided by the square of its mean value. This dispersion is plotted in the same diagram in comparison with the lower estimate in accordance with Theorem 2. The gain factor $$~G_1~$$, the dispersion $$~D_1~$$, and its lower bound $$~E_1~$$ decrease as the amplitude grows, which corresponds ot the squeezing; $$~D_1~$$ remains greater than its lower bound, as must happen.

At moderate input amplitude, the compression of the profile of uncertainty in the region of net compression can be interpreted semiclassically. The mapping corresponding to degenerate parametric amplification does not change an element of the phase volume $$~{\rm d}p\!~{\rm d}x~$$, so it can be carried out without additional degrees of freedom [equation (23) with a C-number instead of $$~b~$$ describes a parametric amplifier with classical pumping]. But compressions or dilations of the coordinate and momentum in which the phase volume is not conserved (as on the right side of Fig.1b) suggest the correlation of field states with additional degrees of freedom. Such a correlation appears as additional quantum noise.

Conclusions
We characterized a single-mode amplifier of general type by its transfer function, which is defined in the phase space. This function maps mathematical expectations of the quadrature components of the initial field onto mathematical expectations of the components of the output field. Saturation of amplification makes this function nonlinear. Such mapping generates quantum noise, and we derived the lower bound of this noise for the total noise and the dispersion of each quadrature component in Theorems 1 and 2. Theorem 1 gives the lower bound of the noise of a depleted parametric amplifier with a squeezed coherent input signal. For such an amplifier, Theorem 2 gives the lower bound of the dispersion of a single quadrature component of the amplified field. These are the first lower bounds of quantum noise of a nonlinear amplifier of general type. Particular examples show that the lower bounds given by Theorems 1 and 2 are valid, although the body of uncertainty and phase space are not always deformed in the same manner.

For simplicity, we considered states of a single mode of the field. Multimode states may also be important in practice. The lower bounds given by Theorems 1 and 2 are not the best possible, because the projector $$~I_0~$$ introduced in (13) is only one of a set of orthogonal projectors in the multimode space. The consideration of additional projectors may improve the lower bounds.