Efficient noiseless linear amplification for light fields with larger amplitudes

We suggest and investigate a scheme for non-deterministic noiseless linear amplification of coherent states using successive photon addition, $(\hat a^{\dagger})^2$, where $\hat a^\dagger$ is the photon creation operator. We compare it with a previous proposal using the photon addition-then-subtraction, $\hat a \hat a^\dagger$, where $\hat a$ is the photon annihilation operator, that works as an appropriate amplifier only for weak light fields. We show that when the amplitude of a coherent state is $|\alpha| \gtrsim 0.91$, the $(\hat a^{\dagger})^2$ operation serves as a more efficient amplifier compared to the $\hat a \hat a^\dagger$ operation in terms of equivalent input noise. Using $\hat a \hat a^\dagger$ and $(\hat a^{\dagger})^2$ as basic building blocks, we compare combinatorial amplifications of coherent states using $(\hat a \hat a^\dagger)^2$, $\hat a^{\dagger 4}$, $\hat a \hat a^\dagger\hat a^{\dagger 2}$, and $\hat a^{\dagger 2}\hat a \hat a^\dagger$, and show that $(\hat a \hat a^\dagger)^2$, $\hat a^{\dagger 2}\hat a \hat a^\dagger$, and $\hat a^{\dagger 4}$ exhibit strongest noiseless properties for $|\alpha| \lesssim 0.51$, $0.51 \lesssim |\alpha| \lesssim 1.05 $, and $|\alpha|\gtrsim 1.05 $, respectively. We further show that the $(\hat a^{\dagger})^2$ operation can be used for amplifying superpositions of the coherent states. In contrast to previous studies, our work provides efficient schemes to implement a noiseless amplifier for light fields with medium and large amplitudes.

In order to implement a nondeterministic noiseless amplifier, state-of-the-art techniques of quantum optics are required. There are two types of noiseless linear amplification schemes for coherent states of small amplitudes. The first type proposed earlier [5,7,8] utilizes quantum scissors [19] to implement a noiseless amplifier. The other employs the photon number operationn as its basic element [6,9], where the photon subtraction [20] and addition [21] operations, represented by the photon annihilation and creation operatorsâ andâ † , are required for an experimental implementation. Recently, the first order approximation,ââ † , of the noiseless amplification was experimentally implemented [9]. This approach enables one to realize a highfidelity (F > 0.9) amplifier with the fixed amplitude gain of g = 2 for coherent states of very small amplitudes (|α| 0.67). Therefore, it is important to develop an efficient amplification scheme for coherent states of larger amplitudes.
Another interesting issue is to apply a noiseless amplifier to superpositions of coherent states (SCSs). It is well known that the free-traveling SCSs of light with large amplitudes are useful for both fundamental studies of quantum mechanics such as Bell-and Legett-type inequality tests [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] [56,57]. It is thus worth investigating whether the fidelities to large SCSs can be enhanced with the use of a nondeterministic noiseless amplifier. We also note that a scheme for deterministic amplification of SCSs in circuit quantum electrodynamics was proposed [60] but this scheme cannot be applied to free-traveling SCSs.
In this paper, we show that the two-photon addition ((a † ) 2 ) works as a more effective noiseless amplifier, compared to the photon-addition-and-subtraction (ââ † ), when amplifying coherent states and SCSs of relatively large amplitudes. Figures of merit examined here are the state fidelity, the amplitude gain, and the equivalent input noise (EIN) [61]. The noiseless property of the amplification is assessed by the EIN of the amplifier [7,9], which is affected by both the state fidelity and the amplitude gain. Our analysis also shows that the amplified squeezed vacuum and single-photon states using the (a † ) 2 operation exhibit higher fidelities to ideal SCSs than the states without the amplification and requires less squeezing for a large range of parameters.
The remainder of our paper is organized as follows. In Sec. 2, we examine one-and twocycle amplifications of coherent states. The fidelity, amplitude gain and noiseless property after the amplifications are investigated for comparisons. Sec. 3 is devoted to the amplification of SCSs. We conclude with final remarks in Sec. 4. The higher amplitude gain is obtained when the amplification is performed byâ †2 rather thanââ † . The gains fromââ † andâ †2 approach 2 and 3 as α i → 0, and are dropped to 1 as α i → ∞. (c) The upper solid and dashed curves represent average EINs, i.e. EINs averaged over all values of λ , while the lower solid and dashed curves correspond to EINs with λ = 0, which gives the lowest EINs. Theâ †2amplification exhibits lower average EINs than theââ † -amplification with large amplitude α i 0.91, while the opposite is true for α i 0.91. As α i → 0, the average EINs approach −3/8 forââ † , and −2/9 forâ †2 . The average EINs approach zero as α i increases for both the cases.

One-cycle amplification
Applying the amplification operatorÂ ∈ {ââ † ,â †2 } to a coherent state of amplitude α i , the amplified state is expressed as where the normalization factors are and α i is assumed to be real without loss of generality. The fidelity between theÂ-amplified coherent state of initial amplitude α i and the coherent state of real amplitude α f is We have obtained analytic expressions of FÂ as We take the maximum fidelity of theÂ-amplified coherent state of initial amplitude α i to the final coherent state as where the maximum is taken over the amplitude α f . The results of numerical maximization by steepest descent [62] are presented in Fig. 1(a). The (ââ † )-amplification always exhibits higher maximum fidelity than the (â † ) 2 -amplification. On the other hand, it is the opposite for the amplitude gain as explained in what follows. The amplitude gain from the amplificationÂ can be defined as the ratio of the expectation values of the quadrature operator with phase λ (x λ ) [9]: of which explicit expressions are obtained as The amplitude gains are independent of λ for the amplifications of coherent states considered here. The gain monotonically decreases to unity with respect to α i ( Fig. 1(b)), sinceââ † orâ †2 merely alters the ratios of the superposition of Fock states for large-amplitude coherent states. The maximum fidelity in Fig. 1(a) reaches unity for large α i with the same reason. We now employ the equivalent input noise (EIN) [61] for comparison between the two amplification schemes. The EIN came from a classical electronics terminology used to quantify the performance of an amplifier considering the amplification gain and the generated noise at the same time. It measures the amount of noise that must be added to the input noise level to mimic the observed output noise for the given gain using a classical amplifier [7]. When a quantum amplifier is used, the EIN can be negative [7,9]. The EIN of an amplifier is defined as where ∆x λ 2 in and ∆x λ 2 out are the expectation values of the quadrature variance operator with phase λ for input and output states. The first term of the right hand side of Eq. (11) represents the level of the input noise to classically mimic the output noise, and the second term represents the actual level of input noise. Thus the difference between the two terms corresponds to the level of noise added into the input signal to mimic the output signal in quadrature x λ . We find its explicit forms as The EINs with λ = 0 (Eââ † 0 (α i ) and E a †2 0 (α i )) and the λ -averaged EINs (Eââ † (α i ) and E a †2 (α i )) are plotted in Fig. 1(c). The parameter λ = 0 is chosen because it gives lower value of EIN than any other λ . All average EINs are negative, which indicates the characteristic of noiseless amplification; negative EIN cannot be obtained by the classical amplification. As theââ †amplification has much higher fidelity thanâ †2 for small α i ,ââ † has lower EIN for small α i , whileâ †2 has lower EIN for large α i due to higher amplitude gains. Theââ † -amplification of the coherent state shows higher fidelities to the coherent states than theâ †2 -amplification. However, when the initial amplitude is large enough to approach a sufficiently high fidelity, theâ †2 -amplification is advantageous in terms of amplitude gain and EIN.
In order to better understand how the amplification processes works, we plot changes of the Wigner functions [68] of coherent states after the photonics operations in Fig. 2. When the operationâ † is applied to a coherent state of α i = 2, the peak of the Wigner function moves from the origin while its shape becomes less circular than the original coherent state as shown in Fig. 2(b). If the operationâ is successively applied toâ † |α i , the shape of the Wigner function (Fig. 2(c)) becomes more circular thanâ † |α i , but the distance from the origin somewhat decreases. In other words, the final state better approximates a coherent state at the price of less amplification. However, if the operationâ † is applied, instead of the second operationâ, to the stateâ † |α i , the Wigner function becomes a little more distorted thanâ † |α i , but its amplitude becomes larger as shown in Fig. 2(d). We provide explicit forms of the Wigner functions in Appendix A.

Two-cycle amplification
We consider four possible combinatorial two-cycle amplifications,Â ∈ {(ââ † ) 2 ,â †4 , a †2ââ † ,ââ †â †2 }, where each of the four processes are a combination from the two basic ampli- fication units,ââ † and (â † ) 2 . We have obtained the fidelities, gains and EINs as explained in the previous section using Eqs. (4), (8) and (11), and the results are presented in Appendix B. The maximum fidelity FÂ max (α i ) in terms of the initial amplitude α i is numerically obtained [62] and plotted in Fig. 3(a). Among the two-cycle amplifications, the (ââ † ) 2 -amplification exhibits the highest maximum fidelity to the coherent state, although the fidelity is slightly lower than one-cycle amplification,ââ † . The order of the fidelity performance is All the gains from the two-cycle amplifications (Eq. (43) of Appendix B) are higher than those from one-cycle amplification, which also monotonically decreases to unity with respect to α i (Fig. 3(b)). The gain from (ââ † ) 2 is the lowest among the two-cycle amplifications, although the fidelity is the highest. For sufficiently large amplitude (α i 0.27), the following relation holds for amplitude gains: Integrating EINs (Eqs. (44) and (45)) to obtain λ -averaged EINs, Fig. 3(c), which are all negative indicating the characteristic of noiseless amplification. The following two-cycle amplifications achieve the lowest EINs, including one-cycle amplifications, in the corresponding regions: (ââ † ) 2 in α i 0.51, a †4 in 0.51 α i 1.05, andâ †2ââ † in α i 1.05.

Success probabilities of the amplification processes
The photon subtraction process uses a beam splitter and a photodetector while the photon addition process relies on a parametric down converter and a photodetector [21]. The success probability of the photon subtraction thus depends on the ratio of the beam splitter, while the success probability of the photon addition is determined by the parametric gain. It should be noted that the reflectivity of the beam splitter should be very small in order to well approximate the photon annihilation operatorâ. Assuming an ideal single-photon detector, the success probability of photon addition using a parametric down converter with parametric gain λ g ≪ 1 is [63] p add ≈ |λ g | 2 (n + 1), (15) where n is the average photon number of the initial state. The success probability of photon subtraction using a beam splitter of reflectivity R ≪ 1 is approximated as For instance, with realistic experimental parameters of λ g ∼ 0.1 [69, 70] and R ∼ 0.05 the realization ofââ † and (â † ) 2 to noiselessy amplify a coherent state of amplitude α i = 2 have estimated probability of ∼ 10 −2 and ∼ 10 −3 , respectively. Such probabilities are very well compatible with the realization of the proposed schemes with pulsed pump lasers where the final heralding rate is enhanced by the high pulse repetition rate of the laser ∼ 10 8 Hz [9,21,64]. However, the implementation of the (â † ) 4 ,ââ †â †2 , andâ †2ââ † with the same experimental parameters of above will results with a probability of ∼ 10 −5 which is more demanding with the present technology.

Ideal even and odd superpositions of coherent states
Even and odd SCSs are defined as where |+ α (|− α ) certainly contains an even (odd) number of photons as implied by its name. The amplified SCSs with the initial amplitude α i are where and α i is assumed to be real without loss of generality. The maximum fidelities of theÂamplified even and odd SCSs with initial amplitude α i are where We observe from the numerical results [62] plotted in Figs. 4(a) and 4(d) that the (ââ † )amplification results in higher maximum fidelities than the (â † ) 2 -amplification. The maximum fidelities of theââ † -amplified even and odd SCSs are higher than F max > 0.97 ( Fig. 4(a)) and F max > 0.98 ( Fig. 4(d)), respectively. Clearly, the maximum fidelities usingââ † approach 1 for small and large values of α i while those usingâ †2 approach 1 only for large values of α i .
Unlike coherent states, which has Gaussian probability distributions in the measurement of x λ , the definitions of the amplitude gain in Eq. (8) and the EIN in Eq. (11) cannot be applied to the even and odd SCSs. We first define the amplitude gain as the ratio between the input amplitude of SCS and the output amplitude of SCS that maximizes the fidelity as where α f maximizes FÂ ± . When this definition is applied to the case of coherent states, we do not obtain exactly the same results with those obtained using Eq. (8), although the differences become negligible when the fidelities are high. As shown in Figs. 4(b) and 4(e), it is numerically [62] verified that the gain fromâ †2 is always higher than that fromââ † .
It is a nontrivial task to develop an equivalent notion and definition of the EIN for SCSs because the definition of noise for SCSs is not clear in this context. We pay attention to the fact that the larger SCSs are more useful for phase estimation [43]. Largely amplified SCSs with high fidelities should become more useful for phase estimation. In fact, the optimal phase estimation is closely related to both of the amplitude gain and the noiseless property of amplifiers [43], and these two quantities are what the EIN quantifies. We thus compare the optimal phase estimations obtained from Cramér-Rao bound [65, 66] for even and odd SCSs, where F ± is quantum Fisher information for pure states.
The analytic results forÂ ∈ {1,ââ † ,â †2 } are presented in Appendix C, where1 denotes the identity operator. The enhancement in phase uncertainty obtained by the two amplification schemes,â †2 andââ † , are shown in Figs. 4(c) and 4(f) as signatures of noiseless amplifications. As expected from the case of coherent states, theâ †2 -amplification is more efficient for phase estimation with SCSs of large amplitudes than theââ † -amplification. Theâ †2 amplification exhibits lower phase uncertainties (i.e., better for phase estimation) thanââ † when applied to even and odd SCSs with large amplitude (α i 0.755 and α i 1.324, respectively), while the opposite is true for smaller α i . It is also clear in Figs. 4(c) and 4(f) that the phase uncertainties decrease afterâ †2 andââ † are applied to the even and odd SCSs of amplitude α i , respectively.

Approximations with squeezed vacuum and squeezed single-photon states
It is known that a squeezed vacuum state and a squeezed single-photon state well approximate the even and odd SCSs with small amplitudes, respectively [56,57], and that multiple applications of the photon addition on the squeezed vacuum produces a squeezed SCS of a very high fidelity [67]. The squeezed vacuum and squeezed single-photon states can be expressed in the number state basis asŜ (2n + 1)! 2 n n! |2n + 1 , whereŜ(r) = exp[−r(â 2 −â †2 )/2] is the squeezing operator with squeezing parameter r. When an even (odd) SCS of amplitude α f is desired, the maximum fidelity which the squeezed vacuum (single-photon) state can achieve is The dot-dashed curve in Fig. 5(a) (Fig. 5(c)) shows that the maximum fidelity of the squeezed vacuum (single-photon) state to the even (odd) SCS of amplitude α f approaches unity, as α f → 0. The dot-dashed curves represent the cases with the squeezed vacuum and single-photon states, respectively, without the amplification methods. In (a) and (c), higher fidelities are obtained for approximating the even and odd SCSs of large amplitudes using the amplification methodsââ † andâ †2 (α f 1.47 and α f 2.04, respectively), compared to the cases without the amplification methods. The amplification methodâ †2 achieves the fidelities to even and odd SCSs up to max Fâ