Efficient noiseless linear amplification for light fields with larger amplitudes

Jinwoo Park; Jaewoo Joo; Alessandro Zavatta; Marco Bellini; Hyunseok Jeong

doi:10.1364/OE.24.001331

1. Introduction

Quantum noise is a fundamental property which forbids quantum cloning [1], superluminal communication [2], and violation of generalized uncertainty principle [3]. Quantum theory prohibits deterministic linear amplification of bosonic system to be noiseless [4]. The one way to circumvent this fundamental restriction is to adopt probabilistic amplification heralded by successful measurements [5–9]. Noiseless amplification can be used for many quantum information tasks such as entanglement distillation [8], continuous variable quantum key distribution [10, 11], loss suppression [12], quantum repeater [13], phase estimation [14], quantum error correction [15], high-accuracy homodyne detection with low-efficiency detector [16], and quantum nonlocality tests [17, 18].

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 operation n̂ as its basic element [6, 9], where the photon subtraction [20] and addition [21] operations, represented by the photon annihilation and creation operators â and â^†, 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 high-fidelity (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 [22–37] and for quantum information applications including precision measurements [38–46], quantum computation [47–52], quantum teleportation [53,54] and quantum key distribution [55]. Squeezed single photon and squeezed vacuum states are often used to approximate SCSs [56–59]. These implementations, however, are good approximations only for SCSs with small amplitudes and they exhibit low fidelities to SCSs with large amplitudes [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 two-cycle 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.

2. Amplification of coherent states

2.1. One-cycle amplification

Applying the amplification operator Â ∈ {ââ^†, â^†2} to a coherent state of amplitude α_i, the amplified state is expressed as

N^{\hat{A}} (α_{i}) \hat{A} | α_{i} 〉,

where the normalization factors are

N^{\hat{a} \hat{a} †} (α_{i}) = {(α_{i}^{4} + 3 α_{i}^{2} + 1)}^{- \frac{1}{2}},

N^{\hat{a} † 2} (α_{i}) = {(α_{i}^{4} + 4 α_{i}^{2} + 1)}^{- \frac{1}{2}},

and α_i is assumed to be real without loss of generality.

The fidelity between the Â-amplified coherent state of initial amplitude α_i and the coherent state of real amplitude α_f is

F^{\hat{A}} = {N^{\hat{A}} (α_{i})}^{2} {| 〈 α_{f} | \hat{A} | α_{i} 〉 |}^{2} .

We have obtained analytic expressions of F^Â as

F^{\hat{a} \hat{a} †} = {N^{\hat{a} \hat{a} †} (α_{i})}^{2} e^{- {(α_{f} - α_{i})}^{2}} {(α_{f} α_{i} + 1)}^{2},

F^{\hat{a} † 2} = {N^{\hat{a} † 2} (α_{i})}^{2} α_{f}^{4} e^{- {(α_{f} - α_{i})}^{2}} .

We take the maximum fidelity of the Â-amplified coherent state of initial amplitude α_i to the final coherent state as

F_{max}^{\hat{A}} (α_{i}) = max_{α_{f}} F^{\hat{A}}

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 amplification Â can be defined as the ratio of the expectation values of the quadrature operator with phase λ(x̂_λ) [9]:

g_{λ}^{\hat{A}} = \frac{{| N^{\hat{A}} (α_{i}) |}^{2} | 〈 α_{i} | {\hat{A}}^{†} {\hat{x}}_{λ} \hat{A} | α_{i} 〉 |}{| 〈 α_{i} | {\hat{x}}_{λ} | α_{i} 〉 |},

of which explicit expressions are obtained as

g_{λ}^{\hat{a} {\hat{a}}^{†}} = {N^{\hat{a} {\hat{a}}^{†}} (α_{i})}^{2} (α_{i}^{4} + 4 α_{i}^{2} + 2),

g_{λ}^{{\hat{a}}^{† 2}} = {N^{{\hat{a}}^{† 2}} (α_{i})}^{2} (α_{i}^{4} + 6 α_{i}^{2} + 6) .

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)), since ââ^† or â^†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.

Fig. 1 (a) Maximum fidelities, (b) amplitude gains, and (c) EINs when the amplification methods ââ^† (solid curve) and â^†2 (dashed curve) are applied to the coherent state of initial amplitude α_i. (a) The fidelities of the ââ^†-amplified coherent states are higher than F_max > 0.98, which are close to 1 for small and large α_i. The fidelity of the â^†2-amplified coherent state approaches 1 for large α_i. (b) The higher amplitude gain is obtained when the amplification is performed by â^†2 rather than ââ^†. The gains from ââ^† and â^†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. The â^†2-amplification exhibits lower average EINs than the ââ^†-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 for ââ^†, and −2/9 for â^†2. The average EINs approach zero as α_i increases for both the cases.

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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

E_{λ}^{\hat{A}} = \frac{{〈 Δ x_{λ} 〉}_{out}^{2}}{{(g_{λ}^{\hat{A}})}^{2}} - {〈 Δ x_{λ} 〉}_{in}^{2},

where

{〈 Δ x_{λ} 〉}_{in}^{2}

and

{〈 Δ x_{λ} 〉}_{out}^{2}

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

\begin{array}{l} E_{λ}^{\hat{a} {\hat{a}}^{†}} (α_{i}) & = \frac{{2 α_{i}^{6} + 11 α_{i}^{4} + 11 α_{i}^{2} + 2 ({α_{i}}^{4} + 5 {α_{i}}^{2} + 3) {α_{i}}^{2} cos (2 λ) + 1}}{2 {N^{\hat{a} {\hat{a}}^{†}} (α_{i})}^{2} {({α_{i}}^{4} + 4 {α_{i}}^{2} + 2)}^{2}} - 2 {α_{i}}^{2} {cos}^{2} λ - \frac{1}{2} \\ E_{λ}^{{\hat{a}}^{† 2}} (α_{i}) & = \frac{{2 N^{{\hat{a}}^{† 2}} (α_{i})}^{2} (α^{2} + 2) {α^{4} + (α^{2} + 6) α^{2} cos (2 λ) + 6 α^{2} + 2} + 1}{{2 N^{{\hat{a}}^{† 2}} (α_{i})}^{4} {(α^{4} + 6 α^{2} + 6)}^{2}} \\ - 2 α^{2} {cos}^{2} λ - \frac{1}{2} \end{array}

The EINs with λ = 0 (

E_{0}^{\hat{a} {\hat{a}}^{†}} (α_{i})

and

E_{0}^{a^{† 2}} (α_{i})

) and the λ-averaged EINs (E^{ââ^†} (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 the ââ^†-amplification has much higher fidelity than â^†2 for small α_i, ââ^† has lower EIN for small α_i, while â^†2 has lower EIN for large α_i due to higher amplitude gains.

The ââ^†-amplification of the coherent state shows higher fidelities to the coherent states than the â^†2-amplification. However, when the initial amplitude is large enough to approach a sufficiently high fidelity, the â^†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 operation â^† 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 operation â is successively applied to â^†|α_i〉, the shape of the Wigner function (Fig. 2(c)) becomes more circular than â^†|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 operation â^† is applied, instead of the second operation â, to the state â^†|α_i〉, the Wigner function becomes a little more distorted than â^†|α_i〉, but its amplitude becomes larger as shown in Fig. 2(d). We provide explicit forms of the Wigner functions in Appendix A.

Fig. 2 Wigner functions of (a) a coherent state of amplitude α_i = 2 and the states after photonic operations, (b) â^†, (c) ââ^†, and (d) (â^†)². The amplitudes of the amplified states, gα_i, are noted for comparison.

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2.2. Two-cycle amplification

We consider four possible combinatorial two-cycle amplifications, Â ∈ {(ââ^†)², â^†4, â^†2ââ^†, ââ^†â^†2}, where each of the four processes are a combination from the two basic amplification 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 $F_{max}^{\hat{A}} (α_{i})$ 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

F_{max}^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i}) > F_{max}^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i}) > F_{max}^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i}) > F_{max}^{{\hat{a}}^{† 4}} (α_{i}) .

Fig. 3 (a) Maximum fidelities, (b) amplitude gains, and (c) average EINs after two-cycle amplifications (ââ^†)² (solid curve), â^†4 (dashed curve), ââ^†â^†2 (dot-dashed curve), and â^†2ââ^† (dotted curve). (a) The maximum fidelity of the (ââ^†)²-amplified coherent state approaches 1 for small and large α_i’s, which is the highest among the two-cycle amplifications. All the maximum fidelities become perfect as the initial amplitude α_i increases. (b) The higher gain is obtained by the two-cycle amplifications compared to the one-cycle amplifications. The gains from (ââ^†)², â^†4, â^†2ââ^†, and ââ^†â^†2 become 4, 5, 4, and 6 as α_i → 0, respectively, and drop to 1 as α_i increases. (c) In the regions of α_i ≲ 0.51, 0.51 ≲ α_i ≲ 1.05, and α_i ≳ 1.05, (ââ^†)², â^†4, and â^†2ââ^† show the lowest EIN, respectively, which are all lower than EINs obtained by the one-cycle amplifications.

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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:

g_{λ}^{{\hat{a}}^{† 4}} (α_{i}) < g_{λ}^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i}) > g_{λ}^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i}) > g_{λ}^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i}) .

Integrating EINs (Eqs. (44) and (45)) to obtain λ-averaged EINs, E^{(ââ^†)²} (α_i), E^{â^†2ââ^†} (α_i), E^{ââ^†â^†2} (α_i), and E^{â^†4} (α_i) are plotted in 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, â^†4 in 0.51 ≲ α_i ≲ 1.05, and â^†2ââ^† in α_i ≳ 1.05.

2.3. 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 operator â. 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} \approx {| λ_{g} |}^{2} (\bar{n} + 1),

where n̄ 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

p_{sub} \approx R \bar{n} .

For instance, with realistic experimental parameters of λ_g ∼ 0.1 [69, 70] and R ∼ 0.05 the realization of ââ^† 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 ∼ 10⁸ Hz [9,21,64]. However, the implementation of the (â^†)⁴, ââ^†â^†2, and â^†2ââ^† with the same experimental parameters of above will results with a probability of ∼ 10⁻⁵ which is more demanding with the present technology.

3. Amplification of superpositions of coherent states

3.1. Ideal even and odd superpositions of coherent states

Even and odd SCSs are defined as

| \pm_{α} 〉 = \frac{1}{\sqrt{2 (1 \pm e^{- 2 {| α |}^{2}})}} (| α 〉 \pm | - α 〉),

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

| \pm_{α_{i}}^{\hat{A}} 〉 = N_{\pm}^{\hat{A}} (α_{i}) \hat{A} | \pm_{α_{i}} 〉,

where

\begin{array}{l} N_{\pm}^{\hat{a} {\hat{a}}^{†}} (α_{i}) = {[2 {\pm e^{- 2 α_{i}^{2}} (α_{i}^{4} - 3 α_{i}^{2} + 1) + (α_{i}^{4} + 3 α_{i}^{2} + 1)}]}^{- \frac{1}{2}}, \\ N_{\pm}^{{\hat{a}}^{† 2}} (α_{i}) = {[2 {\pm e^{- 2 α_{i}^{2}} (α_{i}^{4} - 4 α_{i}^{2} + 2) + (α_{i}^{4} + 4 α_{i}^{2} + 2)}]}^{- \frac{1}{2}} \end{array}

and α_i is assumed to be real without loss of generality. The maximum fidelities of the Â-amplified even and odd SCSs with initial amplitude α_i are

{}_{max}{F_{\pm}^{\hat{A}}} (α_{i}) = max_{α_{f}} {| 〈 \pm_{α_{f}} | \pm_{α_{i}}^{\hat{A}} 〉 |}^{2},

where

\begin{array}{l} F_{\pm}^{\hat{a} {\hat{a}}^{†}} = \frac{2 {N_{\pm}^{\hat{a} {\hat{a}}^{†}} (α_{i})}^{2} e^{- {(α_{f} + α_{i})}^{2}} {e^{2 α_{f} α_{i}} (α_{f} α_{i} + 1) \mp (α_{f} α_{i} - 1)}^{2}}{\pm e^{- 2 α_{f}^{2} + 1}}, \\ F_{\pm}^{{\hat{a}}^{† 2}} = \frac{2 {N_{\pm}^{{\hat{a}}^{† 2}} (α_{i})}^{2} α_{f}^{4} {\pm e^{- \frac{1}{2} {(α_{i} + α_{f})}^{2}} + e^{- \frac{1}{2} {(α_{i} - α_{f})}^{2}}}^{2}}{\pm e^{- 2 α_{f}^{2}} + 1} . \end{array}

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 the ââ^†-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 using ââ^† approach 1 for small and large values of α_i while those using â^†2 approach 1 only for large values of α_i.

Fig. 4 (a) Maximum fidelities, (b) amplitude gains, and (c) optimal phase uncertainties when the amplification methods ââ^† (solid curve) and â^†2 (dashed curve) are applied to the even SCS of initial amplitude α_i. The same functions for the case of the odd SCS are plotted in panels (d), (e) and (f). The dot-dashed curves in panels (c) and (f) represent optimal phase uncertainties of the even and odd SCSs, respectively. The amplitude gains using ââ^† approach $\sqrt{3}$ and $\sqrt{2}$ for even and odd SCSs, respectively, as the initial amplitude α_i → 0. The amplitude gain using either of the amplification methods approaches 1 as α_i → ∞.

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Unlike coherent states, which has Gaussian probability distributions in the measurement of x̂_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

g_{\pm}^{\hat{A}} (α_{i}) = \frac{| α_{f} |}{| α_{i} |},

where α_f maximizes

F_{\pm}^{\hat{A}}

. 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 from â^†2 is always higher than that from ââ^†.

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,

δ ϕ_{\pm} = \frac{1}{\sqrt{ℱ_{\pm}}},

where ℱ_± is quantum Fisher information

ℱ_{\pm} = 4 〈 {(Δ \hat{n})}^{2} 〉

for pure states.

The analytic results for Â ∈ {1̂, ââ^†, â^†2} are presented in Appendix C, where 1̂ denotes the identity operator. The enhancement in phase uncertainty obtained by the two amplification schemes, â^†2 and ââ^†, are shown in Figs. 4(c) and 4(f) as signatures of noiseless amplifications. As expected from the case of coherent states, the â^†2-amplification is more efficient for phase estimation with SCSs of large amplitudes than the ââ^†-amplification. The â^†2 amplification exhibits lower phase uncertainties (i.e., better for phase estimation) than ââ^† when applied to even and odd SCSs with large amplitude (α_i ≳ 0.755 and α_i ≳ 1.324, respectively), while theopposite is true for smaller α_i. It is also clear in Figs. 4(c) and 4(f) that the phase uncertainties decrease after â^†2 and ââ^† are applied to the even and odd SCSs of amplitude α_i, respectively.

3.2. 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 as

\hat{S} (r) | 0 〉 = \sum_{n = 0}^{\infty} \frac{{(- tanh r)}^{n}}{{(cosh r)}^{1 / 2}} \frac{\sqrt{(2 n)!}}{2^{n} n!} | 2 n 〉,

\hat{S} (r) | 1 〉 = \sum_{n = 0}^{\infty} \frac{{(- tanh r)}^{n}}{{(cosh r)}^{3 / 2}} \frac{\sqrt{(2 n + 1)!}}{2^{n} n!} | 2 n + 1 〉,

where Ŝ(r) = exp[−r(â² − 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

{}_{max}{F_{+ S}^{\hat{1}}} (α_{f}) = max_{r} {| 〈 +_{α_{f}} | \hat{S} (r) | 0 〉 |}^{2},

{}_{max}{F_{- S}^{\hat{1}}} (α_{f}) = max_{r} {| 〈 -_{α_{f}} | \hat{S} (r) | 1 〉 |}^{2} .

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.

Fig. 5 (a) Maximum fidelities and (b) required levels of squeezing (dB) when the amplification methods ââ^† (solid curve) and â^†2 (dashed curve) are applied to the squeezed vacuum state to approximate the even SCS of amplitude α_f. The same functions for the cases of the squeezed single-photon state for approximating the odd SCS are plotted in panels (c) and (d). 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 methods ââ^† and â^†2 (α_f ≳ 1.47 and α_f ≳ 2.04, respectively), compared to the cases without the amplification methods. The amplification method â^†2 achieves the fidelities to even and odd SCSs up to ${}_{max}{F_{+ S}^{{\hat{a}}^{† 2}}} ≃ 0.943$ and ${}_{max}{F_{- S}^{{\hat{a}}^{† 2}}} ≃ 0.959$ around α_f ≃ 2.12 and α_f ≃ 2.59, respectively.

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Now, we apply the amplifier Â ∈ {ââ^†, â^†2} to the squeezed vacuum and single-photon states, Ŝ(r)|0〉 and Ŝ(r)|1〉 in order to obtain approximate SCSs. The amplified squeezed vacuum and single-photon states are

| + S_{r}^{\hat{A}} 〉 = M_{+}^{\hat{A}} (r) \hat{A} \hat{S} (r) | 0 〉,

| - S_{r}^{\hat{A}} 〉 = M_{-}^{\hat{A}} (r) \hat{A} \hat{S} (r) | 1 〉,

where

\begin{array}{l} M_{\pm}^{\hat{a} {\hat{a}}^{†}} (r) = {(sech [r])}^{2} {[\frac{{(1 \mp 1) {tanh}^{4} r + (12 \mp 8) {tanh}^{2} r + 5 \mp 3}}{2}]}^{- \frac{1}{2}}, \\ M_{\pm}^{{\hat{a}}^{† 2}} (r) = {(sech [r])}^{2} {[(5 \mp 4) {tanh}^{2} r + 4 \mp 2]}^{- \frac{1}{2}} . \end{array}

The fidelities between the Â-amplified squeezed vacuum (single-photon) state with initial squeezing r and the ideal even (odd) SCS of amplitude α_f are

F_{\pm S}^{\hat{a} {\hat{a}}^{†}} = {| 〈 \pm_{α_{f}} | \pm S_{r}^{\hat{a} {\hat{a}}^{†}} 〉 |}^{2} = \frac{2 {M_{\pm}^{\hat{a} {\hat{a}}^{†}} (r)}^{2} α_{f}^{1 \mp 1} e^{- α_{f}^{2} (tanh r - 1)} {(α_{f}^{2} tanh r - \frac{3 \mp 1}{2})}^{2}}{(\pm 1 + e^{2 α_{f}^{2}}) {cosh}^{2 \mp 1} r},

F_{\pm S}^{{\hat{a}}^{† 2}} = {| 〈 \pm_{α_{f}} | \pm S_{r}^{{\hat{a}}^{† 2}} 〉 |}^{2} = \frac{2 {M_{\pm}^{{\hat{a}}^{† 2}} (r)}^{2} α_{f}^{5 \mp 1} e^{- α_{f}^{2} (tanh r + 1)}}{(1 + e^{- 2 α_{f}^{2}}) {cosh}^{2 \mp 1} r},

where subscript ‘+’ (‘−’) denotes the even (odd) SCS. The maximum fidelity with which Â-amplified squeezed vacuum (single-photon) state can approximate the even (odd) SCS of target amplitude α_f is

{}_{max}{F_{\pm S}^{\hat{A}}} (α_{f}) = max_{r} {| 〈 \pm_{α_{f}} | \pm S_{r}^{\hat{A}} 〉 |}^{2} .

The maximum fidelities

{}_{max}{F_{\pm S}^{\hat{A}}}

are numerically calculated [62] and plotted in Fig. 5 with respect to the desired amplitude α_f of the SCSs. Figures 5(a) and 5(c) show that higher fidelities to the even and odd SCSs can be achieved using the amplification methods ââ^† and â^†2 on the squeezed vacuum and single-photon states, respectively, compared to the cases without the amplification methods. Clearly, the ââ^†-amplification on the squeezed vacuum (single photon) state better approximates the even (odd) SCS of any amplitude with a higher fidelity than the squeezed vacuum (single photon) state itself does. While the ââ^† amplification is more efficient to approximate the even and odd SCSs of small amplitude (α_f ≲ 1.81 and α_f ≲ 2.32, respectively), the â^†2 amplification works better to increase the fidelities to SCSs of larger amplitude as shown in Figs. 5(a) and 5(c).

We also note that the required degrees of squeezing, $r_{\pm S}^{\hat{a} {\hat{a}}^{†}} (α_{f})$ and $r_{\pm S}^{{\hat{a}}^{† 2}} (α_{f})$ , of the ââ^†- and â^†2- amplified squeezed vacuum and single-photon states are significantly lessened as shown in Figs. 5(b) and 5(d), where $r = r_{\pm S}^{\hat{A}} (α_{f})$ maximizes fidelities ${}_{max}{F_{\pm S}^{\hat{A}}}$ in Eq. (34). The amplification method â^†2 requires less squeezing than ââ^† does in order to approximate the even and odd SCSs.

4. Remarks

We have suggested and investigated an efficient noiseless amplification scheme for coherent states with large amplitudes, while previous studies are suitable for amplification of coherent states with small amplitudes [5, 6, 8, 9]. Our scheme may be useful for various quantum information applications that require SCSs of sufficient large amplitudes. As a specific example, this is closely related to efficient realization of fault-tolerant quantum computing [48–50], where superpositions of coherent states (SCSs) with α ≈ 1.6 (or α ≈ 2.26 to generate entangled coherent states of amplitude ≈ 1.6) are required as optimized resources [50]. Our proposal enables one to efficiently amplify SCSs of α > 0.755 (for even SCSs) and α > 1.324 (for odd SCSs) in order to reach this size so that they become useful as optimized resources for quantum computing [48–50].

The proposed amplification scheme utilizes two-photon addition, â^†2, and its performance was compared with that of a previous scheme using the photon addition and subtraction, ââ^†, which is efficient for amplifying small-amplitude coherent states [6,9]. We have found that â^†2 is better than ââ^†, in terms of EIN, as a noiseless amplifier for coherent states when the initial amplitudes are α_i ≳ 0.91. Similarly, when ââ^† and â^†2 are successively applied with different possible orders, â^†2 is found to serve as the best noiseless amplifier as far as the initial amplitude is α_i ≳ 1.05. Our study verifies that â^†2 is also more noiseless for large SCSs than ââ^† in terms of optimal phase uncertainty. In addition, the ââ^† and â^†2 amplifications can be applied to the squeezed vacuum and squeezed single-photon states in oder to approximate even and odd SCSs in a more efficient way. With the help of the amplification methods, SCSs with higher fidelities can be obtained using relatively small degrees of squeezing on the vacuum or single-photon states; the â^†2 amplification can be particularly useful to obtain SCSs of large amplitudes.

Appendix A: Wigner functions of coherent states after photonic operations

The Wigner function of a quantum state ρ̂ is defined as [68]

W (x, y) = \frac{2}{π} Tr [{\hat{D}}^{†} (x + i y) \hat{ρ} \hat{D} (x + i y) {(- 1)}^{\hat{n}}],

where D̂^†(x + iy) is a displacement operator defined as

{\hat{D}}^{†} (x + i y) = exp [(x + i y) {\hat{a}}^{†} - (x - i y) \hat{a}]

The Wigner function of a coherent state with real amplitude α_i is calculated as

W^{\hat{1}} (x, y) = \frac{2}{π} e^{- 2 {(x - α_{i})}^{2} - 2 y^{2}} .

After applying â^†, the Wigner function becomes

W^{{\hat{a}}^{†}} (x, y) = \frac{2}{π} e^{- 2 {(x - α_{i})}^{2} - 2 y^{2}} \frac{{(α_{i} - 2 x)}^{2} + 4 y^{2} - 1}{1 + α_{i}^{2}} .

When â is applied to Eq. (38), the Wigner function transforms into

W^{\hat{a} {\hat{a}}^{†}} (x, y) = \frac{2}{π} e^{- 2 {(x - α_{i})}^{2} - 2 y^{2}} \frac{{α_{i}}^{4} + {α_{i}}^{2} (4 x^{2} + 4 y^{2} - 3) - 4 {α_{i}}^{3} x + 4 α_{i} x + 1}{{α_{i}}^{4} + 3 {α_{i}}^{2} + 1} .

However when â^† is applied to Eq. (38) instead of â, the Wigner function becomes

\begin{array}{l} W^{\hat{a} {\hat{a}}^{†}} (x, y) & = \frac{2}{π} e^{- 2 {(x - α_{i})}^{2} - 2 y^{2}} \\ \times \frac{16 y^{4} + 8 y^{2} {{(α_{i} - 2 x)}^{2} - 2} + {(α_{i} - 2 x)}^{2} {{(α_{i} - 2 x)}^{2} - 4} + 2}{{α_{i}}^{4} + 4 {α_{i}}^{2} + 2} \end{array} .

Appendix B: fidelities, amplitude gains and EINs of amplified coherent states for two-cycle amplifications

We have obtained the normalization factor in Eq. (1) for the amplification of Â ∈ {(ââ^†)², â^†4, ââ^†â^†2, â^†2ââ^†} with an initial coherent state of amplitude α_i as

\begin{array}{l} N^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i}) & = {({α_{i}}^{8} + 10 {α_{i}}^{6} + 25 {α_{i}}^{4} + 15 {α_{i}}^{2} + 1)}^{- \frac{1}{2}}, \\ N^{{\hat{a}}^{† 4}} (α_{i}) & = {({α_{i}}^{8} + 16 {α_{i}}^{6} + 72 {α_{i}}^{4} + 96 {α_{i}}^{2} + 24)}^{- \frac{1}{2}}, \\ N^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i}) & = {({α_{i}}^{8} + 15 {α_{i}}^{6} + 63 {α_{i}}^{4} + 78 {α_{i}}^{2} + 18)}^{- \frac{1}{2}}, \\ N^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i}) & = {({α_{i}}^{8} + 11 {α_{i}}^{6} + 31 {α_{i}}^{4} + 22 {α_{i}}^{2} + 2)}^{- \frac{1}{2}} . \end{array}

The fidelities of the Â-amplified coherent state to the coherent state of amplitude α_f are then calculated as

\begin{array}{l} F^{{(\hat{a} {\hat{a}}^{†})}^{2}} & = {N^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i})}^{2} e^{{(- α_{i} - α_{f})}^{2}} {({α_{i}}^{2} α_{f}^{2} + 3 α_{i} α_{f} + 1)}^{2}, \\ F^{{\hat{a}}^{† 4}} & = {N^{{\hat{a}}^{† 4}} (α_{i})}^{2} {α_{i}}^{8} e^{- {(α_{i} - α_{f})}^{2}}, \\ F^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} & = {N^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i})}^{2} α_{f}^{4} e^{- {(α_{i} - α_{f})}^{2}} {(α_{i} α_{f} + 3)}^{2}, \\ F^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} & = {N^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i})}^{2} α_{f}^{4} e^{- {(α_{i} - α_{f})}^{2}} {(α_{i} α_{f} + 1)}^{2} . \end{array}

The gains from the amplification Â are obtained using Eq. (8) as

\begin{array}{l} g_{λ}^{{(\hat{a} {\hat{a}}^{†})}^{2}} & = {N^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i})}^{2} (α_{i}^{8} + 12 α_{i}^{6} + 38 α_{i}^{4} + 32 α_{i}^{2} + 4), \\ g_{λ}^{{\hat{a}}^{† 4}} & = {N^{{\hat{a}}^{† 4}} (α_{i})}^{2} (α_{i}^{8} + 20 α_{i}^{6} + 120 α_{i}^{4} + 240 α_{i}^{2} + 120), \\ g_{λ}^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} & = {N^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i})}^{2} (α_{i}^{8} + 18 α_{i}^{6} + 96 α_{i}^{4} + 168 α_{i}^{2} + 72), \\ g_{λ}^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} & = {N^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i})}^{2} (α_{i}^{8} + 14 α_{i}^{6} + 54 α_{i}^{4} + 60 α_{i}^{2} + 12), \end{array}

and the EINs are obtained using Eq. (11) as

\begin{array}{l} E_{λ}^{{(\hat{a} {\hat{a}}^{†})}^{2}} (α_{i}) & = - {4 {α_{i}}^{14} + 62 {α_{i}}^{12} + 382 {α_{i}}^{10} + 1101 {α_{i}}^{8} + 1554 {α_{i}}^{6} + 955 {α_{i}}^{4} + 226 {α_{i}}^{2} \\ + 2 (2 {α_{i}}^{12} + 26 {α_{i}}^{10} + 125 {α_{i}}^{8} + 254 {α_{i}}^{6} + 225 {α_{i}}^{4} + 66 {α_{i}}^{2} + 7) {α_{i}}^{2} cos (2 λ) + 15} \\ / {2 {({α_{i}}^{8} + 12 {α_{i}}^{6} + 38 {α_{i}}^{4} + 32 {α_{i}}^{2} + 4)}^{2}}, \\ E_{λ}^{{\hat{a}}^{† 4}} (α_{i}) & = - 4 {{α_{i}}^{14} + 26 {α_{i}}^{12} + 276 {α_{i}}^{10} + 1488 {α_{i}}^{8} + 4344 {α_{i}}^{6} + 6624 {α_{i}}^{4} + 4896 {α_{i}}^{2} \\ + ({α_{i}}^{12} + 24 {α_{i}}^{10} + 228 {α_{i}}^{8} + 1056 {α_{i}}^{6} + 2520 {α_{i}}^{4} + 2880 {α_{i}}^{2} + 1440) {α_{i}}^{2} cos (2 λ) + 1152} \\ / {{({α_{i}}^{8} + 20 {α_{i}}^{6} + 120 {α_{i}}^{4} + 240 {α_{i}}^{2} + 120)}^{2}}, \end{array}

\begin{array}{l} E_{λ}^{\hat{a} {\hat{a}}^{†} {\hat{a}}^{† 2}} (α_{i}) & = - 3 {2 {α_{i}}^{14} + 49 {α_{i}}^{12} + 486 {α_{i}}^{10} + 2421 {α_{i}}^{8} + 6432 {α_{i}}^{6} + 8784 {α_{i}}^{4} + 5688 {α_{i}}^{2} \\ + 2 ({α_{i}}^{12} + 22 {α_{i}}^{10} + 189 {α_{i}}^{8} + 780 {α_{i}}^{6} + 1626 {α_{i}}^{4} + 1584 {α_{i}}^{2} + 648) {α_{i}}^{2} cos (2 λ) + 1188} \\ / {2 {({α_{i}}^{8} + 18 {α_{i}}^{6} + 96 {α_{i}}^{4} + 168 {α_{i}}^{2} + 72)}^{2}}, \\ E_{λ}^{{\hat{a}}^{† 2} \hat{a} {\hat{a}}^{†}} (α_{i}) & = - {6 {α_{i}}^{14} + 103 {α_{i}}^{12} + 714 {α_{i}}^{10} + 2395 {α_{i}}^{8} + 4128 {α_{i}}^{6} + 3344 {α_{i}}^{4} + 1192 {α_{i}}^{2} \\ + 2 (3 {α_{i}}^{12} + 46 {α_{i}}^{10} + 271 {α_{i}}^{8} + 716 {α_{i}}^{6} + 886 {α_{i}}^{4} + 400 {α_{i}}^{2} + 72) {α_{i}}^{2} cos (2 λ) + 124} \\ / {2 {({α_{i}}^{8} + 14 {α_{i}}^{6} + 54 {α_{i}}^{4} + 60 {α_{i}}^{2} + 12)}^{2}} . \end{array}

Appendix C: quantum Fisher information of amplified SCSs

The values of the quantum Fisher information after applying Â ∈ {1̂, ââ^†, â^†2} to SCSs of amplitude α_i are

\begin{array}{l} ℱ_{+}^{\hat{1}} (α_{i}) = & \frac{4 {α_{i}}^{2} (4 e^{2 {α_{i}}^{2}} {α_{i}}^{2} + e^{4 {α_{i}}^{2}} - 1)}{{(e^{2 {α_{i}}^{2}} + 1)}^{2}}, \\ ℱ_{+}^{\hat{a} {\hat{a}}^{†}} (α_{i}) = & 16 {(N_{+}^{\hat{a} {\hat{a}}^{†}})}^{4} e^{- 2 {α_{i}}^{2}} {α_{i}}^{2} [{(N_{+}^{\hat{a} {\hat{a}}^{†}})}^{- 2} e^{α_{i}^{2}} {4 (2 α_{i}^{4} + 1) sinh α_{i}^{2} + (α_{i}^{4} + 14) α_{i}^{2} cosh α_{i}^{2}} \\ - 4 {({α_{i}}^{4} + 4) α_{i} sinh {α_{i}}^{2} + 5 {α_{i}}^{3} cosh {α_{i}}^{2}}^{2}], \\ ℱ_{+}^{{\hat{a}}^{† 2}} (α_{i}) = & 16 {(N_{+}^{{\hat{a}}^{† 2}})}^{4} e^{- 2 {α_{i}}^{2}} [e^{{α_{i}}^{2}} {(N_{+}^{{\hat{a}}^{† 2}})}^{- 2} {(13 {α_{i}}^{4} + 46) {α_{i}}^{2} sinh {α_{i}}^{2} + ({α_{i}}^{8} + 46 {α_{i}}^{4} + 8) cosh {α_{i}}^{2}} \\ - 4 {({α_{i}}^{4} + 14) {α_{i}}^{2} sinh {α_{i}}^{2} + 4 (2 {α_{i}}^{4} + 1) cosh {α_{i}}^{2}}^{2}] \end{array}

\begin{array}{l} ℱ_{-}^{\hat{1}} (α_{i}) = & \frac{4 {α_{i}}^{2} (- 4 e^{2 {α_{i}}^{2}} {α_{i}}^{2} + e^{4 {α_{i}}^{2}} - 1)}{{(e^{2 {α_{i}}^{2}} - 1)}^{2}}, \\ ℱ_{-}^{\hat{a} {\hat{a}}^{†}} (α_{i}) = & 16 {(N_{-}^{\hat{a} {\hat{a}}^{† 4}})}^{4} e^{- 2 {α_{i}}^{2}} [{(N_{-}^{\hat{a} {\hat{a}}^{†}})}^{- 2} e^{α_{i}^{2}} {(α_{i}^{4} + 14) α_{i}^{2} sinh α_{i}^{2} + 4 (2 α_{i}^{4} + 1) cosh α_{i}^{2}} \\ - 4 {({α_{i}}^{4} + 4) α_{i} cosh {α_{i}}^{2} + 5 {α_{i}}^{3} sinh {α_{i}}^{2}}^{2}], \\ ℱ_{-}^{{\hat{a}}^{† 2}} (α_{i}) = & 16 {(N_{-}^{{\hat{a}}^{† 2}})}^{4} e^{- 2 {α_{i}}^{2}} [e^{{α_{i}}^{2}} {(N_{-}^{{\hat{a}}^{† 2}})}^{- 2} {(13 {α_{i}}^{4} + 46) {α_{i}}^{2} cosh {α_{i}}^{2} + ({α_{i}}^{8} + 46 {α_{i}}^{4} + 8) sinh {α_{i}}^{2}} \\ - 4 {4 (2 {α_{i}}^{4} + 1) sinh {α_{i}}^{2} + ({α_{i}}^{4} + 14) {α_{i}}^{2} cosh {α_{i}}^{2}}^{2}], \end{array}

where subscripts + and − correspond to the even and odd SCSs, respectively.

Acknowledgments

The authors thank H. Kwon, C. Park, S. Yang, Y. Lim, D. Oi and T. Spiller for useful discussions. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2010-0018295) and by the EU under the ERA-NET CHIST-ERA project QSCALE.

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Efficient noiseless linear amplification for light fields with larger amplitudes

Abstract

1. Introduction

2. Amplification of coherent states

2.1. One-cycle amplification

2.2. Two-cycle amplification

2.3. Success probabilities of the amplification processes

3. Amplification of superpositions of coherent states

3.1. Ideal even and odd superpositions of coherent states

3.2. Approximations with squeezed vacuum and squeezed single-photon states

4. Remarks

Appendix A: Wigner functions of coherent states after photonic operations

Appendix B: fidelities, amplitude gains and EINs of amplified coherent states for two-cycle amplifications

Appendix C: quantum Fisher information of amplified SCSs

Acknowledgments

References and links

Cited By

Figures (5)

Equations (47)

Optics Express