On nonlinear amplification: improved quantum limits for photon counting

Tzula B. Propp; S. J. van Enk

doi:10.1364/OE.27.023454

1. Quantum amplification and noise

The fundamental relations between quantum noise and quantum amplification are most straight-forwardly derived in the Heisenberg picture. Thus, the standard way [1] to describe linear phase-preserving quantum amplification of a bosonic mode amplitude a is through Caves’ relation for the annihilation operator â,

{\hat{a}}_{out} = \sqrt{G} {\hat{a}}_{in} + \sqrt{G - 1} {\hat{b}}_{in}^{+},

where b̂⁺ is the creation operator corresponding to an independent auxiliary bosonic mode b. Here the input field amplitude of mode a is amplified by a factor of

\sqrt{G}

, but there is a cost: extra noise arising from the additional mode b [2]. If this mode contains (thermal) excitations, mode a after amplification will contain excitations, too, and their number is multiplied by G − 1. Even if mode b is in the vacuum state, it still adds noise [1]. It is clear that this extra noise is due to the additional creation operator term proportional to

\sqrt{G - 1}

in Eq. (1), but since that term is necessary so as to preserve the standard bosonic commutation relation [â_out,

{\hat{a}}_{out}^{+}

] = 𝟙 this tradeoff between linear amplification and added noise is fundamental. Indeed, phase-preserving linear amplification in proposed number resolving platforms using superconducting qubits have noise bounded by the Caves limit [3, 4] and similar bounds are found for linear amplification used in quantum circuits [5].

Recently, there has been some effort to describe all parts of the photo detection process, including amplification [6], fully quantum mechanically [7–10]. One conclusion that may be drawn from that research is that there is no severe amplification-driven tradeoff between efficiency and (thermally induced) dark counts. In particular, even though a few-photon signal must be amplified to a macroscopic level [forcing us to consider G ≫ 1], thermal fluctuations in internal detector modes apparently do not get amplified by the same factor of G. Experiments [11,12] on superconducting nanowires demonstrate that over a wide range of detected wavelengths dark count rates can indeed be extremely low (on the order of one dark count per day). How can we reconcile these results with that of the previous paragraph?

The answer, as we will show, is that amplification is not necessarily linear. That is, in the Heisenberg picture, the transformation of the bosonic annihilation operator can be nonlinear while still preserving the bosonic commutation relation. And, perhaps surprisingly, that way of amplifying can decrease the amount of noise added. Previous models of nonlinear amplification were developed in a different context, that of amplifying superpositions of different number states [13–18]. The relevant figure-of-merit then was the signal to noise ratio of the output relative to that of the input, and noise from the auxiliary modes was ignored. Here, in contrast, we are interested exclusively in the noise in the output and especially in the contributions arising from thermal fluctuations in the auxiliary modes, as functions of the gain factor G. In order to explore this issue, we construct physical transformations that implement nonlinear amplification satisfying the correct mathematical requirements which, to the authors’ knowledge, have not appeared in the literature.

2. Nonlinear amplification

The idea is that for detecting single or few photons it is sufficient to have an output field whose total number of excitations is given by N_out = N_in + Gn_a with n_a the number of input photons we would like to detect, and N_in the (fluctuating) number of excitations initially present in the output mode, which is not amplified. A physically allowed but highly idealized unitary transformation that accomplishes this is easiest written down in the Schrödinger picture (valid for any n [19], even though in practice we will be interested mainly in small values of n, say, n = 0, 1, 2) as

{| n 〉}_{a} {| M 〉}_{1} {| N 〉}_{2} \mapsto {| n 〉}_{a} {| M - G n 〉}_{1} {| N + G n 〉}_{2} .

All states here are number (Fock) states of bosonic modes. The transformation involves two energy reservoirs: energy is transferred from the first reservoir to the second with the amount of energy transferred determined by the number n of input photons in mode a (with nothing happening at all when n = 0). The assumption is that excitations of the two reservoirs have identical energies, ħω′, such that energy is conserved. The input mode can have any frequency ω. The second reservoir ideally starts out with N = 0 excitations—corresponding to the zero temperature limit—such that in the end it would contain exactly Gn excitations if the input field contained n photons. Clearly, this ideal transformation would represent perfect (noiseless) amplification of a photon number state (and G will have to be an integer for this to work).

Now we wish to describe this ideal process in the Heisenberg picture so as to make a direct comparison with Eq. (1). In that picture, the ideal transformation that is linear in the number operator for the excitations in the second reservoir should be [23]

{\hat{b}}_{out}^{+} {\hat{b}}_{out} = {\hat{b}}_{in}^{+} {\hat{b}}_{in} + G {\hat{a}}_{in}^{+} {\hat{a}}_{in} .

We are now going to do three things: (A) we will construct expressions for

{\hat{b}}_{out}^{+}

and b̂_out such that Eq. (3) is reproduced and such that their commutator [

{\hat{b}}_{out}^{+}

, b̂_out] = 𝟙; (B) we will add non-ideal features that make the model more realistic, and (C) we will include fluctuations in the initial number of excitations in the b mode and calculate the signal-to-noise ratio (SNR) for the final number of excitations in the b mode, both for the ideal limit and the more realistic models. A comparison with linear amplification will then show how nonlinear amplification improves upon the former.

To start with part of task (B), we adjust the idealized Schrödinger picture to get rid of two features that make the process in Eq. (2) obviously inapplicable to real detectors, but such that the Heisenberg picture Eq. (3) is still valid. First note that the n photons in the process in Eq. (2) are not destroyed, whereas in a standard detector they are. We fix that by introducing another quantum system S with a continuum of energies E that can absorb the energy nħω of the n photons. This modifies Eq. (2) by adding a step

{| n 〉}_{a} {| E 〉}_{S} \mapsto {| 0 〉}_{a} {| E + n ℏ ω 〉}_{S} .

Since this extra step does not affect the state of reservoir 2, the crucial expression Eq. (3) stays the same. The second change concerns phase: in the Schrödinger picture we can insert random phase factors exp(iϕ^♣) on the right-hand side of Eq. (2). This makes the amplification process irreversible (as any amplification process in a real detector is) and it destroys superpositions of different number states (e.g., coherent states will not be coherently amplified). It destroys any entanglement between the different modes as well [24].

For task (A) we would like to use the polar decompositions of the creation and annihilation operators. That is, in analogy to the polar decomposition of a complex number, $z = exp (i ϕ) \sqrt{{| z |}^{2}}$ , we would like to write

{\hat{b}}_{out} = \hat{S} \sqrt{{({\hat{b}}^{+} \hat{b})}_{in} + G {({\hat{a}}^{+} \hat{a})}_{in}},

where Ŝ is a unitary operator which could be written in the suggestive form exp (iϕ̂) for some hermitian operator ϕ̂. In a finite-dimensional Hilbert space of dimension s + 1 there is no problem defining Ŝ: it is a shift operator that acts on number states |N〉 of the bosonic mode as

\hat{S} | N 〉 = e^{i ϕ^{♣}} | N - 1 〉 for s \geq N > 0,

with Ŝ |0〉 = |s〉 and ϕ^♣ the random phase we introduced earlier. Since Fock space is infinite-dimensional, we use the Pegg-Barnett trick [25] of truncating the Hilbert space at a high excitation number s and only in the end (when calculating physical quantities) taking the limit s → ∞. It is easy to verify that the relation Eq. (5) yields the commutator [b̂_out,

{\hat{b}}_{out}^{+}

] = 𝟙_in − (s + 1) |s〉〈s|, in which the extra Pegg-Barnett term won’t contribute to any physical quantity, while ensuring a traceless commutator, necessary in finite dimensions [26].

Construction of Hamiltonians that implement nonlinear photon number amplification was done elsewhere (see [16,18]), but they are far more complicated than the Hamiltonians implementing linear amplification; in the interaction picture phase-preserving linear amplification is implemented with the simple interaction Hamiltonian

\hat{H} = i ℏ κ (\hat{a} \hat{b} - {\hat{a}}^{+} {\hat{b}}^{+})

with a time-dependent gain factor G = cosh(κt) for a coupling strength κ [27]. In contrast, an interaction Hamiltonian implementing nonlinear amplification even for a modest gain of G = 5 has 40 terms (fewer terms for smaller G, more terms for larger G) [18]. Since we are only interested in the output signal and not the details of the evolution of other operators, we will not construct a system Hamiltonian; indeed, the same transformation on the output signal could be given by different Hamiltonians corresponding to different physical implementations of photon number amplification that give the same overall gain factor.

The nonlinear expression Eq. (5) does not seem to have appeared in the large literature on bosonic amplification (for a review, see, e.g., [4]). [13–15] did discuss photon-number amplifiers (especially in the high-photon number limit) decades ago, but no attempt was made there to find commutator-preserving operator equations.

3. More realistic models for amplification

Continuing with task (B), in a more realistic description the reservoirs consist of many modes. So, instead of having just one bosonic output mode b we really should describe many output reservoir modes. For example, we may have G modes b_k [recall G is an integer now] each one of which satisfies

{\hat{b}}_{k out} = {\hat{S}}_{k} \sqrt{{({\hat{b}}^{+} \hat{b})}_{k in} + {({\hat{a}}^{+} \hat{a})}_{in}}, k = 1 \dots G .

Here the macroscopic signal monitored and analyzed consists of the sum of all detected excitations (since each mode by itself contains just a microscopic number of excitations we cannot simply assume to be able to count those individual numbers: then we would not need amplification at all!). That is, we consider as our macroscopic output signal

{\hat{I}}_{out} = \sum_{k = 1}^{G} {({\hat{b}}^{+} \hat{b})}_{k out} = \sum_{k = 1}^{G} {({\hat{b}}^{+} \hat{b})}_{k in} + G {({\hat{a}}^{+} \hat{a})}_{in} .

Another extension is to “avalanche” photodetection where one small-scale amplification event triggers the next and the process repeats, giving rise to a macroscopic signal. Iterating the transformation Eq. (3) of single mode amplification N times with a gain factor g in each step gives a total gain factor G = g^N and an input-output relation

{({\hat{b}}^{+} \hat{b})}_{N out} = \sum_{k = 1}^{N} g^{N - k} {({\hat{b}}^{+} \hat{b})}_{k in} + G {({\hat{a}}^{+} \hat{a})}_{in}

where mode b̂_k here contains the output of the kth amplification step, and the last mode b_N contains the signal.

Another extension, relevant for n > 1, describes multiplexing: the idea is that n photons are most conveniently detected by n detectors that each detect one (and only one) photon, along the lines of [28,29]. We will not describe this model in any detail, except to state that amplification would in that case be described by Gn modes, each containing exactly one extra excitation.

Lastly, we combine both multi-mode and multi-step extensions above by repeating the process in Eqs. (8) and (9) of amplification into several (g) modes N times, again with a total gain factor defined G = gⁿ and an input-output relation for the macroscopic signal

{\hat{I}}_{out} = \sum_{k_{N} = 1}^{G} {({\hat{b}}^{+} \hat{b})}_{k_{N} out} = \sum_{n = 1}^{N} \sum_{k_{n} = 1}^{g^{n}} {({\hat{b}}^{+} \hat{b})}_{k_{n} in} + G {({\hat{a}}^{+} \hat{a})}_{in}

where the mode b̂_{k_n} is the k_nth mode in the nth step.

Note that in our nonlinear amplification models the amplified signal ends up in a different bosonic mode or modes: indeed, a photodetector typically converts the input signal (light) to an output signal of a physically different type, e.g., electron-hole pairs (which may sometimes be approximated as composite bosons; see also [30–33]).

4. Number fluctuations

We turn to task (C) and calculate the noise in photon number introduced by the amplification process and by the coupling to reservoirs. For the reservoir we monitor, we write

〈 {({\hat{b}}^{+} \hat{b})}_{in} 〉 = {\bar{n}}_{b}; 〈 {({\hat{b}}^{+} \hat{b})}_{in}^{2} 〉 = {\bar{n}}_{b}^{2} + Δ n_{b}^{2}

and make no further assumptions about its initial state.

We assume that there is some (unknown) number of photons in the input mode a that we want to measure. We thus consider input states that are diagonal in the photon number basis, with some nonzero photon number fluctuations Δn_a. (Thanks to the randomized phase assumption we can use this assumption without loss of generality for our nonlinear models.) So, we write

〈 {({\hat{a}}^{+} \hat{a})}_{in} 〉 = {\bar{n}}_{a}; 〈 {({\hat{a}}^{+} \hat{a})}_{in}^{2} 〉 = {\bar{n}}_{a}^{2} + Δ n_{a}^{2} .

In the following we always assume the initial states of modes a and b to be independent, such that

〈 f (\hat{a}, {\hat{a}}^{+}) g (\hat{b}, {\hat{b}}^{+}) 〉 = 〈 f (\hat{a}, {\hat{a}}^{+}) 〉 〈 g (\hat{b}, {\hat{b}}^{+}) 〉

for any functions f and g.

For linear phase-insensitive amplification Eq. (1), we find the following variance in the number of excitations in the amplified signal:

\begin{array}{l} σ_{{({\hat{a}}^{+} \hat{a})}_{out}}^{2} & = & G^{2} Δ n_{a}^{2} + {(G - 1)}^{2} Δ n_{b}^{2} + \\ G (G - 1) (2 {\bar{n}}_{a} {\bar{n}}_{b} + {\bar{n}}_{a} + {\bar{n}}_{b} + 1) . \end{array}

Not only are the fluctuations in the auxiliary mode b amplified [second term in Eq. (15)], there is inherent noise from the amplification process itself even if

Δ n_{b}^{2} = 0

[the second line is strictly positive for G > 1].

We should also consider linear phase-sensitive amplification [1], described by

{\hat{a}}_{out} = \sqrt{G} {\hat{a}}_{in} + \sqrt{G - 1} {\hat{a}}_{in}^{+} .

Here, compared to Eq. (1) the b̂⁺ term is replaced by the â⁺ term, such that the commutator [â_out,

{\hat{a}}_{out}^{+}

] is still preserved. This gives a variance

\begin{array}{l} σ_{{({\hat{a}}^{+} \hat{a})}_{out}}^{2} & = & (6 G (G - 1) + 1) Δ n_{a}^{2} + \\ 2 G (G - 1) ({\bar{n}}_{a}^{2} + {\bar{n}}_{a} + 1) . \end{array}

There is again extra amplification noise for G > 1 [the second line], much like what we found for phase-insensitive amplification.

We compare these two results for linear amplification to the result for the nonlinear amplification process described by Eq. (5). The variance in excitation number is

σ_{{({\hat{b}}^{+} \hat{b})}_{out}}^{2} = Δ n_{b}^{2} + G^{2} Δ n_{a}^{2} .

Here the number fluctuations in the auxiliary mode are not amplified and there is no additional amplification noise either; if the input field is in a number state such that Δn_a = 0, the gain factor G will not effect the noise at all. This is a result of the reservoir and auxiliary mode being coupled conditionally on the presence of the input field; the transformation Eq. (2) always transfers exactly Gn excitations to the auxiliary mode, leaving the noise in the auxiliary mode unchanged when Δn_a = 0. In contrast, linear amplification never adds a definite number of excitations to the auxiliary mode because the input mode and the auxiliary mode are directly coupled; even at zero temperature there is inherent noise from the amplification of the auxiliary mode amplitude by a factor

\sqrt{G - 1}

[see Eq. (1)]. For linear amplification this is necessary so as to preserve the spectrum of the number operator [which non-linear amplification through Eq. (3) does by construction].

Already we are able to see that the scheme of amplification into a single mode is optimal; any transformation that would reduce the prefactor of $Δ n_{b}^{2}$ in Eq. (18) below unity would fail to realize a well-behaved annihilation operator (for details, see again [23])!

For nonlinear amplification into many modes described by the more realistic model in Eqs. (8) and (9), we find

σ_{{\hat{I}}_{out}}^{2} = G Δ n_{b}^{2} + G^{2} Δ n_{a}^{2},

where for simplicity we assumed all reservoir modes to be independent with the same number fluctuations. This shows amplifying according to Eq. (8) is suboptimal; even though it still beats both linear amplification limits Eqs. (15) and (17) the noise in the reservoir modes is still amplified. Similarly, amplification using multiple fermionic degrees of freedom will be sub-optimal; a similar multi-mode description will be necessary [34].

Defining the total gain g^N = G with N the number of steps, we find for our multi-step models that

σ_{{({\hat{b}}^{+} \hat{b})}_{out}}^{2} = \frac{G^{2} - 1}{g^{2} - 1} Δ n_{b}^{2} + G^{2} Δ n_{a}^{2}

for amplification of g excitations into a single mode and

σ_{{\hat{I}}_{out}}^{2} = G \frac{G - 1}{g - 1} Δ n_{b}^{2} + G^{2} Δ n_{a}^{2}

for amplification of a single excitation into g modes. (Note Eqs. (20) and (21) reduce to Eqs. (18) and (19) for g = G.)

5. Signal-to-noise ratios

We can now write down explicit tradeoff relations between amplification and number fluctuations in terms of signal-to-noise ratios for all types of amplification discussed here, for the case where the number of input photons is fixed to be n_a (and so Δn_a = 0). Using the standard signal-to-noise ratio as the number of excitations in the amplified mode minus the background, divided by the standard deviation in the number of excitations, we get

{SNR}_{PhaseInsensitive} \leq \frac{G}{G - 1} \frac{n_{a}}{Δ n_{b}}

{SNR}_{PhaseSensitive} \leq \frac{2 G - 1}{\sqrt{2 G (G - 1)}} n_{a}

{SNR}_{SingleMode} = \frac{G n_{a}}{Δ n_{b}}

{SNR}_{GModes} = \frac{G n_{a}}{\sqrt{G} Δ n_{b}} = \frac{\sqrt{G} n_{a}}{Δ n_{b}} .

The linear amplification mechanisms have increasingly worse signal-to-noise ratios as G increases [35], albeit saturating in the limit G → ∞. In contrast, the signal-to-noise ratios for the nonlinear amplification mechanisms improve with increasing G, with amplification into a single-mode performing best, with an improvement by a factor of G over an unamplified signal [36]. Amplification into G modes performs worse than single mode amplification, showing an improvement over an unamplified signal by a factor of

\sqrt{G}

. In the large G limit, this can be considered the well-known

1 / \sqrt{G}

reduction in the standard error; while there are G noisy modes, they do each carry the signal we wish to measure so the total noise is reduced by a factor of

\sqrt{G}

.

Similarly, we consider multi-step amplification models

{SNR}_{MultiStepSingleMode} = \frac{G \sqrt{g^{2} - 1} n_{a}}{\sqrt{G^{2} - 1} Δ n_{b}}

{SNR}_{MultiStepMultiMode} = \frac{\sqrt{G (g - 1)} n_{a}}{\sqrt{G - 1} Δ n_{b}} .

These intermediate noise limits fill in the space between the optimal SNR in Eq. (24) and linear amplification [37]. (Indeed, multi-step multi-mode amplification performs slightly worse than both linear mechanisms for g = 2!)

In general, nonlinear amplification enables SNRs that, as functions of the total gain factor G, do not saturate with the gain G and outperform what is possible with linear amplification. This is most apparent in the low temperature limit where the lack of zero-temperature noise for non-linear amplification (due to amplification taking place at a different frequency) is most prominent.

6. Single-photon pre-amplification

While our paper focuses on the amplification part of the photo detection process, we very briefly consider the pre-amplification process now. We certainly cannot treat that part in full generality here and we adopt several simplifications in order to arrive at an important result concerning the suppression of thermal noise. First, we assume that we can decouple the amplification stage from the pre-amplification filtering [by having an irreversible step in between the two] such that filtering does not interfere negatively with the absorption/transduction part [8]. We then focus on just the time/frequency degree of one incoming photon [38]. A single absorber with some resonance frequency ω₀ able to absorb that single photon will act as a frequency filter. If the pre-amplification filtering is passive (easy to implement, but we certainly can go beyond this [39]) and unitary (i.e., lossless: we consider this because we are interested in the fundamental limits of photo detection. Internal losses only degrade performance.), then frequency filtering is described by the linear transformation

{\hat{a}}_{out} (ω) = T (ω) {\hat{a}}_{in} (ω) + R (ω) {\hat{c}}_{in} (ω)

where c_in(ω) is yet another internal bosonic detector mode at the same frequency as the input mode [40]. Here T(ω) and R(ω) are “transmission” and “reflection” coefficients which satisfy |T(ω)|² + |R(ω)|² = 1 and which are determined by the resonance structures internal to the photodetector. The amplification process that follows the initial absorption of the photon energy is applied to the operator â_out(ω) of Eq. (28), so that (explicitly displaying the different frequencies of the modes now) ideal amplification (single-mode and single-shot) is described by

{\hat{b}}_{out}^{+} (ω^{'}) {\hat{b}}_{out} (ω^{'}) = {\hat{b}}_{in}^{+} (ω^{'}) {\hat{b}}_{in} (ω^{'}) + G {\hat{a}}_{out}^{+} (ω) {\hat{a}}_{out} (ω) .

This makes rigorous the idea that one can amplify at any frequency, enabling the mantra that one should amplify at high (optical) frequencies [41]. Namely, thermal fluctuations at a frequency ω′ may be suppressed by choosing the reservoir mode frequency ω′ such that ħω′ ≫ kT. This suppression is exponential:

Δ n_{b}^{2} \propto {\bar{n}}_{b} \propto exp (- ℏ ω^{'} / k T)

. Note that number fluctuations in the internal mode c_in(ω) at the input frequency will be amplified by the subsequent amplification process. However, one can in principle construct ideal detectors for light with a particular frequency ω₀ [8], such that |T(ω₀)| = 1 and hence R(ω₀) = 0 [40], avoiding internally generated dark counts at that particular frequency. If instead of a single spectral mode a small range of frequencies is amplified with differing probabilities, matching the amplification spectrum to the filtering spectrum is sufficient for reducing internally generated dark counts, as we will discuss in more detail in work in preparation [42].

7. Further applications

The models for amplification considered here apply to other types of quantum measurement as well. For example, electron-shelving [20–22] is a well-known method to perform atomic state measurements. Here one particular atomic state (e.g., one of the hyperfine ground states of an ion) is coupled resonantly to a higher-lying excited state which can then decay back by fluorescence only to that same ground state. A laser tuned to that transition can then induce the atom to emit a macroscopic amount (visible by eye) of fluorescent light. In the language accompanying Eq. (2), the laser beam forms the first reservoir, while the second reservoir consists of vacuum modes that are filled with fluorescent light as described by Eq. (8). The gain factor G (the number of fluorescence photons) is determined by the ratio of Einstein’s coefficients for spontaneous and stimulated emission and the total integration time. By placing the atom/ion inside a high-Q optical resonator (with resonant frequency ω′) we would reduce the number of output modes and thereby get closer to the optimum. The idea of placing a detector inside a resonant cavity is, of course, not new [43], but that idea is usually associated with increasing the coupling to light. Although we do have that effect as well, the main purpose here is to reduce the number of output modes, and thereby increase the SNR (Fig. 1).

Fig. 1 An input photon with frequency ω undergoes amplification into a macroscopic signal via electron-shelving [20–22]: when an on-resonance photon is absorbed, an atom (modeled here as a three-level system) enters the first excited state and a laser tuned to the second transition frequency ω_L ≈ ω′ induces fluorescence. If there are multiple input photons, they are absorbed by multiple atoms and the fluorescence signal is increased proportionally. The number of fluorescence modes may be reduced by using a high-Q cavity so that amplification moves towards the ideal transformation given in Eq. (2).

Download Full Size | PDF

A method similar to electron shelving (but using a collective metastable state of an atomic gas) was proposed in [44] for implementing high-efficiency photon counting with an, in principle, quantum nondemolition measurement. Our model describes that detector too and clarifies why such a measurement is possible; nonlinear amplification enables low-noise photon counting and does not destroy the input photons without an additional step as in Eq. (4).

In [13], a transformation similar to Eq. (2) is given: n photons are directly converted to Gn photons in a single mode. Though this transformation is unphysical (there is no way to preserve the commutator), a SNR is calculated that increases linearly with G like our Eq. (24). However, the SNR found in [13] diverges for a photo detector with unit efficiency, which is not the case once fluctuations in the reservoir mode are properly taken into account as our results clarify.

In [6], Yang and Jacob propose an interesting model for amplification that makes use of a first-order phase transition for a collection of N interacting spin-1/2 particles. These spins are coupled both to an input photon and to an output bosonic mode. The SNR (as we define it here) for that model scales as $\sqrt{N}$ while the gain G of that model is linear in N. Thus, the SNR scales with $\sqrt{G}$ just as our Eq. (25): the number of spins in [6]’s model plays a similar role as our number of amplification modes.

8. Conclusions

We discussed various linear and nonlinear amplification schemes for bosonic modes. For detecting few photons, we found that the latter add considerably less noise, leading to better signal-to-noise ratios, as exemplified in Eqs. (22)–(27). Unlike for linear amplification, number fluctuations in internal detector modes are not amplified, while the number of photons that we want to detect is amplified. All amplification schemes explicitly preserve the bosonic commutation relations.

While amplification into a single-mode may not be feasible in practice, it provides the fundamental lower limit to noise in photon-number measurements across amplification mechanisms. In practice, one may have many output modes and thus may find a SNR closer to Eq. (25), which is worse by a factor of $\sqrt{G}$ than the fundamental limit (but still better by a factor of $\sqrt{G}$ than linear amplification), or one may have multiple amplification steps as in Eq. (26), or both as in Eq. (27). To test this, we suggest that measurement of the gain dependence of the SNR for a given photo detector should provide a rough but useful indication of the underlying amplification mechanism.

Funding

U.S. Defense Advanced Research Projects Agency (DARPA) (W911NF-17-1-0267).

Acknowledgements

The authors thank Joseph Altepeter and Sae Woo Nam for their useful comments on this project.

References

1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982). [CrossRef]

2. This could be a mode internal to the detector.

3. A. Metelmann and A. Clerk, “Quantum-Limited Amplification via Reservoir Engineering,” Phys. Rev. Lett. 112, 133904 (2014). [CrossRef] [PubMed]

4. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010). [CrossRef]

5. U. Gavish, B. Yurke, and Y. Imry, “Generalized constraints on quantum amplification,” Phys. Rev. Lett. 93, 250601 (2004). [CrossRef]

6. L.-P. Yang and Z. Jacob, “Quantum critical detector: amplifying weak signals using discontinuous quantum phase transitions,” Opt. Express 27, 10482–10494 (2019). [CrossRef] [PubMed]

7. S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018). [CrossRef]

8. S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018). [CrossRef]

9. S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Comm. 1, 045001 (2017). [CrossRef]

10. E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018). [CrossRef]

11. F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013). [CrossRef]

12. E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017). [CrossRef] [PubMed]

13. H. P. Yuen, “Generation, detection, and application of high-intensity photon-number-eigenstate fields,” Phys. Rev. Lett. 56, 2176–2179 (1986). [CrossRef] [PubMed]

14. S.-T. Ho and H. P. Yuen, “Scheme for realizing a photon number amplifier,” Opt. Lett. 19, 61–63 (1994). [CrossRef] [PubMed]

15. H. P. Yuen, “Quantum amplifiers, quantum duplicators and quantum cryptography,” Quantum Semiclass. Opt.: J. Eur. Opt. Soc. Part B 8, 939 (1996). [CrossRef]

16. G. D’Ariano, “Hamiltonians for the photon-number-phase amplifiers,” Phys. Rev. A 45, 3224–3227 (1992). [CrossRef]

17. G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996). [CrossRef]

18. G. Björk, J. Söderholm, and A. Karlsson, “Superposition-preserving photon-number amplifier,” Phys. Rev. A 57, 650–658 (1998). [CrossRef]

19. The transformation in Eq. (2) can be realized only when M ≥ Gn. There is always such a restriction on amplification relations; the energy transferred to reservoir 2 must come from somewhere.

20. H. G. Dehmelt, “Proposed 1014 δv<v laser fluorescence spectroscopy on Tl + ion mono-oscillator II,” Bull. Am. Phys. Soc. 20, 60 (1975).

21. D. Wineland, J. Bergquist, W. M. Itano, and R. Drullinger, “Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions,” Opt. Lett. 5, 245–247 (1980). [CrossRef] [PubMed]

22. J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986). [CrossRef] [PubMed]

23. The construction of physically implementable transformations such as Eq. (3) (and transformations including higher powers of the input photon number operator) are highly constrained by two conditions: that the spectrum of the operator-representation be ℕ⁰ (the natural numbers including zero) and that the commutator be preserved [14]. From these conditions, in Eq. (3) we need at least one term with a number operator on the right with a prefactor of one. Additional reservoirs with arbitrary prefactors are allowed but they will carry additional noise and decrease the SNR.

24. Phase randomization is necessary for optimal amplification and measurement of photon number due to number-phase uncertainty. Indeed, amplification of photon number deamplifies phase and vice versa, see [17].

25. D. Pegg and S. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989). [CrossRef]

26. Note the dimension of both input and output mode Hilbert spaces are s + 1; they necessarily match in the Heisenberg picture.

27. C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012). [CrossRef]

28. R. Nehra, C.-H. Chang, A. Beling, and O. Pfister, “Photon-number-resolving segmented avalanche-photodiode detectors,” e-print arXiv:1708.09015 [physics.ins-det] (2017).

29. Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26, 12499–12505 (2018). [CrossRef] [PubMed]

30. L. V. Keldysh and A. N. Kozlov, “Collective properties of excitons in semiconductors,” Sov. Phys. JETP 27, 521–528 (1968).

31. M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature 406, 1039–1046 (2000). [CrossRef] [PubMed]

32. B. Laikhtman, “Are excitons really bosons?” J. Phys.: Condens. Matter 19, 295214 (2007).

33. M. Combescot, O. Betbeder-Matibet, and R. Combescot, “Exciton-exciton scattering: Composite boson versus elementary boson,” Phys. Rev. B 75, 174305 (2007). [CrossRef]

34. One way around this limitation is for the incident photons to only interact with a single symmetrized collective degree of freedom of many fermions, on which a measurement is then made. In this idealized case, this collective degree of freedom plays the role of a single bosonic mode and amplification could still be described by Eq. (5) and photon number amplification is improved past the limit for linear fermionic amplification [5].

35. The signal-to-noise ratios Eqs. (22) and (23) for linear amplification become infinite at G = 1 simply because there is no noise when both G = 1 and Δn_a = 0.

36. We find the linear dependence on G resulting from single-shot single-mode amplification holds for transformations describing higher-order amplification of photon number operator, again subject to the constraints of [23].

37. The space is further filled in by considering nonlinear amplification where G excitations are distributed into G′ > G modes so that ancillary modes contribute only to the noise and not to the signal. In this case, we find that the SNR goes to 0 as G′ → ∞; the effect of additional noise modes is always to reduce the SNR and move us away from the optimal SNR in Eq. (24).

38. Photon-number resolved photo detection can be achieved by multiplexing an n-photon signal to many (N ≫ n) single photon detectors [28], each satisfying Eq. (28) independently. However, this means an additional noise mode will be added with each splitting of the signal, decreasing the integrated signal-to-noise ratio. To avoid added noise a nonlinear multi-photon filtering process could be used, but for this a full S-matrix treatment must be used, see [45–47].

39. See, for example, [48]. The result is that, instead of certain frequencies, it is certain spectral “Schmidt modes” that are detected perfectly.

40. Tz. B. Propp and S. J. van Enk, “Quantum networks for single photon detection,” e-print arXiv:1901.09974 [quant-ph] (2019).

41. J. Dowling. Private communication.

42. Tz. B. Propp and S. J. van Enk, “POVMs for photo detection,” in preparation.

43. M. S. Ünlü and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995). [CrossRef]

44. A. Imamoḡlu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002). [CrossRef]