Abstract

We investigate coherent perfect absorption (CPA) in quantum optics, in particular when pairs of squeezed coherent states of light are superposed on an absorbing beam splitter. First, by employing quantum optical input–output relations, we derive the absorption coefficients for quantum coherence and for intensity, and reveal how these will differ for squeezed states. Second, we present the remarkable properties of a CPA gate: two identical but otherwise arbitrary incoming squeezed coherent states can be completely stripped of their coherence, producing a pure entangled squeezed vacuum state that with its finite intensity escapes from an otherwise perfect absorber. Importantly, this output state of light is not entangled with the absorbing beam splitter by which it was produced. Its loss-enabled functionality makes the CPA gate an interesting new tool for continuous-variable quantum state preparation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Coherent perfect absorption (CPA) of light [1] is an interference-assisted absorption process that in its simplest form can take place when two coherent beams impinge on the opposite sides of an absorbing beam splitter. With input light from only one side, some light would leave the beam splitter, but no light emerges if there is equal input from both sides. While scattering theory can provide a rigorous mathematical description, in essence, the reflected part of one of the incident beams interferes destructively with the transmitted part of the other (and vice versa), forming an artificial trap for the light that is subsequently dissipated [1]. So unlike the usual absorption, the coherent absorption of light is an emergent property [2] that arises from a specific interplay of interference and dissipation [1,3].

Many realizations of CPA have been proposed, including homogeneously broadened two-level systems [4], epsilon-near-zero metamaterials [5], graphene [6], and heterogeneous metal–dielectric composite layers [7]. CPA has been successfully demonstrated in many setups, e.g., in a silicon cavity with two counterpropagating waves [8], using a pair of resonators coupled to a transmission line [9], and using graphene to observe CPA of optical [10] and of terahertz radiation [11]. Achieving CPA under single-beam illumination with perfect magnetic conductor surfaces has also been reported [12]. A recent study revealed that CPA of light can be used as an unconventional tool to strongly couple light to surface plasmons in nanoscale metallic systems [13]. The fields in which CPA may play a central role include, but are not limited to, photodetection [14,15], sensing [16], photovoltaics [17], and cloaking [18,19]. For further details and examples of CPA realizations, refer to the excellent recent review by Baranov et al. [3].

Investigations of CPA in the quantum regime (also known as quantum CPA [3]) have only recently begun, notably with entangled few-photon input states [2023]. Here instead, we consider squeezed coherent states of light [24] and report the effects of quadrature squeezing on the absorption profile of the system, and vice versa the effects of the absorbing beam splitter on the generated output states of light. To the best of our knowledge, this important class of continuous-variable quantum states of light has not yet been considered in the context of CPA.

Squeezed states of light have no classical analogues. They reduce noise in one field quadrature at the expense of larger noise in the other, such that Heisenberg’s uncertainty relation for their product holds. First realized decades ago [25], squeezed states of light continue to attract attention, due mainly to their indispensable roles in quantum information, communication, and optics protocols. In particular, squeezed states are key for quantum teleportation [2628], quantum key distribution [29], quantum metrology [30], quantum cryptography [31], quantum dense coding [32], quantum dialogue protocols [33], quantum laser pointers [34], and quantum memories [35,36]. They can be used to distinguish quantum states by enhancing quantum interference [37] and for robust electromagnetically induced transparency [38]. They are used to increase the sensitivity of gravitational wave detectors [39] as well. While usually produced in macroscopic setups [25], squeezed light may also be produced by (pairs of) individual emitters in optical nanostructures [40]. Recently, squeezed vacuum was proposed to engineer interactions between electric dipoles [41].

In this contribution, we distinguish between the usual absorption of intensity and that of quantum coherence, the latter measured as the coherent degree of freedom in Glauber’s sense [42]. The latter measure is conveniently chosen such that the two measures are equivalent for coherent states, but for squeezed coherent input states, we show that they differ. For the latter case, we will show that under CPA conditions, a one- and two-mode combined squeezed vacuum state [4346] is produced, a finding that does not rely on our definition of coherence. Meanwhile, all coherence of the input states is transferred to the internal modes of the beam splitter. Importantly, we will show that in the output state, the light is not entangled with the lossy beam splitter. These and further intriguing properties may make the lossy CPA beam splitter a useful element in continuous-variable protocols.

The interference of waves depends on their statistical properties. With quantum states of light as inputs, it is then natural to look for connections between CPA and quantum statistics. Here, we express the coherent absorption coefficient in terms of the fidelity [47] between two incoming coherent states. The requirement for perfect absorption of coherence becomes the complete indistinguishability of the incoming fields. This requirement also holds for squeezed coherent states of light as input.

This paper is organized as follows. In Section 2, we revisit CPA from first principles and distinguish between the absorption of quantum coherence and intensity, and we identify a statistical connection between the coherent absorption coefficient and the fidelity of the input states. In Section 3, we present coherent absorption of squeezed coherent input states. In Section 4, we derive and discuss the remarkable quantum state transformation that is performed by the CPA beam splitter, and we conclude in Section 5.

2. CPA REVISITED: COHERENT STATES

A. Basic Setup and Key Concepts

Let us consider a lossy beam splitter in free space that superposes two incident quantized modes of light and creates two outgoing modes, as shown in Fig. 1. The incident modes are described by the discrete annihilation operators a^1 and a^2. The field operators b^1 and b^2 of the outgoing modes are then given by the relations [4851]

b^1=ta^1+ra^2+L^1,
b^2=ra^1+ta^2+L^2,
where t and r are, respectively, the beam splitter’s transmission and reflection amplitudes that in the case of loss satisfy |r|2+|t|2<1 and tr*+t*r0. The L^1 and L^2 describe Langevin-type noise operators corresponding to device modes of the beam splitter in which the light absorption takes place. Both the input and output operators for the optical modes satisfy the standard bosonic input–output relations, so [a^i,aj]=δij and [a^i,aj]=0for the input operators. The output operators satisfy the analogous standard relations, and consistency is ensured by the Langevin operators for which [L^1,L^2]0. We discuss this quantum noise further in Section 4. We adopted the discrete-mode representation of the quantized fields, but there would be ways to generalize this to full continuum [52].

 figure: Fig. 1.

Fig. 1. Sketch of the model system: two quantum states of light are superposed on an absorbing beam splitter. We consider squeezed coherent states that, e.g., can be prepared by sending coherent laser (L) light through a crystal that produces squeezing (S). Squeezing affects the coherent absorption. In turn, the CPA beam splitter gives rise to dissipation-enabled preparation of pure entangled two-mode output squeezed vacuum states (see main text).

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In the lossy beam splitter, the light typically loses part of its coherence and also part of its intensity, both due to a combination of destructive interference and dissipation. It will be enlightening to distinguish between the absorption of intensity and of coherent amplitudes (our distinction).

We define the total intensities of incoming and outgoing fields in the standard way as

Iina^1a^1+a^2a^2andIoutb^1b^1+b^2b^2,
and the lost intensity as ΔIIinIout. The coefficient of absorption of intensity is
AcohI1Iout/Iin,
being the fraction of intensity that gets lost. Analogously, we choose to quantify the input coherences through the quantity
Cin|a^1|2+|a^2|2,
i.e., as the sum of the absolute values squared of the expectation values , the latter taken with respect to the initial quantum state |ψin. This state describes both the quantum state of light in both arms and of the internal states of the beam splitter. We give a rationale for this measure of coherences below. Correspondingly, we quantify the output coherences, Cout|b^1|2+|b^2|2, and the net loss in coherent amplitudes as ΔCCinCout. We define the coefficient of absorption of coherences as
AcohCΔC/Cin=1Cout/Cin.
This definition is state independent and well defined for systems for which the input–output relations (1) hold, and the input coherence is non-vanishing. We will be especially interested in two specific situations: CPA, corresponding to AcohI1 (no output intensity), as opposed to perfect absorption of coherence, or AcohC1 (vanishing output coherences).

Let us elucidate our measure of coherences (5), and let us first state what it is not. It does not measure quantum correlations between different optical ports. Such correlations of the form a^1a^2 are already encoded in the coefficients Cin,out through the relations (1) of our quantum optical input–output formalism. Second, it does not measure the most general case of quantum coherent resources introduced by arbitrary quantum states [53]. Such generality is not needed here, since we focus on coherent and squeezed coherent states.

So what does Eq. (5) measure? Throughout this paper, we consider coherent states |α as introduced by Glauber [42], and quantum states that can directly be defined in terms of them. The well-known coherent states are eigenstates of the annihilation operator a|α=α|α, with the eigenvalue α being the complex coherent amplitude. Therefore, the magnitude of the coherent amplitude of the fully coherent states |α [42,54] and of quantum states that can directly be obtained through them can be quantified by the absolute expected value |a^| of the corresponding bosonic annihilation operator a^ [54]. We intentionally choose to measure the square of |a^|, so that for coherent states, our measure will numerically coincide with that of intensity (see Section 2B). This enables us to witness the breakdown of this equality in the case of squeezed states of light and thus provides a genuine new perspective to quantum CPA.

Written out in the photon-number state basis, the average intensity depends only on populations (diagonal elements) of the density matrix describing the state of light: a^a^=nρnnn. In contrast, the expectation value of the annihilation operator depends only on off-diagonal matrix elements (also known as quantum coherences [53]) of the density matrix: a^=nρnn1n. In particular, it depends only on the one-photon coherences, being the coherences between states that differ in photon number by one. It follows that the intensity absorption coefficient AcohI of Eq. (3) depends only on the photon-number populations of the density matrix, whereas the absorption coefficient of coherences AcohC in Eq. (5) depends only on its one-photon coherences. In the following, for simplicity, we refer to AcohC as the absorption of (quantum) coherence. The coherences are bounded from above by the average photon number as described by the inequality |a^|2a^a^, which follows from the generalized Cauchy–Schwarz inequality [54].

B. Coherent Absorption of Coherent States

Throughout this paper, we make the common assumption that initially the internal device modes of the beam splitter are in their ground states, denoted as |BS. For the incident optical modes, let us first consider that both are prepared in coherent states, also known as displaced vacuum states [55]. A coherent state (e.g., of mode 1) is defined as |α1=D^1(α)| in terms of the optical vacuum state | and the displacement operator

D^1(α)exp(|α|2/2)exp(αa^1)exp(α*a^1).
The total input state can then be written as |ψin=|α1|β2|BS, with α=|α|exp(iθ1) and β=|β|exp(iθ2). For the input coherence of these coherent states, we then find
Cin=|α|2+|β|2,
and for the output coherence,
Cout=(|α|2+|β|2)(|t|2+|r|2)+2cos(θ)|α||β|(tr*+rt*),
where we used the input–output relations (1) and defined θθ2θ1 as the phase difference between the coherent states. The coefficient of absorption of quantum coherence then reads
AcohC=1[|t|2+|r|2+2|α||β|(tr*+rt*)|α|2+|β|2cos(θ)],
which reduces to the conventional expression for absorption A=1(|t|2+|r|2) when replacing cos(θ) by its average value of zero.

For two coherent incident states, the quantum coherence lost is in fact equal to lost intensity, because the identities Cin=Iin and Cout=Iout hold in that case. This then immediately implies ΔC=ΔI and AcohC=AcohI. At this point, it may seem pedantic that we first distinguished between these two coefficients, but as we shall see in Section 3, this equality is not true for incident squeezed states of light.

C. Fidelity and CPA

Next, we relate the expression (9) for coherent absorption of coherent states to their quantum fidelity. As our starting point, we recall that the inner product of any pair of coherent states |α and |β is [55]

α|β=e12(|α|2+|β|2)+αβ*,
which implies they are never orthogonal. It follows that
αβ*=|α||β|eiθ=12(|α|2+|β|2)+lnα|β,
βα*=|α||β|eiθ=12(|α|2+|β|2)+lnβ|α.
Hence, we obtain
2|α||β|cos(θ)=|α|2+|β|2+ln|α|β|2=|α|2+|β|2+lnF(ρα,ρβ),
where F(ρα,ρβ) is the Ulhmann’s fidelity [47], with ρα|αα| and ρβ|ββ|. Written in terms of the fidelity, the coherent absorption coefficient (9) reads
AcohC,I=1[|t+r|2+(tr*+rt*)|α|2+|β|2lnF(ρα,ρβ)],
which holds for arbitrary lossy beam splitters and for any pair of coherent input states.

Now we turn to the condition of coherent perfect absorption for the specific type of lossy beam splitters with t=1/2 and r=1/2. These values for transmission and reflection amplitudes give rise to maximum incoherent absorption of 1/2 for a thin film (a beam splitter) in a homogeneous background, meaning that for a single incident coherent state, such beam splitters absorb half the light, since |t|2+|r|2=1/2. For the two incident coherent states, we end up with the coefficient for coherent absorption:

AcohC,I=1+lnF(ρα,ρβ)|α|2+|β|2.
Here, the natural logarithm is well defined, since the fidelity of two arbitrary coherent states always is greater than zero due to their non-orthogonality property (10). As a check, we find back AcohC,I=1/2 for a single incident coherent state (take αβ=0).

CPA according to Eq. (14) occurs when F(ρα,ρβ)=1, i.e., when the incoming coherent states are indistinguishable. This is satisfied if and only if the coherent states have the same phases and amplitudes. These are the same well-known requirements of CPA as in classical optics: the expression Eq. (9) for the coherent absorption reduces to that obtained through classical scattering amplitudes for t=1/2 and r=1/2 [56]. Indeed, these values are required for CPA to occur in two-port lossy systems [3,21,56] and will be used subsequently in our analysis of CPA. Thus, with the “quasi-classical” coherent states, we recover the classical condition for CPA, but this time explained in terms of quantum mechanical indistinguishability. Incidentally, while counterpropagating normally incident waves are commonly used in CPA literature, one can have a more general setup, as given in Fig. 1 (see, e.g., Ref. [7]. Decisive for CPA are the values t=r=1/2, not the angle of incidence.

By discussing CPA in quantum rather than classical optics, we replaced interference by quantum interference. The latter is described by transition probabilities between quantum states, which in the case of pure states, as we have here, are given by the fidelity [57]. It is intriguing that by Eq. (13) or Eq. (14), fidelities can be measured in terms of absorption. Thus, practically, one may distinguish two quantum states through a dissipative process, e.g., by using an absorbing metamaterial with known properties.

3. COHERENT ABSORPTION OF SQUEEZED COHERENT STATES

Prepared by the theory and results for coherent states in Section 2, we will now investigate the effects of squeezing on the coherent absorption of light. Mathematically, a squeezed coherent state |α,ζ is obtained by the action of a squeeze operator on a coherent state [55], e.g., for mode 1,

|α1,ζ11=S^1(ζ1)|α1=S^1(ζ1)D^1(α1)|.
Here, the squeeze operator is defined as
S^1(ζ1)e12ζ1*a^1212ζ1a^12
in terms of the mode creation and annihilation operators a1 and a1. The degree of squeezing is determined by the complex coefficient ζ1=ξ1expiϕ1. Here, ξ1 is called the squeezing parameter, while the angle ϕ1 quantifies the amount of rotation of the field quadratures in the corresponding quantum optical phase space.

Let us now assume two squeezed coherent states |α,ζ11 and |β,ζ22 as the input states of a general lossy beam splitter. The input states are then characterized by in total four complex parameters: the coherence parameters α=|α|exp(iθ1) and β=|β|exp(iθ2), and the squeezing parameters ζ1=ξ1exp(iϕ1) and ζ2=ξ2exp(iϕ2), with ξ1,ξ2 real-valued. The total input state has the form |ψin=|α,ζ11|β,ζ22|BS.

A. Perfect Coherent Absorption of Coherence

The expected values of the input operators with respect to the state |ψin read

a^1=|α|(eiθ1cosh(ξ1)eiθ1eiϕ1sinh(ξ1)),
a^2=|β|(eiθ2cosh(ξ2)eiθ2eiϕ2sinh(ξ2)).
For the input coherence (4), i.e., the coherent content within squeezed states introduced by the Glauber’s degree of freedom, it then follows that
Cin=γ12|α|2+γ22|β|2,
where γ12=cosh(2ξ1)cos(η1)sinh(2ξ1), γ22=cosh(2ξ2)cos(η2)sinh(2ξ2), with η12θ1ϕ1 and η22θ2ϕ2. Similarly, by using the input–output relations (1), we obtain the output coherence
Cout=(|t|2+|r|2)Cin+Γ(tr*+rt*),
where
Γ=a^1a^2*+a^2a^1*=2|α||β|(cos(θ)cosh(ξ1)cosh(ξ2)cos(θ1+θ2ϕ2)cosh(ξ1)sinh(ξ2)cos(θ1+θ2ϕ1)sinh(ξ1)cosh(ξ2)+cos(θϕ)sinh(ξ1)sinh(ξ2)),
and where θ=θ2θ1 and ϕ=ϕ2ϕ1. Therefore, by Eq. (5), the fraction of the coherence that gets lost is
AsqC=1(|t|2+|r|2+Γ(tr*+rt*)γ12|α|2+γ22|β|2).
In the following we will analyze whether it is possible that all coherence gets lost at the absorbing beam splitter, i.e., whether perfect absorption of coherent photons (AsqC=1) can be achieved with two incoming squeezed states. We will study the corresponding coherent absorption of intensities in Section 2.B.

We study how squeezing affects the coherent absorption of quantum coherence by exploring Eq. (21) for several parameter regimes. Let us first assume two squeezed beams with unit coherent amplitudes, |α|=|β|=1, with coherent phase angles θ1=θ2=δ and also equal squeezing angles ϕ1=ϕ2=ϕ, such that δϕ. So we take many parameters to be equal, but we do allow the squeezing amplitudes ξ1 and ξ2 to be different. It follows that

AsqC=1(|t|2+|r|2+Ω1Ω2),
where
Ω1=2[cosh(ξ)cos(ϵ)sinh(ξ)](tr*+rt*),Ω2=cosh(2ξ1)+cosh(2ξ2)cos(ϵ)[sinh(2ξ1)+sinh(2ξ2)],
and we defined ξξ1+ξ2 and ϵ2δϕ. We specify again t=1/2 and r=1/2 and then find the condition for complete absorption of coherence to be Ω1/Ω2=1/2, which is satisfied only if ξ1=ξ2. Therefore, for squeezed coherent states, equal coherent amplitudes and phases are not sufficient criteria for perfect absorption of coherence. Equal degree of squeezing is an additional requirement. For coherent states, this is trivially satisfied (ξ1=ξ2=0). For squeezed states, the additional requirement is nontrivial, generalizing the requirement of indistinguishability that we identified for coherent input states in Section 2.C.

Further non-trivial effects of quantum squeezing on the absorption of quantum coherence can be revealed by considering the special case of two input states with equal coherent amplitudes (α=β), a nonvanishing coherent phase difference (θ1=θ and θ2=0), equal squeezing amplitudes ξ1=ξ2=ξ, and vanishing squeezing phases (ϕ1=ϕ2=0). As before, the absorbing beam splitter is characterized by t=1/2 and r=1/2. We obtain

AsqC=12+cos(θ)1+e2ξ[cosh(2ξ)cos(2θ)sinh(2ξ)].
Thus, for all phase differences θ, in the limit ξ+ (amplitude squeezing), the coefficient of absorption of quantum coherence converges to the maximum incoherent absorption of A=1/2. With squeezing, up to ξ3.5 say, (almost) perfect absorption of quantum coherence is recovered for θ equal to a multiple of 2π. In the opposite limit ξ (phase squeezing), the coefficient of absorption becomes AsqC=1/2+(cos(θ)/[1+(1/2)(1+cos(2θ))], i.e., we have a similar behavior as in the case of bare coherent states. An illustration of these analytical results is depicted in Fig. 2.

 figure: Fig. 2.

Fig. 2. Coefficient of coherent absorption AsqC(ξ,θ) of Eq. (23) upon variation of parameters of the squeezed coherent input states, for a beam splitter with t=1/2 and r=1/2. We vary the coherence angle θ=θ1 and the squeezing parameter ξ=ξ1=ξ2, while keeping θ2=ϕ1=ϕ2=0 fixed. Figure is valid for arbitrary equal coherence amplitudes α=β.

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While Eq. (23) was found for different coherent phases but equal squeezing, as a final example to illustrate loss of coherence in the presence of squeezing, we revert the situation and take the coherence phases and amplitudes to be equal, but different squeezing amplitudes (and again δ=ϕ=0 and the same beam splitter with t=1/2 and r=1/2). This gives

AsqC=112(12eξe2ξ1+e2ξ2).
This formula is depicted in Fig. 3 and implies that in the absence of phases, squeezing always works against absorption, provided that ξ1ξ2. Indeed, the perfect absorption of coherent photons occurs only if ξ1=ξ2. Furthermore, for equal squeezing, we regain the symmetry such that in the limits ξ±, the absorption saturates at its classical maximum of 0.5.

 figure: Fig. 3.

Fig. 3. Variations in the coefficient of coherent absorption (a) AsqC(ξ1,ξ2) of Eq. (24) and (b) AsqC(ξ1=ξ,ξ2=0) of Eq. (31a). All other parameters are as explained in the main text.

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B. Coherent Absorption of Intensity

We showed that, for two coherent incident states, the fraction of quantum coherence lost is equal to that of intensity, i.e., AcohC=AcohI or ΔCΔI=0. In the case of two squeezed incident states, the loss of intensity reads

ΔI=Iin(1(|t|2+|r|2)Γ(t*r+r*t)Iin),
where Γ is given in Eq. (20). Hence, the coefficient of absorption of intensity becomes
AsqI=1(|t|2+|r|2)Γ(t*r+r*t)Iin=AΓ(t*r+r*t)Iin,
which can be rewritten as
IinAsqI=IinAΓ(t*r+r*t).
Similarly, from Eq. (21), we have
CinAsqC=CinAΓ(t*r+r*t).
Equations (27) and (28) can be combined with CinAsqC=ΔC and IinAsqI=ΔI to give
ΔIΔC=(IinCin)A.
Important about this identity is its generality: it relates coherence losses and intensity losses for arbitrary lossy beam splitters and arbitrary squeezed coherent input states. It is the generalization of the simple identity ΔI=ΔC that we obtained for coherent states. If AsqI,AsqCA, then Eq. (29) implies that one has equal quantum absorption coefficients AsqI=AsqC if and only if the total input intensity is equal to the total input coherence, i.e., Cin=Iin. But if the latter are not equal, and the coherent part of the squeezed state is completely absorbed (ΔC=Ci), then it follows from Eq. (29) that Iout=(IinCin)(1A)0; in other words, a quantum state with finite intensity survives the coherent absorption process, as the noise parts of the squeezed states are incoherent with respect to each other and thus cannot interfere and create a CPA-like effect. We will analyze this output state in Section 4.

For a fair comparison of coherence and intensity absorption, we now consider the same special cases that we already investigated in our analysis of the coefficient of absorption of quantum coherence. First, we choose θ1=θ, ξ1=ξ2=ξ, θ2=ϕ1=ϕ2=0, and t=1/2 and r=1/2 with equal coherent amplitudes. We obtain

AsqI=12+(cos(θ)/2)e2ξe2ξ+[1cos(2θ)2]sinh(2ξ)+sinh2(ξ)|α|2,
which should be compared with Eq. (23), which gives AsqC for the same input states. In the limit ξ+, we have AsqI=1/2 as before. Thus, in this limit, all the quantum contributions due to quantum coherence and squeezing are lost, and the corresponding absorption coefficients reduce to that of maximum incoherent one, i.e., AsqC=AsqI=A. In the opposite limit of ξ, we obtain AsqI=(1/2)+2cos(θ)/[3+cos(2θ)+(1/|α|2)]<1 for all αC, thus characteristically different from absorption of quantum coherence but complying with our general relation Eq. (29). Figure 4 illustrates that perfect absorption of intensity is possible if and only if there is no squeezing. The discrepancy between absorption of coherence in Fig. 2 and of intensity in Fig. 4 is evident. The quantum coefficient AsqI saturates to its classical incoherent value of 1/2 faster than AsqC for ξ>0. In the opposite regime where ξ<0, we have coherent oscillations similar to the case of absorption of quantum coherence, though always in the interval AsqC[0,1).

 figure: Fig. 4.

Fig. 4. Same as Fig. 2 but for AsqI(ξ,θ) with |α|=|β|=1.

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A crucial difference between the coefficients AsqC and AsqI is that the latter depends explicitly on the mean number of photons in the initial states. This leads to the breaking of the parity symmetry such that AsqI(ξ,θ=0,δ=0)AsqI(ξ,θ=0,δ=0). To clarify this, let ξ2=0 and ξ1=ξ, so that the first beam is prepared in a squeezed state, while the second one is in a coherent state with equal amplitudes |α|. By setting all optical phases to zero, the coefficient of absorption of coherent degree of freedom of squeezed states Eq. (24) is found to be

AsqC=12+eξ1+e2ξ=1+coshξ2coshξ,
while the intensity absorption coefficient becomes
AsqI=12+eξ1+e2ξ+sinh2(ξ)|α|2.
Hence, in this regime, the coefficient AsqC is an even function and thus totally symmetric in the squeezing parameter ξ [Fig. 3(b)]. The symmetry-breaking term in the expression for AsqI is now easily recognized as the ratio R=sinh2(ξ)/|α|2 of the mean photon contributions of independent squeezing and coherent degrees of freedoms to the total intensity of the squeezed coherent state. The maximally asymmetric and symmetric regimes are then identified by R1 and R1, respectively. In Fig. 5, we plot the coefficient of coherent absorption of intensity AsqI(ξ1=ξ,ξ2=0)of Eq. (31b), as a function of the squeezing parameter ξ, for a set of coherent amplitudes |α|2{103,100,103,106}. The figure illustrates clearly that intensity absorption in general is not symmetric in ξ. To restore the symmetry in the considered regime, the minimum number of coherent photons is found to be of order 106.

 figure: Fig. 5.

Fig. 5. Variations in the coefficient of coherent absorption of intensity AsqI(ξ1=ξ,ξ2=0) of Eq. (31b) for |α|2=103 (solid blue), |α|2=100 (dashed orange), |α|2=103 (dotted yellow), and |α|2=106 (dashed-dotted purple). All the other parameters are as explained in the main text.

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4. CONTINUOUS-VARIABLE QUANTUM STATE PREPARATION WITH CPA

We would like to know the quantum states of light produced at the output of a beam splitter that exhibits CPA. For equal coherent states |α1 and |α2 as input, we found in Section 2B that there is zero intensity in the output, meaning that the output state for the two optical output modes has to be the vacuum state. The remarkable robustness of this way of producing the vacuum as output is that the same output state is produced whatever the coherence amplitude |α| of the input state.

Quantum state preparation becomes even more interesting for squeezed coherent input states, which we write in terms of squeezing and displacement operators as

|ψin=|α,ζ11|β,ζ22|BS=S^2(ζ2)D^2(β)S^1(ζ1)D^1(α)||BS=S^1(ζ1)S^2(ζ2)D^1(α)D^2(β)||BS=e12(ζ2*a^22+ζ1*a^12)12(ζ2a^22+ζ1a^12)×D^1(α)D^2(β)||BS,
where in the third and fourth equalities we used that a^1 and a^2 commute. In the previous section, we saw that for equal input amplitudes and squeezing, and for t=r=1/2, all coherence can be coherently absorbed, but some output intensity will always remain. We will now use the input state (32), specify β=α and ζ2=ζ1=ζ, and then determine the output state for this specific case.

In Eq. (1), output operators were defined in terms of input operators. We need to invert this, writing the input operators in terms of the output operators. Thereby we can obtain the sought output state by writing the input state in terms of the output operators. We will use the known quantum optical input–output theory for absorbing beam splitters [4851], in particular Ref. [51], and identify what is special about quantum state transformation by absorbing beam splitters that exhibit CPA. Following Ref. [51], we write the input–output operator relations Eq. (1) in matrix notation as

b^=Ta^+Ag^.
Here, T is the 2×2 transmission matrix. The Langevin noise of the absorbing beam splitter is accounted for by linear combinations of bosonic device input operators g^1 and g^2 that together form the vector g^. The corresponding linear coefficients form the 2×2 absorption matrix A. Besides optical output operators b^1,2, there are device output operators h^1,2. Also the latter pair can be written as a linear combination of all four input operators. The 4×4 matrix that relates all four output operators in terms of the four input operators is restricted by the requirement that output operators satisfy standard bosonic commutation relations and are canonically independent. This restricts A once T is given, for example.

The formalism of Ref. [51] simplifies particularly for the CPA beam splitter because T is a real symmetric matrix in this special case, and the absorption matrix A is then also easily found:

Tcpa=(trrt)=12(1111)=12(1σx),Acpa=12(1111)=12(1+σx)
in terms of both the 2×2 unit matrix 1 and the Pauli matrix σx. This simplifies the input–output relation Eq. (33), and the corresponding relation for the device output operators becomes h^=Acpaa^+Tcpag^, which combined with Eq. (33) provides the full 4×4 input–output operator matrix relation.

For the CPA beam splitter, by matrix inversion, the inverse relationship of Eq. (33) becomes

a^=Tcpab^Acpah^.
Now we can use these relations to write a^1 and a^2 in the input state (32) in terms of the four output operators b^1,2 and h^1,2, and it is easy to show that a^12+a^22=[(b^1b^2)2+(h^1+h^2)2]/2. We thereby obtain as one of our main results the output state
|ψout=e14ζ*(b^1b^2)214ζ(b^1b^2)2|e14ζ*(h^1+h^2)214ζ(h^1+h^2)2|α,αBS,
which is remarkable for several reasons. First, just like the input state, it is a direct-product state of optical output states and beam splitter device states. In other words, the optical output state is not entangled with the absorbing beam splitter that was used to produce it. Tracing out the beam splitter’s internal degrees of freedom therefore leaves the output state of light in a pure state, rather than the usual mixed state that requires a density-matrix description. This remarkable outcome is the main reason that the CPA beam splitter, despite being lossy, can become a useful component in continuous-variable quantum state engineering.

The second remarkable property of the state is the perfect coherent absorption: all coherence of the input state |ψin=|α,ζ1|α,ζ2|BS resided in the optical channels and ends up in the material modes of the beam splitter. There are no coherent photons, in Glauber’s sense, in the optical output state, i.e., it does not depend on the coherence amplitude α at all. This explains that we found AsqC=1 in Secton 3A.

As a special case and check of our results, for vanishing squeezing, we indeed find standard CPA behavior: for the two-mode coherent input state |ψin=|α1|α2|BS, we find from Eq. (36) the corresponding output state |ψout=||α,αBS. This, indeed, is a direct product of the optical vacuum state and coherent states for the device modes of the beam splitter. So for coherent states, the coherent absorption is indeed perfect; no photons leave the CPA beam splitter and AsqI=1.

Returning to the general case of squeezed coherent input, the optical output state e14ζ*(b^1b^2)214ζ(b^1b^2)2| in Eq. (36) is a one- and two-mode combined squeezed vacuum state. The one-mode squeezing corresponds to quadratic operators in the exponent such as b^22, and the two-mode squeezing to the products of different operators such as b^1b^2. Being generalizations of the generic squeezed vacuum, the optical output states can be used for the implementations of quantum teleportation [26], quantum metrology [30], quantum dense coding [32], quantum dialogues [33], and electromagnetically induced transparency protocols [38]. Squeezed vacuum states have non-vanishing intensities, and since a beam splitter under CPA conditions emits squeezed vacuum states of light, coherent perfect absorption of intensity is not possible, and AsqI<1 for non-vanishing squeezing. This optical output state is independent of the coherence amplitude α of the incident squeezed coherent states. This quantum property explains why the coherent absorption coefficient AsqI in Eq. (31b) became dependent on the input intensity via |α|2, while such a nonlinear dependence is absent for AsqC in Eq. (31a).

These one- and two-mode combined squeezed vacuum states have been studied before, albeit in a different setting [43]. The exact form that emerges here was first proposed by Abdalla [44,45] and later studied by Yeoman and Barnett [46]. In the latter contribution, it was identified that these states could be generated by superposing two identical (equally squeezed) single-mode squeezed vacuum states via a 50/50 ideal beam splitter. It was found that |t|2=|r|2 should hold to produce such states on a beam splitter, a condition that also the lossy CPA beam splitter satisfies.

So while the ideal beam splitter requires squeezed vacuum input states, the CPA beam splitter can take any pair of identical squeezed coherent states to distill [58] a two-mode entangled squeezed vacuum state out of it. Moreover, squeezing takes place via an absorption process resulting in beam-splitter internal modes that end up in one- and two-mode combined squeezed coherent states. We find that half the squeezing is absorbed into internal modes of the beam splitter. This leaves the other 50% of the squeezing for the optical output modes, in accordance with a spectral analysis performed for the special case of incoming squeezed vacuum states [20]. The distillation of squeezed vacuum states that we propose is not possible with non-absorbing beam splitters. Thus, our results generalize the previous ones and propose a new engineering procedure to produce pure quantum states via perfect absorption of quantum coherence.

5. CONCLUSION

In conclusion, we investigated the coherent absorption of light when two squeezed coherent beams are superposed on an absorbing beam splitter. We first reconsidered the generic case of two incoming bare coherent states and distinguished two types of absorption, namely, of quantum coherence introduced to the system by the complex amplitude α and of intensity. We showed that the corresponding absorption coefficients are identical for the case of bare coherent state inputs and can be written in terms of quantum fidelity, suggesting a general condition of indistinguishability of input states for CPA to occur that holds for squeezed coherent states as well.

In the case of squeezed coherent beams, the coherent degree of freedom is completely absorbed, provided that the CPA conditions hold. By Eq. (29), we provided a general argument that the input intensity will not be fully absorbed in the presence of quantum squeezing. More specifically, we revealed that an entangled squeezed vacuum state is produced at the output, leaving the absorber in a one- and two-mode combined squeezed coherent state. In some cases, both states might be reused as quantum resources [59,60].

We propose to test and use the lossy CPA gate as a new tool for quantum state preparation. Since quite remarkably the CPA gate produces a direct-product state of an optical output state and an internal beam-splitter state [see Eq. (36)], it does not suffer from the usual disadvantage of lossy optical components in becoming entangled with optical fields, producing mixed reduced quantum states for the light fields. Instead, the optical output states of the CPA gate are pure quantum states. Yet the action of the gate crucially depends on the CPA beam splitter being lossy: all coherence is absorbed.

It is interesting to compare the CPA gate with the usual practical implementation of “phase-space displacement” by which a squeezed vacuum state and a strong coherent state are mixed on a low-reflectivity non-lossy beam splitter [61], resulting in a squeezed coherent output state. Our CPA gate does more or less the reverse, separating squeezing from coherence, but the crucial difference is that it does so for arbitrary (but equal) input coherence amplitudes. This arbitrariness constitutes a potentially useful robustness of this gate. In particular, the CPA gate would work in a small-signal regime where saturation effects in absorption can safely be neglected.

Our proposal of the CPA quantum gate is part of an interesting wider trend to engineer quantum dissipation and to use it for quantum state preparation and other quantum operations (see, e.g., Refs. [6269]). Also for our CPA gate for continuous-variable quantum state preparation, loss is a resource to obtain new functionality: the CPA gate prepares its pure quantum states both despite being lossy and because it is lossy.

Funding

Villum Fonden; Danish National Research Foundation (DNRF) (DNRF103).

Acknowledgment

We thank E. C. André, N. Stenger, and J. R. Maack for useful discussions. A. Ü. C. H. acknowledges the support from the Villum Foundation through a postdoctoral Block Stipend. The Center for Nanostructured Graphene is sponsored by DNRF.

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References

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  1. Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref]
  2. J. H. Holland, Complexity: A Very Short Introduction (Oxford University, 2014).
  3. D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
    [Crossref]
  4. S. Longhi, “Coherent perfect absorption in a homogeneously broadened two-level medium,” Phys. Rev. A 83, 055804 (2011).
    [Crossref]
  5. S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012).
    [Crossref]
  6. F. Liu, Y. D. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” 2D Mater. 1, 031001 (2014).
    [Crossref]
  7. S. Dutta-Gupta, O. J. F. Martin, S. D. Gupta, and G. S. Agarwal, “Controllable coherent perfect absorption in a composite film,” Opt. Express 20, 1330–1336 (2012).
    [Crossref]
  8. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
    [Crossref]
  9. Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
    [Crossref]
  10. G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7, 4810–4817 (2013).
    [Crossref]
  11. N. Kakenov, O. Balci, T. Takan, V. A. Ozkan, H. Altan, and C. Kocabas, “Observation of gate-tunable coherent perfect absorption of terahertz radiation in graphene,” ACS Photon. 3, 1531–1535 (2016).
    [Crossref]
  12. S. Li, J. Luo, S. Anwar, S. Li, W. Lu, Z. H. Hang, Y. Lai, B. Hou, M. Shen, and C. Wang, “An equivalent realization of coherent perfect absorption under single beam illumination,” Sci. Rep. 4, 7369 (2014).
    [Crossref]
  13. H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
    [Crossref]
  14. G. Konstantatos and E. H. Sargent, “Nanostructured materials for photon detection,” Nat. Nanotechnol. 5, 391–400 (2010).
    [Crossref]
  15. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
    [Crossref]
  16. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
    [Crossref]
  17. A. Luque and S. Hegedus, Handbook of Photovoltaic Science and Engineering (Wiley, 2011).
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  19. K. J. Vinoy and R. M. Jha, Radar Absorbing Materials: From Theory to Design and Characterization (Kluwer Academic, 1996).
  20. S. Huang and G. S. Agarwal, “Coherent perfect absorption of path entangled single photons,” Opt. Express 22, 20936–20947 (2014).
    [Crossref]
  21. T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
    [Crossref]
  22. T. Roger, S. Restuccia, A. Lyons, D. Giovannini, J. Romero, J. Jeffers, M. Padgett, and D. Faccio, “Coherent absorption of N00N states,” Phys. Rev. Lett. 117, 023601 (2016).
    [Crossref]
  23. C. Altuzarra, S. Vezzoli, J. Valente, W. Gao, C. Soci, D. Faccio, and C. Couteau, “Coherent perfect absorption in metamaterials with entangled photons,” ACS Photon. 4, 2124–2128 (2017).
    [Crossref]
  24. D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
    [Crossref]
  25. U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
    [Crossref]
  26. G. Milburn and S. L. Braunstein, “Quantum teleportation with squeezed vacuum states,” Phys. Rev. A 60, 937–942 (1999).
    [Crossref]
  27. P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482–3485 (2000).
    [Crossref]
  28. H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
    [Crossref]
  29. N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
    [Crossref]
  30. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
    [Crossref]
  31. M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A 61, 022309 (2000).
    [Crossref]
  32. M. Ban, “Quantum dense coding via a two-mode squeezed-vacuum state,” J. Opt. B Quantum Semiclass. Opt. 1, L9 (1999).
    [Crossref]
  33. N.-R. Zhou, J.-F. Li, Z.-B. Yu, L.-H. Gong, and A. Farouk, “New quantum dialogue protocol based on continuous-variable two-mode squeezed vacuum states,” Quantum Inf. Process. 16, 4 (2017).
    [Crossref]
  34. N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
    [Crossref]
  35. K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. Plenio, A. Serafini, M. Wolf, and E. Polzik, “Quantum memory for entangled continuous-variable states,” Nat. Phys. 7, 13–16 (2011).
    [Crossref]
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2017 (7)

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
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C. Altuzarra, S. Vezzoli, J. Valente, W. Gao, C. Soci, D. Faccio, and C. Couteau, “Coherent perfect absorption in metamaterials with entangled photons,” ACS Photon. 4, 2124–2128 (2017).
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N.-R. Zhou, J.-F. Li, Z.-B. Yu, L.-H. Gong, and A. Farouk, “New quantum dialogue protocol based on continuous-variable two-mode squeezed vacuum states,” Quantum Inf. Process. 16, 4 (2017).
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S. Zeytinoğlu, A. İmamoğlu, and S. Huber, “Engineering matter interactions using squeezed vacuum,” Phys. Rev. X 7, 021041 (2017).
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A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
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D. D. B. Rao, S. Yang, and J. Wrachtrup, “Dissipative entanglement of solid-state spins in diamond,” Phys. Rev. A 95, 022310 (2017).
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B. Vest, M.-C. Dheur, É. Devaux, A. Baron, E. Rousseau, J.-P. Hugonin, J.-J. Greffet, G. Messin, and F. Marquier, “Anti-coalescence of bosons on a lossy beam splitter,” Science 356, 1373–1376 (2017).
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2016 (4)

A. Winter and D. Yang, “Operational resource theory of coherence,” Phys. Rev. Lett. 116, 120404 (2016).
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U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
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T. Roger, S. Restuccia, A. Lyons, D. Giovannini, J. Romero, J. Jeffers, M. Padgett, and D. Faccio, “Coherent absorption of N00N states,” Phys. Rev. Lett. 117, 023601 (2016).
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N. Kakenov, O. Balci, T. Takan, V. A. Ozkan, H. Altan, and C. Kocabas, “Observation of gate-tunable coherent perfect absorption of terahertz radiation in graphene,” ACS Photon. 3, 1531–1535 (2016).
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2015 (4)

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6, 7031 (2015).
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D. Kienzler, H.-Y. Lo, B. Keitch, L. de Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. P. Home, “Quantum harmonic oscillator state synthesis by reservoir engineering,” Science 347, 53–56 (2015).
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A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
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G. Morigi, J. Eschner, C. Cormick, Y. Lin, D. Leibfried, and D. J. Wineland, “Dissipative quantum control of a spin chain,” Phys. Rev. Lett. 115, 200502 (2015).
[Crossref]

2014 (6)

D. D. B. Rao and K. Mølmer, “Deterministic entanglement of Rydberg ensembles by engineered dissipation,” Phys. Rev. A 90, 062319 (2014).
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J. Zhang, C. Guo, K. Liu, Z. Zhu, W. Ye, X. Yuan, and S. Qin, “Coherent perfect absorption and transparency in a nanostructured graphene film,” Opt. Express 22, 12524–12532 (2014).
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S. Huang and G. S. Agarwal, “Coherent perfect absorption of path entangled single photons,” Opt. Express 22, 20936–20947 (2014).
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S. Li, J. Luo, S. Anwar, S. Li, W. Lu, Z. H. Hang, Y. Lai, B. Hou, M. Shen, and C. Wang, “An equivalent realization of coherent perfect absorption under single beam illumination,” Sci. Rep. 4, 7369 (2014).
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F. Liu, Y. D. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” 2D Mater. 1, 031001 (2014).
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Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
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2013 (3)

G. Pirruccio, L. Martín Moreno, G. Lozano, and J. Gómez Rivas, “Coherent and broadband enhanced optical absorption in graphene,” ACS Nano 7, 4810–4817 (2013).
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J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

F. G. Brandao, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, “Resource theory of quantum states out of thermal equilibrium,” Phys. Rev. Lett. 111, 250404 (2013).
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2012 (3)

S. Dutta-Gupta, O. J. F. Martin, S. D. Gupta, and G. S. Agarwal, “Controllable coherent perfect absorption in a composite film,” Opt. Express 20, 1330–1336 (2012).
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S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012).
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H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
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2011 (6)

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
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S. Longhi, “Coherent perfect absorption in a homogeneously broadened two-level medium,” Phys. Rev. A 83, 055804 (2011).
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W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
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K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. Plenio, A. Serafini, M. Wolf, and E. Polzik, “Quantum memory for entangled continuous-variable states,” Nat. Phys. 7, 13–16 (2011).
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M. J. Kastoryano, F. Reiter, and A. S. Sørensen, “Dissipative preparation of entanglement in optical cavities,” Phys. Rev. Lett. 106, 090502 (2011).
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H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011).
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2010 (4)

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
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Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
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N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
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G. Konstantatos and E. H. Sargent, “Nanostructured materials for photon detection,” Nat. Nanotechnol. 5, 391–400 (2010).
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2009 (1)

R. Tualle-Brouri, A. Ourjoumtsev, A. Dantan, P. Grangier, M. Wubs, and A. S. Sørensen, “Multimode model for projective photon-counting measurements,” Phys. Rev. A 80, 013806 (2009).
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2008 (1)

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100, 093602 (2008).
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2006 (1)

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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2004 (2)

D. Akamatsu, K. Akiba, and M. Kozuma, “Electromagnetically induced transparency with squeezed vacuum,” Phys. Rev. Lett. 92, 203602 (2004).
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H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
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2003 (1)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
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2001 (1)

N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
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2000 (2)

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482–3485 (2000).
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M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A 61, 022309 (2000).
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1999 (3)

M. Ban, “Quantum dense coding via a two-mode squeezed-vacuum state,” J. Opt. B Quantum Semiclass. Opt. 1, L9 (1999).
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G. Milburn and S. L. Braunstein, “Quantum teleportation with squeezed vacuum states,” Phys. Rev. A 60, 937–942 (1999).
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L. Knöll, S. Scheel, E. Schmidt, and D.-G. Welsch, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
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1998 (1)

S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
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1996 (1)

S. M. Barnett, C. R. Gilson, B. Huttner, and N. Imoto, “Field commutation relations in optical cavities,” Phys. Rev. Lett. 77, 1739–1742 (1996).
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1995 (1)

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823–4838 (1995).
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1993 (1)

G. Yeoman and S. M. Barnett, “Two-mode squeezed Gaussons,” J. Mod. Opt. 40, 1497–1530 (1993).
[Crossref]

1992 (2)

M. S. Abdalla, “Statistical properties of a new squeezed operator model,” J. Mod. Opt. 39, 771–781 (1992).
[Crossref]

M. S. Abdalla, “The statistical properties of a generalized two-mode squeezed operator,” J. Mod. Opt. 39, 1067–1081 (1992).
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1990 (1)

F. Hong-Yi, “Squeezed states: operators for two types of one-and two-mode squeezing transformations,” Phys. Rev. A 41, 1526–1532 (1990).
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1987 (1)

1983 (1)

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
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1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
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Aasi, J.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Abadie, J.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Abbott, B.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Abbott, R.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Abbott, T.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Abdalla, M. S.

M. S. Abdalla, “Statistical properties of a new squeezed operator model,” J. Mod. Opt. 39, 771–781 (1992).
[Crossref]

M. S. Abdalla, “The statistical properties of a generalized two-mode squeezed operator,” J. Mod. Opt. 39, 1067–1081 (1992).
[Crossref]

Abernathy, M.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Adam, S.

F. Liu, Y. D. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” 2D Mater. 1, 031001 (2014).
[Crossref]

Adams, C.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Adams, T.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Addesso, P.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

Adesso, G.

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

Adhikari, R.

J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, C. Adams, T. Adams, P. Addesso, and R. Adhikari, et al., “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

Agarwal, G. S.

Agio, M.

D. Martín-Cano, H. R. Haakh, and M. Agio, “Quadrature-squeezed light from emitters in optical nanostructures,” in Quantum Plasmonics (Springer, 2017), pp. 25–46.

Akamatsu, D.

D. Akamatsu, K. Akiba, and M. Kozuma, “Electromagnetically induced transparency with squeezed vacuum,” Phys. Rev. Lett. 92, 203602 (2004).
[Crossref]

Akiba, K.

D. Akamatsu, K. Akiba, and M. Kozuma, “Electromagnetically induced transparency with squeezed vacuum,” Phys. Rev. Lett. 92, 203602 (2004).
[Crossref]

Altan, H.

N. Kakenov, O. Balci, T. Takan, V. A. Ozkan, H. Altan, and C. Kocabas, “Observation of gate-tunable coherent perfect absorption of terahertz radiation in graphene,” ACS Photon. 3, 1531–1535 (2016).
[Crossref]

Altuzarra, C.

C. Altuzarra, S. Vezzoli, J. Valente, W. Gao, C. Soci, D. Faccio, and C. Couteau, “Coherent perfect absorption in metamaterials with entangled photons,” ACS Photon. 4, 2124–2128 (2017).
[Crossref]

Alù, A.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Andersen, U. L.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-Gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[Crossref]

Anisimov, P. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[Crossref]

Anwar, S.

S. Li, J. Luo, S. Anwar, S. Li, W. Lu, Z. H. Hang, Y. Lai, B. Hou, M. Shen, and C. Wang, “An equivalent realization of coherent perfect absorption under single beam illumination,” Sci. Rep. 4, 7369 (2014).
[Crossref]

Aoki, T.

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
[Crossref]

Appel, J.

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100, 093602 (2008).
[Crossref]

Bachor, H.-A.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

Balci, O.

N. Kakenov, O. Balci, T. Takan, V. A. Ozkan, H. Altan, and C. Kocabas, “Observation of gate-tunable coherent perfect absorption of terahertz radiation in graphene,” ACS Photon. 3, 1531–1535 (2016).
[Crossref]

Ban, M.

M. Ban, “Quantum dense coding via a two-mode squeezed-vacuum state,” J. Opt. B Quantum Semiclass. Opt. 1, L9 (1999).
[Crossref]

Baranov, D. G.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Barnett, S. M.

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the model system: two quantum states of light are superposed on an absorbing beam splitter. We consider squeezed coherent states that, e.g., can be prepared by sending coherent laser (L) light through a crystal that produces squeezing (S). Squeezing affects the coherent absorption. In turn, the CPA beam splitter gives rise to dissipation-enabled preparation of pure entangled two-mode output squeezed vacuum states (see main text).
Fig. 2.
Fig. 2. Coefficient of coherent absorption AsqC(ξ,θ) of Eq. (23) upon variation of parameters of the squeezed coherent input states, for a beam splitter with t=1/2 and r=1/2. We vary the coherence angle θ=θ1 and the squeezing parameter ξ=ξ1=ξ2, while keeping θ2=ϕ1=ϕ2=0 fixed. Figure is valid for arbitrary equal coherence amplitudes α=β.
Fig. 3.
Fig. 3. Variations in the coefficient of coherent absorption (a) AsqC(ξ1,ξ2) of Eq. (24) and (b) AsqC(ξ1=ξ,ξ2=0) of Eq. (31a). All other parameters are as explained in the main text.
Fig. 4.
Fig. 4. Same as Fig. 2 but for AsqI(ξ,θ) with |α|=|β|=1.
Fig. 5.
Fig. 5. Variations in the coefficient of coherent absorption of intensity AsqI(ξ1=ξ,ξ2=0) of Eq. (31b) for |α|2=103 (solid blue), |α|2=100 (dashed orange), |α|2=103 (dotted yellow), and |α|2=106 (dashed-dotted purple). All the other parameters are as explained in the main text.

Equations (41)

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b^1=ta^1+ra^2+L^1,
b^2=ra^1+ta^2+L^2,
Iina^1a^1+a^2a^2andIoutb^1b^1+b^2b^2,
AcohI1Iout/Iin,
Cin|a^1|2+|a^2|2,
AcohCΔC/Cin=1Cout/Cin.
D^1(α)exp(|α|2/2)exp(αa^1)exp(α*a^1).
Cin=|α|2+|β|2,
Cout=(|α|2+|β|2)(|t|2+|r|2)+2cos(θ)|α||β|(tr*+rt*),
AcohC=1[|t|2+|r|2+2|α||β|(tr*+rt*)|α|2+|β|2cos(θ)],
α|β=e12(|α|2+|β|2)+αβ*,
αβ*=|α||β|eiθ=12(|α|2+|β|2)+lnα|β,
βα*=|α||β|eiθ=12(|α|2+|β|2)+lnβ|α.
2|α||β|cos(θ)=|α|2+|β|2+ln|α|β|2=|α|2+|β|2+lnF(ρα,ρβ),
AcohC,I=1[|t+r|2+(tr*+rt*)|α|2+|β|2lnF(ρα,ρβ)],
AcohC,I=1+lnF(ρα,ρβ)|α|2+|β|2.
|α1,ζ11=S^1(ζ1)|α1=S^1(ζ1)D^1(α1)|.
S^1(ζ1)e12ζ1*a^1212ζ1a^12
a^1=|α|(eiθ1cosh(ξ1)eiθ1eiϕ1sinh(ξ1)),
a^2=|β|(eiθ2cosh(ξ2)eiθ2eiϕ2sinh(ξ2)).
Cin=γ12|α|2+γ22|β|2,
Cout=(|t|2+|r|2)Cin+Γ(tr*+rt*),
Γ=a^1a^2*+a^2a^1*=2|α||β|(cos(θ)cosh(ξ1)cosh(ξ2)cos(θ1+θ2ϕ2)cosh(ξ1)sinh(ξ2)cos(θ1+θ2ϕ1)sinh(ξ1)cosh(ξ2)+cos(θϕ)sinh(ξ1)sinh(ξ2)),
AsqC=1(|t|2+|r|2+Γ(tr*+rt*)γ12|α|2+γ22|β|2).
AsqC=1(|t|2+|r|2+Ω1Ω2),
Ω1=2[cosh(ξ)cos(ϵ)sinh(ξ)](tr*+rt*),Ω2=cosh(2ξ1)+cosh(2ξ2)cos(ϵ)[sinh(2ξ1)+sinh(2ξ2)],
AsqC=12+cos(θ)1+e2ξ[cosh(2ξ)cos(2θ)sinh(2ξ)].
AsqC=112(12eξe2ξ1+e2ξ2).
ΔI=Iin(1(|t|2+|r|2)Γ(t*r+r*t)Iin),
AsqI=1(|t|2+|r|2)Γ(t*r+r*t)Iin=AΓ(t*r+r*t)Iin,
IinAsqI=IinAΓ(t*r+r*t).
CinAsqC=CinAΓ(t*r+r*t).
ΔIΔC=(IinCin)A.
AsqI=12+(cos(θ)/2)e2ξe2ξ+[1cos(2θ)2]sinh(2ξ)+sinh2(ξ)|α|2,
AsqC=12+eξ1+e2ξ=1+coshξ2coshξ,
AsqI=12+eξ1+e2ξ+sinh2(ξ)|α|2.
|ψin=|α,ζ11|β,ζ22|BS=S^2(ζ2)D^2(β)S^1(ζ1)D^1(α)||BS=S^1(ζ1)S^2(ζ2)D^1(α)D^2(β)||BS=e12(ζ2*a^22+ζ1*a^12)12(ζ2a^22+ζ1a^12)×D^1(α)D^2(β)||BS,
b^=Ta^+Ag^.
Tcpa=(trrt)=12(1111)=12(1σx),Acpa=12(1111)=12(1+σx)
a^=Tcpab^Acpah^.
|ψout=e14ζ*(b^1b^2)214ζ(b^1b^2)2|e14ζ*(h^1+h^2)214ζ(h^1+h^2)2|α,αBS,

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