Quantum dense metrology by an SU(2)-in-SU(1,1) nested interferometerWei Du,1, a) J. F. Chen,2 Z. Y. Ou,3, b) and Weiping Zhang4, c)

1)Quantum Institute of Light and Atoms,Department of Physics, East China Normal University, Shanghai, 200241,People’s Republic of China2)Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, China3)Department of Physics, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202,USA4)Department of Physics and Astronomy, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240,People’s Republic of China

With the help of quantum entanglement, quantum dense metrology (QDM) is a technique that can performthe joint estimates of two conjugate quantities such as phase and amplitude modulations of an optical fieldwith an accuracy beating the standard quantum limit simultaneously. SU(1,1) interferometers (SUI) canrealize QDM with detection loss tolerance but is limited in absolute sensitivity. Here we present a QDMscheme with a linear interferometer (SU(2)) nested inside an SU(1,1) interferometer. By using a degenerateSUI and controlling the phase angle of the phase-sensitive amplifiers in SUI, we can achieve the optimumquantum enhancement in the measurement precision of arbitrary mixture of phase and amplitude modulation.

It is well-known1 that the phase uncertainty of an in-terferometer with classical sources is bounded by stan-dard quantum limit (SQL) that scales as 1/

√N with N

as the photon number sensing the phase, which is viewedas classical resource. That is why strong power of light isemployed inside the Laser Interferometer Gravitational-Wave Observatory (LIGO)2 for the highest absolute sen-sitivity. But high power causes various problems such asthermal effects. In the field of biological imaging, lighttoxicity of the probe field is a huge issue3. A possibleway to reach a high absolute sensitivity of phase mea-surement but at a relatively low light level is to reducethe vacuum fluctuation by placing the unused input portof the interferometer in a squeezed state of light1,4–8.

A quantity that is equally important at fundamentallevel but relatively less in practical applications is the am-plitude of a field, whose noise can also be reduced by thesqueezed states9. An example is the radiation pressurenoise in LIGO induced by light intensity fluctuations1.However, because of the Heisenberg uncertainty prin-ciple, it is impossible to beat SQL with a single-modesqueezed state in the measurement of phase and am-plitude simultaneously. In recent years, with the ap-plication of quantum entanglement, two-mode squeezedlight is demonstrated to be capable of embedding twoor more non-commuting observables in information withtheir measurement precision beyond the SQL simultane-ously. This is so-called quantum dense coding in quan-tum information science10 or quantum dense metrology(QDM) in quantum metrology11–15.

An alternative way to achieve quantum-enhanced sen-sitivity is to amplify noiselessly the signal16–18, achievedin the so-called SU(1,1) interferometer (SUI)19–24, whichis firstly proposed by Yurke et al.25 some thirty years ago.

a)Electronic mail: weiduecnu@126.comb)Electronic mail: zou@iupui.educ)Electronic mail: wpz@sjtu.edu.cn

SUIs, while achieving a quantum-enhanced phase mea-surement, possess a number of advantages26 to the con-ventional squeezed state interferometers especially theproperty of detection loss tolerance20,23,27,29. Recently,it was demonstrated that SUI is also capable of measur-ing multiple noncommuting parameters and beating thestandard quantum limit simultaneously14,15. In spite ofso many advantages over the conventional interferome-ters, SUIs suffer low phase sensing photon numbers (Ips)due to saturation of parametric amplifiers. Thus the ab-solute sensitivity of such type of interferometer is still notcomparable to classical interferometers. Moreover, cur-rent schemes of SUI have so far only the phase measure-ment that makes full use of available resource and reachesthe optimum sensitivity but not amplitude14,30,31.

In this paper, we consider a more practical scheme,embedding a linear interferometer (also known as SU(2)interferometer25) inside an SU(1,1) interferometer. Al-though the linear interferometer has strong injection forincreasing absolute sensitivity, it operates in dark fringemode so that the SUI basically works with no injectionand thus low photon numbers inside. We will demon-strate that the scheme is suitable for simultaneous mea-surement of phase and amplitude, thus realizing QDM,but without the limitation on photon number, and itsdegenerate version can be used to measure either phaseor amplitude modulation signal, or any arbitrary mixtureof the two, with an optimum sensitivity that is reachedwith full use of both classical and quantum resources.

Before discussing QDM, let us first define the phase(PM) and amplitude (AM) measurement. A phase mod-ulation signal δ can be added to an incoming probe fieldain by a phase factor: eiδain without adding extra vac-uum noise. But since amplitude modulation ε changesthe intensity or energy of the probe field, which is equiv-alent to a loss, we need to model it with a beam spitter:

a = tain + raν (1)

with t = e−ε and r =√

1− t2. Here, aν is the vacuum

arX

iv:2

002.

0219

5v2

[qu

ant-

ph]

7 M

ay 2

020

2

field coupled in through loss. But if the modulation signalε is very small: ε � 1 so that t ≈ 1 − ε, r ≈

√2ε, we

can neglect the vacuum contribution because its noise〈∆2X〉 = t2〈∆2Xin〉 + r2〈∆2Xν〉 ≈ (1 − 2ε)〈∆2Xin〉 +2ε〈∆2Xν〉 ≈ 〈∆2Xin〉. With this in mind, we will dropthe vacuum term in Eq.(1) in all treatment of amplitudemodulation later. Therefore, for small δ, ε(� 1), we havethe modulated field as a′ ≈ eiδe−εain ≈ (1 + iδ − ε)ain

We start by first considering how to make simultaneousAM and PM measurement with a classical coherent state.The simplest scheme is by direct homodyne measure-ment, as shown in Fig.1(a) and simultaneous measure-ment is achieved by splitting the modulated field with abeam splitter. An alternative way is by a linear or SU(2)interferometer, as shown in Fig.1(b) with similar signalsplitting. Both schemes have been shown29 to achieveoptimum sensitivity in joint measurement by a classicalfield. But the interferometric scheme is advantageous tothe direct homodyne detection scheme because the for-mer can operate at the dark port without a large coher-ent component and it is especially suitable for increasingthe intensity of the probe field for higher sensitivity. Toprepare the discussion on QDM with SU(1,1) interfer-ometers, we analyze in detail the interferometric schemenext.

For the interferometric scheme in Fig.1(b), we can findthe output field by using the following beam splitter re-lations (refer to Fig.1(b) for notations):

A =√T1ain +

√R1bin, B =

√T1bin −

√R1ain,

aout =√T2A−

√R2Be

−εeiϕ,

bout =√T2Be

−εeiϕ +√R2A. (2)

The outputs are related to the input by

aout = (√T1T2 +

√R1R2e

−εei(ϕ+δ))ain

+ (√R1T2 −

√T1R2e

−εei(ϕ+δ))bin

bout = (√T1R2 −

√R1T2e

−εei(ϕ+δ))ain

+ (√R1R2 +

√T1T2e

−εei(ϕ+δ))bin, (3)

where ϕ is the phase added to field B to account foroverall phase difference in the interferometer.

For simplicity without loss of generality, we assumeidentical beam splitters: T1 = T2 ≡ T,R1 = R2 ≡ R.Notice that when ϕ = 0, the interferometer without mod-ulations (δ = 0 = ε) acts as if nothing is there: aout = ainand bout = bin. This is so-called dark fringe operation.We will work at this point throughout the paper. Nowlet input field ain be in a coherent state |α〉 and field

bin be in vacuum. Using small modulations assumption(δ, ε� 1), we obtain for the dark port output:

bout ≈ bin + ain√TR(ε− iδ). (4)

Since coherent state has the same size of noise as vacuumstate and δ, ε � 1, field bout has fluctuations dominated

by that of bin but also contains information about mod-ulations δ, ε.

a b

AM PM

ˆina

inb

ˆina

A

Bˆouta

outbVBSVBSHDX

HDY AMPM

α

υ

HD2

HD1VBS

FIG. 1. Schematic diagram for (a) direct homodyne measure-ment;(b) interferometer with homodyne measurement. VBS:variable beam splitter; HD: homodyne detection; AM: ampli-tude modulation; PM: phase modulation.

Without loss of generality, we assume α = real forthe input coherent state |α〉 at ain. Then homodyne

measurement of quadrature phase amplitudes Xbout≡

bout + b†out, Ybout≡ (bout − b†out)/i gives signal and noise

respectively as

〈Xbout〉 = 2αε√TR, 〈Ybout〉 = 2αδ

√TR,

〈∆2Xbout〉 = 1 = 〈∆2Ybout

〉, (5)

leading to the signal-to-noise ratio (SNR) for phase andamplitude modulations

SNRδ ≡ 〈Ybout〉2/〈∆2Ybout〉 = 4TIpsδ2

SNRε ≡ 〈Xbout〉2/〈∆2Xbout

〉 = 4TIpsε2, (6)

where Ips ≡ Rα2 is the photon number for the probefield to the modulations, which can be considered as theoverall classical resource for the measurement. OptimumSNRs26 of 4Ipsδ

2, 4Ipsε2 are achieved for a very unbal-

anced interferometer with T ≈ 1, R = 1− T � 1.The above is for individual δ or ε measurement alone.

For simultaneous measurement of δ and ε, an extra BS(T3, R3) can be used to split the output but with split

SNR as well: SNR(s)δ = 4T3Ipsδ

2, SNR(s)ε = 4R3Ipsε

2,similar to Fig.1(a). Notice that simultaneous measure-ment of δ and ε needs to share the overall resource of Ips(T3Ips +R3Ips = Ips)

29.When squeezed states are used to replace the vacuum

state at unused input port of bin, the vacuum quantumnoise can be reduced and the SNR in either phase1,4,5

or amplitude measurement9 is enhanced, but not in bothbecause only one quadrature is squeezed at a time. Butwith the employment of quantum entanglement, the tech-nique of quantum dense metrology (QDM) is capable ofmeasuring two non-orthogonal quadratures with quan-tum enhancement simultaneously13. SU(1,1) interferom-eters, which possess the property of detection loss tol-erance, also have the ability to measure simultaneouslymultiple quadrature-phase amplitudes at arbitrary angleswith quantum enhanced precision14,15. However, despiteof its numerous advantages over traditional interferome-ters, SU(1,1) interferometers have limited absolute sensi-tivity due to practical issues. Here, to circumvent theseproblems, we study a variation of SU(1,1) interferometer,which combines a traditional linear or SU(2) interferome-

3

PA1 outbA

B

inb

0b HD2

HD10a

φC

ϕina

2d

1douta

υ

υ

αAM

PM

PA2

FIG. 2. QDM with non-degenerate SU(1,1) interferometry.PA1, PA2: parametric amplifier; HD: homodyne detection.

ter with an SU(1,1) interferometer for QDM and inheritsthe advantages of both interferometers.

Consider the scheme shown in Fig. 2. It is a Mach-Zehnder interferometer (MZI) nested in an SU(1,1) in-terferometer (SUI), which was shown recently32 to pos-sess all the advantages of SUI for phase measurement butwithout the limit on the intensity of the probe field. Toachieve QDM, we make homodyne measurement on bothoutputs of the second parametric amplifier (PA2), onefor phase (d1) and the other for amplitude (d2). If theMZI works at dark fringe point, we have the input-output

relation in Eq.(4). Since bout ≈ bin as if MZI were notthere, the overall performance of the SUI is not affectedby MZI and it does not have any injection at its inputs

(a0, b0) but the modulation signals δ, ε is contained in

bout for SUI to measure.

Using the input-output relation for PA1 and PA2:

bin = G1b0 + g1a†0, C = G1a0 + g1b

†0,

d1 = G2Ceiφ + g2b

†out, d2 = G2bout + g2C

†e−iφ, (7)

we obtain the outputs of the SUI for QDM as

d1 = (G1G2eiφ + g1g2)a0

+(g1G2e−iφ +G1g2)b†0 + g2

√R(ε+ iδ)a†in

d2 =(G1G2 + g1g2e−iφ)b0

+(g1G2+G1g2e−iφ)a†0 +G2

√R(ε− iδ)ain, (8)

where φ is a phase to account for the overall phase ofthe SUI. Here we used Eq.(4) with T ≈ 1 for optimumperformance of the MZI.

With input fields a0 and b0 in vacuum state, R → 0and small modulations (δ, ε� 1), it is straightforward tocalculate the noise of the outputs of the SUI as

〈∆2Yd1〉 = 〈∆2Xd2〉= |G1G2 + g1g2e

−iφ|2 + |g1G2+G1g2e−iφ|2

= (G21 + g2

1)(G22 + g2

2) + 4G1G2g1g2 cosφ, (9)

where Yd1 = (d1 − d†1)/i, Xd2 = d2 + d†2. The noise isminimum when φ = π corresponding to dark fringe. Thesignals are calculated as

〈Yd1〉 = 2g2δ√Ips, 〈Xd2〉 = 2G2ε

√Ips (10)

HD

DPA1 outbA

B

inb DPA2

ϕina

0b

outa

1 2

υ

AMPM

1b

α

FIG. 3. QDM with degenerate SU(1,1) interferometry whenloss is considered. DPA: Degenerate parametric amplifier.

with Ips = Rα2. So, the SNRs for simultaneous mea-surement of δ and ε are

SNRSUI(Yd1) =4g2

2Ipsδ2

(G2G1 − g1g2)2 + (G1g2 −G2g1)2

SNRSUI(Xd2) =4G2

2Ipsε2

(G2G1 − g1g2)2 + (G1g2 −G2g1)2.

Both reach optimum value when G2 ≈ g2 � 1:

SNR(op)SUI(Yd1) = 2Ipsδ

2(G1 + g1)2

SNR(op)SUI(Xd2) = 2Ipsε

2(G1 + g1)2. (11)

When (G1 + g1)2/2 > 1, both measurement of δ and εbeats simultaneously the classical sensitivity expressed inEq.(6) by the same factor of (G1 +g1)2/2, thus achievingQDM. Notice that the total resource of Ips(G1 + g1)2 isshared equally, satisfying the quantum resource sharinglaw29,31. The resource now consists of photon number Ipsand quantum entanglement characterized by (G1 + g1)2.

Since the resource is shared equally between phase andamplitude measurement, when SUIs are used for phaseor amplitude measurement alone, only half the optimumsensitivity is realized. Recently, a variation of SUI30 em-ploys dual beam to sense the modulations for taking allthe resource for phase measurement and achieving opti-mum sensitivity. However, the scheme is only suitable forphase. In practice, the information embedded may not bepurely in phase but could be in amplitude or something ofa mixture of phase and amplitude. To achieve optimumsensitivity for these, we make a variation to the SUI. Weuse a degenerate parametric amplifier (DPA) instead ofthe regular non-degenerate PA, as shown in Fig.3. Bene-fiting from the degenerate PA, which can be regarded asa phase-sensitive amplifier, we can selectively distributeall resources to any mixture of phase and amplitude foroptimum measurement sensitivity, as shown next.

Referring to the notations in Fig.3, we write the input-output relations of DPA as

bin = G1b0 + g1eiθ1 b†0,

b1 = G2bout + g2eiθ2 b†out (12)

where phase θ1(2) is for DPA1(2). Using Eq.(4) with

T ≈ 1 for bout, we find the output of DPA2 as

b1 = G2[G1b0 + g1eiθ1 b†0 + ain

√R(ε− iδ)]

+g2eiθ2 [G1b

†0 + g1e

−iθ1 b0 + a†in√R(ε+ iδ)]

4

= GT b0 + gT b†0 +√RG2(ε− iδ)ain

+√Rg2e

iθ2(ε+ iδ)a†in], (13)

where GT ≡ G1G2 + g1g2ei(θ2−θ1), gT ≡ g1G2e

iθ1 +G1g2

eiθ2 as the overall gains of the degenerate SUI. Since b0is in vacuum, the signal part of the output is then

〈b1〉 = α√R[G2(ε− iδ) + g2e

iθ2(ε+ iδ)]

= α√Reiθ2/2

[ε(g2e

iθ2/2 +G2e−iθ2/2)

+iδ(g2eiθ2/2 −G2e

−iθ2/2)]. (14)

If we measure X ≡ b1e−iθ2/2 + b†1eiθ2/2, Y ≡ (b1e

−iθ2/2 −b†1e

iθ2/2)/i, we find the signal becomes

〈X 〉 = −2(G2 + g2)√Ipsγ−

〈Y〉 = −2(G2 − g2)√Ipsγ+, (15)

where γ− ≡ −ε cos θ22 + δ sin θ22 , γ+ ≡ ε sin θ2

2 + δ cos θ22are two orthogonal modulation signals. Therefore, mea-surement of X , Y gives orthogonal quantities γ∓ with γ−amplified by G2 + g2 but γ+ de-amplified by G2 − g2.

For the noise of the degenerate SUI, since R� 1, it is

governed by GT , gT and with b0 in vacuum, we have

〈∆2X 〉 = |GT e−iθ2/2 + g∗T eiθ2/2|2

= (G2 + g2)2|G1 + g1e−i∆|2

〈∆2Y〉 = |GT eiθ2/2 − g∗T eiθ2/2|2= (G2 − g2)2|G1 − g1e

−i∆|2, (16)

where ∆ ≡ θ1− θ2 is total phase of the SUI. With ∆ = πfor dark fringe (destructive interference), we have

〈∆2X 〉 = (G2 + g2)2(G1 − g1)2

〈∆2Y〉 = (G2 − g2)2(G1 + g1)2. (17)

So, the SNRs for the measurement of X , Y are

SNRX = 4Ipsγ2−(G1 + g1)2

SNRY = 4Ipsγ2+(G1 − g1)2. (18)

Because there is only one output for the degenerate SUI,we cannot make simultaneous measurement of the or-thogonal quantities γ± and so the degenerate SUI is notsuitable for QDM. But a homodyne detection of X at theoutput of the degenerate SUI will give the measurementof γ− = −ε cos θ2/2 + δ sin θ2/2, an arbitrary mixture ofAM (ε) and PM (δ) signals, depending on θ2 with a quan-tum enhancement of (G1 + g1)2. Setting θ2 = 0 gives thefull amplitude modulation (ε) whereas θ2 = π gives thefull phase modulation (δ). The measurement reaches theoptimum measurement sensitivity making full use of thequantum resource of Ips(G1 + g1)2 of the probe field.

The physical meaning of the output signal in Eq.(15)and noise in Eq.(17) for the degenerate SUI is illustratedin the phase space representation of the state evolutionin Fig.4 at each stage of the SUI: (a) shows the initialinput vacuum state (circle); In (b) DPA1 squeezes thevacuum state and amplifies the vacuum noise at the se-lected direction (θ1/2) but de-amplifies in orthogonal di-rection; In (c) the SU(2) linear interferometer encodes

SU(2)DPA1 DPA2 HD

(a) (b)

𝛼|ۄ

𝑣|ۄ

(c) (d)

(e)

𝛼𝛿

(𝐺2+𝑔 2)𝛼𝛿

𝜃1/2 𝜃2/2

(−𝜖)𝛼 (𝐺2+𝑔2)(−𝜖)𝛼

(𝐺2+𝑔2)𝛼γ−

𝜃2/2

𝛼γ−

FIG. 4. State evolution of the degenerate SU(1,1) interferom-etry. (a)-(d) are the phase-space (X-Y) representation of theevolved state at the corresponding position of the interferom-eter shown in (e), where |v〉 represents the vacuum state and|α〉 represents the coherent state.

weak signals of phase αδ and amplitude α(−ε) to theprobe beam; In (d) DPA2 un-squeezes in the selecteddirection θ2/2 and noiselessly amplifies the mixed modu-lation signal αγ−. The actions of the two degenerate PAsare represented by two ellipses. Note that θ1,2/2 gives thedirection of maximum amplification (G1,2 + g1,2)2 of thephase-sensitive DPA1,2. The orthogonal orientations ofthe two ellipses is due to dark fringe operation condition:(θ1− θ2)/2 = π/2 for the SUI. When G1 = G2, the noisein Eq.(17) becomes vacuum noise of one (circle) at theoutput because the actions of DPA1,2 are opposite butequal to each other (squeezing and then un-squeezing),leading to vacuum noise output, but in the same timeDPA2 amplifies the modulation signal αγ−, leading tosensitivity enhancement of (G1 + g1)2.

Different from non-degenerate SUI whose optimumsensitivity is achieved when G2 ≈ g2 � 1, the de-generate SUI reaches optimum sensitivity at any gainG2 for DPA2. On the other hand, to fully utilize thedetection loss-tolerant property of SUI20,27,28, we needto have G2 � G1 so that Eq.(17) gives output noise

〈∆2X 〉 = (G2 + g2)2/(G1 + g1)2 � 1. In this way, anyvacuum noise (size of 1) added through detection losses

will be negligible compared to 〈∆2X 〉.In summary, we have demonstrated that an SU(1,1)

interferometer with a linear interferometer nested insidecan make phase and amplitude measurement simultane-ously with precision beating SQL, thus achieving quan-tum dense metrology. The phase and amplitude measure-ment share equally the overall resource of photon numberand quantum entanglement. The degenerate version ofthe SUI can devote all resource to one measurement ofan arbitrary mix of phase and amplitude with optimummeasurement sensitivity. This should be particularly use-ful if the signal is encoded in both phase and amplitudesuch as LIGO interferometer where both phase and inten-sity fluctuations play important roles. In fact, we noticeda recent work from LIGO Collaboration33 in which theyexperimentally demonstrated quantum enhancement in ajoint measurement of the phase of laser beams and posi-tion of mirrors by choosing proper squeezed angle of theinitial squeezer (similar to θ1 in our work).

5

FundingW.Z. acknowledge support by the National Key Re-search and Development Program of China under Grantnumber 2016YFA0302001, and the National NaturalScience Foundation of China (NSFC) through GrantNo. 11654005, Science and Technology Commission ofShanghai Municipality (STCSM) (16DZ2260200). J.F.C.acknowledge the support from NSFC through GrantNo. 11674100 and Shanghai RisingStar Program No.17QA1401300.Data availability statementData sharing is not applicable to this article as no newdata were created or analyzed in this study.DisclosuresThe authors declare that there are no conflicts of interestrelated to this article.

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