Estimation of gravitational acceleration with quantum optical interferometers

S. Y. Chen1,2 and T. C. Ralph2∗

1.Quantum Institute for Light and Atoms, Department of Physics,East China Normal University, Shanghai 200062, China

2.Centre for Quantum Computation and Communication Technology,School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia

(Dated: July 30, 2018)

The precise estimation of the gravitational acceleration is important for various disciplines. Weconsider making such an estimation using quantum optics. A Mach-Zehnder interferometer in an“optical fountain” type arrangement is considered and used to define a standard quantum limit forestimating the gravitational acceleration. We use an approach based on quantum field theory on acurved, Schwarzschild metric background to calculate the coupling between the gravitational fieldand the optical signal. The analysis is extended to include the injection of a squeezed vacuum tothe Mach-Zehnder arrangement and also to consider an active, two-mode SU(1,1) interferometer ina similar arrangement. When detection loss is larger than 8%, the SU(1,1) interferometer shows anadvantage over the MZ interferometer with single-mode squeezing input. The proposed system isbased on current technology and could be used to examine the intersection of quantum theory andgeneral relativity as well as for possible applications.

I. INTRODUCTION

Precise measurement of Earth’s gravity is of practi-cal importance for many fields, such as navigation, geo-physics and nature resource exploration. Such measure-ments can also be used as tests of general relativity [1].Of considerable interest from both the fundamental andapplied point of view is the coupling of quantum sys-tems to gravity, for example as demonstrated by Colella,Overhauser and Werner [2]. Several proposals have beenmade to extend these types of experiments [3–6] and ap-plications suggested such as enhanced global positioningsystems and telecommunications [7, 8].

Atomic system has been applied to the detection ofEarth’s gravity with high precision [9–19]. In particularatomic fountains can achieve a precision of ∆g ≈ 3 ×10−9g [9, 11, 12]. Such systems may approach quantumlimits for parameter estimation [17–19]. Some authorshave considered the measurement of gravity via quantumoptical methods [20–25]. In order to determine quantumlimits for the estimation of the gravitational field strengthvia such methods it is important to delineate the signalphotons, which acquire a differential phase shift due tothe gravitational field, from reference photons that areused to perform homodyne detection or to pump activemedia, etc.

In this paper, we analyse a quantum optical interfer-ometer in an “optical fountain” arrangement whereby asignal field is generated at some height, sent to a greaterheight where it is delayed, then returned to the originalsource height and interfered. The advantage of this ar-rangement is that all reference beams and optical pumpfields remain at the source height and so do not acquirea signal. Hence the reference and pump fields can be

∗ ralph@physics.uq.edu.au

consistently treated as “free-resourses”, whilst the signalphotons are treated as the quantum resources. In thisway we define a quantum standard limit (SQL) for quan-tum metrology of the gravitational field strength, as afunction of the photon number in the signal beam, basedon a standard Mach-Zehnder (MZ) type arrangement.We then examine surpassing this quantum limit usingsingle mode squeezing in the MZ arrangement and byemploying an active SU(1,1) type interferometer [26–30].The effects of internal and detection losses are included.The calculations are carried out in a general relativisticway using quantum field theory techniques on a curvedbackground [31].

The paper is organized in the following way. In Sec-tion II we describe our various set-ups and derive ana-lytical expressions for the relative sensitivities for esti-mating the gravitational acceleration. Subsection II Areviews phase estimation in a MZ interferometer. Sub-section II B calculates the phase shift acquired by thesignal beam propagating in the Schwarzschild metric asa function of the Schwarzschild radius and hence derivesthe standard quantum limit for estimation of the grav-itational acceleration parameter. Enhancement of theestimation via the inclusion of a squeezed vacuum inputin the presence of losses is analysed in subsection II C.Subsection II D considers the two-mode SU(1,1) interfer-ometer. In section III performance of the various set-upsis analysed numerically and best strategies under differ-ent conditions are proposed. In section IV we concludeand summarise our results.

II. SETUP AND THEORY

We propose the optical interferometer shown in Fig.1to sense the local gravity. The interferometer is placed onthe ground vertically along the radial direction of Earth.Due to space-time curvature, when the proper lengths of

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the two arms in the interferometer at different heights(upper path b1 and lower path a1) are the same, lighttravelling along the two arms will experience differentlocal time, leading to different phases of the two opti-cal paths. This phase difference can be detected by theintensity detection of the interferometer output.

In Fig.1, a coherent field a0 is injected into the firstbeam splitter BS1 while the other input port is injectedwith vacuum b0, or the squeezed vacuum c1 from theprocess in the dashed rectangle. After the BS1, the lighttravels along two paths. The signal beam b1 travels verti-cally up to radius coordinate R1, then reflected by mirrorto travel along a horizontal path with a local distanceL, finally reflected downwards vertically to the secondbeam splitter BS2 on the ground. Meanwhile, the refer-ence beam a1 travels horizontally along the lower pathon ground. After a time delay device, such as a laser de-lay line [32, 33], reference beam a1 arrives at the secondbeam splitter BS2 and combines with signal beam b1.The final outputs for detection after BS2 are light fieldsa2 and b2. The total proper length of the upper signalpath is 2H + L, where H is the vertical proper distancefrom R1 to R2, L is the proper length of the upper hor-izontal arm. For simplicity, we assume the delay devicemakes the proper length of the lower path equal to theproper length of the upper path 2H+L. The gray cubeswith labels t1, t2 in Fig.1 represent internal and externalamplitude loss respectively. For simplicity, the losses onboth signal beam b1 and reference beam a2 are the same.The dashed rectangle labeled with “Squeezed vacuum” issingle-mode squeezed vacuum production. Vacuum c0 isinjected into squeezer S, and a phase sensitive squeezedsingle-mode vacuum field c1 is produced, with a phasemodulator ξ. Single-mode squeezed vacuum c1 = b0can be injected into BS1 in the vertical interferometerto decrease the detection noise and improve the sensitiv-ity [21].

A. Interferometer theory

The intensity transmittance of beam splitters BS1 andBS2 in the interferometer is T and and reflectivity is 1−T .

With the coherent state a0 and vacuum state b0 as inputs,the input-output relations for the interferometer are,

a2 =(T − (1− T )eiε

)a0 +

√T (1− T )(1 + eiε)b0 (1)

b2 =−√T (1− T )(1 + eiε)a0 +

(Teiε − (1− T )

)b0(2)

Here we set ε = ε0 + εG with ε0 = π a constant off-set,and if the gravity induced phase εG � π, the outputs a2and b2 in Eq.2 can be approximated as

a2 =(1 + (1− T )iεG)a0 −√T (1− T )iεGb0 (3)

b2 =√T (1− T )iεGa0 − (TiεG + 1)b0 (4)

From homodyne detections at the outputs a2 and b2,the phase quadrature of both outputs X−a = i(a2 −a†2), X−b = i(b2 − b†2) are,

r

R2

L

H

Delaya0

b0

t1

t2

c0 c1Squeezed vacuum

S

a2

b2

b1

signal

referencet1

t2

BS1 BS2

R1

a1

H

M MV1b

V1a

V2a

V2b

x

ξ

FIG. 1. The optical interferometer is placed vertical to theground from radial coordinate R2 to coordinate R1, with alocal height difference H. For the losses, t1 is the transmit-tance of the optical amplitude for internal loss, and t2 is thetransmittance for the external loss. The input states for BS1are coherent light a0 in one port and vacuum b0 or squeezedvacuum c1 in the other port. After BS1, the reference beama1 goes along the lower path and signal beam b1 goes alongthe upper path. Finally, the outputs at the detection portsare a2 and b2. ε is the phase acquired by the signal beamof the interferometer. In the dashed rectangle is the single-mode squeezer. S: squeezer; ξ: phase modulator for single-mode squeezed vacuum; c0: input vacuum field; M: mirror;c1: single-mode squeezed vacuum.

X−a =√T (1− T )εGX(b0)− (1− T )εGX(a0)

+ Y (a0)

X−b =−√T (1− T )εGX(a0) + TεGX(b0)− Y (b0)

(5)

Here X(b0) = b0 + b†0, Y (b0) = i(b0 − b†0), X(a0) =

a0 + a†0, Y (a0) = i(a0 − a†0). Focusing on the output b2,

the expectation and variance of quadrature Xb are⟨X−b

⟩= − 2

√T (1− T )εG|α| (6)⟨

∆(X−b )2⟩

=T (1− T )ε2G + T 2ε2G + 1 ≈ 1 (7)

So we get the phase sensitivity of single output detec-tion of X−b :

(∆εG)b =

⟨∆(X−b )2

⟩d⟨X−b

⟩dεG

=1√

T2√Nsig

(8)

Here Nsig = (1 − T )N0 is the photon number of thesignal beam b1 which senses the phase change throughthe upper path and N0 = |α|2 is the initial input photonnumber of the coherent state a0. The phase sensitivityof single output detection of the interferometer is the

best (∆εG)b =1

2√Nsig

when T ≈ 1. T ≈ 1 means the

3

photon number of the reference beam is much larger thanthe signal beam, which is an unbalanced interferometer,and also analogous to a homodyne detection apparatuswith much stronger local oscillator field [34].

Another choice is to take joint homodyne detectionsof both outputs: Xj = X−a + X−b [35]. Under the samephase situation, we get the phase sensitivity of the inter-ferometer from joint quadrature Xj :

(∆εG)j =

⟨∆X2

j

⟩d⟨Xj

⟩dεG

=

√2

(√T +√

1− T )2√Nsig

(9)

When T =1

2, the phase sensitivity from the joint ho-

modyne detection is the best: (∆ε)j =1

2√Nsig

. Thus

the optimal phase sensitivities from the two differentmethods of detections achieve the same result with par-ticularly chosen transmittance value T . For simplicity,we focus on the single output detection to define the stan-dard quantum limit (SQL) for the gravity signal in thenext step.

B. Gravity phase and SQL definition

According to the Schwarzschild solution to general rel-ativity equations, the space time around a massive bodycan be approximately described by the metric [36]

ds2 =− (1− rsr

)dt2 + (1− rsr

)−1dr2

+ r2(dθ2 + sin2θdφ2)(10)

Here t is the time coordinate as seen by a far-away clock,r is the radial coordinate defined as the circumferenceat that radius divided by 2π, θ is the colatitude, φ is thelongitude and rs is the Schwarzschild radius of the Earth.We work in units c = 1, where c is the speed of light. In

standard units, rs =2GM

c2, where G is the gravitational

constant. Here we neglect the rotation and ellipticity ofEarth.

In the interferometer, the proper lengths of the uppersignal path and lower reference path are set the same2H+L. The observer is assumed standing on the groundbeside the interferometer. Because the horizontal size ofthe interferometer L is much smaller than the radius ofEarth Re ≈ 6.7 × 106m, the deflection of geodesic lineof light can be neglected and the horizontal paths canbe thought of as a geodesic line with constant radius r.For simplicity and to avoid the influence of the longitudeφ change, the interferometer is devised to sit with thehorizontal arm along the latitude line. When light travelsalong the horizontal path, the change of the longitude isdφ = 0. The vertical paths are taken to be approximatelyradial.

The proper length of the vertical path for signal beamb1 can be calculated according to the Schwarzschild met-ric in Eq.10 by substituting dt = 0, dθ = 0, dφ = 0 to

obtain the metric function ds2 =1

1− rsr

dr2. Hence, we

get the proper distance of the vertical path is

H =

∫ R1

R2

ds =

∫ R1

R2

1√1− rs

r

dr

≈ R1 −R2 +rs2lnR1

R2

(11)

The Taylor series expansion1√

1− rsr

≈ 1+rs2r

is applied

here. So the proper length of the whole upper path for

signal b1 is L+ 2H ≈ L+ 2(R1 −R2 +rs2lnR1

R2).

The observed travelling time of the signal b1 along theupward vertical path and downward vertical path is thesame. The coordinate time of signal b1 travelling alongone vertical path is given by assuming a null geodesic

0 = ds2 = −(1− rsr

)dt2 +1

1− rsr

dr2 (12)

So we get the coordinate time of the signal travellingalong the vertical arm from R2 to R1,

tv =

∫dt =

∫ R1

R2

1

1− rsr

dr ≈ R1 −R2 + rslnR1

R2(13)

For the signal b1 travelling along the upper horizontalpath L at coordinate R1, we derive

0 = ds2 = −(1− rsr

)dt2 + r2dθ2 (14)

Thus, we get the coordinate time of the signal beam b1horizontally travelling,

th =L√

1− rsR1

(15)

Here we use∫rdθ ≈ L with the approximation sinθ ≈ θ,

because the angle θ is small with L� Re. The local timeobserved by the observer on ground is

τ =

√(1− rs

R2)t (16)

where t is the coordinate time as seen by the far-away ob-server. So, from the view of the observer on the ground,according to Eq.16, the total local time of the signal beamb1 travelling along the whole upper path is,

τb =

√(1− rs

R2)

L√1− rs

R1

+ 2(R1 −R2 + rslnR1

R2)

(17)

4

For the reference beam a1 on the ground, the travellingcoordinate time along the lower path can be obtainedfrom Eq.14. Because the proper length of the lower pathis 2H + L, the coordinate time for the reference beam is

ta =L+ 2(R1 −R2 +

rs2lnR1

R2)√

1− rsR2

(18)

From Eq.16, the local time for the reference beam a1 bythe observer on the ground is

τa = L+ 2(R1 −R2 +rs2lnR1

R2) (19)

The local time difference between the reference beama1 and the signal beam b1 along different paths inside theinterferometer is obtained from Eq.17 and Eq.19,

∆τ = τa − τb =rs2

(∆RL

R1R2+

∆R2

R22

) (20)

Here ∆R = R1 −R2 is the coordinate separation.

From Eq.11, ∆R = R1 − R2 = H − rs2lnR1

R2, and

substituting this into Eq.20,

∆τ ≈ rs2

(∆RL

R22

+∆R2

R22

)

=rs2

((H − rs

2lnR1

R2)L

R22

+(H − rs

2lnR1

R2)2

R22

)

≈ rs2

(HL

R22

+H2

R22

)

(21)

Finally at BS2, the frequency of the b1 and a1 is thesame ω for the observer, but the travelling time by eachbeam is not the same, so a gravitational induced phaseshift appears between the two beams.

Supposing the phase of a1 is (k (L+ 2H)− ωτa) = 0,the gravitationally induced phase difference between a1and b1 is,

ψ = ω∆τ = ωrs2

(HL

R22

+H2

R22

) = ωg

c3(HL+H2) (22)

where g =rs

2R22

is the local gravitational acceleration at

coordinate R2.So in the MZ interferometer theory, replacing εG with

this specific gravitationally induced phase shift ψ inEq22, we can define the standard quantum limit (SQL)for the relative sensitivity in the estimate of the gravita-

tional acceleration∆g

g. According to

∆g

g=

√⟨∆X−2b

⟩d⟨X−b⟩

dψ

dψ

dgg

we find (∆g

g

)SQL

=1

2√Nsig

gω(H2 + LH)

c3

(23)

In this gravity measurement situation, we regard thephoton number Nsig acquiring the relativistic phase shiftas the photon number that counts in the SQL. The refer-ence beam a1 and local oscillator beams remain at coor-dinate R2, thus do not acquire any differential phase dueto the gravitational field. Hence we consider them “free”resources and only count the photons in the signal beamb1.

C. Squeezing and loss

To suppress the noise of the signal and improve thesensitivity of the interferometer, single-mode squeezed

vacuum [37] c1 = (Gc0 + gc†0)eiξ, as produced from thesqueezing source in the dashed rectangle in Fig.1, can beapplied to inject into the vacuum input port b0 in theMZ interferometer. Here G = cosh(r), g = sinh(r) andG2 − g2 = 1 and r is the squeezing parameter of single-mode squeezer S. c0 is the input vacuum for the squeezerS, as in Fig.1. ξ is the phase for adjusting the squeezedvacuum c1. In realistic experiments, losses from the in-ternal optical paths and detections are unavoidable. Sothe internal and external intensity losses of the interfer-ometer are analysed with beam splitter models here. Theamplitude transmittance for the loss model is tj , and lossrate is ηj , with t2j + η2j = 1. (j = 1 is for internal loss;j = 2 is for external loss.) The input-output relations ofthe interferometer are

a2 =t1t2((1 + (1− T )iεG)a0 −√T (1− T )iεGb0)

+ η1t2(√T V1a +

√1− T V1b) + η2V2a

b2 =t1t2(√T (1− T )iεGa0 − (TiεG + 1)b0)

+ η1t2(√

1− T V1a +√T V1b) + η2V2b

(24)

Here V1a, V1b correspond to the vacuum noise from in-ternal loss (V1a for the reference beam a1 and V1b forsignal b1), and V2a, V2b correspond to the vacuum noisefrom external loss (V2a for reference beam a2 and V2b forsignal b2), as in Fig.1. Internal losses for signal beam b1and reference beam a1 are set the same with the trans-mittance t1 to keep the two beams noise balanced, whichcan be realized by adjusting the reference beam loss rate.It is the same for the external losses.

Substituting squeezed vacuum c1 into the vacuum in-put port b0, and introducing the gravity phase ψ in Eq.22,the relative sensitivity of g with single-mode squeezed in-put and losses is,(

∆g

g

)sq

=

√t21t

22e−2r + η21t

22 + η22

t1t22√T√Nsig

gω(H2 + LH)

c3

(25)

5

The photon number of the signal beam is still taken tobe Nsig ≈ (1−T )N0, where we assumed (1−T )N0 � g2.When T ≈ 1, the interferometer has the best sensitivityfor g. From Eq.25, we can find the input squeezed state

can improve the relative sensitivity (∆g

g)sq, while the

losses are detrimental for the sensitivity. When there isno squeezing input r = 0 and losses are ignored t1 = t2 =1, we recover Eq.23.

D. Two-mode SU(1,1) interferometer

As suggested by some work [26–30], the two-modeSU(1,1) interferometer [34, 38, 39] made up of two para-metric amplification processes instead of two beam split-ters, may remedy the sensitivity reduction from intensitylosses with the advantage of two-mode squeezing. Sothe application of the SU(1,1) interferometer for gravityfield measurement is also analysed here. We introducea two-mode SU(1,1) interferometer as in Fig.2, which isthe analogue to the MZ interferometer in Fig.1, wherethe signal beam b1 goes along the upper path and theidler beam a1 and the pump beam goes along the lowerpath as reference. The AP1 and AP2 in Fig.2 repre-sent the two parametric amplification processes, which

work with the input-output relation: aout = Gain+gb†in,

bout = Gbin + ga†in. G is the gain of the amplitude, andG2 − g2 = 1. A linear phase of π is placed on the pumpbeam. A small gravitational phase shift, εG � π on thesignal beam b1 is again assumed.

The input-output relations of this scenario are

a2 =(G− + g1g2iεG)a0 + (g− + g2G1iεG)b†0

b2 =− (G− +G1G2iεG)b0 − (g− +G2g1iεG)a†0(26)

Here G− = G1G2−g1g2, g− = G2g1−g2G1. The best rel-ative sensitivity of gravity acceleration in the two-modeSU(1,1) interferometer in Fig.2 by single homodyne de-tection at output b2 is

(∆g

g

)2sq

=

√t21t

22

(g−)2 + (G−)2

G22

+ η21t22(1 +

g22G2

2

) +η22G2

2

2t1t2√Nsig − g21

gω(H2 + LH)

c3(27)

The first and second amplification process have gainsGi = cosh(ri) and gi = sinh(ri) with G2

i − g2i = 1(i = 1, 2). r1, r2 are the squeezing parameters in the firstand second amplifications in the SU(1,1) interferometer.Here the photon number of the gravitational inducedphase sensing signal beam is Nsig = g21(N0 + 1) ≈ g21N0,where we have assumed the photon number from the co-herent source is large, N0 � 1.

If the two gains of the two amplifications are set the

r

x

R2

L

H

Delaya0

b0

t1

t2

a2

b2

b1

signal

reference

t1t2

AP1

R1

a1

H

M M

pump

V1a

V1b

V2b

V2a

AP2π

2H+L

b1

FIG. 2. The optical interferometer is placed vertical to theground from local height R2 to local height R1, with a properlength H. The input states are coherent light a0 in one portand vacuum b0 in the other port, aligned with pump. Af-ter the first amplification process (AP1), signal beam b1 goesalong the upper path and the reference beam a1 goes alongthe lower path. And they combine at the second amplificationprocess (AP2). Finally, the outputs at the detection ports area2 and b2, For the losses, t1 is the internal amplitude transmit-tance of the signal b1 and reference a1, and t2 is the externalamplitude transmittance for signal output b2 and referenceoutput a2. Loss rates are η1 =

√1 − t21 and η2 =

√1 − t22.

V1a, V1b, V2a, V2b are induced vacuum noise during the lossychannel. A delay device is used to make the proper lengthsof the upper path and lower path the same 2H + L.

same, r1 = r2, we have a sensitivity with a simpler form

(∆g

g

)2sq

=

√t21t

22

G22

+ η21t22(1 +

g22G2

2

) +η22G2

2

2t1t2√Nsig

gω(H2 + LH)

c3

(28)

When the gains for both amplification processes areunity, G1 = G2 = 1, the sensitivity is the same as theMZ interferometer in Eq.25 with no squeezing. If lossesare also neglected, t1 = t2 = 1, the gravity accelerationsensitivity reaches the SQL in Eq.23.

To avoid losing information, joint homodyne detectionof Xj = Xa + Xb [28, 29, 35] using both outputs of theSU(1,1) interferometer is analysed. After calculation, thebest relative sensitivity of gravity acceleration from thejoint homodyne detection is(

∆g

g

)2jq

=

√2t21t

22e−2r1 + 2η21t

22 + 2e−2r2η22

2t1t2√Nsig

gω(H2 + LH)

c3

(29)

We compare this joint detection result in Eq.29 to thesingle detection result in SU(1,1) interferometer in Eq.27.The joint detection result shows the squeezing parame-ter 2e−2r1 for the first term in the numerator in Eq.29,while the single detection result Eq.27 shows a factorcosh−2(r2). These two functions cross at certain squeez-ing parameters, which will be shown in the next section.

6

We also compare this two-mode squeezing result Eq.29to the sensitivity with single-mode squeezing in the MZinterferometer as Eq.25. The first and second terms onthe numerator in Eq.29 are twice as large as the MZ in-terferometer, with the common squeezing benefit terme−2r1 on the first term. For the third term in the nu-merator in Eq.29, the loss term η2 in two-mode SU(1,1)interferometer joint detection is decreased by a factor of2e−2r2 than the MZ interferometer in Eq.25.

The best strategy between these three situations willbe analysed using the numerical comparisons in the nextsection.

III. PARAMETERS ANALYSIS

The relations between∆g

gand different parameters in

the optical interferometer are analysed here consideringparticular experimental conditions.

1010 1012 1014 1016 1018 10 20

10-5

10-4

0.001

0.010

0.100

1

10

Nsig

L=100mL=500mL=1000m

g/g

FIG. 3.∆g

gagainst Nsig. The proper length of upper hor-

izontal arm is set L = 100m (green), L = 500m (cyan),L = 1000m (blue). The height of the vertical arms of theinterferometer in Fig.1 is set H = 50m. Input coherent lighthas the frequency ω = 2.82×1014Hz and works in continuouslight mode. The reference gravity acceleration is g = 9.8m/s2,and the speed of light is c = 3 × 108m/s.

First we estimate the required signal photon numberNsig and upper horizontal arm length L in the MZ in-terferometer according to Eq.23 , to determine whichparameters are good to choose when operating at thestandard quantum limit (SQL). The result is shown asFig.3. The x-axis is the photon number of the signalbeam Nsig which experiences the gravity phase shift,

and the y-axis is∆g

g. The three lines with different

colors are corresponding to different lengths L of theupper horizontal arm, with fixed vertical arm heightof H = 50m. Input coherent light is chosen with thewavelength of λ = 1064nm, corresponding frequencyω = 2.82 × 1014Hz. We assume the interferometer isoperated in continuous light mode.

From the Fig.3, with a proper length of upper hori-

zontal arm L = 100m, the relative sensitivity of gravity

acceleration can reach∆g

g= 5 × 10−3 with signal pho-

ton number Nsig = 1018, corresponding to a 1s detectionwith a continuous wave of power 1W . When the length is

L = 1000m, the relative sensitivity can reach∆g

g= 10−4

under the same power conditions.

no loss0.0 0.5 1.0 1.5 2.0

2.×10- 5

5.×10- 5

1.×10-4

r

t1=0.9, t2=0.9t1=1, t2=0.9t1=0.9, t2=1g/

g

SQL

FIG. 4.∆g

gagainst the single-mode squeezing parameter

r of the squeezed input vacuum in the MZ interferometer.Different lines are sensitivities with different transmittancevalues t1, t2. Other parameters are H = 50m,L = 1000m,ω = 2.82 × 1014Hz, c = 3 × 108m/s, g = 9.8m/s2 and signalbeam photon number Nsig = 1018.

We now consider the inclusion of squeezing to improvethe sensitivity and the detrimental effect of losses in the

MZ interferometer. The relation between∆g

gand the

single-mode squeezing parameter r is analysed in Fig.4.Different lines correspond to the different levels of in-ternal and external losses. The green dashed line is the

sensitivity∆g

gwithout losses, t1 = t2 = 1. The red solid

line is with internal transmittance t1 = 0.9 and no ex-ternal loss t2 = 1, or only external loss t2 = 0.9 andno internal loss t1 = 1. According to Eq.25, the effects

on∆g

gfrom internal loss and external loss are the same

when only internal loss or only external loss exists. Theblue line is corresponding to when both internal loss andexternal loss exist, t1 = t2 = 0.9. Squeezing improvessensitivity but its effectiveness is reduced by the losses.The effective SQL with losses are given by the startingpoints at r = 0 in different lines.

Finally, the performance of the two-mode SU(1,1) in-terferometer with transmittance parameters t1, t2 andsqueezing parameters r1, r2 are analysed in Fig.5.

In Fig.5(a), the x-axis is the internal transmittancet1, and the external transmittance is set perfect t2 = 1.

For the whole range of t1, relative sensitivities∆g

gfrom

two-mode SU(1,1) interferometer (blue and green) arenot as good as results from the MZ interferometer withsingle-mode squeezing input (magenta). When t1 < 0.70,

7

, single detection, joint detection

, single mode squeezing0.0 0.2 0.4 0.6 0.8 1.0

5.×10-5

1.×10- 4

5.×10-4

0.001

0.005

t2

SQLr1=r2=1r1=r2=1r=1

g/g

(b)

0.0 0.2 0.4 0.6 0.8 1.0

5.×10-5

1.×10-4

5.×10-4

0.001

0.0050.010

g/g

t1

, single detectionr1=r2=1SQL

, single mode squeezingr=1, joint detectionr1=r2=1

(a)

0.860.70

0.920.81

FIG. 5. (a)∆g

gagainst the internal transmittance t1; (b)

∆g

gagainst the external transmittance t2. In either (a) or

(b), the red dashed line represents the effective SQL. Theblue and green lines are corresponding to the sensitivity ofSU(1,1) interferometer with single homodyne detection (blue)and with joint homodyne detection (green) with t1 = 1, andwith squeezing parameters r1 = r2 = 1; magenta solid lineis corresponding to the sensitivity of MZ interferometer withsingle-mode squeezing parameter r = 1. Other parameters areH = 50m,L = 1000m, ω = 2.82 × 1014Hz, c = 3 × 108m/s,g = 9.8m/s2 and signal beam photon number Nsig = 1018.

the∆g

gfrom taking single detection or joint detection in

the SU(1,1) interferometer are both not good enough tobreak the SQL. For the comparison between the two dif-ferent detection methods in SU(1,1) interferometer, whent1 > 0.85, joint detection result (green) is lower thansingle detection result (blue). When t1 < 0.85, singledetection is better than joint detection.

In Fig.5(b), the effect from external loss is analysedand shown. The x-axis is the external transmittancet2, and the internal transmittance is set t1 = 1. Asfor the whole range of 0 ≤ t2 ≤ 1, the results fromtwo-mode SU(1,1) interferometer (blue and green) andfrom MZ interferometer with single-mode squeezing (ma-genta) are all better than the effective SQL (dashed red),which shows that the internal loss is more detrimentalthan the external loss for the sensitivity, compared tothe Fig.5(a). When only external loss exists, the jointdetection of SU(1,1) interferometer (blue) shows bettersensitivity than the single detection (green). When tak-ing joint detection with t2 < 0.92 or taking single de-tection with lower than t2 < 0.81 in the SU(1,1) in-

terferometer, the sensitivity∆g

gin SU(1,1) interferome-

ter is better than the result in MZ interferometer withsingle-mode squeezing (magenta). But when external loss

t2 > 0.92, MZ interferometer with single-mode squeezingis the best.

In summary, when only internal loss exists, the SU(1,1)interferometer shows no advantage over the MZ interfer-ometer with single-mode squeezing input. Only when theexternal transmittance is t2 < 0.92, the SU(1,1) inter-ferometer has an advantage over the MZ interferometerwith single-mode squeezing input. In the realistic mea-surement, both internal and external losses exist, so thebest strategy depends on the conditions. It is currentlyquite possible to control the external loss within 10%, andthe detector efficiency has been reported as good as 98%[40, 41]. In that situation, the MZ interferometer withsingle-mode squeezing input and single output homodynedetection is easier to operate and has better sensitivity.

IV. CONCLUSION

We have discussed a system for precision measurementof the gravity acceleration constant g, based on the in-terferometry of optical fields interacting with the spacetime curvature of Earth. The output of the interferom-eter is sensitive to the phase shift of the signal beam,which contains the information about the gravity. Thussuch an optical interferometer can be used for gravityestimation.

The photon number of the signal beam that can becarried up, and experience the gravity phase shift is lim-ited by current technology. So we define the standardquantum limit (SQL) of gravity acceleration g based onthe photon number limit of the signal beam without any

loss or squeezing,

(∆g

g

)SQL

=1

2√Nsig

gω(H2 +HL)

c3

.

The resources operated on ground, such as the pump andreference beams, are regarded as “free resources”. Withsingle-mode squeezing input into the MZ interferometer,the relative sensitivity of g on Earth can be measuredwith a better sensitivity than SQL. With a squeezing pa-

rameter r = 1, the sensitivity∆g

gcan be improved by a

factor of1

eof magnitude. The effects from losses are also

analysed.Two-mode SU(1,1) interferometer is introduced to find

the best strategy for the experiment when losses are in-evitable. We show that with detection loss rate η2 > 8%,the two-mode SU(1,1) interferometer will show improve-ment for the sensitivity compared to MZ interferometerwith single-mode squeezing input. But for small externalloss or the internal loss dominating, two-mode SU(1,1)interferometer has no advantage, and MZ interferometerwith single-mode squeezing input is suggested as the firstchoice.

In principle, this system can be used for precision mea-surement of Earth’s gravity. One could imagine deploy-ing such a system between satellites and extending it to

8

the gravity of other planets or celestial bodies. Sensitiv-ity is improved by larger power of laser, shorter wave-length, larger size or longer probing time.

Comparing to the atomic system, in order to achievethe same level of sensitivity ∆g = 3 × 10−9g, the op-tical system would need for example Nsig = 7.1 × 1024

(Mega Watt power), L = 5km, r = 1, H = 50m for 1sdetection. On the other hand, observation of the gravita-tionally induced phase shift in quantum optics does seemwithin the reach of current technology with this type ofsystems.

ACKNOWLEDGMENTS

We acknowledge the support from the Australian Re-search Council Centre of Excellence for Quantum Com-putation and Communication Technology (Project No.CE170100012) and the China Scholarship Council pro-grams. We also would like to extend our gratitude to theuseful discussions with Magdalena Zych, Austin Lund,S.P. Kish and Marco Ho from the University of Queens-land; and to Daiqin Su from Xanadu located in Torontofor plenty of mindful communications about this work.

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