Time-Reversal and Super-Resolving Phase Measurements

doi:10.1103/PHYSREVLETT.98.223601

K. J. Resch,

1,3

K. L. Pregnell,

1,2,4

R. Prevedel,

1,5

A. Gilchrist,

1,2

G. J. Pryde,

1,2,6

J. L. O’Brien,

1,2,7

and A. G. White

1,2

1

Department of Physics, University of Queensland, Brisbane QLD 4072, Australia

2

Centre for Quantum Computer Technology, University of Queensland, Brisbane QLD 4072, Australia

3

Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1 Canada

4

Institute for Mathematical Sciences, Imperial College, London, SW7 2PG Unied Kingdom

5

Institute for Experimental Physics, University of Vienna, Vienna, A-1090 Austria

6

Centre for Quantum Dynamics & Centre for Quantum Computer Technology, School of Biomolecular and Physical Sciences,

Grifﬁth University, Nathan, Brisbane, QLD 4111 Australia

7

H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol,

Bristol BS8 1UB United Kingdom

(Received 7 August 2006; published 31 May 2007; corrected 2 August 2007)

We demonstrate phase super-resolution in the absence of entangled states. The key insight is to use the

inherent time-reversal symmetry of quantum mechanics: our theory shows that it is possible to measure,as

opposed to prepare, entangled states. Our approach is robust, requiring only photons that exhibit classical

interference: we experimentally demonstrate high-visibility phase super-resolution with three, four, and

six photons using a standard laser and photon counters. Our six-photon experiment demonstrates the best

phase super-resolution yet reported with high visibility and resolution.

DOI: 10.1103/PhysRevLett.98.223601 PACS numbers: 42.50.Dv, 03.67.a, 42.50.St

Common wisdom holds that entangled states are a nec-

essary resource for many protocols in quantum informa-

tion. An example is quantum metrology, which promises

superprecise measurement, surpassing that possible with

classical states of light and matter [1,2]. In the last 20 years

quantum metrology schemes have been proposed for im-

proved optical [3–8] and matter-wave [9] interferometry,

atomic spectroscopy [10], and lithography [11–13]. The

entangled states in these schemes give rise to phase super-

resolution, where the interference oscillation occurs over a

phase N-times smaller than one cycle of classical light

[14,15] and phase supersensitivity, a reduction of phase

uncertainty.

Many quantum metrology schemes are based on path-

entangled number states, e.g., the NOON-state [1], a two-

mode state with either N particles in one mode and 0 in the

other or vice-versa, jN0ij0Ni=



2

p

. A deterministic

optical source of path-entangled states is yet to be realized,

requiring optical nonlinearities many orders of magnitude

larger than those currently possible. However, entangled

states can be made nondeterministically using single-

photon sources, linear optics, and photon-resolving detec-

tors [16]: leading to a ﬂurry of proposals to generate path-

entangled states [17–21]. While phase super-resolution

with two photons has been demonstrated often since

1990 [22–25], phase super-resolution was experimentally

demonstrated for 3-photon [14] and 4-photon [15] states

only recently. As efﬁcient photon sources and photon-

number resolving detectors do not yet exist, all demonstra-

tions to date necessarily used multiphoton coincidence

postselection [26]. In this Letter we introduce a time-

reversal technique that eliminates the need for exotic

sources and detectors, achieving high-visibility phase

super-resolution with a standard laser and photon de-

tectors.

Figure 1(a) depicts a method for probabilistically gen-

erating NOON states via linear optics and postselection.

Single-photon states are prepared in each of the N input

modes, j



i

ij11...1i

12...N

, of a linear optical multiport

interferometer, U

multi

[27]. With probability 

p

, no pho-

tons are found in modes 3 to N, heralding the NOON state

in modes 1 and 2, jN0i

12

j0Ni

12

=



2

p

. A relative phase

shift, , between modes 1 and 2 introduces an N shift

between the terms in the state–phase super-resolution.

Maximum fringe visibility will be achieved when the

system is measured in a state h

N

j which has equal over-

lap, 

N

jh

N

jN0ij

2

jh

N

j0Nij

2

, with both compo-

nents of the NOON state. The probability of detecting a

ﬁnal state h

f

jh

N

j

12

h0...0j

3...N

after propagating

(iii

(iv

FIG. 1 (color online). Nondeterministic (a) preparation and

(b) measurement of NOON-states for phase super-resolution,

as described in the text. (i) Photon counting after a 50% beam

splitter to measure hNj

1

& h0j

2



N

 1=2

N

. (ii) Coherent-state

detection, hj

1

& hj

2

, via a 50% beam splitter and two homo-

dyne detectors to measure amplitude and phase 

ii

N



i

N

=



2N

p

.

(iii) and (iv) are the corresponding time-reversed processes.

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through the multiport and phase shifter U



is P 

jh

f

jU



U

multi

j

i

ij

2

 

p



N

1  cosN. This probabil-

ity exhibits phase super-resolution since the fringes com-

plete N oscillations over a single cycle of 2.

Probabilities in quantum mechanics are invariant under

time reversal [28–30], i.e., if we swap the input and mea-

sured states and suitably time reverse the operation of the

multiport, as shown in Fig. 1(b), the probability is un-

changed, P jh

i

jU

y

multi

U

y



j

f

ij

2



p



N

1 cosN.

In the time-reversed picture, the interferometer no longer

plays the role of probabilistic NOON-state generator, but

rather constitutes a probabilistic NOON-state detector:

since the probability, P, is invariant under time reversal,

phase super-resolution is also invariant. Experimentally,

detecting NOON states is much easier than creating them:

time reversing turns the difﬁcult generation of N single

photons into straightforward detection of N photons in

coincidence, and turns the problematic detection of the

vacuum into vacuum inputs which are automatically avail-

able with perfect ﬁdelity. Successful detection of a NOON

state is signaled by the coincident detection of a single pho-

ton at each output detector; this is the time reverse of the

coincident creation of a single photon at each input. This

time-reversal technique is a simple example of a more gen-

eral measurement technique introduced by Pregnell and

Pegg [30,31].

The theory implicitly assumes that all of the photons

have the same polarization, spectral, and transverse spa-

tial mode properties; i.e., they are indistinguishable. A sig-

niﬁcant advantage of our approach is that it is robust: phase

super-resolution can occur even when the photons are dis-

tinguishable. Creating NOON states relies on nonclassical

interference [32] (interference between multiphoton am-

plitudes) which is unaffected by the degree of distinguish-

ability between the photons or photon arrival statistics. As

we will explain in more detail, phase super-resolution can

manifest through multiple classical interferences.

Experimentally, there is a trade-off between temporal

distinguishability and counting rate: photons become dis-

tinguishable as the coincidence-window time is increased

above the input light coherence time, but this increases the

counting rate. We run in the high counting rate limit to

achieve the best statistics, limited only by saturation effects

in our coincidence-counting electronics.

In our experiments the two bright inputs to the multiport,

modes 1 and 2, are the vertical and horizontal polarization

modes of the one spatial mode from a laser. We use an

attenuated He:Ne laser (Uniphase 1135P) and set the po-

larization with a half-wave plate (HWP) followed by a

quarter-wave plate (QWP) at an angle of 45



. Changing

the angle of the HWP by =4 changes the relative phase

between the modes by , while ensuring the vertical and

horizontal modes are the same amplitude. In classical

interferometry, this yields one oscillation for 0 <<2.

Multiports can be symmetric (every input mode is con-

verted into an equal superposition of N output modes [19])

or asymmetric (not every input satisﬁes this condition

[18]). Scaling up symmetric multiports beyond N  2

can be done either with a polynomial number of nested

standard interferometers [27], which would be arduous to

phase lock, or a single N  N fused ﬁber, except that it is

not known how to control the large set of internal phases

[19]. Fortunately, symmetric multiports are not required

for phase super-resolution: an asymmetric multiport sufﬁ-

ces for even N. Figure 2 shows our symmetric N  3, and

asymmetric N  4, 6 multiports (the N  4 multiport was

independently proposed in [33]): all designs are passively

stable and do not require active phase locking.

In Fig. 2(c) the output modes are sent to three pinhole

photon-counting detectors, D1–D3, where the small aper-

ture is a single-mode ﬁber without a coupling lens; in

Fig. 2(d) each output mode is ﬁrst passed through a polar-

izing beam splitter and then detected. The singles rate is

the number of photons per second detected by an individ-

ual detector: for N  3 the maximum was 5  10

4

Hz; for

N  4, 6 the maximum singles rate was 1:3  10

5

Hz. The

N singles rates are recorded individually. For N  3, the

N-fold coincidence rate is measured using two Time-to-

Amplitude Converter/Single Channel Analyzer (TAC/

SCA) each with a 1:5 s coincidence window; for

N 4, 6 coincidence counting was performed using up to

FIG. 2 (color online). Ideal multiports, U

multi

: (a) symmetric

3  3 and (b) asymmetric 4  4 (6  6), constructed from 2 2

beam splitters with reﬂectivities as shown [polarizing beamsplit-

ter (PBS)]. Laser is input to modes 1, 2, with no light to modes

3–6. In (a) the internal phase, , ensures each input transforms

to an equal superposition of the three outputs; in (b) polarization

rotations U

1

to U

3

set the phases of the singles fringes.

(c),(d) Experimental realizations of (a),(b). In (c) the indicated

HWP’s are set to 22.5



to form 1=2 beam splitters with the beam

displacers, the third is set to 17.6



so that 2=3 of the light

intensity takes the upper path; the angle of the tilted HWP sets

, its optic axis is at 0



. In (d), the beam splitter for modes 3, 4 is

a pellicle; for modes 5, 6 it is a microscope slide set at a small

angle of incidence, to avoid polarization effects. Reﬂectivities

are not ideal, rates are equalized with lossy coupling. U

1

to U

3

,

are realized using wave plates: the orientation and tilt of each

wave plate was adjusted so that one detector reaches an inter-

ference minimum every 22.5



for N  4; every 15



for N  6.

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3 TAC/SCAs and a quad logic unit. For N  4 (6) all pulse

length inputs to the quad were 1:5 s (5 s) as were the

coincidence windows on the TAC/SCA. In all cases, due to

a restricted number of recording channels, the singles were

measured immediately after a coincidence run. To avoid

saturation in the coincidence electronics, the mean number

of photons per coincidence window must be  1: for N 

3, 4, and 6 it was up to 0.07, 0.15, and 0.48.

Figure 3 shows the coincidence and singles rates for the

N  3 symmetric, and the N  4, 6 asymmetric experi-

ments of Figs. 2(c) and 2(d). Figures 3(a)–3(c) show the

three-, four-, and sixfold coincidence rates as a function of

the phase, , with three, four, and six distinct oscillations

within a single phase cycle. This is in contrast to the fringes

observed in the singles rates, Figs. 3(d)– 3(f ), which un-

dergo only a single oscillation over the same range. This is

the experimental signature of phase super-resolution. We

emphasize that this was achieved without production of a

path-entangled state, which would have had the signature

of ﬂat singles rates over an optical cycle [15,25].

As discussed above, time-reversed phase super-

resolution does not rely on nonclassical interference: the

coincidence rate is determined entirely by the product of

the singles rates. Consider an N  N multiport set up so

that the detection probability in the k

th

output mode is P

k

/

1 cos  2k=N  ’, where ’ is a constant phase

offset. The N-fold coincidence probability is then simply

the product of the single-mode probabilities, i.e., P

11...1

/

1 cosN  N; ’, which clearly exhibits N oscil-

lations per cycle, where  is the offset. Applying this to

Figs. 3(a)–3(c), we, respectively, ﬁt a product of 3, 4, and 6

sinusoidal fringes, s

i

 c

i

v

i

sin  

i

c

i

, where v

i

is

the visibility, and c

i

and 

i

are amplitude and phase offsets,

of the ith fringe. The resulting ﬁts in Figs. 3(a)–3(c) are

very good, with reduced 

2

of 1.6, 6, and 1.7, respectively.

(The high value in the N  4 case is most likely due to

observed amplitude instability of the D3 signal during the

course of the coincidence measurement.) The coincidence

fringes for all three experiments differ from a pure sinusoid

due to small variations in the underlying singles rates, and

become more pronounced for larger values of N. The solid

lines in Figs. 3(d)–3(f) are the individual sinusoidal

fringes, s

i

, scaled by a constant factor that matches the

amplitude to the data but does not alter the visibility or

phase of each fringe. Again the agreement is very good.

The N  6 case matches the largest phase super-resolution

reported to date, obtained in an ion-trap system [34], but

has signiﬁcantly better visibility.

Our results clearly show that path-entangled states are

not required for phase super-resolution [35]. Previous op-

tical demonstrations used nonclassical light sources, which

are notoriously dim, limiting the threefold and fourfold

coincidence rates to 5 Hz [14] and 0.1 Hz [15], respec-

tively. We signiﬁcantly improve on this, achieving phase

super-resolution with a six-photon coincidence rate of

about 2.7 Hz (cf. 0.012 Hz in [36]); furthermore, owing

to the high-visibility singles and extremely stable construc-

tion of the multiports, our fringe patterns all exhibited high

visibility. Fitting a single sinusoid, and without any back-

ground subtraction, the fringe visibilities for the N  3,4,

and 6 cases are respectively 81  3%, 76  2%, and

90  2%—well exceeding previously reported raw visi-

bilities of 42  3% for N  3 [14] and 61% for N  4

[15]. An alternative technique for realizing phase super-

resolution sums multiple occurrences of a fringe pattern

narrowed by nonlinear detection, either spatial [37]or

temporal [38]. This suffers from exceedingly low visibil-

ity: when the number of exposures equals the number

of fringes, V  2N!

2

=2N!, for N  6 this predicts

V  0:2%.

Phase supersensitivity occurs when there is a reduction

of the phase uncertainty as compared to that possible with

classical resources. Unlike phase super-resolution, phase

supersensitivity cannot be determined solely from the

fringe pattern: careful accounting is needed to determine

the resource consumption required to achieve the measured

signal. For small variations in phase around a given value,

FIG. 3 (color online). (a)–(c) N-fold coincidence rates as a

function of phase, , respectively, exhibiting 3, 4, and 6 distinct

oscillations within a single phase cycle. The main source of

uncertainty is Poissonian statistics: error bars represent the

square root of the count rate. The solid line is a ﬁt to a product

of N sinusoidal fringes, as explained in the text. (d)–(f) Corre-

sponding singles rates, each exhibiting only one oscillation per

phase cycle. Error bars are contained within the data points; solid

lines are the individual sinusoidal fringes obtained from (a)–(c).

In (d) D1 to D3 are, respectively, indicated by black, blue, and

red; in (e)–(f) D1 to D6 are indicated by black, gray, blue, cyan,

red, and pink. Ideally, the phase differences between adjacent

fringes is 2=N, we ﬁnd: N  3, {122



,119



}; N  4,

{92



,90



,90



}; and N  6, {55



,66



,56



,62



,56



}.

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the phase uncertainty is   A=j

dhAi

d

j, where A and A

are an observable and its associated uncertainty [3]. All

other things being equal, the slope in the denominator is

increased by phase super-resolution, reducing . Phase

supersensitivity is achieved when  is less than the clas-

sical limit, 

class

 1=



N

tot

p





=N

p

, where  allows

for nonideal efﬁciency in using N

tot

resources to estimate

. Phase super-resolution produces normalized fringes

of the form 1  V cosN=2, where V is the fringe visi-

bility, and the slope is dhAi=d 

1

2

NV sinN. Beating

the classical limit requires V

2

> 4A

2

=Nsin

2

N.

Consider A to be a projector with measurement outcomes

bounded by 0 and 1, the worst case is A  1=2, at the

point of minimum phase uncertainty, the above reduces to,

V

2

N>1: (1)

By this criterion, although several experiments have dem-

onstrated phase super-resolution, there has been no unam-

biguous demonstration of phase supersensitivity.

The best known preparation efﬁciency in nondetermin-

istic linear optical schemes is   2N!=N

N

[17–19]. All

known single photonic schemes, including the one pre-

sented here, have an exponentially decreasing count rate

with increasing N: in the ideal limit, Eq. (1)gives

2N!=N

N1

> 1, which is true only for N  2, 3. Using

single-photon sources and linear optics, phase supersensi-

tivity cannot be achieved in any described nondeterminis-

tic linear optical scheme for N  4 [39]. This argument is

based on comparing the phase estimation of a given NOON

scheme versus the straightforward classical scheme that

consumes the same amount of energy. Instead one might

consider only counting the resources which actually pass

through the phase shifter; this is clearly a less stringent

deﬁnition forphase super-resolution.This alternative makes

arguable sense when the goal is not to consume as little

energy as possible, but rather to subject the phase shifter

(such as a biological sample) to as little light as possible.

Under this deﬁnition, phase supersensitivity can, in prin-

ciple, be achieved for all N since in principle time-forward

schemes can be heralded with perfect efﬁciency [17–19].

Phase super-resolution was recently observed for 4, 5,

and 6 ions with respective visibilities of 69:8  0:3%,

52:7  0:3%, and 41:9  0:4% [34]. A cannot be deter-

mined from the published data, but in the worst case,

Eq. (1) shows that phase supersensitivity was achieved if

the overall efﬁciencies were, respectively, 51:3 0:4%,

72:0  0:8%, and 95  2%.

We have used a time-reversal analysis to show that it is

not necessary to produce path-entangled states to achieve

phase super-resolution, nor to have nonclassical interfer-

ence. We derive the necessary conditions to claim phase

supersensitivity from phase super-resolution. Using a stan-

dard laser we obtain high-visibility and contrast phase

super-resolution of up to 6 oscillations per cycle in a six-

photon experiment: equivalent to using 105.5 nm in a

standard interferomentric setup. Inverting the roles of state

production and measurement is an application of a more

general time-reversal analysis technique [30,31]: given the

dramatic improvement demonstrated here, it remains an

interesting open question as to which other quantum tech-

nologies will beneﬁt from this technique.

This work was supported by the Australian Research

Council and the DTO-funded U.S. ARO Contract

No. W911NF-05-0397. We thank T. Ralph, G. Milburn,

and H. Wiseman, D. Pegg, and P. Meredith for discussions.

After completion of this work, proposal [33] was demon-

strated experimentally for

N  4 [40].

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j3;0ij0;3i=



2

p

[14]orj2;2;0;0ij0;0;2;2i=



2

p

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[39] The three-photon phase super-resolution experiment [14]

did not exhibit phase supersensitivity as the reported

visibility of 42 3% is below V

thr

 1=



3

p

 57:7%.

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PRL 98, 223601 (2007)

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Time-Reversal and Super-Resolving Phase Measurements

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