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Bragg spectroscopic interferometer and quantum measurement-induced correlations in atomic

Bose–Einstein condensates

View the table of contents for this issue, or go to the journal homepage for more

2012 New J. Phys. 14 073057

(http://iopscience.iop.org/1367-2630/14/7/073057)

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T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Bragg spectroscopic interferometer and quantum

measurement-induced correlations in atomic

Bose–Einstein condensates

M D Lee1,3, S Rist2,3and J Ruostekoski1

1School of Mathematics, University of Southampton, Southampton,

SO17 1BJ, UK

2CNR-SPIN, Corso Perrone 24, I-16152 Genova and NEST, Scuola Normale

Superiore, I-56126 Pisa, Italy

E-mail: mark.lee@soton.ac.uk

New Journal of Physics 14 (2012) 073057 (13pp)

Received 5 April 2012

Published 31 July 2012

Online at http://www.njp.org/

doi:10.1088/1367-2630/14/7/073057

Abstract.

of two spatially separated atomic Bose–Einstein condensates that was

experimentally realized by Saba et al (2005 Science 307 1945) by continuously

monitoring the relative phase evolution. Even though atoms in the light-

stimulated Bragg scattering interact with intense coherent laser beams, we show

that the phase is created by quantum measurement-induced backaction on the

homodyne photocurrent of the lasers, opening the possibilities for quantum-

enhanced interferometric schemes. We identify two regimes of phase evolution:

a running phase regime observed in the experiment of Saba et al, which is

sensitive to an energy offset and suitable for an interferometer, and a trapped

phase regime, which can be insensitive to the applied forces and detrimental to

interferometric applications.

We theoretically analyse the Bragg spectroscopic interferometer

3These authors contributed equally to this work.

New Journal of Physics 14 (2012) 073057

1367-2630/12/073057+13$33.00© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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Contents

1. Introduction

2. The model and the effective Hamiltonian

3. Continuous homodyne measurement

4. Numerical results

5. Concluding remarks

Acknowledgments

References

2

4

6

8

11

12

12

1. Introduction

Bragg spectroscopy has become an established spectroscopic tool in ultracold atom

experiments [1–6]. In typical setups an intersecting pair of low-intensity pulsed laser beams

is used to excite atoms to higher momentum states. The momentum kick experienced by the

atoms corresponds to the recoil of a photon upon light-stimulated scattering between the two

laser beams. As the spontaneous scattering for off-resonant lasers is negligible and the photons

are only exchanged between the directed coherent laser beams, the momentum transfer of the

atoms can be measured for specific values of the energy and the momentum. In particular, in a

spectroscopic analysis of the many-particle properties of ultracold atoms it is sufficient in the

scattering process to describe the light beams classically.

In the experiments by Saba et al [7] the relative phase coherence between two

Bose–Einstein condensates (BECs) was measured by Bragg scattering atoms between two

condensate fragments. Previous Bragg spectroscopy experiments based on time of flight had

concentrated on directly detecting the atoms that were transferred to higher momentum states

by the laser beams. In the experiment by Saba et al [7], however, the strength of the Bragg

scattering was measured by monitoring the variations of light intensity in the laser beams

by homodyne detection. Due to the correspondence between the light-stimulated scattering of

photons between the laser beams and the atoms scattered between two momentum states, the

intensity fluctuations are directly proportional to the number of atoms scattered between the

condensate fragments.

Saba et al [7] measured the light intensity variations of the Bragg beams, which revealed

relative phase coherence between the condensates even when the BECs were independently

produced and possessed no a priori phase information. By theoretically analysing a continuous

atom detection process, it has been previously shown that the backaction of quantum

measurement of the atomic correlations [8–15], and analogous photon correlations [16, 17],

can establish a relative phase between two BECs even when they have ‘never seen each

other’ before. It has also been suggested that phase-coherent states of condensates may

naturally emerge as robust state descriptions due to dissipative interaction with the environment

[18, 19]. With regard to the Bragg spectroscopic interferometer of [7], the question we raise is:

how is the phase coherence between the two BECs created, given that the condensates interact

with coherent laser beams that can usually be described classically?

Here we analyse a model of the experimental detection scheme [7], illustrated in figure 1,

and show that the phase coherence can be built up by continuously monitoring the photocurrent

obtained by a homodyne measurement that describes the intensity fluctuations of the laser

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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Figure 1. Our model of the Bragg interferometric measurement of the

relative phase between two distant BECs. The two BECs are described by

the macroscopic wavefunctions φb(r) and φc(r), and are illuminated by two

coherent laser beams. Bragg scattering imparts momentum to atoms from the

left condensate, transferring them to the state described by φk(r). After an

appropriate time the outcoupled atoms will overlap with the right condensate,

and the Bragg beams will drive Rabi oscillations between the two atomic clouds.

This establishes an optical weak link between the two BECs, and continuous

monitoring of the intensity fluctuations in the laser beams measures the phase

coherence between the BECs.

beams. We show the rapid establishment of a well-defined relative phase between two

independently produced BECs. We identify two distinct regimes of subsequent phase evolution:

a running phase and a trapped phase regime. In the running phase regime, the relative phase

grows linearly in proportion to the energy offset between the two condensate wells and could

be suitable for a weak force detection in interferometric applications [7]. In the trapped

phase regime, in the case of a very weak energy offset, the measurement process drives the

system close to a dark state where a destructive interference between different scattering paths

suppresses the intensity fluctuations of the lasers. In the trapped phase regime, the effect of the

energy offset on phase evolution is suppressed, potentially to the detriment of interferometric

applications.

Our analysis demonstrates how Bragg spectroscopy can be sensitive to subtle quantum

features of ultracold atom systems. Quantum measurement-induced backaction of photocurrent

detection on the relative phase coherence of BECs represents a spatially nonlocal entanglement

of the laser beams and the relative many-particle state of the atoms. Indeed, the location of

the photocurrent detection can be far away from the region of interaction between the coherent

laser beams and the atoms. Moreover, one Bragg pulse can be used to entangle the two spatially

isolated BECs. A second pulse may then be employed in optical readout of the subsequent

evolution dynamics of the measurement-established relative phase coherence between the

condensates. An energy offset between the two condensate wells between the subsequent pulses

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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would result in a detectable phase shift providing potential interferometric applications [7].

Here the phase is determined by a continuous quantum measurement process opening the

possibilities for quantum feedback and control methods, e.g. in the generation of sub-shot-noise

phase-squeezed states [20, 21]. Such states may be useful in quantum-enhanced metrology for

the realization of a high-precision quantum interferometer overcoming the standard quantum

limit of classical interferometers [22–25]. Probe field response was also measured recently

in the Bragg spectroscopy of condensate excitations in a heterodyne-based detection system,

which was able to reach the shot-noise limit [26]. Previous theoretical studies of the effects of

continuous monitoring on light scattered from BECs have considered photon counting [16, 17],

e.g. in the preparation of macroscopic superposition states [17], and dispersive phase-contrast

imaging [27, 28], e.g. in the suppression of heating [28].

This paper is organized as follows. In section 2, we give a short review of the experimental

setup of Saba et al [7] and the relevant results for this work. We then introduce our basic

theoretical model. In section 3, we derive a stochastic differential equation which describes

the evolution of the system under the continuous measurement of the scattered light intensity.

In section 4, we present our numerical results with a physical interpretation. Finally, some

concluding remarks are made in section 5.

2. The model and the effective Hamiltonian

An interferometric scheme between two spatially isolated BECs was experimentally realized

in [7] without the need for splitting or recombining the two condensate atom clouds. The

methodwasbasedonthestimulatedlightscatteringofasmallfractionoftheatoms,onlyweakly

perturbing the condensates and therefore representing an almost nondestructive measurement.

Two isolated BECs were prepared in the sites of an unbalanced double-well potential, and

illuminated by the same pair of Bragg beams. These beams outcoupled atoms from each

well, and the interference between such atoms provided a coupling between the BECs. When

outcoupled atoms from one condensate spatially overlapped the second, measurement of the

Braggbeamintensitywasshowntobesensitivetotherelativephase?betweenthecondensates.

In addition, the potential offset between the two wells gave rise to a difference in energies

δµ, which in turn led to a relative phase evolution ?(t) = ?(0)+δµt/¯ h. This was observed

as oscillations in the Bragg beam intensity of frequency ωosc= δµ/¯ h, demonstrating that

monitoring the Bragg beam intensity directly measured the dynamical evolution of the relative

phase between the macroscopic wavefunctions.

In the experiment, a single Bragg pulse established a random relative phase between the

two independently produced BECs. If two successive Bragg pulses were applied to the same

BEC pair, the relative phase measured by the second pulse was correlated with that detected

by the first pulse, indicating that the interaction of the first Bragg beam with the atoms had

projected the system into a state with a well-defined relative phase between the condensates.

The key to the method is the weak link established between the BECs by the Bragg laser

beams that couple out small atomic samples from the condensates [29]. The coherently driven

population dynamics between the BECs is influenced by the relative phase coherence [30, 31],

and Bragg scattering may be understood as an interference in momentum space [32]. The

specific advantage of the Bragg spectroscopic interference scheme [7] is the nondestructive

nature of the detection process, potentially constituting a major advance for interferometric

applications since it allows one to probe the evolution of the phase coherence in time by a

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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continuous measurement process [33–35]. It has also been argued that this setup can be viewed

as an analogue of homodyne detection for matter waves [36].

In order to analyse the continuous measurement process of Bragg spectroscopy, we

consider the system depicted in figure 1, which is analogous to the experimental setup of

Saba et al [7]. We assume that the two condensates are initially uncorrelated and that there

is no tunnelling between the two spatial regions. As in the experiment, an offset in the trapping

potential between the two condensates is accounted for by a difference in chemical potential,

δµ. The condensates are illuminated by two Bragg beams, which impart momentum, kicking

atoms out of the traps. The outcoupled atoms propagate from the left condensate to the right

and establish an optical weak link between the two macroscopic wavefunctions [29].

For simplicity, in the theoretical analysis we use a single-mode approximation for the

condensates and assume that all atoms in the left (right) condensate are in the state |b? (|c?). The

atoms in the left (right) condensate are then described by the second quantized field operators

ˆψL(r) = φb(r)ˆb (ˆψR(r) = φc(r)ˆ c), which fulfil the usual bosonic commutation relations. Here,ˆb

(ˆ c) annihilates an atom in the state |b? (|c?) and φb,c(r) obey the Gross–Pitaevskii equation [37].

Atoms from the BEC in the left well in the state |b? are transferred by the Bragg beams

to the momentum state k = k1−k2, where kj are the wavevectors of the Bragg beams. The

outcoupled atoms propagate with momentum k towards the right BEC in the state |c?. We

take the wavefunction of the outcoupled atoms φk(r) to be the momentum shifted original

wavefunction

φk(r) = φb(r−rL)eik·(r−rL),

where rL (rR) gives the position of the centre of the left (right) trap. We assume that the

momentum kick of the atoms is sufficiently strong, so that the essential characteristics of the

continuous quantum measurement process are not obscured by collisions with the remaining

trapped atoms, collisions among the outcoupled atoms, and the effect of the trapping potential.

We assume that enough time has passed such that the outcoupled atom cloud from the left

condensate completely overlaps the right condensate. We therefore neglect the time evolution

of the outcoupled cloud while flying from the left to the right trap. In our model, this evolution

leads to an additional phase factor which is inconsequential to our findings. We also take the

same functional form of the trapping potential for the atoms in the left and right condensates

such that φb(r−l) = φc(r) ≡ φ(r), where l = rR−rLis the distance vector between the two

potential minima. With these assumptions we find for the effective Hamiltonian

Heff= HA+ HAL+ HEM,

where

?

describes the Rabi oscillations between the outcoupled atoms and the atoms in the right

condensate due to the Bragg beams. Here ?jare the Rabi frequencies of the Bragg beams, ? is

the detuning from the excited state |e? which couples the two-photon Raman transition between

|bk? ↔ |c? and the operatorˆbkannihilates an outcoupled atom in the momentum shifted state

|bk? with wavefunction φk(r). Hamiltonian (2) is written in the reference frame of the Bragg

beams where we assume the two laser frequencies to be equal ω1≈ ω2= ωL. The term

HEM=¯ h

λ

(1)

(2)

HA=

δµ+¯ h?2

1

?

?

ˆ c†ˆ c+¯ h?2

2

?

ˆb†

kˆbk+¯ h?1?2

?

(ˆ c†ˆbk+ˆb†

kˆ c)

(3)

?

?λˆ a†

λˆ aλ

(4)

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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takes into account the electromagnetic (EM) vacuum energy, where we have used the standard

plane wave decomposition for the EM-field modes. Specifically, the positive frequency

component of the vacuum electric field amplitude reads

?

δˆE+(r,t) =

?

λ

¯ hωλ

2ε0Vˆ eλˆ aλ(t)eikλ·r.

(5)

Here λ labels a mode of the EM field at wavevector kλ, polarization ˆ eλ⊥ kλand frequency

ωλ= c|kλ|. The velocity of light is denoted by c, the quantization volume by V, the vacuum

permittivity is ε0and ?λ= ωλ−ωL. The operator ˆ aλannihilates a photon in mode λ. The

total electric field is the sum of the coherent Bragg laser fields E+

δˆE+(r). The coherent part is responsible for the driving terms in (3), while δˆE+(r) provides the

coupling of the vacuum modes with the atomic dipoles. We consider off-resonant scattering

where the scattering rates for sufficiently large condensates are proportional to the amplitudes

of the macroscopically occupied modes due to Bose enhancement, and we neglect scattering to

other motional states of the atoms. The coupling between the vacuum modes and the atoms is

then given by

?

where we have introduced the operator

in,j(r) and the vacuum fields

HAL=¯ h

λ

(ˆ a†

λˆBλ+ˆB†

λˆ aλ),

(6)

ˆBλ= [(Aλ

1)∗ˆ σ1+(Aλ

2)∗ˆ σ2],

(7)

with

ˆ σ1= ˆ c†ˆ e,

ˆ σ2=ˆb†

kˆ e,

(8)

and, after adiabatic elimination, the excited state annihilation operator can be written as

??1

We have defined

?

2ε0V

where the factor outside the integral is the coupling strength between the atomic dipoles and

the EM-field mode λ [38]. Here the matrix elements of the dipole moment operatorˆd for the

transition are denoted by d−

ˆ e =

?ˆ c+?2

?

ˆbk

?

.

(9)

Aλ

j=

¯ hωλ

?d−

j· ˆ eλ

??

dr|φ(r)|2e−i(kj−kλ)·r,

(10)

1= ?c|ˆd|e?,d−

2= ?bk|ˆd|e?.

3. Continuous homodyne measurement

We consider the condensate and the outcoupled atomic cloud together with the driving fields as

an open quantum system and eliminate the vacuum EM-field modes. The aim of our treatment

is to compute the evolution of the reduced system under continuous measurement of the light

intensity of the Bragg beams. The intensity of the beam j is given by

Ij= 2cε0?ˆE−

j(r,t)ˆE+

j(r,t)?.

(11)

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Here the total electric field amplitude of each Bragg beam is given by the sum of the coherent

driving laser field and the field δˆE+

ˆE+

Assuming that the amplitude of the scattered field is small compared to the applied laser field,

the measured intensity is approximately

j(r,t) due to scattering in the direction of the beam j

j(r,t) = E+

in,j(r)+δˆE+

j(r,t).

(12)

Ij? 2cε0(?E−

in,jE+

in,j?+?E−

in,jδˆE+

j?+?δˆE−

jE+

in,j?),

(13)

where the last two terms give rise to fluctuations in the intensity incident on the detector j.

We may now solve the intensity fluctuations by calculating the scattered field amplitude

from the effective system Hamiltonian. From the Heisenberg equation of motion for ˆ aλ, one

finds that

?t

with ?ˆ aλ(0)? = 0. Inserting (14) into (5) and defining qj= kLn−kj as the change of the

wavevector of light upon scattering with kL= ωL/c [39], we then obtain two contributions from

the vacuum field, one for each beam

ˆ aλ(t) = ˆ aλ(0)e−i?λt−i

0

dt?e−i?λ(t−t?)?(Aλ

1)∗ˆ σ1(t?)+(Aλ

2)∗ˆ σ2(t?)?,

(14)

δˆE+

j(r,t) ?k2

LeikLrD

4πε0rDn×(n×d−

j)ˆ σj(t)

?

dr?|φ(r?)|2eiqj·r?.

(15)

The spatial integral over the wavefunction φ(r?) enforces an approximate momentum

conservation, so that the photons are dominantly scattered into a cone centred at qj= 0 in the

direction of the laser beam j. In deriving (15) we made the expansion |r−r?| = rD−n·r?, with

n being the unit vector that points from the scattering region to the detector, rDis the distance

between the detector and a representative point at the origin of the scattering region. Due to

the normalization of the wavefunction we finally find for the scattered electric field in the two

outgoing beams

δˆE+

j(r,t) =k2

LeikLrD

4πε0rDn×(n×d−

j)ˆ σj(t).

(16)

The atomic operator associated with the spontaneous emission of a photon into beam j in (16)

is given by ˆ σj. The master equation which describes the evolution of the reduced density matrix

after elimination of the vacuum field modes then reads [40]

˙ˆ ρ(t) =i

j=1,2

Here γjis the rate of spontaneously scattered photons, and is related to the total spontaneously

scattered light intensity δI = 2cε0?δˆE−

1

¯ hkLc

where the angular integral is over the scattering cone of beam j. The operators associated with

the light field amplitude of the beam j read

ˆCj=√γj(αj+ ˆ σj),

¯ h[ˆ ρ,ˆHA]−

?

γj

2(ˆ σ†

jˆ σjˆ ρ + ˆ ρ ˆ σ†

jˆ σj−2ˆ σjˆ ρ ˆ σ†

j).

(17)

j(r,t)δˆE+

j(r,t)? via [17]

?

d?r2

DδIj= γj?ˆ σ†

jˆ σj?,

(18)

(19)

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where αjis proportional to the amplitude of the coherent laser beam with wavevector kj. The

intensity is then proportional to ?ˆC†

intensity ∝ γj|αj|2(corresponding to the first term in (13)) and the intensity fluctuations are

dominated by the terms γj(αj?ˆ σj?+c.c.) (corresponding to the second and the third term

in (13)). Extending the treatment of Wiseman and Milburn [41] to our setup, one finds that

the evolution of the system under the continuous monitoring of light intensity can be described

by the stochastic differential equation

¯ h

where

Xj=1

Here, dWjis a Wiener increment with zero mean ?dWj? = 0 and ?(dWj)2? = dt, which appears

as a result of the continuous measurement process. Keeping terms to lowest order in the

fluctuations, one finds an expression for the photocurrent in essence equivalent to (13)

iphot

j

(t) = γjα2

Here,

ξj(t) =dWj

dt

represents Gaussian white noise [41] and arises from the open nature of our quantum system.

jˆCj?. The leading contribution comes from the coherent

|ψ(t +dt)? =

1−i

ˆHAdt +

?

j

?γj

2ˆ σ†

jˆ σjdt +2γjXjdt + ˆ σj√γjdWj

?

|ψ(t)?,

(20)

2(ˆ σj+ ˆ σ†

j).

(21)

j+αj(2γj?Xj?+√γjξj(t)).

(22)

(23)

4. Numerical results

In order to study the effect of the homodyne photocurrent measurements on the system, we

numerically integrate (20) using the Milstein algorithm [42]. As the initial state in the numerical

simulations we take a pure number state in each condensate, with no well-defined phase

between them, and the incident Bragg laser beams are taken to be classical coherent states.

The relative phase between the condensates as a function of time may then be calculated as

?(t) = arg(?ˆ c†ˆb?). We define a measure for the strength of the phase coherence between the

condensates by the absolute value of the normalized phase coherence

g(t) =

|?ˆ c†ˆb?|

?ˆ c†ˆ c??ˆb†ˆb?

?

.

(24)

A value of g(t) close to 1 indicates a high degree of relative phase coherence, while condensates

with no relative phase information have g(t) ? 0.

In figure 2, we plot the time evolution of the coherence and the relative phase for two

different values of the detuning ?. No well-defined relative phase exists at early times, the

coherence starts at zero and ?(t) shows large random fluctuations with time. As the continuous

measurement proceeds the coherence builds rapidly, leading to a well-defined relative phase

with a stable value. Once established, we then see two different regimes of behaviour at

longer times. For large values of the detuning ? = 100γ1we see a running phase behaviour:

once well established with a value which is random for each individual run, the phase grows

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Figure 2. Coherence g(t) as a function of time t in units of t0= 2π¯ h/δµ. The

simulation is done for a total of N = 100 atoms which are initially distributed

equally between the states |bk? and |c?. Parameters were chosen to be ?1= ?2=

105/t0and√?1?2= 103/t0. The black solid line corresponds to ? = 100?1and

the red dashed line to ? = 10?1. In the inset we show the evolution of the phase

?(t) as a function of time t in units of t0.

linearly in time with a rate proportional to the difference in energies between the condensates

?(t) ∼ δµt/¯ h. From (22) we note that the measured photocurrent from the two Bragg beams

is essentially proportional to the quadrature ?Xj? after subtracting the background current, and

the corresponding time evolution of ?Xj? is shown in figure 3. In the running phase regime, the

quadrature exhibits well-defined oscillations with frequency ωosc= δµ/¯ h, and this corresponds

to the experimental measurements obtained by Saba et al [7]. Such oscillations thus give an

interferometric measurement of the relative phase evolution, sensitive to any accumulated phase

shift due to an energy offset between the distant condensates. An interferometer of this type

could be used, for example, to detect a weak force applied to one of the condensates.

Choosing a smaller value for the detuning ? = 10γ1, we find a very different long-time

behaviour. Once again the phase fluctuates as coherence is established, although this occurs on

a much faster timescale. This can be understood from the fact that the phase is established as

a result of the intensity fluctuations in the laser beams, which are enhanced by decreasing the

detuning ?. Unlike the running phase case, once firmly established the phase now locks to an

almost constant value near π. This trapped phase state has the two condensates almost entirely

out of phase, leading to destructive interference in the oscillations between the states |bk? and |c?

and resulting in a state analogous to a dark state. The corresponding quadrature ?X1? therefore

exhibits merely random fluctuations which would not be suited to an interferometric type

experiment. Note that although one would expect the amplitude of these random fluctuations

to be suppressed compared to the coherent oscillations of the running phase regime, and this

is indeed the case, the two different values of detunings used here do not allow such a direct

comparison in figure 3.

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Figure 3. Mean value of the quadrature ?X1? as a function of time t. The black

solid line corresponds to ? = 100γ1and the red dashed line to ? = 10γ1. The

other parameters are the same as in figure 3. The inset shows a magnified view

of the initial behaviour.

The two different regimes of behaviour resemble the ac-Josephson and self-trapping

behaviours seen in double-well condensates [43–45], although we emphasize that here coupling

occurs due to the nonlocal measurement process induced by the Bragg beams. The trapped

phase behaviour is more akin to a dark state however, due to the lack of any nonlinearity in

the Hamiltonian which is required for macroscopic self-trapping. The different regimes may be

understood if we assume that a well-defined phase and population can be associated with each

condensate, and neglect any processes other than those included in HA, leaving a two-mode

model similar to that considered in [43]. The trapped phase regime then occurs with a stable

relative phase difference of π when

δµ = 2¯ h?1?2

?

|z|

√1−z2,

(25)

where z = (Nk− Nc)/(Nk+ Nc) is the relative population difference, where Nk(c) is the

population in state |bk?(|c?). The trapped phase regime therefore requires either δµ/¯ h ∼

?1?2/? or a large population imbalance. This is in agreement with our results, where the

larger value of detuning has δµ/¯ h ? ?1?2/?, and the initial population balance is not extreme.

The trapped phase condition is then not satisfied and we observe a running phase behaviour akin

to the ac-Josephson effect. Note, however, that dissipation is vital for establishing the relative

phase in the first place, and has the effect of shifting the relative phase in the trapped regime

away from π in figure 2. The model (25) does not specify the role of spontaneous scattering

which determines the rate at which the system is driven towards the trapped phase state.

IntheexperimentbySabaetal[7],thesystemtypicallycontainedoftheorderof106atoms,

although the numbers outcoupled would be only a small fraction of this. Numerical simulations

in our basis for such large numbers are prohibitively slow, so here we have typically used a total

atom number of 100. Our results show no significant dependence on atom number however,

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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and we expect our results to give a good qualitative comparison with the physics exhibited in

the experiment. We have so far discussed our results in dimensionless units; in order to give a

specific example we may take γ1,2to be of the order of 2π ×10MHz, for instance. Our results

then correspond to the values δµ/¯ h ∼ 2π ×630Hz and√?1?2∼ 2π ×0.45MHz.

Parameters used in the experiment by Saba et al [7] were δµ/¯ h ∼ 2π ×1kHz, ? =

2π ×1GHz,√?1?2∼ 2π ×0.45MHz. This yields a ratio η ≡ 2¯ h?1?2/(?δµ) ∼ 0.4. In order

for the trapped phase regime to be observed the population imbalance would then be required

to satisfy z ≈ 0.92. In the experiment, the actual population in the momentum-shifted state

(|bk?) that overlapped the second condensate (|c?) was of the order of 2×104atoms during the

coupling. The corresponding population imbalance was in excess of 0.96, and hence did not

satisfy condition (25) for the trapped phase behaviour. The observed running phase behaviour

in the experiment is therefore consistent with our model.

5. Concluding remarks

Bragg spectroscopy was used in [7] to measure the relative phase between two initially

uncorrelated BECs. By studying a simplified model containing the essential ingredients of

the experiment, we have demonstrated how the homodyne measurement process builds up

a coherent relative phase between the two condensates. This quantum measurement-induced

backaction entangles the two macroscopic many-body states even though the measurement

location can be far away from the region of interactions.

Following the establishment of a coherent phase, we have identified two distinct behaviours

under continual subsequent measurement. With a larger atom–laser detuning ? or a large initial

energy imbalance δµ, we reproduce the experimental findings of [7], with the measured photon

flux exhibiting oscillations at a frequency corresponding to the energy offset of the separated

condensates. In this case, once the coherence and a random-valued phase are established it

evolves linearly in time ?(t) ∼ δµt/¯ h. Measurable oscillations in the laser beam intensity

mean that this state has applications in quantum-enhanced interferometry, and the measurement

backaction could potentially be used further to implement feedback mechanisms [21]. By

choosing a smaller atom–laser detuning, we found instead that the system stabilized to a trapped

phase state with the condensate relative phase fixed at almost π, while the scattered light

intensity showed only random fluctuations. A semiclassical model can qualitatively describe the

difference between these two regimes, with the trapped phase behaviour occurring when (25)

was satisfied.

When atoms from two initially uncorrelated condensates overlap, we have shown that

Bragg coupling and continuous homodyne measurement can rapidly establish a well-defined

relative phase. A closely related experiment [46] has been performed using ultra-slow light

pulses, in which optical information was coherently transported between two spatially separated

condensates by a travelling matter wave. An ultra-slow light pulse was stopped in the first

condensate, creating a dark-state superposition between two atomic internal states. Upon

stopping the pulse, one of the internal states received a momentum kick, and outcoupled atoms

in this state passed through a second distant condensate. By illuminating the second condensate

with a coupling laser it was possible to revive the initial light pulse even when the BECs were

independently produced. In the case when the condensates had been prepared separately, the

rapid establishment of a coherent phase in a manner similar to that described in this paper

explains the recovery of the light pulse.

New Journal of Physics 14 (2012) 073057 (http://www.njp.org/)

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Acknowledgments

The authors acknowledge G Morigi and S Pugnetti for stimulating discussions and helpful

comments. This work was supported by the European Commission (EMALI) and the

Leverhulme Trust. The research leading to these results has received funding from the European

Union FP7/2007–2013 under grant agreement N. 234970-NANOCTM.

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