Measurement induced dynamics and stabilization of spinor condensate domain walls
Hilary M. Hurst and I. B. Spielman

arXiv:1809.08257v2 [cond-mat.quant-gas] 26 Apr 2019

Joint Quantum Institute, National Institute of Standards and Technology,
and University of Maryland, Gaithersburg, Maryland, 20899, USA
Weakly measuring many-body systems and allowing for feedback in real-time can simultaneously
create and measure new phenomena in quantum systems. We theoretically study the dynamics ofa continuously measured two-component Bose-Einstein condensate (BEC) potentially containing adomain wall, and focus on the trade-off between usable information obtained from measurement
and quantum backaction. Each weakly measured system yields a measurement record from which
we extract real-time dynamics of the domain wall. We show that quantum backaction due tomeasurement causes two primary effects: domain wall diffusion and overall heating. The system
dynamics and signal-to-noise ratio depend on the choice of measurement observable. We propose a
feedback protocol to dynamically create a stable domain wall in the regime where domain walls are
unstable, giving a prototype example of Hamiltonian engineering using measurement and feedback.

I.

INTRODUCTION

Understanding system-reservoir dynamics in many
body physics is a new frontier. An external bath canbe thought of as a ‘measurement reservoir’ from which
the environment extracts information about the system [1, 2]. From this perspective, minimally destructive (i.e. backaction-limited) measurements constitutea controlled reservoir that also provides a time-resolved
but noisy record of system evolution [3–6]. Weak measurement has long been implemented in quantum-opticalsystems to monitor and control nearly-pure quantum
states [2, 7], or in spin ensembles to create squeezed
states [8–10]. Extending this understanding to interacting many-body systems opens the door to measurement and quantum control of new, otherwise inaccessible
strongly-correlated matter.
We theoretically investigate weakly measured spinor
Bose-Einstein condensates (BECs), an experimentally
accessible system for which closed system dynamics arewell known [11]. We explore measurement protocols sensitive to domain walls in two-component BECs, where
the resulting measurement record tracks the domain wall
over time. Furthermore, we show that classical feedback
based upon the measurement record can create and stabilize domain walls. This process of ‘stochastic stabilization’ via feedback from a noisy environment occurs inmany other contexts, such as cell differentiation in biology whereby environmental noise can stabilize specificcell characteristics [12, 13].
Spinor condensates are predicted to host exotic spin
texture defects such as skyrmions and non-abelian vortices [11, 14–19]. These defects interact with local exictations and undergo diffusion; in real systems the excitations further destabilize many exotic spin textures [20–23]. Stabilizing non-abelian excitations using current
techniques has proven difficult, but might be possibleusing weak measurement and feedback, similar to our
proposed approach for stabilizing a domain wall.
Domain walls in two-component BECs provide a test
platform to understand the effects of repeated weak mea-

surement on the stability and dynamics of topological defects. By combining quantum trajectory techniques (foropen-system physics [24, 25]) with Gross-Pitaevskii simulations (for closed system dynamics [26, 27]), we study
the interplay of measurement, coherent evolution, andclassical feedback. We propose two measurement protocols sensitive to the domain wall position and find thatthe choice of measurement observable affects both theheating rate and the dissipative dynamics of the domain
wall.
II.
A.

MODEL

Measurement

We model spin-resolved dispersive imaging of a quasione-dimensional (1D) multicomponent condensate along
ex which interacts with a brief pulse of far detuned laser
light of wavelength λ and duration δt traveling alongez [28, 29]. Here, the condensate is the system, and thelight pulse is the ‘environment’, which is then subject tostrong quantum measurement. We describe the opticalPfield by the spatial mode basis n χ∗n (z)â†nj where â†nj
describes the creation of a photon at xj (along the longaxis of the 1D BEC) in spatial mode n and χ∗n (z) is a normalized mode function (along the direction of the probe’s
propagation). We model the incoming probe beam as acoherent state with amplitude |α| and phase φ = π/2 ina single spatial mode χ0 (z) = (cδt)−1/2 , where c is the
speed of light.
Atoms interact locally with the light via an interaction Hamiltonian described by a spin-dependent ac Stark
shift [30],
SR
Ĥrj
=

~γ
Ŝrj ⊗ n̂j ,
cδt

(1)

where the reservoir operator n̂j = â†j âj counts thephoton number at xj . The system operators Ŝrj =b̂†σj [τ · r]σσ0 b̂σ0 j measure the spin in the direction r,

2where m(xj ) is a vector describing quantum projection
noise with momentum-space Gaussian statistics m̃k = 0
and m̃k m̃k0 = δkk0 Θ (|k| − kc ) /2, where m̃k denotes theFourier transform of m(xj ), Θ is the Heaviside function,and kc = 2π/λ denotes a momentum cutoff due to finite
resolution. The momentum cutoff is implemented to account for the fact that the environment can only resolve
information within a finite length scale λ.
A measurement with outcome Mr (xj ) transforms thesystem wavefunction to |Ψ|m i = K̂r|m |Ψi where |Ψi isthe system state before measurement andn [arb. units]



↑

↓

Mz [arb. units]K̂r|m ≈ 1 +X
jϕmj δ Ŝrj −2
ϕ2 kc 
δ Ŝrj4 kM(3)is a Kraus operator corresponding to a global measurement of Ŝr , where kM = π/∆x is the maximum momentum in the simulation for grid spacing ∆x and δ Ŝrj =Ŝrj − hŜrj i.

M⊥ [arb. units]



B.

-20

-10
x

0



h̄/ma ωt



10

20

FIG. 1. (a) Computed ground state system with a single domain wall and schematic illustration of phase contrast imaging
layout. The system is weakly coupled to an array of homodyne detectors, where l.o. indicates a strong local oscillator.
The BEC is phase separated into spin up (red/left) and spin
down (blue/right); the black line indicates total density. (bc) Measurement outcome M of a single weak measurementwith strength ϕ = 0.1 of (b) Mz and (c) M⊥ (defined intext).

where b̂†σj describes the creation of an atom of spinσ ∈ {↑, ↓} at xj and τ = (τx , τy , τz ) is the vector of Paulimatrices. The system-reservoir interaction strength γ isset by the atomic transition strength and the detuningfrom resonance.
Just prior to measurement, the system and reservoir
mode evolve together for theh pulse time δt undertheiPinteraction unitary Ûr = exp −iϕ j Ŝrj ⊗ Q̂j , which

is a local displacement operator for √the X̂j quadratureof the optical field at xj whereϕ=hi2γ|α|/c is a small

dimensionless parameter and X̂j , Q̂j 0 = iδjj 0 [? ]. The
outcome of a single measurement for the full detector
array isMr (xj ) = hŜrj i +m(xj ),
ϕ(2)

System Dynamics

We describe the condensate in the mean-field approximation by a complex order parameter Ψj = (ψ↑j , ψ↓j )Twhere ψσj is the coherent state amplitude of each spin (orpseudospin) component σ ∈ {↑, ↓} at xj . The closed system evolves under the Gross-Pitaevskii equation (GPE)hii~∂t Ψj = Ĥ0 + u0 nj Ψj + u2 Szj τ z Ψj ,(4)

where Ĥ0 = p̂2 /2ma + ma ωt2 x2j /2 is the single particle Hamiltonian for atoms of mass ma in a harmonic
trap with frequency ωt , nj = |ψ↑j |2 + |ψ↓j |2 is theatom number at site j, and Szj = |ψ↑j |2 − |ψ↓j |2 is theatom number difference (magnetization) at site j. We
work in p
units defined by the trap with t → t/ωt andxj → xj ~/ma ωt , and the wavefunction
P is normalized
to the total number of atoms, N =j nj . The spinindependent and spin-dependent interaction strengths
u0 = 2π~2 (a+a↑↓ )/ma ∆x and u2 = 2π~2 (a−a↑↓ )/ma ∆xderive from the 1D intraspin and interspin scattering
lengths a and a↑↓ [31, 32]. We fix the total atom number
to be N = 104 , and use u0 ∆x = 0.1 and u2 = ±0.05u0 ,numbers which are representative of alkali atoms. For
u2 < 0 domain walls are stable, while for u2 > 0 domain
walls are unstable. The initial condition for all measurement simulations is the ground state of the GPE foundby imaginary time evolution [? ].
We calculate the Kraus operator’s impact on the initialcoherent state by assuming the system is well-describedby a new mean-field coherent state after measurement,
conditioned on the measurement result [33–36]. To orderϕ2 the coherent state


ϕ2 kc
Ψj|m = 1 −1Ψj + ϕmj [τ · r] Ψj
(5)
4 kM

3
0.2

E [h̄ωt ]50

x2w (t) [h̄/ma ωt ]Mz
M⊥M⊥ adj.

55

45

Mz
M⊥
M⊥ adj.
0.1

40
0.0
0

1

2
t/2π [ωt−1 ]3

4

FIG. 2. System heating for 48 total measurements as shownby increasing energy. E for 128 trajectories is plotted and theshaded area denotes the variance. Each measurement adds
energy to the system, thereby heating the system. The solidlines indicate ϕ = 0.1 while the dotted line indicates M⊥ withadjusted coupling ϕ ≈ 0.13 which adds the same amount ofenergy per measurement as Mz . The shaded area indicatestime in which no measurements are taken, and energy is conserved. The dotted black lines show the analytical prediction
for E from Eqn. (6).

maximally overlaps with K̂r|m |Ψi, thereby defining the
updated coherent state. We numerically implement
Eq. (4) using a second-order symplectic integration
method [27]. For each measurement, we apply Eq. (5) to
the wavefunction with a randomly generated noise vectorm(xj ) leading to a stochastic GPE [26, 33]. We assume
that the system dynamics evolve on a longer timescalethan the duration δt of each probe pulse.
III.

MEASUREMENT BACKACTION ON A
STABLE DOMAIN WALL

For u2 = −0.05u0 we initialize a single stable domain wall and compare two measurement signals: Mz
as in Fig. 1(b) and M⊥ as in Fig. 1(c) where M⊥ =q

M2x + M2y . The M⊥ measurement is implemented in
two steps, one√ measurement along x and one along y,with ϕ → ϕ/ 2 to give the same overall coupling as the
single z measurement; each separate measurement imparts backaction onto the condensate. The signals differ
greatly; Mz gives a large signal everywhere atoms arepresent except at the domain wall, while M⊥ is non-zero
only within the domain wall. The domain wall
√ width isapproximately the spin healing length ξs = ~/ 2ma nu2 .
By fitting the Mz , M⊥ to a tanh and cosh function
respectively, we extract the domain wall width ξw andposition xw over time from the measurement signal.
The two main effects of measurement backaction are
overall system heating and domain wall diffusion. Fig. 2
summarizes heating, which we quantify in terms of the

0

1

2

3

t/2π [ωt−1 ]FIG. 3. Variance x2w (t) for 128 trajectories. Mz shows clear
diffusive behavior while M⊥ does not. As in Fig. 2, the solidlines indicate ϕ = 0.1 while the dotted line indicates M⊥
with adjusted coupling ϕ ≈ 0.13 that gives the same heatingrate. The gray area shows the best-fit and uncertainty for the
diffusion model in Eq. (7).

energy change per measurement δE = E Ψ|m − E [Ψ],where E is the GPE energy functional. From the updated
amplitude in Eqn. (5), we calculate
ϕ2 kc X kc22
δEz ≈nj + u0 Szj+ u2 n2j
(6)
kM j
12
for a single measurement of Ŝz , where nj and Szj denotethe atom number and magnetization of the system before measurement. The first term is from the increase in
kinetic energy due to measurement backaction, while theother two terms describe the change inPinteraction energy.
22− u2 Szj,
For a measurement of Ŝ⊥ , δE⊥ ∝ ϕ2 j u0 S⊥jwhich has a smaller contribution to the overall energy at
equal ϕ (for the domain wall) as verified numerically in
Fig. 2. Fig. 2 also shows the predicted energy increase
from δEz,⊥ (dotted black lines) which agrees well with
the numerical result. Adjusting the coupling for the M⊥
measurement to ϕ ≈ 0.13 leads to the same energy added
per measurement as for Mz with ϕ = 0.1. Thus, thechoice of measurement observable affects overall system
heating.
Measurement backaction also leads to diffusion effects,
similar to the case of a particle coupled to a fluctuating
reservoir. The domain wall is a localized, heavy object
whose motion can be described by a classical Langevintheory [23, 37]. In this case the ‘reservoir’ is the stochastic measurement backaction, which adds energy to thesystem after each measurement without a mechanism fordissipation.
Measurement backaction can impart noise on both themomentum (p) and position (x) of the domain wall.
Fluctuations in x correspond to measurement backaction directly changing the local spin via the Kraus operator, while fluctuations in p correspond to changes in4fq2 + fp2
fq2 − fp2
fp fq
2
t+
sin 2ωt+sin2 ωt+Dm
(7)
2
4ω
ω

where ω is the domain wall’s oscillation frequency. The
2
constant Dm
accounts for initial measurement uncertainty.
Fig. 3 shows x2w (t) extracted from Mz and M⊥ . For
Mz the domain wall undergoes diffusion with ω ≈ 1.5ωt
and the noise strengths fx,p scale linearly with ϕ. In
the case of M⊥ the measurement result stays relatively
flat until t ≈ 4π, indicating that backaction due to theM⊥ measurement does not cause diffusion of the domain
wall. At longer times, x2w (t) does begin to increase, which
we attribute to overall heating rather than measurement
backaction. This shows that measurement backaction
due to Mz is more disruptive to the domain wall because each measurement imparts backaction noise acrossthe whole atom cloud, whereas the backaction for theM⊥ measurement occurs only at the domain wall centerand does not affect the density away from the domain
wall.
IV.

FEEDBACK-STABILIZED DOMAIN WALL

We now turn to creating and stabilizing a domain wallusing a measurement of Ŝz followed by classical feedback.
We start with a condensate with u2 = 0.05u0 which forms
a uniform condensate polarized in xy (easy-plane) withhŜz i = 0, where in the closed system a domain wall is notenergetically favorable [? ]. We derive a feedback signal
1 Xw=
sgn(xj )Mz (xj )
N j

(8)

from each measurement Mz , where on average w̄ = 0
for a uniformly easy-axis or easy-plane polarized phaseand approaches w̄ = ±1 for a domain wall centered at
x = 0; the sign identifies the orientation. For example,
the domain wall signal in Fig. 1(b) has w = −0.99. We
then apply a magnetic field gradient Vz (xj ) = gwτ z xj
with strength proportional to w and gain g.
Figure 4 summarizes the results of feedback. Initiallythe condensate is spin-unpolarized and w randomly fluctuates about zero. After a few measurements the sign
stabilizes and |w| increases, signifying domain wall formation with a stable orientation as shown by the two
branches of w in Fig. 4(a). The average w for ± orientations is calculated by binning the trajectories by thesign of w at the final timestep. Here, the band indicates

0

-1
0

1

2
t/2π [ωt−1 ]

3

4

(b)

+
−

hSz i [arb. units]

x2w =

(a) 1
w [arb. units]

the superfluid velocity caused by density fluctuations,which create a gradient in the overall condensate phaseas the system evolves in time after measurement. We account for both effects by considering a two-noise modelwith strengths fx , fp respectively, which we assume tobe anti-correlated such that hfp (t)fx (t0 )i = −fx fp δ(t −2t0 ) and hfp,x (t)fp,x (t0 )i = fp,xδ(t − t0 ). We quantify
measurement-induced diffusion by tracking the variance,

-20

-10
x

0

hp

h̄/ma ωt

i

10

20

FIG. 4. (a) Domain wall signal w for ϕ = 0.01 and g = −5.
The solid lines are the average over trajectories in the + or
− branch and the semi-transparent area indicates the variance. A domain wall is formed within 3 trap periods with theorientation spontaneously determined based on the first few
measurements of the system. (b) Final value of hSz i averaged
over all trajectories for the + (black) and − (pink) domain
wall orientations; the variance is same as the linewidth. Theinternal spin-dependent interaction parameter is u2 = 0.05u0 .

the variance of all trajectories on each branch. The process is nearly symmetric; out of 256 total trajectories,
122 evolved to the ‘+’ orientation with w = 0.78 and 134
to the ‘−’ orientation with w = −0.8. This bistability isreminiscent of spontaneous symmetry breaking in ferromagnets, but here quantum measurement and feedback“spontaneously” broke the initial symmetry [? ].
In Fig. 4(b) we show hSz i for each final orientation,
which clearly shows the presence of a domain wall. This isreminiscent of the ground state of a two-component BEC
in the immiscible regime with u2 < 0, even though the internal interaction parameter is u2 = 0.05u0 . This showsthat measurement and feedback can be used to stabilize
phases that would not be stable in equilibrium. However,our demonstration protocol is not quite the same as tuning interactions locally because w in Eq. (8) is not spatially dependent. This type of feedback could not lead tothe formation of multiple domains, which happens when5
u2 is rapidly quenched [38, 39].
V.

OUTLOOK

We outlined a new way to dynamically create stablespin textures in cold gases that is directly applicable toother systems such as Fermi gases or atoms in opticallattices. Repeated weak measurements eventually heat
the system, which can be mitigated in experiment by
evaporation, or even by suitable local feedback [34, 40].
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ACKNOWLEDGMENTS

This work was partially supported by the Air Force
Office of Scientific Research’s Quantum Matter MURI,NIST, and NSF (through the Physics Frontier Centerat the JQI). HMH acknowledges the support of the
NIST/NRC postdoctoral program.

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10.1103/PhysRevA.93.023610.

SUPPLEMENTARY MATERIAL: MEASUREMENT INDUCED DYNAMICS AND STABILIZATION OF
SPINOR CONDENSATE DOMAIN WALLS
A. MEASUREMENT MODEL DETAILShiSR
Just prior to measurement, the system and light pulse at xj each evolve together under Ûrj = exp −iδtĤrj
/~
where
SR
Ĥrj
=

~γ
Ŝrj ⊗ n̂j .
cδt

(9)

We take the probe field amplitude to be strong enough that the light is still nearly a coherent
state √
after interacting√X̂
+2ImαQ̂j − |α|2with the atoms such that âj ≈ hâj i + δâj . To first order in δâj , we then have n̂j h≈ 2Reαji√√where X̂j = (âj + â†j )/ 2 and Q̂j = (âj − â†j )/ 2i are quadrature variables with X̂j , Q̂j 0 = iδjj 0 . Thus, up to a
i
h P

√
with couplings ϕx = 2γ|α| cos φ/c,global phase the evolution operator is Ûr = exp −i j Ŝrj ⊗ ϕx X̂j + ϕq Q̂j√√ϕq = 2γ|α| sin φ/c. We then set φ = π/2 which gives ϕx = 0 and ϕq → ϕ = 2γ|α|/c. The beam is homodyne
detected on an array of detectors; during homodyne detection the reservoir state is strongly measured in the eigenbasisof the X̂j operators with eigenvalues X̂j |mj i = mj |mj i. The reservoir state |αi is assumed to be Gaussian over the |mj istates (suitable for a coherent state of light), leading to Gaussian-distributed measurement outcomes mj . Thus, themeasurement outcome for the full detector array is a vector m(xj ) = (m1 , m2 , . . . , mj ). When coupled to the quantum
system, Ûr locally shifts the |mj i states by ϕhŜrj i. The system wavefunction after measurement is |Ψ|m i = K̂r|m |Ψiwhere K̂r|m = hm|Ûr (δt)|αi is a Kraus operator corresponding to a specific measurement outcome m(xj ) and |Ψi isthe system state before measurement. We present the functional form of K̂r|m in the main text by expanding theformal expression to O(ϕ2 ).
B. SIMULATION PARAMETERS

For each simulation the internal dynamics of the system (Eq. (4) in the main text) were modeled via a GrossPitaevskii equation (GPE) using the split-step integration method in Ref. [27]. First, we found the ground state of
the GPE via imaginary time t → −iτ , using the strong convergence critertion in Ref. [41] to test for convergence.
Then, we studied the effect of measurement by running the GPE in real time to account for internal dynamics, andapplying the Kraus operator (Eq. (5) in the main text) each time we ‘measured’ the system. We studied the effect ofmeasurement backaction in the regime where domain walls are stable (u2 /u0 < 0) and we studied measurement andfeedback in the regime where they are unstable (u2 /u0 > 0). The number of particles was fixed to N = 104 , the timeincrement was dt = 3.8 × 10−4 ωt−1 , and the spatial increment was ∆x ≈ 0.02.
In the measurement backaction section of the main manuscript, we study the measurement backaction on the BEC
in the regime where domain walls are stable. These simulations (Figs. 1. 2. and 3 in the main text) were run with
u2 /u0 = −0.05, u0 = 0.1∆x and the initial condition is given in Fig. 5

7

n↑
n↓
n [arb. units]

n

-20

-10
x

hp

0
h̄/ma ωt

i

10

20

FIG. 5. Computed ground state system in the immiscible regime with a single domain wall. Initital condition for the ‘measurement backaction’ section.

In the feedback-stabilized domain wall section of the paper we started the measurement and feedback from theground state of a spin-unpolarized system. These simulations (Fig. 4 in the main text) were run with u2 /u0 = 0.05,
u0 = 0.1∆x and the initial condition is given in Fig. 6.

n↑n↓n [arb. units]

n

-20

-10
x

hp

0
h̄/ma ωt

i

10

20

FIG. 6. Computed ground state system in the miscible regime with equal, evenly distributed spin population. Initial condition
for the ‘feedback’ section.

8
C. BEHAVIOR OF INDIVIDUAL TRAJECTORIES UNDER FEEDBACK

w

1

0

-1

0

1

2
t/2π [ωt−1 ]3

4

FIG. 7. Example of two individual system trajectories under measurement and feedback for ϕ = 0.01 and g = −5. The dottedlines indicate the measurement result (with measurement noise) and the solid lines are calculated using the wavefunction only.
Notice at short times (t/2π < 1) the measurement trajectories oscillate around zero and the solid lines change sign before
stabilizing around w ≈ ±1.

Under measurement and feedback, individual system trajectories show signatures of spontaneous symmetry breaking. The sign of the feedback signal w (defined in the main text) determines the orientation of the domain wall. Fig. 7
shows the evolution of w for two system trajectories under measurement and feedback, showing that the sign of w
does not stabilize for t/2π < 1. The average w for ± orientations is calculated by binning the trajectories based on
the sign of w at the final timestep.