This is the accepted manuscript made available via CHORUS. The article has been
published as:

Dissipation-Assisted Coherence Formation in a Spinor
Quantum Gas
Yujiro Eto, Hitoshi Shibayama, Kosuke Shibata, Aki Torii, Keita Nabeta, Hiroki Saito, andTakuya Hirano
Phys. Rev. Lett. 122, 245301 — Published 21 June 2019
DOI: 10.1103/PhysRevLett.122.245301

Dissipation-assisted coherence formation in a spinor quantum gas
Yujiro Eto1,2 , Hitoshi Shibayama3 , Kosuke Shibata4 , Aki Torii4 , Keita Nabeta4 , Hiroki Saito5 , and Takuya Hirano4
1

National Institute of Advanced Industrial Science and Technology (AIST), NMIJ, Tsukuba, Ibaraki 305-8568, Japan
2
JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
3
College of Industrial Technology, Nihon University, Narashino 275-8576, Japan
4
Department of Physics, Gakushuin University, Toshima, Tokyo 171-8588, Japan
5
Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan(Dated: May 21, 2019)
We report the formation of magnetic eigenstates assisted by naturally occurring particle dissipation in a Bose–Einstein condensate of spin-2 87 Rb atoms. Although the atomic interaction energetically favors the non-ferromagnetic state, we observed the spontaneous evolution of an unpolarized
spin state into the transverse ferromagnetic state. Under such dynamics, the spin-dependent dissipation of atoms enhances the synchronization of the relative phases among five magnetic sublevels to
promote magnetization. Through numerical simulations based on mean-field theory, we show that
another exotic magnetic eigenstate, the cyclic state, can also be formed through the spin-dependentdissipation of atoms.

Clarifying the various effects of energy and particle
dissipation coupling to the environment will lead to a
deeper understanding of open quantum systems and will
expand the applications of quantum physics. Dissipation
often reduces the quantum coherence, which is a major
problem in quantum information science, such as quantum computation and quantum simulation [1]. On theother hand, dissipation sometimes has the opposite effect of promoting quantum coherence, and such dissipation can be used as a new control strategy for quantum
systems and as a useful resource for quantum computations [2]. Through proper design of the coupling between
the system and the surrounding environment, it is possible to prepare a desirable pure quantum state in an open
quantum system to explore quantum simulation using
strongly correlated many body states [3]. The stabilization of entangled qubits in superconductor and atomic
ions [4, 5] and the control of quantum phase transitions
in cold atoms [6] have been experimentally demonstrated
by exploiting controllable dissipation.
Beneficial effects of dissipation are also caused by uncontrollable naturally occurring dissipation. Natural dissipation has been used to protect quantum states in optical applications. Optical loss maintains a coherent stateor brings other states closer to a coherent state [7]. Under such energy dissipation, photons are lost, but thephases between the photon-number bases are not randomized, enabling the transmission of light while maintaining the phase of the complex amplitude of the lightfield. Recently, prolonging the coherence time under natural dissipation has been discussed in biological systems
to understand efficient processes in nature [8]. In photosynthetic systems, such an effect is related to efficientelectron transport [9, 10].
In this Letter, we investigate the magnetization dynamics in dissipative Bose–Einstein condensates (BECs)of spin-2 (hyperfine spin F = 2) 87 Rb atoms. We
experimentally observed the emergence of symmetry-breaking magnetization from an unpolarized spin state.
Symmetry-breaking magnetization has been observed ina spin-1 87 Rb BEC [13], in which the magnetization is induced by the ferromagnetic interactions between atoms.
In contrast, in a spin-2 87 Rb BEC [11], although the corresponding interactions are not ferromagnetic, the unpolarized spin state evolves into a transverse ferromagnetic
state.
We found from the comparison of experimental andtheoretical results that spin-dependent particle dissipation plays an essential role in the formation of a ferromagnetic state. In this system, the F = 2 state of 87 Rb
atoms undergoes inelastic collisional loss, in which two
F = 2 atoms collide and escape from the trap by a transition from the F = 2 state to the F = 1 states [12]. Although the total spin F of the two colliding F = 2 atoms
can be F = 4, 2, or 0, decay through the F = 4 channel
is prohibited by angular-momentum conservation, since
the F = 1 state cannot participate in the F = 4 channel.
In other words, inelastic collisional loss of two atoms is
suppressed when they are fully magnetized in the same
direction. As a consequence, the magnetized state accumulates and is purified in the trap by the dissipationof other states. This magnetization dynamics is in stark
contrast to the conventional dynamics, in which the spin
vectors align in the same direction to lower the interaction energy [13].
A schematic of the observed spinor dynamics is illustrated in Fig. 1. We performed the experiments using a spin-2 BEC consisting of the five magnetic sublevels, m = −2, −1, 0, +1, and +2. The magnetization,
Si , along the i-axis was obtained from the population ineach magnetic sublevel, ρm,i , with the i-axis as the quantization axis (Si = Σm mρm,i ). The atoms were initiallyprepared in the |F = 2, m = 0i state with the quantization axis along a magnetic bias field on the z-axis. This
state is the completely unpolarized spin state with rotational symmetry around the z-axis [Fig. 1(a)]. After time2
sin Fm(b) With dissipation(a) Initial state

sin FmBEC(c) Without dissipation

FIG. 1: Conceptual diagram of coherence formation assistedby spin-dependent particle dissipation. (a) The as-prepared
unpolarized spin state of |F = 2, m = 0i, where F and m
are the quantum numbers for the total angular momentum
and the Zeeman sublevel, respectively. The populations in
the |2, mi states are depicted in the grey box. (b) and (c)show the spin states after time evolution with and without dissipation, respectively. The sinusoidal curves represent
the phases sin Φm = sin (mωt + φm ), where ω is the linear
Zeeman frequency under a magnetic field in the z-direction.
The phase offset, φm , is expressed by different colors. The
atoms populate all five components in both (b) and (c). The
fully polarized transverse magnetization is generated only
in a dissipative system, namely Φm obeys the relationshipΦ2 − Φ1 = Φ1 − Φ0 = · · · = Φ−1 − Φ−2 due to the dissipation.

evolution, the atoms were distributed into all m components by two-body elastic collisions while preserving alongitudinal magnetization Sz of zero. Similar dynamics of ρm,z have previously been investigated experimentally [14–16]. The most significant difference from theprevious studies is that we also measured the transverse
magnetization, S⊥ = S cos α, orthogonal to the z-axis,
making it possible to q
obtain information on the phasecoherence, where S = Sx2 + Sy2 is the magnitude of thetransverse magnetization and α is the azimuth angle ofthe spin vector. We found that the transverse magnetization is almost fully polarized, S ≃ 2, after time evolution.
Transversely fully polarized state rotating on the x − yplane is expressed by the two-axis rotation of the longitudinally fully polarized state as follows:πR̂z (α)R̂y ( ) |m = 2i =
2
r
e−iα
eiα
e2iα
3
e−i2α
|2i +
|1i +
|0i +
|−1i +|−2i ,
42
8
2
4
where the operator R̂i (θ) rotates the spin state aroundthe i-axis by an angle θ. In such state, the phases in themagnetic sublevels, Φm , satisfy Φ2 − Φ1 = Φ1 − Φ0 =
Φ0 − Φ−1 = Φ−1 − Φ−2 . Therefore, the generation of thetransverse magnetization with S ≃ 2 implies the formation of phase coherence. The atoms distributed in thefive m components therefore acquire phase correlation,and the phases rotate with the Zeeman frequency synchronously [Fig. 1(b)]. On the other hand, phase synchronization does not occur in a numerical simulation
without dissipation [Fig. 1(c)]; thus, it can be concluded
that the phase synchronization is due to dissipation.
We used a BEC trapped in a crossed far-off-resonanceoptical trap (FORT). The axial and radial frequencies of
the FORT are ωz /(2π) = 64 Hz and ωr /(2π) = 190 Hz,
respectively. The external magnetic field of Bz = 200 mG
is aligned with the axis of the trap (z direction), which
produces a quadratic Zeeman shift, |q|/h ≃ 3 Hz. The
pure |2, 0i state is produced from the state occupying themultiple m levels by using microwave pulses and a blasted
beam [see Supplemental Material (SM)]. The purity of
the |2, 0i state in this study was greatly enhanced relative
to that in our previous experiment [16], to remove othercomponents. The BEC prepared in the |2, 0i state wasthen held in the optical trap for a variable time of Thold .
To measure Sz , the BEC was released from the trap, andeach m component was spatially separated along the z
direction using the Stern-Gerlach method. On the other
hand, to measure S⊥ orthogonal to the z direction, we
irradiated the BEC with a π/2 radio frequency (rf) pulse
just before releasing it from the trap. The π/2 rf pulse
effectively rotated the measurement axis by π/2. After atime of flight for 15 ms, the spatial distribution of each mcomponent was measured using absorption imaging. The
number of atoms in each m component was estimated by
performing the bimodal fitting of the atomic distribution.
The longitudinal or transverse magnetization per atom,
Si , was calculated from ρm,i .
We first experimentally investigate the spinor dynamics along the longitudinal axis (z-axis). Figure 2(a) shows
the dependence of ρm,z on the free-evolution time, Thold .
The m = ±1 components grow after Thold ≃ 30 ms, followed by a delayed growth in the m = ±2 components
after Thold ≃ 50 ms. The initial slow rise of the m = ±1
components in Fig. 2(a) reflects the metastability of them = 0 state [14, 17]. Indeed, in the case where theinitial state includes components other than the m = 0
state, the initial change is faster (see SM). The longitudinal magnetization, Sz , calculated from the results
shown in Fig. 2(a) remains almost stable around zero,as shown in Fig. 2(b). As shown in the inset of Fig.
2(a), the total number of atoms is reduced, mainly due
to spin-dependent particle dissipation by inelastic collisional loss. The degree of loss by other processes suchas photon scattering from the dipole trap was estimatedusing the |2, −2i state, which is immune to inelastic collisional loss [18]. In this case, the total loss of atoms at
Thold = 150 ms is less than 10%.
We next investigate the dynamics of S⊥ . As for the dynamics measured along the z-axis, no change is seen up to

3

(a)

(b)
FIG. 3: Evolution of the magnetization from numerical simulation. The blue and red curves are obtained by solving the
3D GP equation with and without the dissipation of atoms,
respectively. In the green curve, the quadratic Zeeman effectis excluded.

(c)

(d)

Values estimated from S=1.9

FIG. 2: Observation of the spinor dynamics starting from an
unpolarized spin state |2, 0i. (a) Time evolution of the populations ρm,z in the magnetic sublevels m. The inset shows
the total number of atoms. (b) Dynamics of the longitudinalmagnetization, Sz . (c) Dynamics of the transverse magnetization, S⊥ . (d) Normalized occurrence distribution of |S⊥ |obtained during Thold = 120 - 150 ms. The histogram with abin width of 0.2, was constructed from 60 experimental data
points in (c), and the red empty histogram indicate the fitting
results.

Thold ≃ 30 ms. However, unlike for the dynamics of ρm,z ,the shot-to-shot variations in ρm,⊥ for each Thold greatly
increase after Thold ≃ 30 ms, which results in S⊥ varying between −2 and +2, as shown in Fig. 2(c). We now
consider the cause of the shot-to-shot variations in S⊥
observed in Fig. 2(c). Since the initial m = 0 state hasrotational symmetry around the z-axis, the occurrenceof transverse magnetization should be a consequence of
spontaneous breaking of the rotational symmetry. There-

fore, the azimuth angle of the spin vector, α, becomes
random for each measurement, resulting in variation ofS⊥ = S cos α. The change of α corresponds to adding thephase −mα to each Φm . In addition, α fluctuates due tothe temporal fluctuation of the magnetic bias field.
We evaluate the magnitude of S that best fits the
60 experimental data points of |S⊥ | obtained during
Thold = 120–150 ms in Fig. 2(c). When S reaches a certain value and assuming α is random and uniform over[0, 2π], the probability distribution of |S⊥ | is representedby p(|S⊥ |) ∝ (S 2 − |S⊥ |2 )−1/2 , where 0 ≤ |S⊥ | ≤ S.
The red outline histogram in Fig. 2(d) shows p(|S⊥ |)integrated within the bin width for S = 1.9.
We perform three-dimensional numerical simulationsof the Gross-Pitaevskii (GP) equation (see SM). Figure 3 shows a typical example of the time evolution of
the transverse magnetization S. Magnetization is neverproduced under a condition of zero magnetic field, corresponding to the green curve in Fig. 3. When thequadratic Zeeman energy is included, as shown by the
red curve, the magnetization initially grows but then decreases. The magnitude of the magnetization is limited
to no more than S ≃ 1, and thus this result disagrees
with the experimental findings. The initial growth in themagnetization is due to the combined effect of the s-wave
spin-exchange interaction and the quadratic Zeeman effect. The energies of the m = ±1 and m = ±2 states arelowered by the quadratic Zeeman energy qm2 (q < 0), andthe transitions from the m = 0 state to these states areenhanced. The quadratic Zeeman effect also rotates Φmby (qm2 /~)t, which results in temporal modulation of S.
Thus, the quadratic Zeeman effect produces magnetization, but never synchronizes the phase Φm to enhance
the magnetization.
Next we solve the GP equation by taking into accountspin-dependent particle dissipation. As shown in the blue
curve in Fig. 3, the increase in the magnetization up to
Thold ≃ 60 ms is similar to that for the case without par-4


7g0 − 10g2 + 3g4
g4 − g2
2s·s+|A0 | ,
Espin = dr
14
14
(1)
where gF is the interaction coefficient for the colliding
channels of total spins F = 4, 2, and 0; s is the magnetization density; and |A0 | is the spin-singlet density. The
spin-dependent particle dissipation is expressed by the
non-positive imaginary part of the interaction coefficientgF [12]. The imaginary part of the first term represents
the relative enhancement or suppression of the magnetization, depending on its sign. As mentioned above, decay through the F = 4 channel is prohibited (Img4 = 0)
and the imaginary part of the coefficient of s · s is always non-negative, which enhances |s| relative to the total density. The magnetized state is thus more likely tosurvive than any other state under spin-dependent particle dissipation, which leads to the formation of the ferromagnetic eigenstate. In other words, spin-dependentdissipation assists the synchronization of Φm to promote
the transverse magnetization. This is quite different fromthe phase decoherence due to the dissipative environment; the present effect of dissipation protects the specific eigenstate.
From the form of Eq. (1), we find that the singlet-pairdensity |A0 | is also enhanced or suppressed by the spindependent particle dissipation, which suggests that othermagnetic eigenstates can emerge depending on the valuesof Im gF . One such eigenstate is the cyclic state, which
is a BEC of singlet trios of spin-2 atoms [11, 19–21]. The
cyclic state emerges when the singlet-pair density |A0 |and the magnetization |s| are both suppressed. As mentioned above, the imaginary part of g4 − g2 in Eq. (1) isalways non-negative, which causes |s| to increase. However, if |A0 | decreases before |s| grows, the cyclic state
can be formed. This is achieved when |Img0 | is large and|Img2 | is small. Figure 4 shows the time evolution of |s|and |A0 | for such a case for Img0 and Img2 . The initialstate is R̂x (π/2)|2, 0i. The blue curve in Fig. 4 showsthat |A0 | decays to almost zero in the presence of particle dissipation while |s| remains small, indicating that thecyclic state is formed. The spherical harmonic representations of the spin states also show that the initial polarstate evolves into the cyclic state, as shown in the inset
Z



|s|

ticle dissipation. Unlike the non-dissipative system, themagnetization in the dissipative system continues to increase, which is in qualitative agreement with the experimental result in Fig. 2. This result indicates that thegrowth of magnetization after Thold ≃ 60 ms is driven
mainly by the spin-dependent dissipation. However, theblue curve in Fig. 3 does not reach the values of S ≃ 1.9
obtained in Fig. 2(d), which may be due to effects nottaken into account in the simulations, such as thermal
effects [22].
The growth of magnetization by particle dissipation is
described by the spin-dependent part of the energy:

0 ms
Cyclic state
150 ms≒FIG. 4: Formation of the cyclic state (magnetic eigenstate).
Numerical simulation of the time evolution of |A0 | and |s|,when Img0 and Img2 are ten times and one-tenth their values
for 87 Rb, respectively. The initial spin state is R̂x (π/2) |2, 0i.
The size of the magnetic field, Bz , is 50 mG. The spherical harmonic representations of the spin states at the centreof the condensate are shown in the insets (see Supplemental
Material).

of Fig. 4. It is notable that such tetrahedral symmetry arises from particle dissipation. Both the initial state
and the cyclic state have populations in the m = 0 and
m = ±2 states, with relative phases of φ2 +φ−2 −2φ0 = 0
for the initial state and π for the cyclic state. Thus the
spin-dependent particle dissipation also affects the phases
Φm in the generation of the cyclic state.
We have experimentally and theoretically investigated
the role of dissipation in a BEC of spin-2 87 Rb atoms andfound that particle dissipation can give rise to quantum
coherence. Although the interactions between the atomic
spins are not ferromagnetic, we nonetheless observed theemergence of transverse magnetization, in which the relative phases among the five magnetic sublevels are synchronized. Numerical simulations revealed that this phenomenon is mainly due to spin-dependent particle dissipation. It has also been shown that with appropriate loss
parameters, such dissipation can lead to the formation ofa cyclic magnetic state. These results indicate that naturally occurring dissipation gives rise to robustness ofquantum coherence.
This work was supported by JSPS KAKENHI(grant nos. JP17K05595, JP17K05596, JP16K05505,
JP15K05233, and JP25103007) and JST PRESTO (grant
no. JPMJPR17G3). Y.E. acknowledges support fromthe Leading Initiative for Excellent Young Researchers(LEADER).

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