Parametric excitation and squeezing in a many-body spinor condensate

Abstract

Atomic spins are usually manipulated using radio frequency or microwave fields to excite Rabi oscillations between different spin states. These are single-particle quantum control techniques that perform ideally with individual particles or non-interacting ensembles. In many-body systems, inter-particle interactions are unavoidable; however, interactions can be used to realize new control schemes unique to interacting systems. Here we demonstrate a many-body control scheme to coherently excite and control the quantum spin states of an atomic Bose gas that realizes parametric excitation of many-body collective spin states by time varying the relative strength of the Zeeman and spin-dependent collisional interaction energies at multiples of the natural frequency of the system. Although parametric excitation of a classical system is ineffective from the ground state, we show that in our experiment, parametric excitation from the quantum ground state leads to the generation of quantum squeezed states.

ARTICLE

Received 7 Aug 2015 | Accepted 3 Mar 2016 | Published 5 Apr 2016

Atomic spins are usually manipulated using radio frequency or microwave elds to excite Rabi oscillations between different spin states. These are single-particle quantum control techniques that perform ideally with individual particles or non-interacting ensembles. In many-body systems, inter-particle interactions are unavoidable; however, interactions can be used to realize new control schemes unique to interacting systems. Here we demonstrate a many-body control scheme to coherently excite and control the quantum spin states of an atomic Bose gas that realizes parametric excitation of many-body collective spin states by time varying the relative strength of the Zeeman and spin-dependent collisional interaction energies at multiples of the natural frequency of the system. Although parametric excitation of a classical system is ineffective from the ground state, we show that in our experiment, parametric excitation from the quantum ground state leads to the generation of quantum squeezed states.

DOI: 10.1038/ncomms11233 OPEN

Parametric excitation and squeezing in a many-body spinor condensate

T.M. Hoang1, M. Anquez1, B.A. Robbins1, X.Y. Yang1, B.J. Land1, C.D. Hamley1 & M.S. Chapman1

1 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA. Correspondence and requests for materials should be addressed to M.S.C. (email: mailto:[email protected]

Web End [email protected] ).

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Parametric excitation of an oscillating physical system can be achieved by periodically varying one of its parameters to modulate the natural frequency of the oscillator, f0; a

textbook example is a simple pendulum excited by modulating its length, , such that f0t / 1=

t p

(ref. 1). A fundamental distinction between parametric excitation and direct excitation by periodic forcing is shown in Fig. 1, which shows instantaneous phase space orbits of a simple oscillator for the two cases. For direct excitation, the applied force periodically displaces the equilibrium position of the oscillator, leaving the orbits otherwise unchanged, and efcient excitation occurs when the excitation frequency matches the natural frequency of the oscillator, f f0.

For parametric excitation, the parameter modulation leaves the equilibrium location unchanged but instead periodically distorts the phase orbits; in this case, efcient excitation occurs for excitation frequencies f 2f0/n, n 1,2,3y

Ultracold atomic gases with well-characterized collisional interactions allow new explorations of non-equilibrium dynamics of quantum many-body physics and for synthesis of strongly correlated quantum states including spin-squeezed25 and non-Gaussian entangled states68 relevant for quantum sensing9 and quantum information10. In ultracold atom traps, parametric excitation of the atomic motion, achieved by modulating the trapping potential, is used to measure the trap frequency as well

as in a variety of studies including the excitation of BoseEinstein condensate collective density modes1114, controlling the superuid/Mott insulator transition15,16 and photon-assisted tunnelling in modulated optical lattices and super-lattices1722.

In this work, we demonstrate parametric excitation of the internal states of a collection of atomic spins. The spins are coherently excited to non-equilibrium states by a simple modulation of the magnetic eld magnitude at very low frequencies (o200 Hz) compared with the energy difference of the Zeeman states (DE/h 0.7 MHz, where h is Plancks

constant). The excitation spectrum is fully characterized and compares well to theoretical calculations. Parametric excitation of the ground state is also investigated. Classically, parametric modulation of an oscillating system does not excite the ground state1. Here, we show that the nite quantum uctuations of the collective spin leads to parametric excitation of the ground state, which manifest as exponential evolution of the uctuations and the generation of non-classical squeezed states. The exponential evolution and squeezing of the spin uctuations are measured and agree qualitatively with theory. Finally, we discuss how these techniques can be applied to related systems including the double-well BoseHubbard model and interacting (psuedo) spin-1/2 ensembles.

ResultsExperimental system. The experiments use 87Rb Bose condensates with N 40,000 atoms in the F 1 hyperne level

tightly conned in optical traps such that spin domain formation is energetically suppressed and dynamical evolution of the system occurs only in the internal spin variables. The Hamiltonian describing the evolution of this collective spin system in a bias magnetic eld B along the z-axis is5,2325:

^

H

^S2

1

2 q

a b

p (a.u.)

p (a.u.)

^

Qz 1

where ^S2 is the total spin-1 operator and ^

Qz is proportional to the spin-1 quadrupole moment, ^

Qzz. The coefcient is the collisional spin interaction energy per particle integrated over the condensate and q qzB2 is the quadratic Zeeman energy per

particle with qz 72 Hz G 2 (hereafter, h 1). The longitudinal

magnetization ^Sz

x (a.u.) x (a.u.)

c d

Qz

1

1

2

Q

is a constant of the motion ( 0 for these

experiments); hence, the rst-order linear Zeeman energy p^Sz with ppB can be ignored. The spin-1 coherent states can be represented on the surface of a unit sphere shown in Fig. 1(c) with axes {S>,Q>,Qz}, where S> is the transverse spin,

S2? S2x S2y, Q> is the transverse off-diagonal nematic

moment, Q2? Q2xz Q2yz, and Qz 2r01, where r0 is the

fractional population in the F 1, mF 0 state. In this

representation, the dynamical orbits are the constant energy contours of H 12 cS2? 12 qQz, where c 2N.

The experiment is conducted at high elds where the Zeeman energy dominates the spin interaction energy, q/|c| 10. In this

regime, the lowest energy state is the polar state (r0 1) located

at the top of the sphere, and the dynamical orbits of the excited states to leading order are simple rotations about the Qz axis with a frequency f0Eq cQz (see the Methods for details). Despite the

small relative magnitude of the spin interaction term, it has the important effect of breaking the polar symmetry and thereby slightly distorting the orbits from the latitudinal lines of the sphere. As the state orbits the sphere, the population r0 undergoes small periodic nutations at twice the orbit frequency, as shown in the r0, ys projection in Fig. 1(c). The maximum nutation amplitude is Dr0E0.02 for r0 0.5 and goes to zero for

r0

0,1. Measurements of these distortions are shown in Fig. 1(d) for different initial values of r0.

S

1

[afii9845]0

[afii9845]0

[afii9843]

[afii9843] 0

0 0 5 10 15

[afii9835]S

Time (ms)

Figure 1 | Experimental concept. Momentum-position phase space of a harmonic oscillator under (a) parametric and (b) direct excitation. The original orbits are shown in black, and the modied orbits are shown in colour at different instants of the periodic excitation. (c) The phase space of the condensate. The collective states (normalized to N) lie on a unit sphere with axes S>, Q>, Qz. The orbits of constant energy for non-interacting (c 0) spins are lines of latitude shown in black and the orbits for

interacting spins with q 10|c| are shown in colour. The r0, ys diagram is a

Mercator projection of the hemisphere. ys y

1 y

1

2y0 is the relative phase of the three Zeeman sub-levels, mF 0,1. (d) Measurements

of the natural oscillations of r0 at B 1 G for different initial states

r0(t 0)A[0, 1]. The experimental data (markers) are compared with

simulations (lines). Unless otherwise indicated, the uncertainty in the measurement of r0 is o1%.

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Parametric excitation. Parametric excitation requires periodic modulation of one of the parameters of the Hamiltonian; in a spin-1 condensate described by equation (1), this is conveniently achieved by modulating the bias magnetic eld and hence the quadratic Zeeman energy term, q(t)pB2(t). The condensate is rst prepared in a coherent state with r0, ys (0.5, p) at a eld of

1 G, corresponding to an initial quadratic Zeeman energy q0 72 Hz. The spinor dynamical rate is c 7(1) Hz,

determined from measurements of coherent oscillations at low elds. To parametrically excite the spins, the magnetic eld is modulated for a duration of time, after which the spin populations are measured to determine the nal value of r0.

The applied modulation is harmonic in q and has the form q(t) q0[1 Esin (2pftf0)]. The measured excitation spectrum

versus modulation frequency is shown in Fig. 2(a). The spectrum shows the characteristic features of parametric excitation, namely strong excitation at 2f0 142 Hz and weaker excitation at f0.

Other resonances are theoretically observable at smaller f 2f0/n

values; however, they are dominated by the tails of the more prominent peaks making them difcult to detect. The experimental data (marker) are compared with a simulation using equation (1) (solid line) and show good agreement overall.

Beyond comparing the experimental results to numerical solutions of the quantum Hamiltonian, insight into the parametric excitation is obtained by considering the mean-eld dynamical equations for r0 and the quadrature angle y ys/2

(ref. 26):

r0 4pcr01 r0sin2y_

_ y 2pq c1 2r01 cos2y

2

These equations are similar to bosonic Josephson junction equations describing the double-well condensate27,28 and can be solved like-wise by integrating the phase and using the Jacobi-Anger expansion (see the Methods for details),

_

r0 4pcr01 r0

XnJn 4pqm o

sinno 2o0t fn 3

where Jn is the Bessel function of order n, qm Eq0 is the

modulation strength and fn depends on the initial conditions including the phase of the modulation (see the Methods for details). The parametric resonance frequencies are obvious from this solution because the time-average of _

r0 is zero unless

o 2o0=n.

In the q c

j j high eld regime of these experiments, the

collisional interactions shift the natural oscillation frequency as f0

_

y

=2p

q c2r0 1 to lowest order. This shift is

investigated in Fig. 2(b), where the excitation is measured for different initial values of r0. The hue colours correspond to the initial r0 values using the same scale as the coherent oscillation data in Fig. 1(d), and the square markers indicate the positions of the measured resonance frequencies where Dr0E0. The measured resonance frequencies are in good overall agreement with the expected dependence on the initial value of r0 shown with the dashed linethe small discrepancy is attributed to an inductive delay in the excitation of B1 ms (see the Methods for details) that creates a phase offset df0 0:9 rad in the excitation. Indeed, the

experimental data compare very well to simulations (solid line) that include this phase offset.

The dependence of the excitation amplitude on the drive strength qm Eq0 is reected in the Bessel function Jn4pqm=o.

In Fig. 2(c), the excitation is measured for different modulation amplitudes. The experimental data (markers) are compared with the simulations (solid line) and show good agreement (see the Methods for details). As expected, the modulation amplitude does not affect the resonance frequency of the parametric excitation; however, increasing the modulation amplitude results in larger excitation of r0.

Parametric excitation is a coherent process, and hence it can be employed as a tool for quantum control of the collective spin. However, accurate control requires detailed knowledge of the system response to the excitation parameters. In Fig. 3b, we present a parameter variation map that shows the excitation in the neighbourhood of the 2f0 resonance for different values of excitation time t, phase f0 and frequency f. For each measurement, we prepare the initial state r0, ys (0.5, p) using an radio

frequency (rf) pulse and modulate the quadratic Zeeman energy at different frequencies and initial phases. The change in population Dr0 is measured after excitation times of 40, 100, 160 and 220 ms as shown in Fig. 3a (four vertical slices). The white/orange (black/green) regions represent the positive (negative) changes in population. These two regions evolve and spiral to form a distinctive yin-yang pattern. This pattern in the measurements agrees well with theoretical calculations of the excitation (see the Methods for details). At the centre of the pattern (f, f0) (2f0, p) where 2f0E143 Hz, r0 remains

unchanged and the excitation shows anti-symmetry about this point (diamond marker). Figure 3b shows the temporal evolution

a

0.8

0.6

[afii9845]0

[afii9845]0

[afii9845]0

0.2 0 50 100 150 200 250 300

0.4

2f0/3 f0 2f0

f (Hz)

f (Hz)

b

1.0

0.8

0.6

0.4

0.2

0.0100 120 140 160 180

f (Hz)

c

+ + +

+

+

+

+

0.6

0.5

+ * * * * * * * * * * * * * * * * * * * * * * * * * *

+ [afii9830] = 0.61

[afii9830] = 0.51

+

+ +

+ +

* [afii9830] = 0

[afii9830] = 0.1[afii9830] = 0.21

0.4

[afii9830] = 0.34

+ + + + + +

+

+ +

+ + + +

120 130 140 150 160 170

Figure 2 | Demonstration of parametric excitation. (a) Population r0 after 100 ms of parametric excitation for (r0(0), f0, E) (0.5, p, E 0.5).

Data (markers) are plotted with simulation (solid line) for comparison. The excitation spectrum shows clear resonances at frequencies 2f0 and f0.

(b) Populations r0 after 40 ms of parametric excitation for different initial r0 populations for f0 p and E 0.5. Data (markers) are compared with

simulation (solid lines), and hue colours correspond to the initial r0A[0, 1]. The dashed line is the theoretical prediction for 2f0 resonance, andthe square boxes indicate the location of the measured resonance.

(c) Population r0 after 40 ms of parametric excitation for different modulation amplitudes E.

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a

220

0.3

0.3

[afii9845]0

Time (ms)

160

100

40

0 2[afii9843]

0

125

140

155

Phase [p10] 0

[afii9843]

Time (ms)

0 100 200 300 400 500

b

0.7

f (Hz)

[afii9845]0

0.5

0.3

Figure 3 | Excitation parameter map. (a) The data show the excitation of the condensate, Dr0 r0 r0(0) following parametric excitation as a function

of the excitation frequency f and initial phase f0 of the modulation. The initial state is r0(0),ys (0.5,p) and the modulation strength is E 0.5.

The four vertical slices show Dr0 after 40, 100, 160 and 220 ms of parametric excitation. The horizontal plot shows the evolution of Dr0 for an initial phase f0 0. (b) The temporal evolution of r0 for different f, f0 pairs indicated in the vertical slice in a. The markers on the inset correspond to the

markers on the main diagram.

30

20

of r0 for different f, f0 pairs indicated in the vertical slice. Although in each case the initial state is identical, the nal state after excitation shows a strong dependence on both the frequency and phase of the applied modulation. In addition, we show a map of the population dynamics for the initial phase f0 0 (Fig. 3a,

horizontal slide). The two distinguishable domains, white (orange) and black (green), are separated by the resonance frequency. The population dynamics exhibit oscillations during the excitation process, and, approaching the resonance frequency, both the oscillating period and the amplitude (Dr0) increase.

Generation of squeezed states. We now turn to measurements of uniquely quantum features of the excitation. For a classical oscillator prepared in its stable equilibrium conguration, an important distinction between direct excitation and parametric excitation is that former can efciently excite the oscillator while the latter cannot. The equilibrium (ground) state is a stable xed point in phase space and if the oscillator is perfectly initialized, it will remain unexcited by parametric modulation. However, for a quantum system prepared in its ground state, intrinsic Heisenberg-limited uctuations of the state still allow for parametric excitation. In the semi-classical picture, the quantum uctuations populate a family of orbits about the equilibrium point in the phase space that can be parametrically excited.

In Fig. 4, we investigate parametric excitation from the quantum ground state of the condensate located at r0 1 and

demonstrate that parametric excitation can be used to generate quadrature squeezed states. Although the population, r0, is

largely insensitive to parametric excitation from the r0 1 state

(in contrast with the r0a1 initial states), the uctuations in the transverse coordinates, S>, Q>, evolve exponentially with time and show quadrature squeezing in the spin-nematic phase space.

In contrast to our previous demonstration of squeezing5,

in which the squeezing was generated by free dynamical evolution following a quench that localized the state at a unstable (hyperbolic) xed point, here, the squeezing is generated near a stable xed point by periodic distortion of the phase space orbits produced by the modulation of the quadratic

Min

Max

2 dB

(S )

10

0

Detection limit

Detection limit

10 0 0.1 0.2

Time (s)

0.3 0.4

Figure 4 | Parametric excitation from q0 1. Evolution of the minimum

and maximum values of the transverse spin uctuations, DS2?, following

parametric excitation with E 0.56 and f 134 Hz starting from the initial

state r0 1. The measured maximum (Max; red markers) and minimum

(Min; blue markers) variance of the transverse magnetization S> are compared with simulation (solid red and solid blue). The error bars indicate the s.e. of the measured variances determined from 30 repeated measurements per data point.

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Zeeman energy. The essential difference is the time dependence in the Hamiltonian.

In Fig. 4, measurements of the minimum and maximum values of the quadrature uctuations of the transverse spin, DS>, are shown and compared with a quantum simulation. As evidenced by both the measurements and the calculations, the uctuations in the initial state evolve exponentially at early evolution times and develop into quadrature squeezed states. The maximum squeezing measured is 5 dB, which is close to the detection-

limited ceiling of 6 dB because of the photo-detection shot-

noise and background scattered light5. The simulations suggest that with technical improvements, the system is capable of generating squeezing at the 20 dB level. Although the

experimental data show the main effects predicted by theory, the agreement of the measured uctuations with the theory is not perfect, particularly at longer evolution times. This is possibly due to effects of atom loss from the condensate, which has a lifetime of 1.5 s for these experiments, and we plan to further investigate this question in the future work.

In summary, we have demonstrated a mechanism for control and excitation of an ensemble of spins based on parametric excitation. This is a many-body control technique that relies on spin-dependent collisional interactions, which we have characterized for a wide range of control parameters. We have shown that this method, when applied to the ground state, can be used to generate squeezed states.

DiscussionThe parametric excitation can also be understood as transitions between eigentates of the many-body Hamiltonian, which can be calculated by diagonalizing the tridiagonal matrix6,29

Hk;k0f2k2N 2k 1 2qkgdk;k 0

2fk0 1

N 2k0 1N 2k0 2 p

dk;k0 1

N 2k0N 2k0 1 p

dk;k0 1

4

written in the Fock basis |N, M; ki, where k is the number of

pairs of mf 1 atoms, N is the total number of atoms and M is

the magnetization; both N and M are conserved by the Hamiltonian. Treating as a perturbation, the eigenenergies are Ek 2qk 2k4N 4k 1 and the energy difference between

Fock states is qEk/qk 2q 2c(2r0 1) (see the Methods

for details). Using this picture, we note that the parametric excitation spectrum excitation frequencies f 2f0/n, n 1,2,3y

corresponds to many photon excitations of the system with f 2f0

being the single photon transition.

It is interesting to contrast parametric excitation with the usual Rabi excitation of 2-level atomic spins using rf or microwave magnetic elds. In parametric excitation, the time variation of a parameter modies the Hamiltonian without displacing the equilibrium (ground) state of the system. As we have demonstrated that this can be achieved in a spin-1 condensate by simply modulating the magnitude of the bias magnetic eld, which modulates the quadratic Zeeman energy term in the Hamiltonian. As the eld strength is varied, the ground state remains at the pole of the sphere, while the shape of the orbits is modulated; this is similar to the case shown in Fig. 1(a). Rabi excitation of 2-level atomic spins using rf or microwave magnetic elds, on the other hand, is direct excitation rather than parametric excitation. Although in both cases the excitation employs time-varying magnetic elds, in the Rabi case, the oscillating magnetic eld is transverse to the bias eld, which leads to a oscillation of the orientation of the total eld. Because the ground state of the spin aligns along the eld direction, the addition of the time-varying transverse eld leads to a periodic displacement of the ground

state away from the pole in the usual Bloch sphere picture of the spin vector, similar to the case shown in Fig. 1(b).

We point out that the techniques we have demonstrated are applicable to related many-body systems including the double-well Bose-Hubbard (or Bosonic Josephon junction model), collisionally interacting pseudo-spin 1/2 two component condensates and ensembles of spin-1/2 atoms with photon-mediated interactions. Each of these systems can be described by a version of the LipkinMeshkovGlick model Hamiltonian30, H US2x KSz, whose the mean-eld phase space is functionally

identical to the spin-nematic phase space shown in Fig. 1(c)5,28. At their heart, these systems feature competing energy terms (one of which is non-linear) that give rise to a quantum critical point. The q42|c| polar phase that we explore in this work corresponds to the Rabi regime (K4U) in the Bosonic Josephon junction system, which is the tunnelling-dominated regime perturbed by the interactions US2x, and the modulation of q corresponds to a modulation of the tunnelling coefcient K. Indeed, there have been numerous theoretical proposals for excitations of these systems using periodic modulations (for example, see refs 3134 for details), and many of the lattice-based experimental demonstrations mentioned previously realize closely related ideas generalized to multi-site systems1522.

Methods

Experimental concept. The experiment utilizes 87Rb atomic BoseEinstein condensates created in an optical trap containing N 40,000 atoms initialized in

the |F 1, mF 0i hyperne state in a high magnetic eld (2 G). To prepare the

initial spin state, the condensate is rapidly quenched to a magnetic eld of 1 G, and a Rabi rf pulse resonant with the F 1 Zeeman transition is applied to

prepare the desired initial state r0, ys (r0(t 0), p). The nominal value of the

magnetic eld B 1 G is determined by using rf and microwave spectroscopy,

and the spinor dynamical rate is determined by measuring coherent spin dynamics oscillations using states prepared near the ferromagnetic ground state (c 7(1) Hz).

Parametric excitation of the system is implemented by sinusoidally modulating q, which is implemented by time-varying the magnetic eld using external coils. Owing to induced eddy currents in the metal vacuum chamber and the inductance of the magnetic eld coils, the applied modulation is time-delayed relative to the intended control by 1 ms and reduced in amplitude by 15%. These effects are measured directly using magnetically sensitive rf spectroscopy of the atoms, and they are incorporated in all the simulations.

The nal spin populations of the condensate are measured by releasing the trap and allowing the atoms to expand in a SternGerlach magnetic eld gradient to separate the mF spin components. The atoms are probed for 400 ms with three pairs of counter-propagating orthogonal laser beams, and the uorescence signal is collected by a CCD camera is used to determine the number of atoms in each spin component.

To measure the transverse spin uctuations DS>, an rf Rabi p/2 pulse is applied during the expansion to rotate the transverse spin uctuations (which are in the x,y plane) into the z measurement basis. The uctuations are then determined from 30 repeated measurements of hSzi, the difference in the number of atoms measured in

the mF 1 and mF 1 spin components.

Coherent oscillation dynamics. We rst discuss parametric excitation using semi-classical mean eld theory. The excitation occurs when the quadratic Zeeman energy is modulated at integer divisors of twice the natural coherent oscillation frequency in the (y, r0) phase space. The dynamics of the system are governed by a set of differential equations for the fractional population r0 and the phase y from refs 35,36

_

r0 4pcr0

k0

q

1 r02 m 2

sin 2y 5

_

y 2pq c1 2r0

2pc 1 r01 2r0 m2

cos 2y: 6

q

1 r02 m 2

and we have taken h-1. The spinor energy of the system is given by

E cr01 r0

where m

^Sz

q

1 r02 m 2

cos 2y q1 r0 7

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and has an oscillation period35 of the formT

1 p

with the most dominant excitation frequency corresponding to n 1

f

2f0

2

p

K

q

x x

x x

qc

p x3 x1

8

where K(k) is the elliptic integral of the rst kind and xi are the roots of the differential equation

_

r02 4p2E q1 r0 2cr0 q1 r0 E cr0m2 9 For a condensate prepared in mF 0, with magnetization conserved m 0, as is

the case in our experiment, the roots xi are

xi 2

2c q

(

p 4c

p 4c )

10

4c2 8cE 4cq q

2 ; q E

q ;

2c q

4c2 8cE 4cq q

2

1 2 128:8; 157:6 Hz 8r0 2 0; 1 : 15

Parametric excitation theory. In our experiment, the system is prepared in the mF 0 state with magnetization m 0 and allowed to evolve at sufciently high

elds, where q/|c| 10. Under these conditions, equation (5) simplies to_

r0 4pcr01 r0sin 2y 16 _

y 2pq c2r0 1 17 Parametric excitation is then applied by modulating the quadratic Zeeman energy

q q0 Eq0 sin2pfmt f0Ht f0=2pfm 18 The Heaviside function and phase f0 imply that the modulation starts at q0 and is active for t4f0=2pfm f0=om, where om 2pfm. We prepare an initial r0 at

high eld using an rf pulse, which initializes the quadrature phase y0 p/2.

The system then freely evolves for t f0=om, advancing the quadrature phase

Dy 2pq0 2cxf0=om o0f0=om, followed by modulation of q, where

we have implicitly assumed that for evolution at high eld we can set r0 to be a constant so that integration of _

y is straightforward. We therefore have two initial contributions to the quadrature phase before modulation, namely y0 and Dy.

For evolution times t4f0=om, we can integrate _

y in equation (16) giving:

y y0 Dy

p

In order to calculate the period T, we rst calculate

qc

p x3 x1 q

p

2 q2 4qc2r0 1 4c2

1=4

q2 2qcQz p 2

p

2

p

11

1

c21 Q2z q 2cQz2

where Qz 2r0 1. The elliptical integral part of the period in equation (8)

K

x2 x1x3 x1

Z

t

0 dt0

_

y 19

p=2 o0f0=om o0t

2pEq0

om cosomt 1

2 12

Substituting these results back into the equation for the period, we obtain the natural coherent oscillation frequency in quadrature phase (y, r0)

f0

1T

K0:01

p

p

q2 2qcQz p 2

1

2 p

c21 Q2z q 2cQz2

13

q2 2qcQz p

q cQz

20

where o0 2pq0 c2r0 1. Substituting the phase into the population

dynamical equation (16), we obtain

_

r0 4pcr01 r0

sin 2y0 2o0f0=om 2o0t

4pEq0

om cosomt 1

where we have that q 10 c

j j in our experiment.

Parametric excitation of the system is achieved by periodically modulating the quadratic Zeeman term q in the Hamiltonian. Efcient excitation occurs when the modulation frequency is an integer divisor of twice the natural frequency of the system f 2fn , where n 2 N. In our system, the coherent oscillations occur at a

magnetic eld B 1 G and spinor dynamical rate c 7.2(5) Hz. These

parameters yield a range of natural frequencies

f0 71:6 12 7:2x 2 64:4; 78:8 Hz 8r0 2 0; 1 : 14

21

4pcr01 r0

XnJn 4pEq0

om

sinnom 2o0t d np=2

22

where the phase d 2y0 2o0f0=om 4pEq0=om. In equation (22), we have

made use of the identity sin(A B) sin(A)cos(B) cos(A)sin(B) along with the

JacobiAnger expansions

cosz sina

X1n 1Jnzcosna 23

sinz sina

a b

2[afii9843]

[afii9843]

2[afii9843]

[afii9843]

Phase [p10] 0

Phase [p10] 0

X1n 1Jnzsinna 24

Analysing equation (22) gives us some insight into the population dynamics as a function of the modulation parameters. When om 6 2o0=n, the time average of

r0 /

_ Pn sinOt is zero. When om 2o0=n, the time average of the nth term in the expansion _r0;n 4pcr01 r0Jn4pEqo sind p=2 is non-zero. Therefore, only for the case when om 2o0=n is there sufcient coupling from the

modulation to parametrically excite the system. The behaviour of the Bessel functions Jn4pEqo also indicates that the strongest coupling occurs for n 1 or

om

2o0, a signature of parametric excitation.

The strength of the excitation is controlled by tuning E. When the system is modulated at om o0=2n we can focus on short-time dynamics om 2o0top,

as the higher order terms are negligible becuase of time averaging, and expand equation (22) about E 0. It can be shown that the coefcient in the expansion is

exact up to OEjn

j . Using this fact, the population dynamical equation for _

r0

can be rewritten as:

r0 4pcr01 r0

X1n0 1nx2n

[afii9843] [afii9843]

Phase [p10] 0

0 120 140 160

0 120 140 160

f (Hz) f (Hz)

f (Hz) f (Hz)

[afii9845]0 0.3 0.3

c d

2[afii9843]

2[afii9843]

Phase [p10] 0

_ " #

4pcr01 r0 cos x sin d sin x cos d

4pcr01 r0 sin d x cos d Ox2

0 120 140 160 120 140 160

X1n0 1nx2n1

0

2n !

sind

2n 1 !

cosd

Figure 5 | Excitation parameter map simulation. Semi-classical simulations demonstrating parametric excitation for varying modulation phase f0 and modulation frequency f corresponding to evolution times of 40 ms (a), 100 ms (b), 160 ms (c) and 220 ms (d) running clockwise.

25

where x 4pEq0=om. Depending on the value of d, increasing the strength of the

modulation E will either enhance or reduce the response of the system to the modulation for short times. Furthermore, it is also apparent from equation (25), along with the expression for d, that the response of the system is periodic with respect to the initial phase of the modulation f, and has xed points for r0 0.5

6 NATURE COMMUNICATIONS | 7:11233 | DOI: 10.1038/ncomms11233 | http://www.nature.com/naturecommunications

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11233 ARTICLE

and d np, where n is an integer, both of which agree with the experimental data

shown in Fig. 3 of the main paper.We now consider the strong coupling case where n 1 or om 2o0, and the

higher order terms |n|41 of Jn4pEqo are negligible, so that_

r0 4pcr01 r0 J 1

4pEq0

2o0 sin 4o0t 2y0 f0

J0

4pEq0

2o0 sin 2o0t 2y0 f0

4pEq0

2o0

p

2

References

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4pEq0

2o0

26

J1

4pEq0

2o0 sin2y0 f0

4pEq0

2o0

p

2

and after integration

r0 4pcr01 r0J 1

4pEq0

2o0

J0

4pEq0

2o0

14o0 cos 4o0t 2y0 f0

4pEq0

2o0

p

2

1

2o0 cos 2o0t 2y0 f0

4pEq0

2o0

27

J1

4pEq0

2o0 sin2y0 f0

4pEq0

2o0

p

2t

r00

The population r0(t)Er0(0) if f0 1.37p and om 2o0.

Quantum parametric excitation. An alternative way to view the parametric excitation is by considering transitions between the eigenstates of the many-body Hamiltonian. The energy corresponding to the oscillation frequency matches no single atom transition. Rather, it approximately matches the energy difference between two atoms in the mF 0 state and a pair of atoms in the mF 1 states.

Yet, even this is not precise enough as this energy separation also depends on the collective state of the system varying from DE 2q c for all atoms in the

mF 0 state to DE 2q c for all atoms in the mF 1 states. These energy

separations can be calculated by diagonalizing the tridiagonal matrix given by

Hk;k

f2k2N 2k 1 2qkgdk;k 2fk0 1

N 2k0N 2k0 1 p

dk;k 1

28

where c=2N and k is the number of pairs of mF 1 atoms in the enu

meration of the Fock basis. The Fock basis, |N, M, ki, is also enumerated with N the

total number of atoms, and M the magnetization, both of which are conserved by the Hamiltonian leaving all dynamics in k. The off-diagonal contributions in equation (28) are due to the many-body interaction given by the^S2 term of the Hamiltonian. This interaction results in mixing of the Fock states, even in the high eld limit. Without this interaction, there would be no transitions as the magnetic interactions, both linear and quadratic Zeeman, are diagonal in the Fock basis. In this picture, the integer divisor frequencies of the spectrum correspond to many photon excitations of the system.

In the regime where q qZB242c, we treat as a perturbation and H0k;k 2qk

with expansion coefcients

E0k 2qk

E1k k H0

j jk

h i 2k 2N 2k 1

29

k0

N 2k0 1N 2k0 2 p

dk;k 1g

and eigenenergy of the system

Ek E0k E1k O2: 30 The resonance frequency between Fock states is the energy difference between each Fock state

f

@Ek

@k 2q 22N 8k 1

2q cx

31

where the last line corresponds to kooN. To rst order, the resonance frequency between Fock states is the same as the frequency obtained from mean eld theory equation (13). The factor of two arises from the denition of the resonant frequency f 2f0.

Parameter variation map simulation. To compare the experimental data shown in Fig. 3 of the main paper to theory, we perform four semi-classical simulations at xed evolution times that demonstrate excitation in the neighbourhood of the 2f0 resonance for different values of the modulation phase f0 and modulation frequency f. For each simulation, we initialize the state (r0, ys) (0.5, p) and

modulate the quadratic Zeeman energy q at different frequencies and phases. Details of the simulation method can be found in refs 5,6.

The change in population Dr0 is calculated after excitation times of 40, 100, 160 and 220 ms, as shown running clockwise in Fig. 5. These simulations correspond to the vertical slices shown in Fig. 3 of the main paper, and agree quite favourably.

The white/orange (black/green) regions represent the positive (negative) changes in population, which evolve and spiral to form a distinctive yin-yang pattern. At the centre of the plots (f, f0) (2f0, p), where 2f0E143 Hz, r0 remains unchanged and

the excitation shows anti-symmetry about this point.

NATURE COMMUNICATIONS | 7:11233 | DOI: 10.1038/ncomms11233 | http://www.nature.com/naturecommunications

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11233

34. Jager, G., Berrada, T., Schmiedmayer, J., Schumm, T. & Hohenester, U. Parametric-squeezing amplication of Bose-Einstein condensates. Phys. Rev. A 92, 053632 (2015).

35. Zhang, W., Zhou, D. L., Chang, M.-S., Chapman, M. S. & You, L. Dynamical instability and domain formation in a spin-1 Bose-Einstein condensate. Phys. Rev. Lett. 95, 180403 (2005).

36. Chang, M.-S., Qin, Q., Zhang, W., You, L. & Chapman, M. S. Coherent spinor dynamics in a spin-1 Bose condensate. Nature Phys 1, 111116 (2005).

Acknowledgements

We acknowledge support from the National Science Foundation Grant PHY1208828.

Author contributions

T.M.H., C.D.H. and M.S.C. jointly conceived the study. T.M.H., M.A., B.A.R. and X.Y.Y. performed the experiment and analysed the data. T.M.H., B.J.L. and C.D.H. developed essential theory and carried out the simulations. M.S.C. supervised the work.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Hoang, T. M. et al. Parametric excitation and squeezing in a many-body spinor condensate. Nat. Commun. 7:11233 doi: 10.1038/ncomms11233 (2016).

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