institute for quantum electronics, eth zurich, otto-stern
TRANSCRIPT
Spin-motion entanglement and state diagnosis with squeezed oscillator wavepackets
Hsiang-Yu Lo∗, Daniel Kienzler, Ludwig de Clercq, Matteo Marinelli, Vlad Negnevitsky, Ben C. Keitch,
Jonathan P. Home∗
Institute for Quantum Electronics, ETH Zurich, Otto-Stern-Weg 1, 8093 Zurich, Switzerland
Mesoscopic superpositions of distinguishablecoherent states provide an analog to theSchrodinger’s cat thought experiment [1, 2]. Formechanical oscillators these have primarily beenrealised using coherent wavepackets, for whichthe distinguishability arises due to the spa-tial separation of the superposed states [3–5].Here, we demonstrate superpositions composedof squeezed wavepackets, which we generate byapplying an internal-state dependent force to asingle trapped ion initialized in a squeezed vac-uum state with 9 dB reduction in the quadraturevariance. This allows us to characterise the ini-tial squeezed wavepacket by monitoring the on-set of spin-motion entanglement, and to verifythe evolution of the number states of the oscil-lator as a function of the duration of the force.In both cases, we observe clear differences be-tween displacements aligned with the squeezedand anti-squeezed axes. We observe coherent re-vivals when inverting the state-dependent forceafter separating the wavepackets by more than 19times the ground state root-mean-square extent,which corresponds to 56 times the r.m.s. extent ofthe squeezed wavepacket along the displacementdirection. Aside from their fundamental nature,these states may be useful for quantum metrol-ogy [6] or quantum information processing withcontinuous variables [7–9].
The creation and study of nonclassical states of spinsystems coupled to a harmonic oscillator has providedfundamental insights into the nature of decoherence andthe quantum-classical transition. These states and theircontrol form the basis of experimental developments inquantum information processing and quantum metrology[1, 2, 10]. Two of the most commonly considered statesof the oscillator are squeezed states and superpositions ofcoherent states of opposite phase, which are commonlyreferred to as “Schrodinger’s cat” (SC) states. Squeezedstates involve reduction of the fluctuations in one quadra-ture of the oscillator below the ground state uncertainty,which has been used to increase sensitivity in interfer-ometers [11, 12]. SC states provide a complementarysensitivity to environmental influences by separating thetwo parts of the state by a large distance in phase space.These states have been created in microwave and optical
∗Electronic address: [email protected]; Electronic address:[email protected]
cavities [2, 13], where they are typically not entangledwith another system, and also with trapped ions [1, 3–5],where all experiments performed have involved entangle-ment between the oscillator state and the internal elec-tronic states of the ion. SC states have recently been usedas sensitive detectors for photon scattering recoil eventsat the single photon level [14].
In this Letter, we use State-Dependent Forces (SDFs)to create superpositions of distinct squeezed oscillatorwavepackets which are entangled with a pseudo-spin en-coded in the electronic states of a single trapped ion. Wewill refer to these states as Squeezed Wavepacket Entan-gled States (SWES) in the rest of the paper. By mon-itoring the spin evolution as the entanglement with theoscillator increases [15–17], we are able to directly ob-serve the squeezed nature of the initial state. We ob-tain a complementary measurement of the initial stateby extracting the number state probability distributionof the displaced-squeezed states which make up the su-perposition. In both measurements we observe clear dif-ferences depending on the force direction. We show thatthe SWES are coherent by reversing the effect of the SDF,resulting in recombination of the squeezed wavepackets,which we measure through the revival of the spin coher-ence.
The squeezed vacuum state |ξ〉 is defined by the ac-
tion of the squeezing operator S(ξ) = e(ξ∗ a2−ξ a†2)/2 onthe motional ground state |0〉, where ξ = reiφs with rand φs real parameters which define the magnitude andthe direction of the squeezing in phase space. To pre-pare squeezed states of motion in which the variance ofthe squeezed quadrature is reduced by around 9 dB rel-ative to the ground state wavepacket we utilize reservoirengineering, in which a bichromatic light field is usedto couple the ion’s motion to the spin states of the ionwhich undergo continuous optical pumping. This dissipa-tively pumps the motional state of the ion into the desiredsqueezed state, which is the dark state of the dynamics.More details regarding the reservoir engineering can befound in [18]. This approach provides a robust basis forall experiments described below, typically requiring nore-calibration over several hours of taking data. In theideal case, the optical pumping used in the reservoir engi-neering results in the ion being pumped to |↓〉. To createa SWES, we apply a SDF to this squeezed vacuum stateby simultaneously driving the red |↓〉 |n〉 ↔ |↑〉 |n− 1〉and blue |↓〉 |n〉 ↔ |↑〉 |n+ 1〉 motional sidebands of thespin flip transition [3]. The resulting interaction Hamil-tonian can be written in the Lamb-Dicke approximation
arX
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7100
v2 [
quan
t-ph
] 2
6 M
ay 2
015
2
as
HD = ~Ω
2σx
(a†e−iφD/2 + aeiφD/2
), (1)
where Ω is the strength of the SDF, φD is the relativephase of the two light fields, and σx ≡ |+〉 〈+| − |−〉 〈−|with |±〉 = (|↑〉±|↓〉)/
√2. For an ion prepared in |+〉, this
Hamiltonian results in displacement of the motional statein phase space by an amount α(τ) = −iΩe−iφD/2τ/2which is given in units of the r.m.s. extent of the har-monic oscillator ground state. An ion prepared in |−〉will be displaced by the same amount in the opposite di-rection. In the following equations, we use α in place ofα(τ) for simplicity. Starting from the state |↓〉 |ξ〉, appli-cation of the SDF ideally results in the SWES
|ψ(α)〉 =1√2
(|+〉 |α, ξ〉 − |−〉 |−α, ξ〉) , (2)
where we use the notation |α, ξ〉 = D(α)S(ξ) |0〉 with the
displacement operator D(α) = eα a†−α∗ a. A projective
measurement of the spin performed in the σz basis givesthe probability of being |↓〉 as P (↓) = (1 + X)/2, whereX = 〈α, ξ| − α, ξ〉 = 〈−α, ξ|α, ξ〉 gives the overlap be-tween the two displaced motional states, which can bewritten as
X(α, ξ) = e−2|α|2(exp(2r) cos2(∆φ)+exp(−2r) sin2(∆φ)) (3)
P
X
a c b
0 20 40 60 80 100 1200.4
0.5
0.6
0.7
0.8
0.9
1
P(↓
)
SDF duration (µs)
0 100 200 3000.4
0.6
0.8
1
SDF phase (degree)
P(↓
)
Ground state = /2 = 0
a
b
c
P
X
P
X
FIG. 1: Spin-population evolution due to spin-motionentanglement: Projective measurement of the spin in theσz basis as a function of SDF duration. a, Forces parallel tothe squeezed quadrature (red triangles). b, An ion initiallyprepared in the motional ground state (blue circles). c, Forcesparallel to the anti-squeezed quadrature (green squares). Theinset shows a scan of the phase of the SDF for an initialsqueezed state with the force duration fixed to 20 µs. Eachdata point is the result of > 300 repetitions of the experimen-tal sequence. The given error bars indicate one standard errorof the mean, and are generated under the assumption that thedominant source of fluctuations is quantum projection noise.
where ∆φ = arg(α) − φs/2. When ∆φ = 0, the SDF isaligned with the squeezed quadrature of the state, whilefor ∆φ = π/2, the SDF is aligned with the anti-squeezedquadrature. At displacements for which X gives a mea-surable signal, monitoring the spin population as a func-tion of the force duration τ for different choices of ∆φ al-lows us to characterise the spatial variation of the initialsqueezed wavepacket [15–17]. For values of |α|2 which aregreater than the wavepacket variance along the directionof the force, the state in equation (2) is a distinct su-perposition of squeezed wavepackets which have overlapclose to zero and are entangled with the internal state.For r = 0 (no squeezing) the state reduces to the familiar“Schrodinger’s cat” states which have been produced inprevious work [1, 3–5]. For r > 0 the superposed oscilla-tor states are the displaced-squeezed states [19, 20].
The experiments use a single trapped 40Ca+ ion, whichmechanically oscillates on its axial vibrational modewith a frequency close to ωz/(2π) = 2.1 MHz. Thismode is well resolved from all other modes. We encodea pseudo-spin system in the internal electronic states|↓〉 ≡
∣∣S1/2,MJ = 1/2⟩
and |↑〉 ≡∣∣D5/2,MJ = 3/2
⟩.
All coherent manipulations, including the squeezed statepreparation and the SDF, make use of the quadrupoletransition between these levels at 729 nm, with a Lamb-Dicke parameter of η ' 0.05 for the axial mode. This issmall enough that the experiments are well described us-
P
X
a b = 0 = /2
P
X
0 50 100 150 200 2500.4
0.5
0.6
0.7
0.8
0.9
1 τ1 = 60 µs
|∆α| ≃ 4.99
120 µs
9.99
160 µs
13.3
200 µs
16.6
250 µs
20.8
0 50 100 150 2000.4
0.5
0.6
0.7
0.8
0.9
1 τ1 = 60 µs
|∆α| ≃ 4.99
120 µs
9.99
P(↓
)
Recombined SDF duration (µs)
FIG. 2: Revival of the spin coherence. Spin populationsas a function of the duration of the second SDF pulse with thespin phase shifted by π relative to the first pulse. a, Forcesparallel to the squeezed quadrature. b, Forces parallel to theanti-squeezed quadrature. In all cases an increase in the spinpopulation is seen at the time when the two motional statesare overlapped, which corresponds to the time τ1 used forthe first SDF pulse. The value of τ1 and the corresponding|∆α| calculated from the measured Rabi frequency are writtenabove the revival of each dataset. The fractional error on themean of each of the estimated |∆α| is approximately 3%. Thesolid lines are fitted curves using the same form as using inthe fits in Fig. 1 with the overlap function X(δα, ξ). Theobtained values of r are consistent with the data in Fig. 1.The definition of error bars is the same as in Fig. 1.
3
ing the Lamb-Dicke approximation (a discussion of thisapproximation is given in the Methods) [21].
We apply the SDF directly after the squeezed vacuumstate has been prepared by reservoir engineering and theinternal state has been prepared in |↓〉 by optical pump-ing (in the ideal case, the ion is already in the correctstate and this step has no effect). Figure 1 shows theresults of measuring 〈σz〉 after applying displacementsalong the two principal axes of the squeezed state along-side the same measurement made using an ion preparedin the motional ground state. In order to extract rele-vant parameters regarding the SDF and the squeezing,we fit the data using P (↓) = (A + BX(α, ξ))/2, wherethe parameters A and B account for experimental imper-fections such as shot-to-shot magnetic field fluctuations(Methods). Fitting the ground state data with r fixedto zero allows us to extract Ω/(2π) = 13.25 ± 0.40 kHz(here and in the rest of the paper, all errors are given ass.e.m.). We then fix this in performing independent fitsto the squeezed-state data for ∆φ = 0 and ∆φ = π/2.Each of these fits allows us to extract an estimate for thesqueezing parameter r. For both the squeezed and anti-squeezed quadratures we obtain consistent values witha mean of r = 1.08 ± 0.03, corresponding to 9.4 dB re-duction in the squeezed quadrature variance. The insetshows the spin population as a function of the SDF phaseφD with the SDF duration fixed to 20 µs. This is alsofitted using the same equation described above, and weobtain r = 1.13± 0.03.
The loss of overlap between the two wavepackets indi-cates that a SWES has been created. In order to verifythat these states are coherent superpositions, we recom-bine the wavepackets by applying a second “return” SDFpulse for which the phase of both the red and blue side-band laser frequency components is shifted by π relativeto the first. This reverses the direction of the force ap-plied to the motional states for both the |+〉 and |−〉 spinstates. In the ideal case a state displaced to α(τ1) by afirst SDF pulse of duration τ1 has a final displacementof δα = α(τ1) − α(τ2) after the return pulse of durationτ2. For τ1 = τ2, δα = 0 and the measured probabilityof finding the spin state in |↓〉 is 1. In the presence ofdecoherence and imperfect control, the probability withwhich the ion returns to the |↓〉 state will be reduced. InFig. 2 we show revivals in the spin coherence for the sameinitial squeezed vacuum state as was used for the data inFig. 1. The data include a range of different τ1. Forthe data where the force was applied along the squeezedaxis of the state (∆φ = 0), partial revival of the coher-ence is observed for SDF durations up to 250 µs. Forτ1 = 250 µs the maximum separation of the two distinctoscillator wavepackets is |∆α| > 19, which is 56 timesthe r.m.s width of the squeezed wavepacket in phasespace. The amplitude of revival of this state is similarto what we observe when applying the SDF to a groundstate cooled ion. The loss of coherence as a function ofthe displacement duration is consistent with the effectsof magnetic-field induced spin dephasing and motional
heating [14, 22]. When the force is applied along theanti-squeezed quadrature (∆φ = π/2), we observe thatthe strength of the revival decays more rapidly than fordisplacements with ∆φ = 0. Simulations of the dynam-ics using a quantum Monte-Carlo wavefunction approachincluding sampling over a magnetic field distribution in-dicate that this is caused by shot-to-shot fluctuations ofthe magnetic field (Methods).
We are also able to monitor the number state distri-butions of the motional wavepackets as a function ofthe duration of the SDF. This provides a second mea-surement of the parameters of the SDF and the initialsqueezed wavepacket which has similarities with the ho-modyne measurement used in optics [23, 24]. In or-der to do this, we optically pump the spin state into|↓〉 after applying the SDF. This procedure destroys thephase relationship between the two motional wavepack-ets, resulting in the mixed oscillator state ρmixed =(|α, ξ〉 〈α, ξ|+ |−α, ξ〉 〈−α, ξ|) /2 (we estimate the photonrecoil during optical pumping results in a reduction inthe fidelity of our experimental state relative to ρmixed
by < 3%, which would not be observable in our measure-ments). The two parts of this mixture have the samenumber state distribution, which is that of a displaced-squeezed state [19, 20]. In order to extract this distri-bution, we drive Rabi oscillations on the blue-sidebandtransition [25] and monitor the subsequent spin popu-lation in the σz basis. Figure 3 shows this evolutionfor SDF durations of τ = 0, 30, 60 and 120 µs. Forτ = 30 and 60 µs, the results from displacements ap-plied parallel to the two principal axes of the squeezedstate are shown (∆φ = 0 and π/2). We obtain thenumber state probability distribution p(n) from the spinstate population by fitting the data using a form P (↓) =bt+ 1
2
∑n p(n)(1+e−γt cos(Ωn,n+1t)), where t is the blue-
sideband pulse duration, Ωn,n+1 is the Rabi frequency forthe transition between the |↓〉 |n〉 and |↑〉 |n+ 1〉 statesand γ is a phenomenological decay parameter [25, 26].The parameter b accounts for gradual pumping of pop-ulation into the state |↑〉 |0〉 due to frequency noise onour laser [18, 27]. It is negligible when p(0) is small.The resulting p(n) are then fitted using the theoreticalform for the displaced-squeezed states (Methods). Thenumber state distributions show a clear dependence onthe phase of the force, which is also reflected in the spinpopulation evolution. Figure 4 shows the Mandel Q pa-rameters of the experimentally obtained number statedistributions, defined as Q = 〈(∆n)2〉/〈n〉 − 1 in which〈(∆n)2〉 and 〈n〉 are the variance and mean of p(n) re-spectively [28]. The solid lines are the theoretical curvesgiven by Caves [19] for r = 1.08, and are in agreementwith our experimental results. For displacements alongthe short axis of the squeezed state (Fig. 3), the col-lapse and revival behaviour of the time evolution of P (↓)is reminiscent of the Jaynes-Cummings Hamiltonian ap-plied to a coherent state [29], but it exhibits a highernumber of oscillations before the “collapse” for a stateof the same 〈n〉. This is surprising since the statistics of
4
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
P(↓
)a
r = 1.08 ± 0.03
0 5 10 150
0.2
0.4
0.6
0.8
n
p(n
)
τ = 0 µs
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Blue sideband pulse duration (µs)
c
r = 1.03 ± 0.04, α = 1.09 ± 0.04
0 5 10 15 200
0.2
0.4
0.6
n
p(n
)
τ = 30 µs, ∆φ = π/2
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
e
r = 1.05 ± 0.05, α = 2.12 ± 0.07
0 5 10 15 20 250
0.1
0.2
0.3
0.4
n
p(n
)
τ = 60 µs, ∆φ = π/2
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
P(↓
)
b
r = 1.10 ± 0.06, α = 1.18 ± 0.02
0 5 10 150
0.1
0.2
0.3
0.4
n
p(n
)
τ = 30 µs, ∆φ = 0
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Blue sideband pulse duration (µs)
d
r = 1.03 ± 0.08, α = 2.38 ± 0.02
0 5 10 150
0.1
0.2
0.3
n
p(n
)
τ = 60 µs, ∆φ = 0
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
f
r = 1.15 ± 0.29, α = 4.58 ± 0.04
0 5 10 15 20 25 300
0.1
0.2
0.3
n
p(n
)
τ = 120 µs, ∆φ = 0
FIG. 3: Evolution of displaced-squeezed state mixtures: The observed blue-sideband oscillations and the correspondingnumber state probability distributions for the SDF applied along the two principal axes of the squeezed state and with differentdurations. a, Initial squeezed vacuum state. b, d, f, Forces parallel to the squeezed quadrature. c, e, Forces parallel to theanti-squeezed quadrature. For τ = 30 µs the obtained parameters are consistent within statistical errors. For τ = 60 µs thedisplacement along the anti-squeezed quadrature (e) results in a large spread in the number state probability distribution, withthe result that in the fitting r and α are positively correlated - the errors stated do not take account of this. We think thatthis accounts for the apparent discrepancy between the values of r and α obtained for τ = 60 µs. The green-dashed line in theinset of d and f is the Poisson distribution for the same 〈n〉 as the created displaced-squeezed state mixture, which is given by〈n〉 = |α|2 + sinh2r [19]. The definition of error bars is the same as in Fig. 1.
the state is not sub-Poissonian. We attribute this to thefact that this distribution is more peaked than that of acoherent state with the same 〈n〉, which is obvious whenthe two distributions are plotted over one another (Figs.3(d) and 3(f)). The increased variance of the squeezedstate then arises from the extra populations at high n,which are too small to make a visible contribution tothe Rabi oscillations. For the squeezing parameter inour experiments sub-Poissonian statistics would only beobserved for |α| > 3. For τ = 120 µs we obtain a con-sistent value of r and |α| = 4.6 only in the case wherewe include a fit parameter for scaling of the theoreti-cal probability distribution, obtaining a fitted scaling of0.81±0.10 (Methods). The reconstruction of the numberstate distribution is incomplete, since we cannot extractpopulations with n > 29 due to frequency crowding in the√n+ 1 dependence of the Jaynes-Cummings dynamics.
As a result, we do not include these results in Fig. 4.Measurement techniques made in a squeezed-state basis[18] could avoid this problem, however these are beyondour current experimental capabilities for states of this
size.
We have generated entangled superposition statesbetween the internal and motional states of a singletrapped ion in which the superposed motional wavepack-ets are of a squeezed Gaussian form. These states presentnew possibilities both for metrology and for continuousvariable quantum information. In an interferometerbased on SC states separated by |∆α|, the interferencecontrast depends on the final overlap of the re-combinedwavepackets. Fluctuations in the frequency of theoscillator result in a reduced overlap, but this effect can
be improved by a factor exp[− |∆α|2 (e−2r − 1)/2
]if
the wavepackets are squeezed in the same direction asthe state separation (Methods). In quantum informationwith continuous variables, the computational basisstates are distinguishable because they are separatedin phase space by |∆α| and thus do not overlap [7–9].The decoherence times of such superpositions typicallyscale as 1/|∆α|2 [22]. The use of states squeezedalong the displacement direction reduces the required
5
0 0.5 1 1.5 2 2.5 3 3.5−1
0
2
4
6
8Q
par
amet
er
|α|
FIG. 4: Mandel Q parameter for the displaced-squeezed states: Shown are the results for displacementsalong the squeezed quadrature (red triangles) and the anti-squeezed quadrature (green squares). All the values are cal-culated from the experimental data given in Fig. 3, takingthe propagation of error into account. The solid lines aretheoretical curves for displacements along the squeezed (red)and anti-squeezed (green) quadratures of an initial state withr = 1.08. The values of |α| are obtained from fits to the re-spective p(n) (Fig. 3), with error bars comparable to the sizeof the symbol. The point at |α| = 0 is the squeezed vacuumstate.
displacement for a given overlap by er, increasing theresulting coherence time by e2r which is a factor of 9 inour experiments. We therefore expect these states toopen up new possibilities for quantum state engineeringand control.
Acknowledgement: We thank Joseba Alonso andFlorian Leupold for comments on the manuscript andthank Florian Leupold, Frieder Lindenfelser, JosebaAlonso, Martin Sepiol, Karin Fisher, and ChristaFluhmann for contributions to the experimental appara-tus. We acknowledge support from the Swiss NationalScience Foundation under grant number 200021 134776,and through the National Centre of Competence inResearch for Quantum Science and Technology (QSIT).
Author Contributions: Experimental data weretaken by H.Y.L., D.K., and L.d.C., using an apparatusprimarily built up by D.K., H.Y.L., and B.C.K., andwith significant contributions from L.d.C., V.N., andM.M.. Data analysis was performed by H.Y.L. andJ.P.H.. The paper was written by J.P.H. and H.Y.L.,with input from all authors. The work was conceived byJ.P.H..
The authors have no competing financial interests.
Methods
Experimental details. The experiments make useof a segmented linear Paul trap with an ion-electrode
distance of ≈ 185 µm. Motional heating rates from theground state for a calcium ion in this trap have beenmeasured to be 10 ± 1 quanta s−1, and the coherencetime for the number state superposition (|0〉 + |1〉)/
√2
has been measured to be 32 ± 3 ms.The first step of each experimental run involves cool-
ing all modes of motion of the ion close to the Dopplerlimit using laser light at 397 and 866 nm. The laserbeam used for coherent control of the two-level pseudo-spin system addresses the narrow-linewidth transition|↓〉 ≡
∣∣S1/2,MJ = 1/2⟩↔ |↑〉 ≡
∣∣D5/2,MJ = 3/2⟩
at729 nm. This transition is resolved by 200 MHz from allother internal state transitions in the applied magneticfield of 119.6 Gauss (G). The State-Dependent Forces(SDFs) and the reservoir engineering [18] in our experi-ment require the application of a bichromatic light field.We generate both frequency components using Acousto-Optic-Modulators (AOMs) starting from a single laserstabilized to an ultra-high-finesse optical cavity with aresulting linewidth < 600 Hz (at which point magneticfield fluctuations limit the qubit coherence). We applypulses of 729 nm laser light using a double-pass AOM towhich we apply a single radio-frequency tone, followed bya single-pass AOM to which two radio-frequency tonesare applied. Following this second AOM, both frequencycomponents are coupled into the same single-mode fibrebefore delivery to the ion. The double-pass AOM is usedto switch on and off the light. Optical pumping to |↓〉is implemented using a combination of linearly polarizedlight fields at 854 nm, 397 nm and 866 nm. The internalstate of the ion is read out by state-dependent fluores-cence using laser fields at 397 nm and 866 nm.
The 729 nm laser beam enters the trap at 45 degreesto the z axis of the trap resulting in a Lamb-Dicke pa-rameter of η ' 0.05 for the axial mode. For this Lamb-Dicke parameter, we have verified whether for displace-ments up to |α| = 9.75 the dynamics can be well de-scribed with the Lamb-Dicke Approximation (LDA). Wesimulate the wavepacket dynamics using the interactionHamiltonian with and without LDA. In the simulation,we apply the SDF to an ion prepared in |↓〉 |ξ〉. The in-teraction Hamiltonian for a single trapped ion coupled toa single-frequency laser field can be written as [26]
HI =~2
Ω0σ+expiη( ae−iωzt + a†eiωzt)ei(φ−δt) + h.c.,
where Ω0 is the interaction strength, σ+ = |↑〉 〈↓|, a anda† are motional annihilation and creation operators, ωzis the vibrational frequency of the ion, φ is the phaseof the laser, and δ = ωl − ωa the detuning of the laserfrom the atomic transition. In the laboratory, the appli-cation of the SDF involves simultaneously driving boththe blue and red sideband transitions resonantly result-ing in the Hamiltonian Htot = Hbsb +Hrsb, where δ = ωzin Hbsb and δ = −ωz in Hrsb. Starting from |↓〉 |ξ〉, theevolution of the state can not be solved analytically. Weperform a numerical simulation in which we retain onlythe resonant terms in the Hamiltonian. Figure 5 shows
6
Re(α)
Im(α
)
a
τ = 60 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
Re(α)
c
τ = 120 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
Re(α)
e
τ = 250 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
Re(α)
Im(α
)
b
τ = 60 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
Re(α)
d
τ = 120 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
Re(α)
f
τ = 250 µs
−10 −5 0 5 10
−10
−5
0
5
100
0.02
0.04
0.06
0.08
FIG. 5: Quasi-probability distributions for displaced-squeezed states in phase space using LDA and non-LDA: a,c, e, The simulation results using LDA with different SDF durations. b, d, f, The results simulated using the full Hamiltonian.
the quasi-probability distributions in phase space for cho-sen values of the SDF duration τ . These are compared toresults obtained using the LDA. For τ = 60 µs both casesare similar, resulting in |α| ' 2.4. For τ = 250 µs thesqueezed state wavepackets are slightly distorted and thedisplacement is 4% smaller for the full simulation thanfor the LDA form. Considering the levels of error arisingfrom imperfect control and decoherence for forces of thisduration, we do not consider this effect to be significantin our experiments.
Simulations for the coherence of SWESs. Aftercreating SWESs, we deduce that coherence is retainedthroughout the creation of the state by applying a sec-ond SDF pulse to the ion, which recombines the two sep-arated wavepackets and disentangles the spin from themotion. The revival in the spin coherence is not perfectdue to decoherence and imperfect control in the experi-ment. One dominant source causing decoherence of thesuperpositions is spin decoherence due to magnetic fieldfluctuations. We have performed quantum Monte-Carlowavefunction simulations to investigate the coherence ofthe SWES in the presence of such a decoherence mecha-nism. We simulate the effect of a sinusoidal fluctuationof the magnetic field on a timescale long compared to theduration of the coherent control sequence, which is con-sistent with the noise which we observe on our magneticfield coil supply (at 10 and 110 Hz) and from ambientfluctuations due to electronics equipment in the room.The amplitude of these fluctuations is set to ' 2.2 mG,giving rise to the spin coherence time of 180 µs which we
have measured using Ramsey experiments on the spinalone. Since the frequency of fluctuations is slow com-pared to the sequence length, we fix the field for eachrun of the simulation, but sample its value from a prob-ability distribution derived from a sinusoidal oscillation.In Fig. 6 we show the effect of a single shot taken at a
0 20 40 600.4
0.6
0.8
1
P(↓
)
a
0 20 40 600.4
0.6
0.8
1
0 30 60 90 1200.4
0.6
0.8
1
SDF duration (µs)
P(↓
)
b
0 30 60 90 1200.4
0.6
0.8
1
Recombined SDF duration (µs)
FIG. 6: Coherence of cat states with fixed magneticfield noise: The magnetic-field-induced energy-level-shift of1.5 kHz is used in this simulation. a, The duration of bothSDF pulses is 60 µs. b, The duration of both SDF pulsesis 120 µs. Red-dashed and green-dash-dot curves show theSDF aligned along the squeezed and anti-squeezed quadra-tures. The blue trace is for the SDF applied to a groundstate cooled ion.
7
0 20 40 600.4
0.6
0.8
1 P
(↓)
a
0 20 40 600.4
0.6
0.8
1
0 30 60 90 1200.4
0.6
0.8
1
SDF duration (µs)
P(↓
)
b
0 30 60 90 1200.4
0.6
0.8
1
Recombined SDF duration (µs)
FIG. 7: Coherence of cat states with a magnetic fieldfluctuation distribution: Assuming the magnetic field ex-hibits a 50-Hz sinusoidal pattern with an amplitude of 2.2 mG,this plot shows the simulation results by taking an averageover 100 samples on the field distribution. a, The durationof both SDF pulses is 60 µs. b, The duration of both SDFpulses is 120 µs. Definitions of the curve specification are thesame as Extended Data Fig. 2.
fixed qubit-oscillator detuning of 1.5 kHz, while in Fig.7 we show the average over the distribution. In bothfigures results are shown for the SDF applied along thetwo principal axes of the squeezed vacuum state as wellas for the motional ground state using force durations of60 and 120 µs. We also show the results of applying thesecond SDF pulse resulting in partial revival of the spincoherence. It can be clearly seen that when the SDF isapplied along the anti-squeezed quadrature, the strength
of the revival decays more rapidly, and P (↓) oscillatesaround 0.5. This effect can be seen in the data shown inFig. 2 of the main article.
Number state probability distributions for thedisplaced-squeezed state. For Fig. 3 in the mainarticle, we characterise the probability distribution forthe number states of the oscillator. This is performed bydriving the blue-sideband transition |↓〉 |n〉 ↔ |↑〉 |n+ 1〉and fitting the obtained spin population evolution using
P (↓) = bt+1
2
∑n
p(n)(1 + e−γt cos(Ωn,n+1t)), (4)
where t is the blue-sideband pulse duration, p(n) are thenumber state probabilities for the motional state we con-cern, and γ is an empirical decay parameter [25, 26].In the results presented here we do not scale this de-cay parameter with n as was done by [25]. We havealso fitted the data including such a scaling and seeconsistent results. The Rabi frequency coupling |↓〉 |n〉to |↑〉 |n+ 1〉 is Ωn,n+1 = Ω0| 〈n| eiη( a†+ a) |n+ 1〉 | =
Ω0e−η2/2ηL1
n(η2)/√n+ 1. For small n, this scales as√
n+ 1, but since the states include significant popula-tions at higher n we use the complete form including thegeneralized Laguerre polynomial L1
n(x). The parameterb in the first term accounts for a gradual pumping of pop-ulation into the state |↑〉 |0〉 which is not involved in thedynamics of the blue-sideband pulse [18, 27]. This effectis negligible when p(0) is small.
After extracting p(n) from P (↓), we fit it using thenumber state probability distribution for the displaced-squeezed state [30],
p(n) = κ( 1
2 tanh r)n
n! cosh rexp
[−|α|2 − 1
2(α∗2eiφs + α2e−iφs) tanh r
] ∣∣∣∣Hn
[α cosh r + α∗eiφssinh r√
eiφs sinh 2r
]∣∣∣∣2 ,
where κ is a constant which accounts for the infidelity ofthe state during the application of SDF and the Hn(x)are the Hermite polynomials. The direction of the SDFis aligned along either the squeezing quadrature or theanti-squeezing quadrature of the state. Therefore, we setarg(α) = 0 and fix φs = 0 and π for fitting the data ofthe short axis and the long axis of the squeezed state,respectively. This allows us to obtain the values of rand |α| for the state we created. For the cases of smallerdisplacements (from Figs. 3(a) to (e) in the main article),we set κ = 1. For the data set of |α| ' 4.6 (Fig. 3(f)in the main article), κ is a fitting parameter which givesus a value of 0.81 ± 0.1. We note that in this case 4%of the expected population lies above n = 29 but we arenot able to extract these populations from our data.
The Mandel Q parameter [28], defined as
Q =〈(∆n)2〉 − 〈n〉
〈n〉.
where 〈n〉 and 〈(∆n)2〉 are the mean and variance of theprobability distribution. For a displaced-squeezed statethese are given by Caves [19] as
〈(∆n)2〉 =∣∣α cosh r − α∗eiφssinh r
∣∣2 + 2 cosh2r sinh2r,
〈n〉 = |α|2 + sinh2r.
These forms were used to produce the curves given inFig. 4 of the main article.
Applications of SWESs. The SWES may offer newpossibilities for sensitive measurements which are robust
8
∆𝜃
P
X
a
X X
P P
b P
X X X
P P
∆𝜃
After 1st SDF pulse Phase shift
induced by noise
Apply 2nd SDF pulse
FIG. 8: Possible application of using SWESs for inter-ferometry: a, Use of squeezed state wavepackets. b, Useof ground state wavepackets. The first SDF pulse is used tocreate a spin-motion entangled state. In the middle, a smallphase shift ∆θ is induced by shot-to-shot fluctuation in theoscillator frequency before the application of the second SDFpulse, which recombines the two distinct oscillator wavepack-ets.
against certain types of noise. An example is illustratedin Fig. 8 where we compare an interferometry experi-
ment involving the use of a SWES versus a more standardSchrodinger’s cat state based on coherent states. In bothcases the superposed states have a separation of |2α| ob-tained using a SDF. For the SWES this force is alignedalong the squeezed quadrature of the state. The inter-ferometer is closed by inverting the initial SDF, resultingin a residual displacement which in the ideal case is zero.One form of noise involves a shot-to-shot fluctuations inthe oscillator frequency. On each run of the experiment,this would result in a small phase shift ∆θ arising be-tween the two superposed motional states. As a result,after the application of the second SDF pulse the resid-ual displacement would be αR = 2iα sin(∆θ/2), whichcorresponds to the states being separated along the Paxis in the rotating-frame phase space. The final state ofthe system would then be |ψ(αR)〉 with a correspondingstate overlap given by X(αR, ξ). Therefore the contrastwill be higher for the SWES (Extended Data Fig. 4(a))than for the coherent Schrodinger’s cat state (ExtendedData Fig. 4(b)) by a factor
exp[−2 |αR|2 (e−2r − 1)
].
While in our experiments other sources of noise domi-nate, in other systems such oscillator dephasing may bemore significant.
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