Tomography of Entangled Macroscopic Mechanical Ob-jects

Shlomi Kotler1,2, Gabriel A. Peterson1,2, Ezad Shojaee1,2, Florent Lecocq1,2, Katarina Cicak1, Alex

Kwiatkowski1,2, Shawn Geller1,2, Scott Glancy1, Emanuel Knill1,3, Raymond W. Simmonds1, Jose

Aumentado1, & John D. Teufel1

1National Institute of Science and Technology, Boulder, CO 80305, USA.

2Department of Physics, University of Colorado, Boulder, CO 80309, USA.

3Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA.

Observing quantum mechanics at the macroscopic scale has captured the attention of scien-

tists and the imagination of the public for more than a century. While quantum mechanics

was conceived in the context of electrons and atoms, the ability to observe its properties on

ever more macroscopic systems holds great promise for fundamental research and techno-

logical applications. Therefore, researchers have been preparing larger material systems

in interesting quantum states and, in particular, entangled states of separate mechanical

oscillators 1–3. As these quantum devices move from demonstrations to applications, their

full potential can only be realized by combining entanglement generation with an efficient

measurement of the joint mechanical state. Unfortunately, such a high level of control and

measurement can expose the system to undesired interactions with its environment, a prob-

lem that becomes more pronounced at the macroscopic scale. Here, using a superconducting

electromechanical circuit and a pulsed microwave protocol, we ground-state cool, entangle

and perform state tomography of two mechanical drumheads with masses of 70 pg. Entan-

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glement is generated deterministically and is followed by a nearly quantum-limited measure-

ment of the positions and momentums of both mechanical oscillators. From these efficient

measurements, the resulting tomography demonstrates entanglement without noise subtrac-

tion. Highly entangled, massive quantum systems, as demonstrated here, are uniquely poised

to address fundamental tests of quantum mechanics 4, 5, enable force sensing beyond the stan-

dard quantum limit 6, and possibly serve as long-lived nodes of a future quantum network 7, 8.

The experimental pursuit of macroscopic quantum phenomena straddles the line between

seemingly contradicting constraints. Since quantum states are very fragile, usually they can only

persist when well isolated from their surroundings. However, a perfectly isolated system cannot be

seen, measured, or manipulated since these actions involve interaction with either an experimental

meter or control. Therefore some interaction is needed, especially if the quantum state is to be

used as a resource for applications in quantum information processing. This problem becomes

more pronounced for larger systems that tend to interact more strongly with their environments,

which causes quantum coherence to vanish more quickly.

The competing requirements of environmental isolation and experimental control and mea-

surement are well demonstrated by considering two leading platforms for quantum information

processing: trapped ions 9 and superconducting qubits 10. In a trapped-ions system, quantum infor-

mation is stored in the internal atomic state of individual ions. Due to their small dipoles, trapped

ions can be well-isolated from their environment. The challenge is to strongly and efficiently in-

terface many ions in a programmable and scalable manner. In contrast, superconducting qubits are

designed to have large electrical dipoles, which can easily be controlled using standard microwave

2

engineering techniques. Therefore, it is challenging to isolate them from their environment and

allow them to retain quantum coherence for long enough to allow for processing.

Hybrid quantum systems have the potential to alleviate the inherent tension between isola-

tion and interaction 11. In a hybrid system, two (or more) degrees of freedom inhabit a single

platform. Typically, one degree of freedom is used as a quantum memory while another is used to

interface the experimental apparatus. In order to measure or manipulate the memory, one turns on

an interaction between the two degrees of freedom for as short as possible while still allowing full

control. In this paper, we describe a hybrid system composed of mechanical oscillators that are

well isolated from their environments and each other, and an electrical resonator that can efficiently

interface the mechanical oscillators with the experimental instrumentation. Since the mechanical

and electrical counterparts are sensitive to different interactions, they can be optimized separately

for their different purposes. The mechanical oscillators are long-lived and can retain energy for

milliseconds, while the electrical resonator can be controlled and measured strongly in a fraction

of a microsecond 12. Moreover, the mechanical and electrical degrees of freedom only interact

when a microwave pulse is applied. This separation of roles has potential quantum and classical

applications, for example, in sensing, communication and storage 8.

Some future applications rely on continuous variable (CV) entanglement 13 and CV quantum

information processing 14. Originally conceived by Einstein, Podolsky and Rosen 15, CV entan-

glement has been demonstrated in various systems, such as: optical fields 16, 17, spin ensembles 18,

optical polarization 19, microwave fields 20, 21, mechanical motion and microwave fields 22, and re-

cently using a mechanical oscillator as a mediator to entangle two microwave modes 23 and optical

3

modes 24.

For entanglement to be useful, generation alone is not enough; one also needs efficient mea-

surements. Consider the following simplified model of an experiment (see Fig.1a). First, entan-

glement is generated between two mechanical oscillators. Ideally, the entangled state can then be

used for some quantum information task by a hypothetical user that measures state observables.

In practice, however, loss mechanisms degrade the state by mixing it with noise 25, 26, quantified

by the measurement efficiency parameter 0 ≤ η ≤ 1. If η = 1, then all of the entanglement is

available for use. If η = 0, then only un-entangled noise remains. Typical low efficiencies can still

be sufficient for state characterization, i.e. to infer what the original entangled state was before

loss mechanisms intervened, but this requires long averaging times and, in many cases, noise sub-

traction. Unfortunately, quantum information protocols, such as teleportation and entanglement

swapping, require high efficiency, as demonstrated for optical modes 27, 28.

One approach to overcoming the difficulty of entangling macroscopic mechanical oscillators

is to use the electromagnetic field as a mediator 4, 29–36. Indeed, two experiments demonstrated

entanglement of massive mechanical oscillators, one using pulsed heralded optical photons 2, and

another using a steady-state scheme with microwave fields 3. However, due to various constraints

of the experimental designs used, a significant portion of the information was lost before it could

reach a hypothetical user. Separately, other experiments, using pulsed interactions, showed that a

single mechanical oscillator can be independently cooled to the ground state 37, 38 and measured

with high efficiency 22, 38–40. A clear next step towards CV quantum information processing 14 has

been to combine deterministic entanglement generation with high detection efficiency in a single

4

platform. Here we strongly entangle two massive mechanical oscillators and perform efficient

tomography that completely characterizes their joint covariance matrix. Ground-state cooling,

entanglement and readout are separately optimized through a combination of frequency and time-

domain multiplexed microwave pulses. Moreover, we measure the positions and momentums of

both mechanical oscillators in every single realization of the experiment. We demonstrate clear

evidence of entanglement in the measurement signals, without noise subtraction. We quantify

entanglement using the Simon-Duan criterion 41–43 and find νmeas = 0.44+0.004−0.004 (stat)+0.033

−0.028 (sys) <

0.5. Accounting for noise to correct for measurement inefficiency, we infer the original entangled

state and show that it has ν = 0.15+0.03−0.02 (stat)+0.19

−0.05 (sys).

Our two mechanical oscillators are made of lithographically-patterned thin-film aluminum

that form drum-like membranes, each with a mass of ≈ 70 pg, suspended above a sapphire sub-

strate, as shown in Fig. 1b. We use the fm,1 = 10.9 MHz mode of the left drum in Fig. 1b and

the fm,2 = 15.9 MHz mode of the right drum. By design, the drums have no acoustic interaction

with each other. Here we use electro-mechanics 44 in order to mediate the interaction between the

drums. Below each suspended drum, we fix an aluminum bottom plate to the substrate, so each

drum is the top plate of a parallel-plate capacitor. A change in the position of each drumhead

changes the capacitance of its corresponding capacitor. Both capacitors are shunted by a shared

superconducting aluminum inductor, as shown in Fig. 1c. The capacitors and inductor form a sin-

gle microwave resonator, known as the ‘cavity’, whose resonance frequency depends on the drums

positions, and is centered at fc = 6.0806 GHz. Thus, information about the drums can be encoded

into the microwave field. We uniquely associate an acoustic frequency with a specific drum since

their eigenmode frequencies are not degenerate and their bottom plates are split along the long and

5

short axes of the drums (see Fig. 1b). The latter changes the transduction strength between the

drums’ motions and the microwave cavity in a way that depends on the spatial overlap between the

bottom plate and the acoustic mode shape, and therefore is drum-dependent. The cavity is coupled

to the outside world using an inductive coupling as shown in Fig. 1d, at a rate of≈ 800 kHz. A mi-

crowave pulse, reflected off the cavity, imparts forces on the drums and encodes the amplitudes of

their positions and momentums into Doppler-shifted sidebands of the microwave pulse. The carrier

frequency of the microwave pulse determines the nature of the interaction. A red-sideband (RSB)

pulse is sent with a carrier frequency of fd = fc−fm, where fm is the mechanical frequency of the

drum of interest. It generates an effective beam-splitter interaction 37, Hrsb = hgr(t)(ab† + ba†),

where gr(t) is proportional to the pulse power at time t, h is the Planck constant divided by 2π, a

and a† are the annihilation and creation operators of the microwave cavity photons and b and b† are

the annihilation and creation operators of the drum’s phonons, satisfying the canonical commuta-

tion relations of [b, b†] = [a, a†] = 1. We use the RSB interaction to cool the mechanical mode

nearly to the ground state 45. In contrast, a blue-sideband (BSB) pulse has a carrier frequency of

fd = fc + fm, resulting in an effective two-mode squeezing interaction, Hbsb = hgb(t)(a†b† + ba).

These pulses have been shown to entangle the motion of a mechanical oscillator with the itinerant

reflected microwave field 22. As shown in Fig. 1e, an experimental sequence includes an initial

ground-state cooling pulse, followed by a middle entangling pulse and a final readout pulse, as

detailed below.

To entangle the two drums, we use a single microwave pulse, that combines a BSB with

respect to drum 1, and an RSB with respect to drum 2. Specifically, the BSB entangles the cavity

with drum 1, generating correlated photon-phonon pairs. At the same time, the RSB swaps some

6

of the cavity photons with phonons in drum 2. Therefore, some of the correlations that originated

from the BSB interaction become distributed between the two drums. The two-drum system can

be characterized by the first and second moments of the unit-less positions Xi = (b†i + bi)/√

2 and

momentums Pi = i(b†i − bi)/√

2 for i = 1, 2. Our entangled state should exhibit four features:

(1) The joint probability distribution of position and momentum of each individual drum should

be approximately thermal. (2) The drum that receives a BSB interaction should have a higher tem-

perature than the drum that receives an RSB interaction. (3) The positions should be correlated,

and the momentums should be anti correlated. (4) The angle determining the squeezed and anti-

squeezed joint quadratures of the two-drum system depends on the duration of the entanglement

pulse. However, these features could be consistent with classical correlations. To verify that the

correlations signal entanglement, we use the Simon-Duan criterion 42, 46, which can be derived by

applying the positive partial transpose criterion 47 to oscillator states. The Simon-Duan criterion

is calculated from the covariance matrix C of ~S = {X1, P1, X2, P2}. Element i, j of the 4 × 4

covariance matrix is Cij = 〈(Si − 〈Si〉)(Sj − 〈Sj〉)〉 where 〈. . .〉 denotes the expectation value.

The smallest symplectic eigenvalue ν of the covariance matrix of the partially-transposed density

matrix, quantifies the entanglement of the system 43, 48 (see Methods Summary for explicit expres-

sions). The two-drum state is entangled if ν < 12, where the zero-point fluctuations have variance

12.

In practice, proving entanglement with minimal assumptions comes with a heavy price, as

it requires knowledge of the full covariance matrix. To estimate it, we measure the position and

momentum of the two drums in each single experiment, analogously to a heterodyne measurement

of electromagnetic radiation. As argued theoretically 49, our strategy of measuring both positions

7

and momentums allows more accurate covariance matrix estimation (for a given number of state

preparations) than a strategy based on homodyne detection in which each drum’s phase is rotated

and then only one quadrature of each drum is measured. Crucially, this concurrent measurement

improves substantially if it is efficient. The inefficiency of our measurement apparatus can be

modeled as an effective beam-splitter 22, 37, 40, where each variable describing the mechanics Si ∈

{X1, P1, X2, P2}, becomes mixed with vacuum noise such that, si =√ηiSi + (

√1− ηi)ξi, where

ηi is the efficiency of the measurement of drum i, ξi is a Gaussian random variable with zero mean

and vacuum variance 〈ξ2i 〉 = 12

and ξi and ξj are independently distributed for i 6= j. Therefore,

following calibration of the measurement chain, we have direct access to the measured variables si.

For a two-mode thermal squeezed state with minimal symplectic eigenvalue ν, we observe νmeas:

νmeas = ην + (1− η)12, (1)

where η =√η1η2 is the geometric mean of the two efficiencies. If efficiencies are low, νmeas will

approach 12, and it will be more difficult to certify that νmeas <

12

with high confidence.

We are able to perform efficient measurements of the drums by using a pulsed, parametric

amplification of their variables. Measurement is performed by monitoring the reflected microwave

pulse after it has interacted with the drums. Since the device must be well-isolated from the en-

vironment, it is cooled inside a low-temperature dilution refrigerator (< 20 mK). Therefore, the

microwave signal originating from the quantum device has to pass through multiple stages of

transmission, isolation and amplification before it is recorded. Typically, this results in an effective

measurement efficiency that is ∼ 0.01. Here we achieve higher effective efficiency by using the

8

BSB interactions native to the device as a preamplifier, following techniques that were developed

for single drum readout 39, 40. Using frequency multiplexing we have extended these methods to im-

prove our measurement efficiency by more than an order of magnitude. We employ simultaneous

amplified readout of the two drums with total efficiencies of η1 = 0.29(6) and η2 = 0.156(5), and

a geometric mean of η = 0.21(4). Characterizations of the efficiencies are explained in the Supple-

mentary Information. Notice that for an ideal concurrent measurement of position and momentum

η, η1 and η2 are less than 0.5, as constrained by the Heisenberg uncertainty principal.

Figure 2 shows tomography of the two-drum system, as characterized by its covariance ma-

trix, for different protocols. First, we prepare a low temperature thermal state by applying a pulsed

sequence of ground state cooling followed by readout, rendering a single concurrent measurement

of x1, p1, x2, p2. In Fig. 2a we show the experimental distribution of the measured variables for

10, 000 repetitions of the experiment. Indeed the scatter plots show no correlation between any

of the measured variables. This is reaffirmed in Fig. 2b by the covariance matrix of the state,

which is diagonal to a good approximation. From the magnitude of the diagonal elements, we

infer nearly ground-state occupancies of n1 = 0.79(1) and n2 = 0.60(1) for drum 1 and 2 re-

spectively. We now turn to a pulse sequence of ground state cooling, entanglement and readout.

Fig. 2c exhibits all the expected features of a highly correlated state. First, the position-momentum

distribution of each drum is consistent with an effective thermal state of occupancy n1 = 11.5(1)

and n2 = 4.42(4), respectively. As expected, drum 1, which undergoes a BSB interaction, is hotter

than drum 2, which undergoes an RSB interaction. Second, a clear signature of drum-drum interac-

tion is demonstrated by the correlation of x1, x2 and the anti-correlation of p1, p2. The covariance

matrix of the measured variables in Fig. 2d displays a dominant diagonal and four off-diagonal

9

elements C1,3 ∼ −C2,4 and C3,1 ∼ −C4,2 (See Supplementary Information for how the covariance

matrix is estimated). Indeed, clear correlations are directly observable in the measured variables.

Delineating those from classical correlations requires an application of the Simon-Duan criteria.

Figure 3 shows the evolution of the two-drum state for different entangling pulse durations,

all of which are kept significantly shorter than 100 µs to avoid the thermal decoherence of either

drum. First, we focus on the individual variances of each drum (Fig. 3a). Recall that drum 1 un-

dergoes a BSB amplification whereas drum 2 undergoes a RSB cooling. At short times < 1 µs,

the drums show no evidence of interaction. Drum 2 cools while drum 1 heats, the same behavior

that would have been expected if the drums did not interact with one another. The dashed lines in

Fig. 3a extrapolate this non-interacting behavior. Entangling pulse durations of > 1 µs, result in

both drums deviating from this independent evolution; their variances now grow together in time

with similar rates. Recall that in Fig. 2c, the scatter plot exhibited a correlation between x1 and

x2. Theory predicts that the elliptical (x1, x2) distribution’s major axis makes with the horizontal

will evolve as shown by the solid theory line in Fig. 3b (see Supplementary Information). We now

focus on the measured variable νmeas, shown in Fig. 3c. Again, with short pulses the drums are

only classically correlated as the initial state is close to but not exactly the ground state, and the

interaction does not yet entangle the drums. At ∼ 4 µs, νmeas crosses below 12, indicating entan-

glement. At 16.8 µs interaction time we observe νmeas = 0.44+0.004−0.004 (stat)+0.033

−0.028 (sys), where “sys”

indicates systematic uncertainty mainly caused by uncertainty in the measurement efficiencies, and

“stat” indicates statistical uncertainty estimated using bootstrap as described in the Supplementary

Information. This is a direct measurement of entanglement for a bipartite system of macroscopic

objects. Although νmeas is an important quantity for future quantum information applications, it is

10

still natural to estimate the entanglement before noise processes intervened during the measure-

ment. To that end we separate state preparation and measurement by accounting for noise in order

to correct for measurement inefficiency. Doing that with a linear transformation of the observed

covariance matrix could produce covariance matrices that violate the Heisenberg uncertainty prin-

ciple. To avoid this, we use a semi-definite program to find the closest (in l2 distance) physical

covariance matrix consistent with the data. From that we estimate the entanglement criterion ν,

which is shown in Fig. 3d. Indeed, we see more entanglement for the same interaction time, with

ν = 0.15+0.03−0.02 (stat)+0.19

−0.05 (sys). Such a level of entanglement might be useful for exploration of

mesoscopic Einstein-Podolsky-Rosen non-locality 5 and fundamental tests of quantum mechan-

ics 4.

This work demonstrates pulsed, time-domain control of three important building blocks

for CV quantum information processing and quantum communication: state initialization, en-

tanglement and measurement. Pulsed control played a key role as it allowed optimization of

each piece separately, and improve our measurement efficiency by more than an order of mag-

nitude compared to traditional steady-state operation. As a result, we generated a highly entan-

gled state of two macroscopic mechanical oscillators, surpassing the entanglement threshold by

5.2+0.7−0.7 (stat)+1.9

−3.5 (sys) dB. Most excitingly, we even observe entanglement directly in the mea-

sured variables. This is important for future applications that require decisions based on measure-

ment outcomes. We therefore expect the methods described here to serve as a stepping stone for

teleportation and entanglement swapping of states of massive objects.

11

Methods summary

To quantify the Simon-Duan criteria, we calculate the minimal symplectic eigenvalue ν of the

covariance matrix Cpt of the partial transposition of the oscillator’s density matrix. When a density

matrix is partially transposed, the momentum of the transposed mode is reversed, so Cpt = ΛCΛ,

for Λ =

(1 0 0 00 1 0 00 0 1 00 0 0 −1

). The value of ν can be calculated from the 4 × 4 covariance matrix C, that

corresponds to the original un-transposed density matrix, by subdividing it as C =(

A CCt B

)where

A,B,C are 2 × 2 sub-matrices, and using the formula ν =√

12

(∆−

√∆2 − 4 det C

), where

∆ = det(A) + det(B)− 2 det(C) (see 41, 42).

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Acknowledgements We thank Boaz Katz and Danny Ben-Zvi for feedback and insight on data taking and

analysis. We thank Bradley Hauer and Adam Sirois for their careful reading of the manuscript. We thanks

Konrad Lehnart and Robert Delaney for useful discussions on measurement efficiency.

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Figure 1: Experiment overview. a, Concept. Two mechanical harmonic oscillators (pendulums),characterized by their respective positions and momentums X1, P1, X2, P2, are placed in an entan-gled state |ψent〉. A user illuminates the pendulums using an electromagnetic source and detectsDoppler-shifted pulses carrying information about the system, x1, p1, x2, p2. The latter encode themechanical variables mixed with noise due to inevitable loss effects. b, Scanning electron mi-crograph. Two aluminum drums are suspended above a sapphire substrate (false colored in blue)resulting in well-defined harmonic modes in the direction perpendicular to the substrate. Eachdrum forms the top plate of a capacitor, along with the bottom plate which is fixed to the substrate.The latter are split in half along the long and short drum dimensions, respectively, to allow singledrum addressing (see text). c, Device optical image. The parallel capacitance of the two drums isshunted by a spiral inductor, forming a microwave cavity at a frequency of fc . An input line (topright) inductively couples to the microwave resonator. d, Circuit schematics. Mechanical motionof the drums (dashed) modulates the frequency of the microwave cavity. Therefore, an incomingmicrowave pulse is Doppler-shifted as it reflects off the microwave cavity, encoding informationabout the drums’ positions and momentums. Incoming and reflected pulses are separated usinga circulator. e, Experimental sequence of incoming pulses. The modulation of the incoming mi-crowave pulses around fc determines the nature of their interaction with the drums. State initial-ization is achieved by sideband cooling each drum mode close to its ground state. This is followedby a short entangling pulse. A readout pulse imprints an amplified record of the mechanical statesonto the reflected microwave pulse.

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Figure 2: Tomography of two mechanical oscillators. a, Scatter plots of a two-oscillatorsideband-cooled state. Each experiment records the four system variables x1, p1, x2, p2 concur-rently. Blue and red points display the position-momentum scatter plots of 10, 000 experimentrepetitions for drums 1 and 2 resp. Green points display the inter-drum correlations. All variablesare normalized according to the canonical commutation relation [xi, pi] = i, so the vacuum statehas variance 1

2. The drums’ individual mean phonon numbers are n1 = 0.79(1) and n2 = 0.60(1).

b, Covariance matrix of the data in a. c, Scatter plots of a two-mode thermal squeezed state. Aftersideband-cooling as in a, a 16.8 µs entangling pulse generates position-position (x2, x1 panel) cor-relation and momentum-momentum (p2, p1 panel) anti-correlation. Since the entanglement pulsepumps energy into the two-drum system, each drum’s individual variance grows from their groundstate cooled value to n1 = 11.5(1) and n2 = 4.42(4), respectively. d, Covariance matrix of the datain c. The correlations and anti-correlations are apparent in the off-diagonal terms of the covariancematrix.

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Figure 3: Entanglement of two drums versus pulse duration. a, Individual drum variances (av-erage of the position variance and momentum variance). Entangling is comprised of an amplifyingpulse for drum 1 and a cooling pulse for drum 2. The dashed lines show a theoretical predictionof the drums’ individual variances if amplifying and cooling where employed separately. Blueand red marks are the measured variances for drums 1 and 2, respectively, when amplifying andcooling are employed simultaneously, i.e. when the entangling protocol is used. Solid lines inall panels show theory lines with no fit parameters (see Supplementary Information). b, Angleof the position-position (x1,x2) correlation that determines the squeezed and anti-squeezed jointquadratures of the bipartite system. c, Entanglement in the measured variables, after loss, quan-tified by νmeas. Points below 0.5 indicate entanglement of the two drums. Error bars are 1-sigmabias-corrected bootstrapping confidence intervals 50. Shaded gray area corresponds to a 1-sigmauncertainty region in the location of the black theoretical prediction curve caused by measurementefficiency uncertainties. Inset shows systematic error for the last measured point in the main graph,due to the uncertainty in the measurement efficiencies. Upper and lower points (triangles) corre-sponds to 1-sigma systematic uncertainty. Statistical uncertainties of 1-sigma are present but maybe smaller than the marker sizes. The last point attains νmeas = 0.44+0.004

−0.004 (stat)+0.033−0.028 (sys). d,

Entanglement in the mechanical variables, prior to loss, quantified by ν. Uncertainties and insetplot are similar to c. The last point attains 0.15+0.03

−0.02 (stat)+0.19−0.05 (sys).

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