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PHYSICAL REVIEW LETTERS

Heisenberg-Limited Qubit Read-Out with Two-Mode Squeezed Light
Nicolas Didier,1,2 Archana Kamal,3 William D. Oliver,3,4 Alexandre Blais,2,5 and Aashish A. Clerk1
1

Department of Physics, McGill University, 3600 rue University, Montreal, Quebec H3A 2T8, Canada
Départment de Physique, Université de Sherbrooke, 2500 boulevard de l’Université, Sherbrooke, Québec J1K 2R1, Canada
3
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4
MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA
5
Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada(Received 6 February 2015; published 27 August 2015)2

We show how to use two-mode squeezed light to exponentially enhance cavity-based dispersive qubit
measurement. Our scheme enables true Heisenberg-limited scaling of the measurement, and crucially, it is
not restricted to small dispersive couplings or unrealistically long measurement times. It involves couplinga qubit dispersively to two cavities and making use of a symmetry in the dynamics of joint cavity
quadratures (a so-called quantum-mechanics-free subsystem). We discuss the basic scaling of the scheme
and its robustness against imperfections, as well as a realistic implementation in circuit quantum
electrodynamics.
DOI: 10.1103/PhysRevLett.115.093604

PACS numbers: 42.50.Dv, 03.65.Ta, 03.67.-a, 42.50.Lc

Introduction.—Research in quantum metrology has
established that squeezed light and entanglement are key
resources needed to approach truly fundamental quantum
bounds on measurement sensitivity [1]. Perhaps the best
known application is interferometry: by injecting squeezed
light into the dark port of an interferometer, one dramatically enhances its sensitivity to small phase shifts [2,3],
reducing the imprecision below the shot-noise limit. Manyof these ideas for squeezing-enhanced measurement werefirst motivated by gravitational wave detection [4–6], andthey have recently been implemented in current-generation
detectors [7,8]. More generally, squeezed light has beenused to enhance the measurement sensitivity in optomechanics [9], and even in biology [10].
Ultrasensitive detection is also essential for quantum
information processing where fast, high-fidelity qubit
read-out is required to achieve fault-tolerant quantum
computation [11]. A ubiquitous yet powerful approach isdispersive read-out, where a qubit couples to a cavity suchthat the cavity frequency depends on the qubit state; see,e.g., Ref. [12]. The read-out consists in driving the initiallyempty cavity with a coherent tone, resulting in a qubit-state
dependent cavity field which is displaced in phase spacefrom the origin [see Fig. 1(a)]. High-fidelity read-out canthen be obtained by measuring the output field quadratures.
This is the standard approach used in state-of-the-art
experiments with superconducting qubits; see, e.g.,
Refs. [13–15].
As with interferometry, one might expect that dispersive
qubit measurement could be enhanced by using squeezed
light. The most obvious approach would be to squeeze thephase quadrature of the incident light [i.e., Y in Fig. 1(b)],thus reducing the overlap between the two pointer states.
As was discussed recently in Ref. [16], the situation is not
0031-9007=15=115(9)=093604(5)so simple, as the dispersive interaction will lead to a qubitdependent rotation of the squeezing axis. Unlike standard
interferometry, this rotation is a problem, as optimal
dispersive qubit read-out involves large couplings andhence large rotations. Further complexity arises from thefact that this rotation is frequency dependent. The upshot isthat measurement always sees the amplified noise associated with the antisqueezed quadrature of the incident

FIG. 1 (color online). Phase-space representation of dispersive
qubit read-out for different input states: (a) coherent state,(b) single-mode phase-squeezed state, (c) amplitude-squeezed
state, (d) two-mode squeezed state in the QMFS ðX − ; Y þ Þ. The
purple dashed lines represent the input state and the blobs
represent the output fields. The input state is displaced along
the X axis and the signal is encoded in the quadrature corresponding to the Y axis with homodyne detection; as depicted in
the leftmost panel, the read-out error corresponds to the overlapof the two marginals. Dispersive interaction with the qubit rotates
the output field by the angle φqb for the ground state j0i (in blue)and −φqb for the excited state j1i (in red). Ideally, one wants theoutput state to be phase squeezed regardless of qubit state [thedotted “desiderata” states in (b)]; this is not possible when using
single-mode squeezing due to the qubit-induced rotation. Our
new QMFS scheme [panel (d)] does not suffer from this problem.

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light, limiting the fidelity improvement from using squeezing to modest values and preventing true Heisenberg
scaling [16].
Despite the above difficulties, we show in this Letter that
it is indeed possible to substantially improve dispersive
qubit measurements using squeezed input states. Our
proposed scheme involves using two-mode squeezed statesin a two-cavity-plus-qubit system (see Fig. 2), which canlead to exponential enhancement of the signal-to-noise ratio(SNR) in dispersive measurement and achieves true
Heisenberg-limited scaling. This is possible even for large
qubit-induced phase shifts and is thus in stark contrast to
previous schemes using two-mode squeezing for interferometry [3] or qubit read-out [16].
The key to our scheme is the use of a special dynamical
symmetry, whereby two commuting collective quadratures
exhibit a simple rotation as a function of time. As thesequadratures commute, they constitute a so-called quantummechanics-free subsystem (QMFS) [17] and can be simultaneously squeezed. The upshot is that one can effectively
make a dispersive qubit measurement where the uncertainties now associated with the two pointer states are notlimited by the uncertainty principle [see Fig. 1(d)]. Though
the scheme is extremely general, for concreteness we
explicitly discuss an implementation in circuit quantum
electrodynamics (QED) using a transmon qubit [18], as
depicted in Fig. 2.
The dynamical symmetry used in our two-mode scheme
crucially relies on one of the cavities having an effective
negative frequency; it is thus related to an idea first
discussed in the context of measurement by Tsang andCaves [19] and Wasilewski et al. [20], and which has since
been applied to other systems [17,21,22]. While many
applications use the idea to suppress the effects of backaction [20–22], we instead use it as an effective means to
exploit squeezed input light. Unlike previous studies, we
calculate here the scaling of the resulting measurement
sensitivity, showing that one obtains Heisenberg-limited
scaling with incident photon number.
Dispersive measurement and standard squeezing.—Westart by reviewing the simplest setup where a qubit dispersively couples to a single-sided cavity (frequency ω1 )FIG. 2 (color online). Circuit QED implementation of QMFS
dispersive qubit read-out with two-mode-squeezed states produced by a nondegenerate paramp (NDPA). The signal is encodedin the joint quadrature Y þ;out ∝ Y 1;out þ Y 2;out ; see the text for
details.

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with the Hamiltonian H ¼ ðω1 þ χ σ̂ z Þâ† â [12]. Standard
dispersive read-out involves driving the input port of thecavity with a coherent tone at the cavity frequency (photon
flux n̄0 κ=4, with κ being the cavity damping rate). As
illustrated in Fig. 1(a), as a consequence of the dispersive
coupling, the output field is rotated by the angle φqb ¼
2 arctanð2χ=κÞ if the qubit is in the ground state j0i andby −φqb for the excited state j1i. Writing the output field asâout ðtÞ ¼ e−iω1 t ðX̂out þ iŶ out Þ=2, for a displacement along
the real axis Xout , the signal of the qubit state is encoded in
the phase quadrature Y out ; this quadrature is then recordedwith homodyne detection.
Measuring Y out for an integration time τ corresponds toevaluatingR the dimensionless measurement operatorpﬃﬃﬃ
M̂ ¼ κ 0τ dtŶ out ðtÞ. The signal is the qubit-state dependent expectation value M S ¼ hM̂i and is the same for all of
the injected states depicted in Fig. 1. The imprecision noiseis the variance of the noise operator M̂N ¼ M̂ − MS. The
signal-to-noise ratio SNR ≡ jMS;j0i − M S;j1i j=ðhM̂ 2N;j0i i þ
hM̂2N;j1i iÞ1=2 is, for this coherent state dispersive read-out,
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
SNRα ðτÞ ≃ j sin φqb j 2n̄0 κτ [23,24]. As expected, the
SNR is maximized for a phase φqb ¼ π=2; it also scales
pﬃﬃﬃﬃﬃ
as n̄0 , akin to standard quantum-limit scaling in interferometry [1].
Next, consider what happens if we instead inject a
displaced squeezed state (squeeze parameter r) into thecavity. As already discussed, this is not as beneficial as one
would hope, as one always sees the noise of the antisqueezed quadrature (∝e2r ) [16,25]. Consider the optimalcase φqb ¼ π=2 which maximizes the signal. For large τ,
the noise behaves ashM̂2N i ≃ κτ½sin2 ðθÞe−2r þ cos2 ðθÞe2r pﬃﬃﬃ
þ 2 2 sinhð2rÞ cosð2θ − 3π=4Þ;ð1Þ

where we have dropped terms that decay exponentiallywith κτ. The first line of Eq. (1) dominates in the longtime
limit and represents the contribution from zero-frequency
noise in the output field. For this line, the choice θ ¼ π=2cancels the contribution from the amplified quadrature andleads to an exponential reduction in the noise compared to acoherent state drive [16]. In contrast, the second line of
Eq. (1) describes the contribution from initial short-time
fluctuations; the noise from the antisqueezed quadrature
here remains, even if θ ¼ π=2. As a result, increasing r
indefinitely does not improve the SNR; for a given τ, there
is an optimal value [see Figs. 3(b) and 3(c)]. This then leads
to a generally modest enhancement of SNR compared to asimple coherent state drive [16]; in particular, there isalmost no improvement in the most relevant case whereτ ∼ 1=κ [the shaded region in Fig. 3(a)]. Optimized
squeezing leads, at best, to the scaling N 3=4 with input

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ðX− ; Y þ Þ. Such a dynamics is generated by the simple
Hamiltonian [19,20]
1
H ¼ χðX̂þ X̂− þ Ŷ þ Ŷ − Þσ̂ z ¼ χðâ†1 â1 − â†2 â2 Þσ̂ z :2FIG. 3 (color online). (a) SNR as a function of integration timeτ for different protocols: coherent state drive (the black line),
displaced single-mode squeeze state (the blue line), two-mode
squeezed QMFS setup (the red line). We assume an optimal
dispersive shift χ ¼ κ=2; for the QMFS setup, the cavity ispresqueezed (t0 ≪ −1=κ) with squeeze strength e2r ¼ 100. The
coherent drive is turned on at τ ¼ 0. For the single-mode case, foreach τ we optimize the squeeze strength e2r ∈ ½1; 100 and angle(see Ref. [25]). The QMFS scheme gives an exponential SNR
enhancement, especially in the most interesting regime whereτ ∼ 1=κ (the shaded region). (b) Integration time τ required toachieve a fidelity F ¼ 99.99%, as a function of e2r ; parameters asin (a), except that n̄0 ¼ 100. Black lines correspond to an
unsqueezed drive, where the drive strength is increased suchthat the intracavity photon number is the same as in the QMFS
scheme, i.e., n̄0 → n̄0 þ 4 sinh2 r. The solid curves correspond tothe case of no photon losses (efficiency η ¼ 1), while the dashedcurves correspond to η ¼ 0.9. (c) Total intracavity photon number
needed to achieve F ¼ 99.99% in a measurement time τ. Even
with nonzero photon losses, the use of squeezing can dramatically reduce the number of intracavity photons.

photon number, similar to a Mach-Zehnder interferometer
driven with squeezed light [1,2,25].
Negative freqencies and two-mode squeezing.—To avoid
having the measurement corrupted by the antisqueezed
quadrature, one ideally wants to squeeze both quadraturesof the input light. While this is impossible with a single
cavity, it becomes conceivable using joint quadratures of
two cavities. If âj ¼ ðX̂j þ iŶ j Þ=2 (j ¼ 1; 2) are theannihilation operators for the two cavities (in an interaction
picture with respect to thecavity Hamiltonians),
wepﬃﬃfreeﬃ
pﬃﬃﬃ
define X̂ ¼ ðX̂1  X̂2 Þ= 2, Ŷ  ¼ ðŶ 1  Ŷ 2 Þ= 2. Since
X− and Y þ commute, they can be squeezed simultaneously,
resulting in a two-mode-squeezed state [28]. The
relevant nonzero input-field noise correlators arehX̂∓ ðtÞX̂∓ ðt0 Þi ¼ hŶ  ðtÞŶ  ðt0 Þi ¼ e∓2r δðt − t0 Þ. We stressthat such states have already been produced in circuit
QED [29,30].
This squeezing by itself is not enough: we also need the
dynamics of these joint quadratures to mimic the behaviorof X̂ and Ŷ in a single cavity, such that the two qubit statesstill give rise to a simple rotation of the vector formed byð2Þ

The qubit thus needs to couple dispersively to both cavities,
with equal-magnitude but opposite-signed couplings. The
resulting dynamics is illustrated in Fig. 1(d): an incident
field with hŶ þ i ¼ 0; hX̂− i ≠ 0 is rotated in a qubit-state
dependent manner, resulting in an output field withhŶ þ i ≠ 0 (i.e., the measurement signal). Note that thesqueezed quadratures X− ; Y þ are never mixed with theantisqueezed quadratures Xþ ; Y − ; hence, this amplification
will not limit our scheme. We also stress that the twocavities need not have the same frequency.
The measurement protocol involves first turning on thevacuum two-mode squeezed drive at a time t ¼ t0 ≤ 0, andthen turning on the coherent cavity drive(s) at t ¼ 0. This
coherent drive (which displaces along X− but not Y þ ) could
be realized by driving one or both of the cavities. We takethe optimal case where both cavities are driven and letn̄0 κ=8 denote the photon flux incident on each cavity due tothe coherent drives. The measurement signal in Y þ can beconstructed from the quadratures Y j;out of the output fieldleaving each cavity. In what follows, we consider the limit
κt0 ≪ −1, such that the measurement is not corrupted byany initial nonsqueezed vacuum in the cavity[25].
pﬃﬃﬃ R
The measurement operator is now M̂ ¼ κ 0τ dtŶ þ;out ðtÞ.
As expected, one finds that this output quadrature is always
squeezed, and hence the imprecision noise is always
described by hM̂ 2N i ¼ e−2r κτ, independent of χ. As desired,
the noise is now exponentially reduced with respect to a
standard dispersive read-out, leading to an exponential
improvement of SNR; i.e., SNRr ðτÞ ¼ er SNRα ðτÞ for all
integration times τ. This is in stark contrast to the singlemode approach, where such an enhancement was only
possible at extremely long times, κτ ≳ e4r [cf. Eq. (1)]. The
SNR is plotted in Fig. 3(a) as a function of integration time
τ, with comparisons against the single-mode squeezing andno-squeezing cases; our two-mode scheme realizes dramatic improvements in the most interesting regime where τ
is not much larger than 1=κ. The integration time τ required
to achieve a measurement fidelity F ¼ 1 − erfcðSNR=2Þ=2
of 99.99% is plotted against the squeezing strength in
Fig. 3(b). Again, the QMFS scheme results in dramatic
improvements.
Heisenberg-limited scaling.—We now show that the
SNR scales as the number of photonspﬃﬃﬃﬃ
N used for themeasurement rather than its square root N , as is the casefor the standard dispersive read-out [1]. For this, we define
pﬃﬃﬃ R
the temporal mode Â ¼ ð1= τÞ 0τ dt½d̂in;1 ðtÞ þ d̂in;2 ðtÞ[3], where the operator d̂in;j describes fluctuations in theresonator-j input field. The total number of input photons

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N ¼ N s þ N d has a contribution from squeezing N s ¼
hÂ† Âi ¼ 2 sinh2 r and N d from the coherent displacement.
Focusing on times τ ≫ 1=κ, we can ignore the transient
response to the coherent drive, and hence N d ¼ 14 n̄0 κτ.
Fixing N and taking t0 ≪ −1=κ, the optimal SNR is
obtained for N s ¼ N 2 =½2ðN þ 1Þ, and it ispﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
SNRopt ¼ 2j sin φqb jN 1 þ 2=N → 2j sin φqb jN;

an unwanted coupling between the squeezed quadraturesðX̂− ; Ŷ þ Þ and the antisqueezed quadratures ðX̂þ ; Ŷ − Þ. The
structure of the QMFS can persist in the presence ofasymmetries for long measurement times κτ ≫ 1, under
the condition [25]
χ 1 þ χ 2 κ1 − κ2¼:
χ 1 − χ 2 κ1 þ κ2

ð3Þ

where we have taken the large N limit. Equation (3)corresponds to true Heisenberg scaling for any value ofthe dispersive coupling. Such scaling is not possible using
single-mode squeezed input light (see the Supplemental
Material [25]).
Our QMFS scheme also shows an improved,
Heisenberg-like scaling of the SNR with the intracavity
photon number n̄. Note that the SNR for the QMFS scheme
has the same form as the SNR for a standard (r ¼ 0)
dispersive read-out made using a larger drive flux n̄0 e2r . If
we fix the intracavity photon number n̄ ¼ n̄0 cos2 ðφqb =2Þ þ
2sinh2 r and optimize r,pthe
ﬃﬃﬃﬃﬃ resulting SNR scales as
SNRopt ≃ 2j sinðφqb =2Þjn̄ κτ, as opposed to the convenpﬃﬃﬃ
tional SNRα ∝ n̄.
Robustness against imperfections.—Our discussion of
the QMFS scheme so far has assumed a broadband, puresqueezing source. The purity of the squeezing is, however,not crucial; our scheme is insensitive to the antisqueezed
quadratures, and hence it is not essential that their variances
be as small as possible. For a finite squeezing bandwidth Γ,
the input squeezing spectrum will typically have a
Lorentzian line shape [31]. We find that the effects of afinite bandwidth are equivalent to an effective reduction of
the squeezing strength; thepSNR
for the scheme is simply
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
reduced by a prefactorΓτ=½Γτ þ ðe2r − 1Þð1 − e−Γτ Þ
[25]. One thus only needs a modest bandwidth; e.g.,Γ ∼ 10κ is enough for κτ ∼ 10 and e2r ∼ 10.
The lack of any enhanced Purcell decay is also crucial, as
in our protocol the squeezing is turned on well before thecoherent measurement tone. Having a finite squeezing
bandwidth can, in fact, be an advantage, as it helps suppress
the Purcell decay of the qubit. This decay corresponds torelaxation of the qubit by photon emission from the cavity[32]. As typical detunings Δ ≫ κ, there is a wide range ofideal squeezing bandwidths satisfying κ ≪ Γ ≪ Δ. Suchbandwidths are large enough to allow a full enhancement of
the SNR (with τ ≳ 1=κ), and small enough that thesqueezing does not appreciably modify cavity-induced
Purcell decay (see the Supplemental Material [25]).
Another nonideality is asymmetry in the system parameters. While the two cavity frequencies can differ, we haveassumed so far that they have identical damping rates(κ 1 ¼ κ 2 ¼ κ) and that the dispersive coupling strengths
satisfy χ 1 ¼ −χ 2 ¼ χ. Deviation from either of these
conditions breaks the symmetry yielding a QMFS, causing

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ð4ÞThe SNR enhancement can, however, be preserved formeasurement times τ ∼ 1=κ by optimizing δκ=δχ, asillustrated in Fig. 4(a). Although this might not be necessary in practice, all parameters in Eq. (4) can be tuned
in situ [18,33,34], thereby greatly relaxing the constraints
on the system.
Finally, like any scheme employing squeezing, photon
losses effectively replace squeezed fluctuations with an
ordinary vacuum, causing the SNR improvement to saturate as a function of squeezing strength [25]. Despite this,our scheme still yields considerable advantages for finite
loss rates; see Figs. 3(b) and 3(c).
Implementation in circuit QED.—We now turn to apossible realization of this protocol in circuit QED. All
parameters discussed here are readily achievable experimentally. As illustrated in Fig. 2, a transmon qubit iscoupled to two resonators, one in the usual dispersive
regime (Δ > EC ) and the other in the “straddling” regime(Δ < EC ) [18,35]. Here, Δ is the qubit-resonator detuning
and EC the transmon anharmonicity. This yields dispersive
couplings χ having opposite signs, as required; see
Fig. 4(b). An alternative strategy is to use a fluxonium
or a flux qubit which exhibits a richer dispersive shiftprofile [36]. Note that either approach does not entail asacrifice of qubit coherence via enhanced Purcell decay

FIG. 4 (color online). (a) SNR enhancement as a function of thedispersive shift asymmetry (χ 1;2 ¼ δχ  χ̄) for different resonatorlinewidth asymmetries (κ 1;2 ¼ κ̄  δκ) calculated for χ̄ ¼ κ̄=2,κτ ¼ 10 and e2r ¼ 100. The dashed line is the maximal SNR
obtained by optimizing δκ. (b) Calculated dispersive shifts as afunction of transmon anharmonicity EC from a numerical
diagonalization of a transmon-resonator system for each of theresonators. The parameters are EJ =h ¼ 25 GHz, ω1 =2π ¼
7.6GHz, ω2 =2π ¼7.9 GHz, g1 =2π ¼8 MHz and g2 =2π ¼15 MHz.
The vertical dashed line shows a typical value of EC that leads toequal and opposite dispersive shifts.

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PHYSICAL REVIEW LETTERS[25]. The displaced two-mode-squeezed state required at
the input can be generated by either a NDPA such as the
Josephson parametric converter [29], a Josephson paramp[37], or the Bose-Hubbard dimer [30].
Conclusion.—We have presented a realistic measurement protocol that allows one to exponentially enhancedispersive measurement using two-mode squeezed light,
enabling Heisenberg-limited scaling even with large dispersive couplings. Our scheme crucially makes use of a
special symmetry in the dynamics of joint cavity quadratures, a so-called quantum-mechanics-free subsystem. It
could be straightforwardly generalized to allow
Heisenberg-limited scaling in any interferometric setup
having large signal phase shifts.
We thank Irfan Siddiqi, Michel Devoret, Shabir
Barzanjeh, Samuel Boutin, Shruti Puri, Baptiste Royer,
and Amit Munje for many useful discussions. This work
was supported by the Army Research Office under Grant
No. W911NF-14-1-0078, INTRIQ, NSERC and CIFAR.

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