1. technology and computing

Heisenberg-limited qubit readout with two-mode squeezed light
Nicolas Didier,1, 2 Archana Kamal,3 Alexandre Blais,2, 4 and Aashish A. Clerk1
1
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, is not restricted to small dispersive couplings or unrealistically long measurement times.
It involves coupling a qubit dispersively to two cavities, and making use of a symmetry in the
dynamics of joint cavity quadratures (a so-called quantum-mechanics free subspace, QMFS). We
discuss the basic scaling of the scheme and its robustness against imperfections, as well as a realistic
implementation in circuit quantum electrodynamics.
PACS numbers: 42.50.Dv, 03.65.Ta, 42.50.Lc, 03.67.-a, 42.50.Pq
1
er
Y+,out
desiderata
e
'qb
Xout
Yout
r
Xout
e
Xout
X
r
,out
s
ic
m
Given the connection to interferometry, one might expect that dispersive qubit measurement could be similarly enhanced by using squeezed light. The most obvious
approach would be to squeeze the phase quadrature of the
incident light [i.e. Y in Fig. 1(b)], thus reducing the overlap between the two pointer states. As discussed recently
in Ref. 16 the situation is not so simple, as the dispersive interaction will necessarily lead to a qubit-dependent
rotation of the squeezing axis. Unlike standard interfer-
Yout
na
dy
Ultra-sensitive detection is also essential for quantum
information processing where fast, high-fidelity qubit
readout is required to achieve fault-tolerant quantum
computation [11]. A ubiquitous yet powerful approach is
dispersive readout, where a qubit couples to a cavity such
that the cavity frequency depends on the qubit state, see
e.g. [12]. The readout consists of driving the initially
empty cavity with a coherent tone, resulting in a qubitstate dependent cavity field which is displaced in phase
space from the origin [see Fig. 1(a)]. High-fidelity readout can then be obtained by measuring the output field
quadratures, and by optimizing parameters to minimize
the overlap between the two displaced pointer states.
This is the standard approach used in state-of-the-art
experiments with superconducting qubits, e.g. [13–15].
Yout
al
re
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.
Many of these ideas for squeezing-enhanced measurement were first motivated by gravitational wave detection
[4–6], and have recently been implemented in currentgeneration detectors [7, 8]. More generally, squeezed light
has been used to enhance the measurement sensitivity in
optomechanics [9] and even biology [10].
p n¯ 0
arXiv:1502.00607v1 [quant-ph] 2 Feb 2015
Department of Physics, McGill University, 3600 rue University, Montreal, Quebec H3A 2T8, Canada
2
D´epartment de Physique, Universit´e de Sherbrooke,
2500 boulevard de l’Universit´e, Sherbrooke, Qu´ebec J1K 2R1, Canada
3
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
4
Canadian Institute for Advanced Research, Toronto, Canada
(a) coherent
state
(b) phase
squeezing
(c) amplitude (d) QMFS
squeezing
setup
FIG. 1. Phase-space representation of dispersive qubit readout for different input states: (a) coherent state, (b) singlemode phase-squeezed state, (c) amplitude-squeezed state,
(d) two-mode squeezed state in the QMFS (X− , Y+ ). 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 readout error corresponds to the
overlap of the two marginals. Dispersive interaction with the
qubit rotates the output field by the angle ϕqb for the ground
state |0i (in blue) and −ϕqb for the excited state |1i (in red).
Ideally, one wants the output state to be phase squeezed regardless of qubit state [dotted “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, achieving exponentially reduced measurement noise and Heisenberg-limited scaling.
ometry, this rotation is a problem, as optimal dispersive
qubit readout involves large couplings and hence large
rotations. Moreover, the rotation of the incident field
by the dispersive interaction is a dynamic process, implying that the phase shift will be frequency-dependent.
The upshot is that measurement always sees the amplified noise associated with the anti-squeezed quadrature
of the incident light, limiting the fidelity improvement
from using squeezing to modest values and preventing
true Heisenberg-scaling [16].
2
Y1,out
NDPA
two-mode
squeezed
state
1
>0
2
<0
Y2,out
FIG. 2. Circuit QED implementation of QMFS dispersive
qubit readout with two-mode-squeezed states produced by a
non-degenerate paramp (NDPA). The signal is encoded in the
joint quadrature Y+,out ∝ Y1,out + Y2,out ; see text for details.
Despite the above difficulties, we show in this Letter
that it is indeed possible to improve dispersive qubit
measurements using squeezed input states. Our proposed scheme involves using two-mode squeezed states
in a two-cavity-plus-qubit system (see Fig. 2), which can
lead to exponential enhancement of the signal-to-noise
ratio (SNR) in dispersive measurement and achieve true
Heisenberg-limited scaling. This is possible even for large
qubit-induced phase shifts, and is thus in stark contrast
from previous schemes using two-mode squeezing for interferometry [3] or qubit-measurement [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 these quadratures commute, they constitute a socalled “quantum-mechanics-free subspace” (QMFS) [17]
and can both be simultaneously squeezed. The upshot
is that one can effectively make a dispersive qubit measurement where now the uncertainties associated with
the two pointer states are not limited at all 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.
We also show that it is robust against a number of imperfections likely to occur in experimental implementations.
Note that the dynamical symmetry used in our twomode scheme crucially relies on one of the cavities having an effective negative frequency; it is thus a manifestation of an idea first discussed in the context of measurement by Tsang and Caves [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 (as is also discussed in Refs. [20, 23]). Unlike previous studies, we calculate here the scaling of the resulting
measurement sensitivity, showing that one indeed recovers Heisenberg-limited scaling with incident photon number. Our scheme also differs from previous work, as the
signal to be detected is not an applied linear force on one
of the cavities, but is rather the cavity frequency itself.
Dispersive measurement and standard squeezing– We
start by reviewing the simplest setup where a qubit dispersively couples to a single-sided cavity (frequency ω1 )
with the Hamiltonian H = (ω1 + χˆ
σz ) a
ˆ† a
ˆ [12]. Standard dispersive readout involves driving the input port
of the cavity with a coherent tone at the cavity frequency (photon flux n
¯ 0 κ/4, κ is 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 = 2 arctan(2χ/κ) if the qubit is in
the ground state |0i and by −ϕqb for the excited state
|1i. Writing the output field in terms of quadratures
ˆ out + iYˆout )/2, for a displacement
as a
ˆout (t) = e−iω1 t (X
along the real axis Xout , the signal of the qubit state
is optimally encoded in the phase quadrature Yout ; this
quadrature is then recorded with homodyne detection.
Measuring Yout for an integration time τ corresponds
to evaluating
R the dimensionless measurement operator
ˆ = √κ τ dtYˆout (t). The signal is the qubit-state
M
0
ˆ i and is the
dependent expectation value MS = hM
same for all the injected states depicted in Fig. 1. On
the other hand, the imprecision noise is the variance
ˆN = M
ˆ − MS . The signal-to-noise raof the noise M
ˆ 2 i + hM
ˆ 2 i)1/2 is,
tio SNR ≡ MS,|0i − MS,|1i /(hM
N,|0i
N,|1i
for this √
coherent state dispersive readout, SNRα (τ ) '
| sin ϕqb | 2¯
n0 κτ [24, 25]. As expected, the SNR
√ is max¯ 0 , akin
imized for a phase ϕqb = π/2; it also scales as n
to standard quantum-limit scaling in interferometry [1].
Next, consider what happens if we instead inject a
displaced squeezed state (squeeze parameter r) into the
cavity. As already discussed and sketched in Fig. 1 (b–
c), this is not as beneficial as one would hope, as one
always sees the noise of the anti-squeezed quadrature
(∝ e2r ) [16, 26]. Consider the optimal case ϕqb = π/2
which maximizes the signal and, for simplicity, consider
the asymptotic long-τ behaviour of the noise. We find
2
ˆN
hM
i ' κτ [sin2 (θ)e−2r + cos2 (θ)e2r ]
√
+ 2 2 sinh(2r) cos(2θ − 3π/4),
(1)
where we have dropped terms that decay exponentially
with κτ . We have furthermore assumed broadband
squeezing. The first line of Eq. (1) dominates in the longtime limit, and represents the contribution from zerofrequency noise in the output field. For this line, the
choice θ = π/2 cancels the contribution from the amplified quadrature, and leads to an exponential reduction
in the noise compared to a coherent state drive [16]. In
contrast, the second line of Eq. (1) describes the contribution from initial short-time fluctuations; the noise
from the anti-squeezed quadrature here remains even if
θ = π/2. As a result, increasing r indefinitely does not
improve the SNR; for a given τ there is an optimal value
[see Fig. 3 (b–c)]. This then leads to generally modest enhancement of SNR compared to a simple coherent state
drive [16]; in particular, there is almost no improvement
3
(a)
102
6
(b)
vector formed by (X− , Y+ ). Such a dynamics is generated
by the simple Hamiltonian [19, 20]
101
100
10−1 0
10
×er
χ = κ/2
e2r = 102
QMFS setup
single-mode squeezing
no squeezing
101
Integration
time κτ
g
102
Measurement time κτ
√
SNR/ n
¯0
4
ˆ+X
ˆ − + Yˆ+ Yˆ− )ˆ
H = 21 χ(X
σz = χ(ˆ
a†1 a
ˆ1 − a
ˆ†2 a
ˆ2 )ˆ
σz . (2)
2
0
100(c)
3
n
¯ 0 = 10
101
102
2
1
0
100
n
¯ 0 = 100
101
Squeeze strength e2r
102
FIG. 3. (a) SNR as a function of integration time τ for different protocols: coherent state drive (black), displaced singlemode squeeze state (blue), two-mode squeezed QMFS setup
(red). We assume optimal value of dispersive shift χ = κ/2
for maximum signal; the cavity is pre-squeezed (t0 −1/κ)
with squeeze strength set to e2r = 100 for the QMFS setup.
The coherent drive is turned on at τ = 0. For the single-mode
case, for each τ we optimize the squeeze strength e2r ∈ [1, 100]
and the squeeze angle (see [26]). The QMFS scheme gives an
exponential SNR enhancement, especially in the most interesting regime where τ ∼ 1/κ (grey shaded region). (b),(c)
Integration time τ required to achieve a fidelity F = 99.99 %,
as a function of squeeze strength; calculations were performed
with the parameters used in (a), except that we take the drive
strength n
¯ 0 = 10 in (b) and n
¯ 0 = 100 in (c). The dashed lines
correspond to an unsqueezed drive, where we now increase the
drive strength such that the intracavity photon number is the
same as in the QMFS scheme, i.e. n
¯0 → n
¯ 0 + 4 sinh2 r.
in the most relevant case where τ ∼ 1/κ [shaded region in
Fig. 3(a)]. One also finds that optimized squeezing leads
at best to the scaling N 3/4 with input photon number, as
can be obtained in a Mach-Zehnder interferometer driven
with squeezed light [1, 2, 26].
Negative-freqencies and two-mode squeezing– To avoid
having the measurement corrupted by the anti-squeezed
quadrature, one ideally wants to squeeze both quadratures of the input light. While this is impossible with a
single cavity, it becomes conceivable using joint quadraˆ j + iYˆj )/2 (j = 1, 2)
tures of two cavities. If a
ˆ j = (X
are the annihilation operators for the two cavities (in
an interaction picture with respect to the free
√ cavity
ˆ ± = (X
ˆ1 ± X
ˆ 2 )/ 2, Yˆ± =
Hamiltonians),
we
define
X
√
(Yˆ1 ± Yˆ2 )/ 2. Since X− and Y+ commute, they can
be squeezed simultaneously, resulting in a two-modesqueezed state [27]. The relevant non-zero input-field
ˆ ∓ (t)X
ˆ ∓ (t0 )i = hYˆ± (t)Yˆ± (t0 )i =
noise correlators are hX
∓2r
0
e δ(t − t ). We stress that such states have already
been produced in circuit QED [28, 29].
This squeezing by itself is not enough: we also need
the dynamics of these joint quadratures to mimic the
ˆ and Yˆ in a single cavity, such that the
behaviour of X
two qubit states still give rise to a simple rotation of the
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):
ˆ−i =
an incident field with hYˆ+ i = 0, hX
6 0 is rotated in
a qubit-state dependent manner, resulting in an output
field with hYˆ+ i =
6 0 (i.e. the measurement signal). Note
that the squeezed quadratures X− , Y+ are never mixed
with the anti-squeezed quadratures X+ , Y− , hence this
amplification will not limit our scheme. We also stress
that the two cavities need not have the same frequency.
The measurement protocol involves first turning on the
vacuum two-mode squeezed drive at a time t = t0 ≤
0, and then turning on the coherent cavity drive(s) at
t = 0. This coherent drive (which displaces along X−
but not Y+ ) could be realized by driving one or both the
cavities. We take the optimal case where both cavities are
driven and let n
¯ 0 κ/8 denote the photon flux incident on
each cavity due to the coherent drives. The measurement
signal in Y+ can be constructed from the quadratures
Yj,out of the output field leaving each cavity. In what
follows, we consider the limit κt0 −1, such that the
measurement is not corrupted by any initial non-squeezed
vacuum in the cavity. More details on the measurement
protocol are provided in [26].
With the√above
protocol, the measurement operator is
R
ˆ = κ τ dtYˆ+,out (t). As expected, one finds that
now M
0
this output quadrature is always squeezed, and hence the
ˆ 2 i = e−2r κτ ,
imprecision noise is simply described by hM
N
for all τ , independent of χ. As desired, the noise is now
exponentially reduced with respect to standard dispersive
readout, leading to an exponential improvement of SNR,
i.e. SNRr (τ ) = er SNRα (τ ) for all integration times τ .
This is in stark contrast to the single-mode approach,
where such an exponential SNR enhancement was only
possible at extremely long times, κτ & e4r [c.f. Eq. (1)].
The SNR is plotted in Fig. 3(a) as a function of integration time τ , with comparisons against the single-mode
squeezing and no-squeezing cases; our two-mode scheme
realizes dramatic improvements in the most interesting
regime where τ is not much larger than a few times 1/κ.
The integration time τ required to achieve a measurement fidelity F = erfc(SNR/2)/2 of 99.99 % is plotted
against squeezing strength in Fig. 3(b). The measurement time is drastically reduced for the QMFS setup and
can even drop below the cavity photon lifetime 1/κ.
Heisenberg-limited scaling– To show that this readout protocol reaches the Heisenberg bound, we now
show that the SNR scales as the number of photons N
used
for the measurement rather than its square root
√
N as is the case for the standard dispersive readout [1]. For this, we define the temporal mode Aˆ =
Rτ
√1
dt[dˆin,1 (t) + dˆin,2 (t)] [3] where the operator dˆin,j
τ 0
4
where we have taken the large N limit. Eq. (3) corresponds to true Heisenberg scaling for any value of the
dispersive coupling. Such scaling is not possible using
single-mode squeezed input light (see [26]).
Our QMFS scheme also shows an improved,
Heisenberg-like scaling of the SNR with the intracavity
photon number n
¯ . This can be seen from the fact that
the SNR for the QMFS scheme has the same form as the
SNR for a standard (r = 0) dispersive measurement made
using a larger drive flux n
¯ 0 e2r . If we fix the intracavity
photon number n
¯=n
¯ 0 cos2 (ϕqb /2) + 2 sinh2 r and optimize the value of r, the resulting
optimized SNR scales
√
κτ
,
as
opposed to the conas SNRopt ' 2| sin(ϕ
/2)|¯
n
√qb
ventional SNRα ∝ n
¯.
Robustness against imperfections– Our discussion of
the QMFS scheme so far has assumed a broadband, pure
squeezing 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, obtained for instance at the output
of a NDPA, the input squeezing spectrum will not be
flat, but will rather have a Lorentzian lineshape [30]. We
find that the effects of a finite bandwidth are equivalent to an effective reduction of the squeezing strength;
the p
SNR for the scheme is simply reduced by a prefactor Γτ /[Γτ + (e2r − 1)(1 − e−Γτ )] [26]. Only a finite
squeezing bandwidth is thus required, e.g. Γ ∼ 10κ is
enough for κτ ∼ 10 and e2r ∼ 10.
The lack of any enhanced Purcell decay is crucial, as in
our ideal protocol the squeezing is turned on well before
the coherent measurement tone. Having a finite squeezing bandwidth can in fact be an advantage as it helps
suppress Purcell decay of the qubit. This decay corresponds to relaxation of the qubit by photon emission from
the cavity [31]. In the dispersive regime, the Purcell decay rate due to an empty cavity is γκ = (g/∆)2 κ[ωa ]
where κ[ωa ] is the damping rate of the cavity evaluated
at the qubit frequency ωa , ∆ is the qubit-cavity detuning
and g is the resonant qubit-cavity coupling [32]. A simple golden-rule calculation shows that, in the presence
of squeezed light, this rate becomes γs = γκ (2¯
na + 1),
2 sinh2 r
where n
¯ a = 1+(∆/Γ)2 is the photon number evaluated
at the qubit frequency. As expected, the contribution
from n
¯ a to the decay is therefore largely suppressed for
δκ/¯
κ=0
1%
2%
4%
δκopt
8
(b) 10
χ/(2π) [MHz]
(a) 10
SNRr /SNRα
describes quantum fluctuations of the field at the input
of resonator j. Using this definition, the total number
of input photons N = Ns + Nd has a contribution from
ˆ = 2 sinh2 r and Nd from the cosqueezing Ns = hAˆ† Ai
herent displacement. Focusing on times τ 1/κ, we can
ignore the transient response to the coherent drive, and
¯ 0 κτ . Fixing N and taking t0 −1/κ, the
hence Nd = 41 n
optimal SNR is obtained for Ns = N 2 /[2(N + 1)], and is
p
SNRopt = 2| sin ϕqb |N 1 + 2/N → 2| sin ϕqb |N, (3)
6
4
2
0
0.01
0.02
δχ/χ
¯
0.03
0.04
5
0
-5
-10
0.31
χ1
χ2
0.32
0.33
EC /h [GHz]
0.34
FIG. 4. (a) SNR enhancement as a function of the dispersive shift asymmetry (χ1,2 = δχ ± χ)
¯ for different resonator
linewidth asymmetries (κ1,2 = κ
¯ ±δκ) calculated for χ
¯=κ
¯ /2,
κτ = 10 and e2r = 100. The dashed line is the maximal
SNR obtained by optimizing δκ. (b) Calculated dispersive
shifts as a function of transmon anharmonicity EC from a
numerical diagonalization of transmon-resonator system for
each of the resonators. The parameters are EJ /h = 25 GHz,
ω1 /2π = 7.6 GHz, ω2 /2π = 7.9 GHz, g1 /2π = 8 MHz and
g2 /2π = 15 MHz. The vertical dashed line shows a typical
value of EC that leads to equal and opposite dispersive shifts.
∆ Γ and hence pre-squeezing (i.e. t0 6= 0) does not
affect the qubit relaxation rate for Γ ∆. As typical
detunings ∆ κ, there is a wide range of ideal squeezing bandwidths satisfying κ Γ ∆. Such bandwidths
are large enough to allow a full enhancement of the SNR
(with τ & 1/κ), and small enough that the squeezing does
not appreciably modify cavity-induced Purcell decay.
Another non-ideality is asymmetry in the system parameters. While the two cavity frequencies can differ,
we have assumed so far that they have identical damping rates (κ1 = κ2 = κ) and that the dispersive coupling strengths satisfy χ1 = −χ2 = χ. Deviation from
either of these conditions breaks the symmetry yielding a QMFS, causing an unwanted coupling between
ˆ − , Yˆ+ ) and the anti-squeezed
the squeezed quadratures (X
ˆ
ˆ
quadratures (X+ , Y− ). The structure of the QMFS can
persist in the presence of asymmetries for long measurement times κτ 1, under the condition [26]
χ1 + χ2
κ1 − κ2
=
.
χ1 − χ2
κ1 + κ2
(4)
The SNR enhancement can however be preserved for
measurement times τ ∼ 1/κ by optimizing δκ/δχ, as
illustrated 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.
Implementation in circuit QED– We now turn to a
possible realization of this protocol in circuit QED. All
parameters discussed here are readily achievable experimentally. As illustrated in Fig. 4(b), a transmon qubit
is coupled to two resonators, one in the usual dispersive regime (∆ > EC ) while the other in the ‘straddling’ regime (∆ < EC ) [18, 35]. Here, ∆ is the qubitresonators detuning and EC the transmon anharmonic-
5
ity. 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 exhibit
a richer dispersive shift profile [36]. The displaced twomode squeezed state required at the input can either be
generated by a NDPA such as the Josephson parametric
converter [28], a Josephson paramp with sufficient bandwidth to allow frequency-selective separation of signal
and idler photons [37] or the Bose-Hubbard dimer [29].
Conclusion– We have presented a realistic measurement protocol that allows one to exponentially enhance
dispersive 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 subspace”.
Acknowledgements– We thank Irfan Siddiqi, Will
Oliver, 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 W911NF-14-1-0078,
INTRIQ, NSERC and CIFAR.
[1] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306,
1330 (2004).
[2] C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[3] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev.
A 33, 4033 (1986).
[4] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne,
Science 209, 547 (1980).
[5] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52, 341
(1980).
[6] V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, Cambridge, 1995).
[7] The LIGO Scientific Collaboration, Nat. Phys. 7, 962
(2011).
[8] The LIGO Scientific Collaboration, Nat. Photon. 7, 613
(2013).
[9] K. Iwasawa, K. Makino, H. Yonezawa, M. Tsang,
A. Davidovic, E. Huntington, and A. Furusawa, Phys.
Rev. Lett. 111, 163602 (2013).
[10] M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage,
B. H.-A., and W. P. Bowen, Nat. Photon. 7, 229 (2013).
[11] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information (Cambridge University Press,
Cambridge, 2000).
[12] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and
R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
[13] E. Jeffrey, D. Sank, J. Y. Mutus, T. C. White, J. Kelly,
R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,
A. Megrant, P. J. J. O’Malley, C. Neill, P. Roushan,
A. Vainsencher, J. Wenner, A. N. Cleland, and J. M.
Martinis, Phys. Rev. Lett. 112, 190504 (2014).
[14] Y. Liu, S. J. Srinivasan, D. Hover, S. Zhu, R. McDermott,
and A. A. Houck, New J. Phys. 16, 113008 (2014).
[15] Z. R. Lin, K. Inomata, W. D. Oliver, K. Koshino,
Y. Nakamura, J. S. Tsai, and T. Yamamoto, Appl. Phys.
Lett. 103, 132602 (2013).
[16] S. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, Phys.
Rev. B 90, 134515 (2014).
[17] M. Tsang and C. M. Caves, Phys. Rev. X 2, 031016
(2012).
[18] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I.
Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
[19] M. Tsang and C. M. Caves, Phys. Rev. Lett. 105, 123601
(2010).
[20] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema,
M. V. Balabas, and E. S. Polzik, Phys. Rev. Lett. 104,
133601 (2010).
[21] M. J. Woolley and A. A. Clerk, Phys. Rev. A 87, 063846
(2013).
[22] K. Zhang, P. Meystre, and W. Zhang, Phys. Rev. A 88,
043632 (2013).
[23] E. S. Polzik and K. Hammerer, Annalen der Physik 527,
A15 (2015).
[24] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck,
D. I. Schuster, and S. M. Girvin, Phys. Rev. A 77,
012112 (2008).
[25] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt,
and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[26] See Supplemental Material for more information.
[27] P. D. Drummond and Z. Ficek, eds., Quantum Squeezing
(Springer, Berlin, 2004).
[28] E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and
B. Huard, Phys. Rev. Lett. 109, 183901 (2012).
[29] C. Eichler, Y. Salathe, J. Mlynek, S. Schmidt, and
A. Wallraff, Phys. Rev. Lett. 113, 110502 (2014).
[30] D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed.
(Springer, Berlin, 2008).
[31] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow,
J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio,
M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys.
Rev. Lett. 101, 080502 (2008).
[32] M. Boissonneault, J. M. Gambetta, and A. Blais, Phys.
Rev. A 79, 013819 (2009).
[33] M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, Appl.
Phys. Lett. 92, 203501 (2008).
[34] Y. Yin, Y. Chen, D. Sank, P. J. J. O’Malley, T. C.
White, R. Barends, J. Kelly, E. Lucero, M. Mariantoni,
A. Megrant, C. Neill, A. Vainsencher, J. Wenner, A. N.
Korotkov, A. N. Cleland, and J. M. Martinis, Phys. Rev.
Lett. 110, 107001 (2013).
[35] K. Inomata, T. Yamamoto, P.-M. Billangeon, Y. Nakamura, and J. S. Tsai, Phys. Rev. B 86, 140508 (2012).
[36] G. Zhu, D. G. Ferguson, V. E. Manucharyan, and
J. Koch, Phys. Rev. B 87, 024510 (2013).
[37] C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen,
J. M. Fink, S. Filipp, and A. Wallraff, Phys. Rev. Lett.
107, 113601 (2011).
Supplemental Material for
“Heisenberg-limited qubit readout with two-mode squeezed light”
Nicolas Didier,1, 2 Archana Kamal,3 Alexandre Blais,2, 4 and Aashish A. Clerk1
1
Department of Physics, McGill University, 3600 rue University, Montreal, Quebec H3A 2T8, Canada
2
D´epartment de Physique, Universit´e de Sherbrooke,
2500 boulevard de l’Universit´e, Sherbrooke, Qu´ebec J1K 2R1, Canada
3
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
4
Canadian Institute for Advanced Research, Toronto, Canada
I.
QUBIT READOUT WITH QMFS AND TMSS
In this section we detail the derivation of the results presented in the paper for qubit readout in a quantum-mechanics
free subspace (QMFS) and driven by two-mode squeezed states (TMSS).
A.
Noise
We consider a two-cavity setup where a single qubit is dispersively coupled to both cavities with opposite dispersive
shifts, i.e. χ1 = −χ2 ≡ χ. The cavities can have different frequencies. We suppose they have the same damping rate,
i.e. κ1 = κ2 ≡ κ. The cavities are coupled to zero-temperature environments at times t < t0 and, at times t ≥ t0 ,
a TMSS is injected at resonance with the cavities. In the rotating frame of the cavity frequencies, we define four
quadratures for the TMSS, X± , Y± , characterized by the correlations




1 0 0 i
e−2r 0
0 i


 0 1 −i 0 
 0 e−2r −i 0 
0
ˆ in (t)Z
ˆ T (t0 )i = 

hZ
Θ(t
−
t)δ(t
−
t
)
+
(S1)

 Θ(t − t0 )δ(t − t0 ),
0
in
 0 i 1 0
 0
i e2r 0 
−i 0 0 1
−i
0
0 e2r
ˆ = (X
ˆ − Yˆ+ X
ˆ + Yˆ− )T and Θ is the unit step function. From the last term of Eq. (S1), we see that the
where Z
ˆ S = (X
ˆ − Yˆ+ )T and the anti-squeezed subspace
TMSS is composed of two coupled QMSS: the squeezed subspace Z
T
ˆ
ˆ
ˆ
ZA = (X+ Y− ) . In order to involve only the squeezed QMFS in the dynamics, the Hamiltonian of the system is
ˆ = 1 χ(X
ˆ+X
ˆ − + Yˆ+ Yˆ− )ˆ
chosen to be H
σz . As explained in the paper, this is equivalent to having opposite dispersive
2
shifts. The Langevin equations for the intracavity quadratures are found from input-output theory [S1]

 1
0
0
σz
− 2 κ χˆ
 −χˆ
√
σz − 12 κ
0
0 
ˆ

ˆ in (t).
ˆ˙
(S2)
Z(t)
=
 Z(t) − κZ
1
 0
0
− κ χˆ
σz 
2
0
0
−χˆ
σz − 21 κ
The squeezed and anti-squeezed subspaces are indeed clearly decoupled. The output
quadratures are calculated from
ˆ +Z
ˆ in [S1]. The two-time output
ˆ out = √κZ
integrating Eq. (S2) and using the input-output boundary condition Z
correlations of the squeezed QMFS are then equal to
1
0
ˆ S,out (t)Z
ˆ T (t0 )i = e−2r δ(t − t0 )1 + κ(1 − e−2r )e− 2 κ(t+t −2t0 ) R[χˆ
hZ
σz (t − t0 )],
(S3)
S,out
!
cos φ sin φ
where 1 is the 2 × 2 identity matrix and R[φ] =
the rotation matrix. Here t, t0 > t0 and t0 ≤ 0. The
− sin φ cos φ
correlations of the anti-squeezed QMFS
are found by replacing e−2r by e2r in Eq. (S3). The noise associated with the
√ Rτ ˆ
ˆ
measurement operator M (τ ) = κ 0 dtYout,+ (t) then reads
ˆ 2 (τ )i = e−2r κτ + 8 cos2 ( 1 ϕqb )(1 − e−2r )[cosh( 1 κτ ) − cos(χτ )]e− 21 κ(τ −2t0 ) .
hM
2
2
(S4)
The measurement noise is identical for the two qubit states, hˆ
σz i = ±1, for all times τ and t0 . The last terms in
Eqs. (S3)–(S4) are due to the leakage of the initial vacuum fluctuations. As expected, the anti-squeezed quadratures
do not contribute to the measurement noise. When the cavities are sufficiently pre-squeezed, that is when the TMSS
is injected well before the measurement (t0 −1/κ), the noise is exponentially reduced.
7
B.
Asymmetries
To understand the impact of realistic asymmetries in the dispersive shifts, i.e. χ2 6= −χ1 , and damping rates, i.e.
κ2 6= κ1 , we define χ
¯ = (χ1 − χ2 )/2, κ
¯ = (κ1 + κ2 )/2, δχ = (χ1 + χ2 )/2, δκ = (κ1 − κ2 )/2. In the quadrature basis,
†
†
ˆ
ˆ 1 + χ2 a
ˆ2 a
ˆ2 )ˆ
σz reads
the Hamiltonian H = (χ1 a
ˆ1 a
2
2
ˆ = 1 χ(
ˆ ˆ
ˆ ˆ σz + 1 δχ(X
ˆ+
ˆ−
H
+ Yˆ+2 + X
+ Yˆ−2 )ˆ
σz .
2 ¯ X+ X− + Y+ Y− )ˆ
4
(S5)
Including the asymmetries, the Langevin equations are


¯
χˆ
¯σz − 12 δκ δχˆ
σz
− 21 κ
 −χˆ
1
√
σz − 21 δκ 
¯ −δχˆ
ˆ
 ¯ σz − 2 κ
ˆ in (t).
ˆ˙
(S6)
Z(t)
= 1
 Z(t) − κZ
1

 − δκ δχˆ
σ
−
κ
¯
χˆ
¯
σ
z
z
2
2
−δχˆ
σz − 12 δκ −χˆ
¯σz − 12 κ
¯
√ Rτ
ˆ (τ ) = κ
The resulting expression for the noise in the measurement operator M
¯ 0 dtYˆout,+ (t) is quite cumbersome.
One can check that the measurement noise does not depend on the qubit state. The leading term for τ 1/κ is
(χδκ
¯ −κ
¯ δχ)2 (e4r − 1)
ˆ 2 (τ )i = 1 +
hM
e−2r κ
¯ τ.
(S7)
[(χ
¯ + δχ)2 + 41 (¯
κ + δκ)2 ][(χ
¯ − δχ)2 + 14 (¯
κ − δκ)2 ]
To eliminate the contribution from the anti-squeezed QMFS, the asymmetries have to be tuned such that
Under this condition, the full expression for the noise is
δκ
κ
¯
2¯
κ2 δχ2
sinh 2r
(χ
¯2 − δχ2 )( 41 κ
¯ 2 + δχ2 )
1 2
1
κ
¯ 2 δχ
+ 2 1 2 1 2
¯ − χδχ
¯
cos(χ1 τ ) − 12 κ
¯ [χ
¯ + δχ] sin(χ1 τ ) e− 2 κ1 τ sinh 2r
4κ
2
¯ )( 4 κ
¯ + δχ )(χ
¯ + δχ)
(χ
¯ + 4κ
2
− 1 κ2 τ
1 2
κ
¯ δχ
1
+ 2 1 2 1 2
κ
¯
+
χδχ
¯
cos(χ
τ
)
+
κ
¯
[
χ
¯
−
δχ]
sin(χ
τ
)
e 2
sinh 2r.
2
2
4
2
(χ
¯ + 4κ
¯ )( 4 κ
¯ + δχ2 )(χ
¯ − δχ)
=
δχ
χ
¯ .
ˆ 2 (τ )i = e−2r κ
hM
¯τ +
(S8)
ˆ 2 (τ )i are constant or decaying exponentially. For finite integration times τ ∼ 1/κ, the
The remaining terms in hM
SNR can be maximized numerically by optimizing δχ and δκ.
C.
For a finite squeezing

1


ˆ in (t)Z
ˆ T (t0 )i =  0
hZ
in
 0
−i
The noise then reads
Squeezing bandwidth
bandwidth Γ, the input field correlations are



0 0 i
e−2r − 1
0
0
0



1 −i 0 
0
e−2r − 1
0
0

 −Γ|t−t0 |
Θ(t−t0 )Θ(t0 −t0 ), (S9)
 δ(t−t0 )+ 
e
2r



i 1 0
0
0
e −1
0
0 0 1
0
0
0
e2r − 1
Γτ + (e2r − 1)(1 − e−Γτ ) −2r
e κτ
Γτ
1
Γ(Γ − 12 κ) + 8 1 − e−2r cos2 ( 21 ϕqb ) 2
cosh( 12 κτ ) − cos(χτ ) e− 2 κ(τ −2κt0 )
1
2
χ + (Γ − 2 κ)
n
i
h
1 − e−Γτ
2
− 21 κτ
1
1
− 4 1 − e−2r cos2 ( 12 ϕqb ) 2
χ
−
κ(Γ
−
κ)
cos(χ(τ
−
t
))e
−
cos(χt
)
0
0
1
2
2
χ + (Γ − 2 κ)2
h
io
1
1
+Γχ sin(χ(τ − t0 ))e− 2 κτ + sin(χt0 ) e(Γ+ 2 κ)t0 .
(S10)
ˆ 2 (τ )i =
hM
The infinite bandwidth result Eq. (S4) is recovered in the limit Γτ → ∞.
8
II.
QUBIT READOUT WITH SINGLE-MODE SQUEEZED STATES (SMSS)
A.
Noise
We consider a cavity dispersively coupled to a qubit. As described for the QMFS protocol, the cavity is coupled to
a zero-temperature environment at times t < t0 and, at times t ≥ t0 , a SMSS is injected at resonance with the cavity.
We note θ the squeeze angle. In the rotating frame of the cavity frequency, we define two quadratures for the SMSS,
X, Y , characterized by the correlations
!
!
2
2r
2
−2r
1
−i
e
cos
θ
+
e
sin
θ
−i
+
sin
2θ
sinh
2r
ˆ in (t)Z
ˆ T (t0 )i =
hZ
Θ(t0 −t)δ(t−t0 )+
Θ(t−t0 )δ(t−t0 ), (S11)
in
i 1
i + sin 2θ sinh 2r
e−2r cos2 θ + e2r sin2 θ
ˆ = (X
ˆ Yˆ )T . The Langevin equations for the intracavity quadratures are
where Z
!
√
σz ˆ
− 21 κ χˆ
˙ˆ
ˆ in (t).
Z(t) − κZ
Z(t) =
1
−χˆ
σz − 2 κ
(S12)
ˆ out,ϕ = R[ϕh ]Z
ˆ out are
In the Fourier space and in the frame of the homodyne angle ϕh , the output quadratures Z
h
κω
κχˆ
σz
ˆ out,ϕ (ω) = − exp i arctan
ˆ in (ω).
Z
Z
(S13)
R
ϕ
−
θ
+
arctan
h
h
1 2
1 2
2 − ω2
2 + ω2
ω
+
χ
ω
−
χ
4
4
The rotation angle is frequency-dependent, the squeezed and anti-squeezed quadratures are thus simultaneously
involved in the quadratures
of the output field for any value of ϕh . For the qubit state |1i, the noise of the measurement
R
ˆ N (τ ) = √κ τ dtYˆout,ϕ (t) is then equal to
operator M
h
0
2
2
−2r
ˆ
hM
+ sin2 (θ − ϕh − ϕqb )e2r
N,|1i (τ )i = κτ cos (θ − ϕh − ϕqb )e
n
o
1
+ 4 sinh 2r sin ϕqb cos 12 ϕqb sin[2(θ − ϕh ) − 32 ϕqb ] − e− 2 κτ sin[2(θ − ϕh ) − 32 ϕqb − χτ ]
1
+ 16 sinh2 r cos2 12 ϕqb cos χτ − cosh 21 κτ e− 2 κ(τ −2t0 )
1
+ 8 sinh 2r cos3 12 ϕqb cos[2(θ − ϕh ) − 23 ϕqb − χ(τ − 2t0 )](1 − cos χτ cosh 21 κτ )e− 2 κ(τ −2t0 )
1
− 8 sinh 2r cos3 21 ϕqb sin[2(θ − ϕh ) − 23 ϕqb − χ(τ − 2t0 )] sin χτ sinh 21 κτ e− 2 κ(τ −2t0 ) .
(S14)
ˆ 2 (τ )i for the qubit state |0i is obtained by replacing ϕqb → −ϕqb and χ → −χ in Eq. (S14). The
The noise hM
N,|0i
anti-squeezed quadrature cannot be eliminated in the measurement noise. As a consequence, for a given measurement
time τ , there is an optimal value of the squeeze parameter r that minimizes the measurement noise.
B.
Optimal SNR
ˆ 2 i + hM
ˆ 2 i)1/2 , the signal can be readily
To calculate the SNR, defined by SNR = MS,|0i − MS,|1i /(hM
N,|0i
N,|1i
calculated from the Langevin equation Eq. (S12), giving
i
√
1
4 χh
MS,|0i − MS,|1i = 2 n
¯ 0 cos(ϕh ) sin(ϕqb )κτ [1 − F (τ )],
F (τ ) =
sin(ϕqb ) − sin(ϕqb + χτ )e− 2 κτ .
(S15)
κτ κ
The noise contribution is calculated from Eq. (S14), we take the limit t0 → −∞ for simplicity,
ˆ 2 i + hM
ˆ 2 i = 2κτ {cosh(2r) − cos[2(ϕh − θ)] sinh(2r)G(τ )} ,
hM
N,|0i
N,|1i
h
i
1
4
G(τ ) = cos(2ϕqb ) +
sin(ϕqb ) cos( 12 ϕqb ) sin( 23 ϕqb ) − sin( 32 ϕqb + χτ )e− 2 κτ .
κτ
The corresponding SNR is optimized with ϕh = 0; θ = 0 if G(τ ) > 0 and θ =
SNRSMSS (ropt ) =
√
2¯
n0 κτ | sin(ϕqb )|
π
2
(S16)
(S17)
if G(τ ) < 0; ropt = 21 arctanh|G(τ )|,
√
|1 − F (τ )|
|1 − F (τ )|
= 2¯
n0 κτ | sin(ϕqb )|eropt
.
2
1/4
[1 − G (τ )]
[1 + |G(τ )|]1/2
(S18)
The optimal squeeze strength is rather small, leading to a modest improvement of the SNR. For instance, for
χ = 21 κ: at τ = 1/κ, e2ropt ' 1.8 and SNRSMSS (ropt )/SNRSMSS (0) ' 1.1; at τ = 10/κ, e2ropt ' 3.0 and
SNRSMSS (ropt )/SNRSMSS (0) ' 1.3; at τ = 100/κ, e2ropt ' 9.9 and SNRSMSS (ropt )/SNRSMSS (0) ' 2.2.
9
C.
Scaling
We now focus on the situation that maximizes the signal, i.e. χ = κ/2. For κt0 → −∞ and neglecting the transient
dynamics due to the drive as well as the terms exponentially decaying with κτ , the SNR reads
SNR2SMSS =
32Ns (N − Ns )
1+
16Ns2
κτ
=
32Ns (N − Ns )2
,
N − Ns + 4¯
n0 Ns2
(S19)
¯ 0 κτ /4. Unlike the
where we used Ns = sinh2 r ' 41 e2r for large squeeze parameters and N = Nd + Ns with Nd = n
QMFS scheme, the SNR cannot be written only in terms of N and Ns ; the parameter κτ or n
¯ 0 is also involved.
For a fixed drive strength n
¯ 0 , we calculate the optimal value of Ns . The SNR then scales as
p
1/4
SNRSMSS,opt ' 8N 3/4 /¯
n0 ,
n0 ,
(S20)
Ns,opt = 21 N/¯
where we have taken the large N limit. Such a scaling, SNR ∝ N 3/4 , is obtained in a Mach-Zehnder interferometer
driven with squeezed light [S2, S3].
For a fixed integration time κτ , we calculate the optimal value of Ns . The SNR then scales as
p
qp
√
√
16N 2 κτ + (κτ )2 − κτ
1/4
2
SNRSMSS,opt =
(16N + κτ )κτ − κτ ' 2 N (κτ ) ,
Ns,opt =
' 14 κτ ,
(S21)
16N
where we have taken the large N limit. Such a scaling, SNR ∝ N 1/2 , is obtained with a coherent drive without
squeezed light [S3].
D.
Purcell decay
R∞ p
The coupling Hamiltonian between a cavity mode and the environment is Hdiss = 0 dω κ(ω)[ˆb† (ω)+ ˆb(ω)][ˆ
a† +ˆ
a],
ˆ
where a
ˆ is the annihilation operator of the cavity and b(ω) the annihilation operator of the bath at ω. We apply the
dispersive transformation to diagonalize the Jaynes-Cummings interaction between the cavity and the qubit in the
dispersive regime, where the detuning ∆ = ωa −ω1 between the qubit and the cavity is much larger than their coupling
strength g. Under this unitary transformation (noted with the superscript D) and at lowest order in g/∆, the cavity
g
operator acquires a contribution from the qubit: a
ˆD = a
ˆ+ ∆
σ
ˆ− . In the dressed basis, the qubit becomes coupled
to the environment of the cavity. Following Ref. [S4], we apply the rotating-wave approximation in the interaction
picture to obtain the coupling Hamiltonian between the system and the environment in the dressed basis
Z ∞ p
Z
g ∞ p
D
Hdiss
=
dω κ(ω) ˆb† (ω)ˆ
a ei(ω−ω1 )t +
dω κ(ω) ˆb† (ω)ˆ
σ− ei(ω−ωa )t + H.c.
(S22)
∆ 0
0
The last term of the coupling Hamiltonian Eq. (S22) results in a decay channel for the qubit. The corresponding
g 2
damping rate is γκ = ∆
κ(ωa ). For a squeezed environnement of squeeze parameter r at resonance with the cavity
frequency ω1 and with a bandwidth Γ, the bath correlators are
hˆb† (ω)ˆb(ω 0 )i = sinh2 r
(ω 0
Γ2
δ(ω + ω 0 ), hˆb(ω)ˆb(ω 0 )i =
− ω1 )2 + Γ2
1
2
sinh(2r)eiθ
Γ2
δ(ω + ω 0 − 2ω1 ). (S23)
(ω − ω1 )2 + Γ2
Because the qubit is strongly detuned with the cavity, the squeezing terms will only have a negligible effect on the
2
qubit. The qubit is thus coupled to a thermal bath with the population n
¯ a = sinh2 r ∆2Γ+Γ2 and the damping rate γκ .
One can then compute the dynamics of hˆ
σz i
1
1
e−γs t −
hˆ
σz i(t) = hˆ
σz i(0) +
,
γs = γκ (2¯
na + 1).
(S24)
2¯
na + 1
2¯
na + 1
The qubit excited state relaxes with the Purcell decay rate γs .
[S1]
[S2]
[S3]
[S4]
C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, Berlin, 2004).
C. M. Caves, Phys. Rev. D 23, 1693 (1981).
V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).
M. Boissonneault, J. M. Gambetta, and A. Blais, Phys. Rev. A 79, 013819 (2009).