PRL 108, 153603 (2012)

week ending
13 APRIL 2012PHYSICAL REVIEW LETTERS

Using Interference for High Fidelity Quantum State Transfer in Optomechanics
Ying-Dan Wang and Aashish A. Clerk
Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8(Received 26 October 2011; published 13 April 2012)
We revisit the problem of using a mechanical resonator to perform the transfer of a quantum state
between two electromagnetic cavities (e.g., optical and microwave). We show that this system possesses
an effective mechanically dark mode which is immune to mechanical dissipation; utilizing this featureallows highly efficient transfer of intracavity states, as well as of itinerant photon states. We provide
simple analytic expressions for the fidelity for transferring both Gaussian and non-Gaussian states.
DOI: 10.1103/PhysRevLett.108.153603

PACS numbers: 42.50.Wk, 07.10.Cm, 42.50.Ex

Introduction.—The field of quantum optomechanics,
where a mechanical resonator is coupled to photons in acavity, has seen remarkable recent progress. Milestones
include using the backaction of photons to cool a mechanical resonator to near its ground state [1,2], and the observation of strong coupling effects [3–5]. The ability of amechanical resonator to couple to diverse electromagnetic
cavities naturally leads to what is perhaps the most promising application of this field: the possibility of efficiently
transferring a quantum state between photons with vastlydiffering wavelengths [6–9]. Such state transfer would
have direct utility in quantum information processing(e.g., the transfer of quantum information from a superconducting qubit in a microwave circuit QED setup to an
optical photon, or highly nonclassical microwave states as
prepared in Ref. [10] to optical photons).
Previous investigations of this problem have largely
considered schemes based on two successive ‘‘swap’’
operations in a two-cavity optomechanical system[Fig. 1(a)]. One pulses the optomechanical interactions tofirst exchange the states of the first cavity and the mechanical resonator; this is then repeated to exchange themechanical and the second cavity states [6,7,11]. Whileintuitively simple, achieving high fidelity with this protocol is only possible with low levels of cavity and mechanical dissipation; we quantify this below. In particular, one
requires extremely low mechanical bath temperatures. Thisis true even if the mechanics is initially prepared in its
ground state [6,7], as heating during the transfer nonetheless degrades the state. Aspects of this swap scheme wererecently implemented experimentally [12].
Given the above, it would be highly advantageous to find
new state transfer schemes less sensitive to mechanicaldissipation. This is the goal of this Letter. We show thatthe two-cavity optomechanical system possesses a mode
which is delocalized between the two cavities while simultaneously being decoupled from the mechanical dissipation; we term this decoupled mode a ‘‘mechanically dark’’mode, as it is analogous to an atomic state which isprotected against optical excitation by destructive interference [13]. We show that by using this dark mode, one can
0031-9007=12=108(15)=153603(5)perform high-fidelity quantum state transfer of intracavity
states at levels of mechanical dissipation where the conventional double-swap scheme is essentially unusable.
We also show that this dark mode can be used for efficienttransfer of itinerant photons (e.g., transferring the state ofphotons incident on a microwave cavity to the state ofphotons leaving an optical cavity). This approach is particularly attractive, as it does not require any time-dependent
variation of optomechanical couplings. Further, if one iswilling to only consider the transfer of small-bandwidthstates, the scheme can also be used without requiring optomechanical strong coupling. We quantify analytically the
fidelity of this scheme for Gaussian states (in a way thatallows easy comparison against the intracavity transfer
schemes mentioned above), as well as nonclassical states;
we also consider limitations on the bandwidth of the statesthat can be transferred. These analytic expressions yield asimple intuitive picture of the factors limiting fidelity. In thelimit of weak coupling, this itinerant-photon transfer
scheme is equivalent to that described by Safavi-Naeni
et al. [8] (though that work did not discuss fidelities, strong
coupling, or the role of the dark mode).
Model.—We consider an optomechanical system wherea single mechanical resonator is simultaneously coupled toboth an optical cavity and a microwave cavity via dispersive couplings [see Fig. 1(a)]; particular experimental realizations are discussed in Refs. [7,8]. We also focus on thestandard situation where a weak bare optomechanicalcoupling g  , !M is enhanced by strongly driving
each cavity, resulting in effective linear couplings (see,e.g., [14,15]). We work in an interaction picture withrespect to the two cavity drives, and in a displacement
picture with respect to the average (classical) field ineach cavity. The resulting Hamiltonian takes the formX^ þ H^ diss :H^ ¼ !M a^ y a^ 
½i d^yi d^i  Gi ða^ y d^i þ d^yi aÞ
i¼1;2

(1)
^ is the mechanical frequency (annihilatHere, !M (a)ion operator), d^i is the annihilation operator of cavity

153603-1

Ó 2012 American Physical Society

PHYSICAL REVIEW LETTERS

PRL 108, 153603 (2012)
(a)

(b)

1.0

Fidelity F

0.8
0.6
0.4
0.2
0.0

0

2

4
6
8
Temperature T (K)10

FIG. 1 (color online). (a) The two-cavity optomechanical system. (b) Fidelity of the double-swap (DS) protocol (green or
light) and the adiabatic transfer (AT) protocol (blue or dark) totransfer a coherent state j ¼ 1i, where G=2 ¼ 2 MHz (solid),
G=2 ¼ 0:5 MHz (dashed), and  ¼ 2  1 kHz, !M =2 ¼
10 MHz. Cavity 1 (2) is a microwave (optical) cavity: 1 =2 ¼
10 GHz (2 =2 ¼ 100 THz), 1 ¼ 2 ¼ 2  50 kHz. For
DS, G1 ¼ G2 ¼ G and the total transfer time is ts ¼ =G. For
AT, we used an optimal modulation with G21 ðtÞ þ G22 ðtÞ ¼ G2
constant, and an optimal transfer time at each temperature,
which represents a trade-off between heating (via nonadiabatic
transitions) and amplitude decay [27]. The blue dotted line
corresponds to AT of j ¼ 0:1i with G=2 ¼ 2 MHz; as theamplitude-decay effect is negligible, a long transfer time can beused to suppress heating caused by nonadiabatic transitions. For
DS, the fidelity is mainly limited by heating, hence F versus T
for j ¼ 0:1i is almost indistinguishable from j ¼ 1i.

i (i ¼ 1, 2) in the displaced frame, and i is the detuning of
the drive applied to cavity i. The driven optomechanicalcoupling between the mechanical resonator and cavity i isdenoted as Gi ; note that these are proportional to the drive
amplitude applied to cavity i, and thus can be controlled intime. Hdiss describes the damping and driving of the twocavities and mechanical resonator by independent Ohmicbaths. We let  (i ) denote the damping rate of themechanical resonator (cavity i), and let NM (Ni ) denotethe bath temperature (expressed as a number of thermal
quanta). We also assumed the optimal situation where
each cavity is far into the resolved-sideband regime!M  i , and where each cavity is driven near the
red-detuned mechanical sideband (i.e., i  !M ). Thispermits us to make a rotating wave approximation inwriting the optomechanical interactions, resulting in a‘‘beam-splitter’’ form which is optimal for state transfer[11]. We assume below negligible phase noise in the cavity
drives. The effect of such noise on coherent transfer wasstudied in Refs. [16,17]; the resulting requirements are
similar to those needed for cavity-cooling [17], and arethus within reach of experiment.
Double-swap protocol.—The optomechanical interactions in Eq. (1) can be used to swap states between thethree modes of the system [6,7,11]. The swap protocol
involves first turning on the interaction G1 for a time ts1 ¼
=ð2G1 Þ (while G2 ¼ 0), which if  ¼ 1 ¼ 2 ¼ 0
would swap the states of cavity 1 and the mechanical^ si Þ ¼ id^i ð0Þ and d^i ðtsi Þ ¼ iað0Þ].
^
resonator [i.e., aðtweek ending
13 APRIL 2012

One then shuts off G1 and turns on G2 for a time ts2 toswap the mechanical state to cavity 2.
The presence of mechanical and cavity dissipation degrades this protocol’s fidelity. To quantify this, we consider
the simple case of transferring a Gaussian state, and calculate the Uhlmann fidelity F [18] between initial and final
states. Letting 1 (2 ) denote the density matrix of cavity 1(2) at the start (end) of the transfer, we find [19]:
1
2
pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ 1=2 2F  ðTr½ð 1 2 1 Þ Þ ¼
exp :
1 þ n h
1 þ n h(2)
Note that we will optimize the fidelity over simple rotations in phase space (so that if 2 is a rotated version of 1 ,F ¼ 1). F depends on just two parameters: n h represents
the heating of the state during the protocol by noise emanating from cavity and mechanical dissipative baths, while characterizes the decay of the mean value of d^ due tocavity and mechanical damping. Efficient transfer requires
minimizing both these effects. In the double-swap protocol, the amplitude-decay will completely suppress F unless
one is in the strong coupling limit Gi > i . In this relevant
limit, and for the case where the state to be transferred is acoherent state ji, we find the simple result [21]:X NM þ i Ni
X i þ n h ¼
ts ; (3)tsi ;  ¼ jj
4 i
2
i
i
where ðNM þ Ni Þ=2 is the average heating rate andði þ Þ=4 is the average amplitude-decay rate during
each time interval. We have assumed the optimal situation
where the mechanical resonator is initially in its groundstate [6,7]. Despite this precooling, the mechanical contribution to n h can still be large. One thus requires anextremely low mechanical bath temperature to ensure
good fidelity using swap scheme [see Fig. 1(b)].
Effective mechanically dark mode.—From Eq. (3), we
see that the heating n h due to mechanical noise in the
double-swap scheme is simply the heating rate times transfer time, and hence scales as 1=G. We now show thattransfer protocols exist where the mechanical-heating
effect is even more greatly suppressed with increasing G.
This is possible by making use of a mode of the two-cavity
optomechanical system which is simultaneously delocalized between both cavities, but at the same time is largely
immune to mechanical dissipation.
Focusing as before on the case where each cavity isdriven at the red-detuned sideband, we note that the coherP
ent part of the Hamiltonian H^ 0 ¼ H^  H^ diss ¼ j @!j c^ yj c^ jpﬃﬃﬃ^
with j ¼ , dk. c^   ðc^ br  aÞ=
2 describe hybridizedmodes of the bright cavity mode c^ br  ðG21 þ^
G22 ÞÞ1=2 ðG1 d^1 þ G2 d^2 Þ and the mechanical mode a,qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
22
with eigenfrequencies ! ¼ !M  G1 þ G2 . Whereas

153603-2c^ dk  ðG21 þ G22 Þ1=2 ðG2 d^1 þ G1 d^2 Þ(4)describes a delocalized cavity mode which is decoupledfrom the mechanics. We thus refer to c^ dk as a ‘‘mechanically dark’’ mode; its frequency is !dk ¼ !M , independentof coupling. As we now demonstrate, utilizing this mode
allows the efficient transfer of both intracavity anditinerant-photon states.
Adiabatic transfer.—Consider first the same problem
addressed by the double-swap scheme, the transfer of an
intracavity state initially in cavity 1 to cavity 2. This can beaccomplished by using an adiabatic passage approach,similar to the well-known STIRAP scheme [23]. One
modulates G1 ðtÞ and G2 ðtÞ so that the dark mode adiabatically evolves from being d^1 at t ¼ 0 to d^2 at the end of
the protocol at a time t ¼ tf . The cavity state is thustransferred from cavity 1 to cavity 2 using the coupling
to the mechanics, but without actually populating themechanics; the result is a greatly enhanced protection
against mechanical sources of dissipation.
Figure 1(b) shows how such an adiabatic transfer
protocol improves the state transfer fidelity over the
double-swap scheme when the mechanical-heating effectis non-negligible. When transferring a Gaussian state, Fagain takes the general form described by Eq. (2). The
adiabatic ‘‘dark mode’’ transfer protocol dramatically suppresses n h compared to the swap scheme. However, toremain adiabatic, the transfer must ideally occur over atime long compared with 1=G. Thus, similar to the swap
scheme, one needs strong coupling (i.e., i  Gi ) to avoid
the amplitude-decay suppression of F described by  in
Eq. (2). Nonetheless, the greater resilience against mechanical noise presents a strong advantage over the doubleswap scheme. A somewhat related scheme for transferring
atomic motional states was discussed in Ref. [11]; theunidirectional ‘‘cascaded’’ coupling used there is fundamentally different from that considered here.
Itinerant state transfer.—While the previously discussed
transfer schemes require a strong optomechanical coupling(i.e., Gi  i ), mechanically mediated transfer is also
possible in the opposite regime if the goal is to transfer anarrow-bandwidth state of photons incident on cavity 1 tothe state of photons leaving cavity 2 [8,24]. We now showthat the mechanically dark mode discussed above plays animportant role in this itinerant-photon state transfer, andeven allows it to be highly effective in regimes of bothstrong and weak optomechanical coupling. We begin bywriting the Heisenberg-Langevin equations [25,26]:a^_ ¼ i!M a^  ð=2Þa^  iweek ending
13 APRIL 2012PHYSICAL REVIEW LETTERS

PRL 108, 153603 (2012)

X

pﬃﬃﬃﬃ
Gi d^i  a^ in ;pﬃﬃﬃﬃﬃd^_ i ¼ ii d^i  ði =2Þd^i  iGi a^  i d^i;in ;(5)with a^ in and d^i;in representing both input noise (taken
to be white) and signals driving each resonator. Solving
Eq. (5) and using standard input-output relations [25]yield the relation between input and output fields^ out ½! ¼ s½!A^ in ½! with A½!^^A
¼ fd^1 ½!; d^2 ½!; a½!gand s½! is the scattering matrix (see [27] for details).
High-fidelity transfer from d^1;in to d^2;out requires thatover the input signal bandwidth, the transmission coefficient js21 ½!j2  1, as well as that js23 ½!j2  0 (i.e.,
negligible transmission of mechanical noise). To quantify
this, we consider a Gaussian R
input state in a temporal mode
defined by u^ 1 ¼ ð2Þ1=2 d!f½!d^1;in ½! (see, e.g.,
Ref. [26]). f½! describes a wave packet incident on cavity
1R which is localized in both frequency and time;d!jf½!j2 ¼ 1 to ensure that u^ 1 is a canonical bosonicannihilation operator. The fidelity of transferring this itinerant Gaussian state again takes the general form of Eq. (2),and the parameters n h and  can be calculated analytically[22]. For a coherent state input j c in i / expðu^ y1 Þj0i:n h ¼

X Z

d!jf½!s2i ½!j2 Ni ;(6)i¼1;2;M







Z




i!
2


s21 ½!jf½!j 
 ¼ jjmin 1  
d!e
:






(7)

We have optimized the final state 2 in Eq. (2) over a timetranslation , so that if the output pulse is simply a timedelayed copy of the input pulse, F ¼ 1.
To have protection against mechanical dissipation, one
would ideally like the input state incident on cavity 1 to
only excite the dark mode. Without dissipation, the darkmode is energetically separated from the coupled modesc^  , and hence protection is achieved by using an input
signal with mean frequency !M in the displaced frame.
Including dissipation (and the consequent lifetime broadening), the input signal incident on cavity 1 will also excite
the bright cavity mode c^ br as well as the mechanical mode
^ This unwanted excitation is irrelevant as long as thea.
cooperativity of each cavity Ci ¼ G2i =i  1. In thislimit, the bright mode amplitude hc^ br i (average overj c in i) is a factor 1=C smaller than the dark mode
amplitude, due to a destructive interference akin to theoptomechanical analogue of electromagnetic-induced
transparency [28–30]. The mechanical mode amplitude
may be large in the case of weak coupling, but only results
in a small flux to the mechanical bath (and hence
pﬃﬃﬃaﬃ smallloss) due to the smallness of  (i.e., s31  1= C). The
transfer of the input signal thus occurs almost entirely via
the dark mode in this limit (see [27] for more details).
Good fidelity also requires that the dark mode, once
excited by the input state, only leaks out via cavity 2,
ensuring js21 ½!M j  1. This requires a destructive interference between the promptly reflected input signal and thewave leaving the dark mode via cavity 1. For Ci  1, thisinterference cancellation results in the simple impedance
matching condition C1 ¼ C2  C, i.e.,

153603-3

G21 =1 ¼ G22 =2 :(8)Taking our input mode jf½!j2 to have mean frequency !Mand a Gaussian profile with variance !2 , and assuming
C1 ¼ C2  1, we find to leading order in !:
n h







NM
! 2
2
1þ
1
;
G
4C
16G2

(9)




 2 2 
1
2! 2

þ
1þ
:
jj
8C

8G2

(10)

Good fidelity requires a high cooperativity C  jj, NM .
In
pﬃﬃﬃﬃﬃﬃthe
ﬃ weak-coupling regime G < , one also needsjj!  ðG2 =Þ, which reflects the width of thes21 ½! transmission resonance.
pﬃﬃﬃﬃﬃﬃIn
ﬃ the opposite regimeG  , one needs ! = jj as shown in Fig. 2(a).
Further, we see that in comparison against the double-swap
scheme, the mechanical-heating effect described by n h is
reduced by a factor =G. The expression of n h is the usual
weak-coupling expression for the mechanical temperaturecavity cooling [14,15]; unlike cavity cooling, it describesn h in both weak and strong coupling regimes.
Transfer of nonclassical itinerant states.—Given theadvantages of the itinerant transfer scheme, it is alsointeresting to consider how well it is able to transfer nonclassical states. While in general it is difficult to obtain
analytic expressions for the evolution of non-Gaussianstates, we show that here, one can obtain useful and reliable
analytic approximations.
We again consider an input mode in a given temporalmode u^ 1 ; we take this mode to be centered on !M , and forsimplicity, to have a vanishingly small bandwidth !.
Suppose now the input state incident on cavity 1 is a
Fock state of this mode jni / ðu^ y1 Þn j0i. We also take thenoise driving both cavities to be zero temperature (N1 ¼
N2 ¼ 0), but allow the mechanical resonator to be driven
by thermal noise. Letting pth ðq; NM Þ be the probability of
having q thermal quanta incident on the mechanical resonator, the fidelity can be decomposed as

Fidelity F

0.40

1.0

0.32

0.8

0.6

0.24

0.4

0.16

0.2

0.08

0.0

0

5 10 15 20 25 30 35
Bandwidth ∆ω/κ0.00

Fidelity F

(a)

0.8

Heating n h

1.0

week ending
13 APRIL 2012PHYSICAL REVIEW LETTERS

PRL 108, 153603 (2012)

(b)

F¼

1
X
r¼0

Pðr; nÞ ¼

1 X
1X

pth ðq; NM Þjfqðr;nÞ j2 ;(11)

r¼0 q¼0

where Pðr; nÞ is the probability of having r outgoing
photons leaving cavity 1 and n photons leaving cavity 2,
and the expression for fqðr;nÞ can be found in the
Supplemental Material [27].
Note that in the regime of optimal state transfer C1 ¼
C2  C  1, the probability of having photons leave cavity 1 is small: the dark mode effectively prevents mechanical photons from contributing, and Eq. (8) ensures minimalreflection of signal photons. One can thus get a good
approximation by simply retaining the r ¼ 0 and r ¼ 1
term in Eq. (11): F is approximately the probability ofobtaining n photons in the cavity 2 output mode and at
most one photon leaving cavity 1. This is a rigorous lower
bound on the exact fidelity, and is exact to order 1=C.
In the limit C  1, one finds that to leading order in 1=C
the fidelity for transferring the n-photon itinerant Fockstate is F ’ 1  ½NM ð3 þ 2nÞ þ n=4C. For NM  1, thecondition for a near-unity fidelity is thus C  NM n; for alarge-n Fock state, this is more stringent than the condition
for havingpﬃﬃﬃ a large fidelity transfer of a coherent state withjj  n [cf. Eqs. (9) and (10)].
Finally, we note that the same approach can be used tocompute the fidelity of transferring
an arbitrary pure inputPstate of the form j1 i ¼ m cm jmi; the full expression isprovided [27]. The transfer fidelity of different nonGaussian states together with a coherent state (for realistic
parameters) is shown in Fig. 2(b).
Conclusions.—In this Letter, we have proposed using amechanically dark delocalized mode in a two-cavity optomechanical system for quantum state transfer. We havedemonstrated that both intracavity states and itinerantphoton states can be transferred with high fidelity, usingparameters within reach of current experiments.
We thank S. Chesi, K. Lehnert, O. Painter, C. Regal, and
A. Safavi-Naeini for useful discussions. This work was
supported by the DARPA ORCHID program under a grantfrom the AFOSR.
Note added.—See the related Letter by Tian in thisissue [31].

0.6
0.4
0.2
1

2

5 10 2050 100Co-operativity C x10 3

FIG. 2 (color online). Fidelity for transferring itinerant-photon
states at T ¼ 2 K. (a) Fidelity (red, solid) and heating (blue,
pﬃﬃﬃ
dashed) versus input bandwidth for transferring j ¼ 3icoherent state. (b) Fidelity versus cooperativity C ¼ G2 =pﬃﬃﬃin
the narrow-bandwidth limit. The input states are j ¼ 3i
(green, dash-dotted),
jn ¼ 3i Fock state (blue, dashed), andpﬃﬃﬃ
ðj1i þ j3iÞ= 2 (red, solid). Unless specified here, the systemparameters are the same as the dashed lines in Fig. 1(b).

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