Quantum State Orthogonalization and
a Toolset for Quantum Optomechanical Phonon Control
Abstract
We introduce a method that can orthogonalize any pure continuous variable quantum state, i.e. generate a state from where , which does not require significant a priori knowledge of the input state. We illustrate how to achieve orthogonalization using the Jaynes-Cummings or beam-splitter interaction, which permits realization in a number of systems. Furthermore, we demonstrate how to orthogonalize the motional state of a mechanical oscillator in a cavity optomechanics context by developing a set of coherent phonon level operations. As the mechanical oscillator is a stationary system such operations can be performed at multiple times, providing considerable versatility for quantum state engineering applications. Utilizing this, we additionally introduce a method how to transform any known pure state into any desired target state.
A qubit basis formed by a pair of orthogonal quantum states is central to quantum information processing. Currently there is considerable effort towards implementing quantum information processing with two-level systems. For such systems, an intriguing and fundamental fact is that quantum mechanics prohibits the construction of a universal-NOT gate that would produce an orthogonal qubit from any input qubit ref:qubitOrthog . This quantum mechanical property is closely related to the quantum no-cloning theorem Scarani2005 , however, faithful cloning can be achieved probabilistically provided that the set of input states is linearly independent Duan1998 . Similarly, using such an input set of states, it is possible to construct a probabilistic NOT operation for qubits Yan2012 . A qubit basis may, however, also be formed using two orthogonal continuous variable states. Thus far, efforts to construct such a basis have mainly concentrated on using a superposition of coherent states Jeong2002 . Also, recently a qubit basis was realized using photon subtraction from squeezed vacuum Neergaard-Nielsen2010 .
In this Letter, we introduce a method for quantum state orthogonalization for continuous variable quantum systems. Notably, the method only requires knowing the angle made by the state’s mean amplitude , where is the annihilation operator, and hence the scheme is magnitude independent. Furthermore, our method is readily extended to generate an arbitrary superposition of the initial state and an orthogonal counterpart to allow the encoding of quantum information. The orthogonalizer i.e. . Thus, is a quadrature operator that is perpendicular to note:SpinHalfAnalog . when is formed by a linear superposition of the bosonic annihilation and creation operators and generates a state orthogonal to any pure state
The orthogonalizer can be realized with interactions that are available in many physical systems, e.g., to realize in cavity-quantum-electrodynamics Kimble1998 ; Raimond2001 , one prepares an input qubit in the state which then weakly interacts via the Jaynes-Cummings Hamiltonian , where is the coupling rate and are the raising and lowering operators. A controllably weighted superposition of addition and subtraction is achieved by projective measurement of the qubit onto . The measurement operator is then , see Fig.1 (a). With this interaction, optical Kimble1998 or microwave Raimond2001 fields in a cavity, or the motional state of trapped ions Leibfried2003 , can be orthogonalized by appropriately setting and . Similarly, a pure state of a traveling optical field can be orthogonalized by interaction on a beam-splitter and then measurement of an optical qubit comprising a superposition of zero and one photons Babichev2003 , see Fig.1 (b). As these interactions are common throughout quantum optics, adaptations of this orthogonalization protocol to other physical systems can be readily achieved. Moreover, a different scheme to perform a superposition of photon subtraction and addition was recently proposed LeeNha2010 , which could also be used to realize state orthogonalization.
The tools we introduce for orthogonalization can also be utilized for quantum state engineering applications. Currently, single-quanta-manipulation techniques performed on traveling light fields Kim2008 have prepared superposition states via photon subtraction ref:photonSubtraction , observed the bosonic commutation relation ref:bosonComm , and engineered arbitrary quantum states up to the two-photon level Bimbard2010 . Much progress has also been made for arbitrary quantum state preparation of the motion of trapped ions and microwave field states ref:arbQSP . As mechanical elements are now also considered for quantum applications, experimental tools are required for the coherent manipulation of phononic modes. Examples of progress in this direction are the observation of the ground state of motion OConnell2010 ; TeufelGS ; Chan2011 , steps towards single-phonon manipulation by coupling to a superconducting phase qubit OConnell2010 , strong coupling ref:strongCoupling , and mechanical mode thermometry via sideband asymmetry Safavi2012 . Also, recently the lattice vibrations of two diamonds were entangled by coherently distributing one quanta across the two vibrational modes Lee2011 .
Coherent phonon manipulation.— In this section we demonstrate how to perform an arbitrary coherent superposition of phonon subtraction, addition and the identity operation to a mechanical oscillator using cavity optomechanics. The prototypical optomechanical system is a Fabry-Pérot cavity where one of the mirrors is sufficiently compliant that the reflection of light can modify the mirror momentum via radiation-pressure. Concurrently, the motion of the moving mirror modulates the optical phase and generates sidebands. To realize phonon subtraction (addition) one can optically drive an optomechanical cavity at the red (blue) sideband and then perform single photon detection on the field scattered onto cavity resonance. Provided that the sidebands are well resolved and the optical phase shifts are small allowing linearization, the red-detuned drive gives rise to a beam-splitter interaction and the blue-detuned drive gives rise to a two-mode-squeezing interaction. This linearization procedure was discussed, for optomechanics, in Ref. Zhang2003 , where quantum state transfer between light and mechanics was proposed. Drive on the blue sideband has also been considered for continuous-variable teleportation from light to the mechanics ref:teleportation . Some other applications utilizing these sidebands are reviewed in Refs. ref:reviews .
Our proposed setup for coherent phonon control uses two orthogonally polarized optical fields to interact with the mechanical resonator, see Fig. 1(c). We consider a pulsed protocol where the conditional mechanical state following the pulsed interaction and measurement is determined. The optomechanical Hamiltonian Law1995 for the two independent optical modes in the optical rotating frame at the drive frequencies is
(1) |
where is the drive term, the subscripts label the two orthogonally polarized modes , and () is the cavity (mechanical) annihilation operator. (, mechanical angular frequency; , optical detuning; , optomechanical coupling rate; , cavity amplitude decay rate; , photon number per pulse; , drive amplitude, where
(2) |
A photon detection gives a measurement operator of the same form, however, with a phase shift on the identity.
Applications.— provides a method to prepare and manipulate quantum coherence between the mechanical energy levels. Setting , and
A mechanical resonator is a stationary system that allows to be conveniently performed multiple times. Moreover, as the superposition weightings can be changed between applications this provides considerable versatility for quantum state engineering and quantum control protocols. For instance, one could realize the protocol by Dakna et al. Dakna1999 to synthesize an arbitrary mechanical motional state. As another application, here we show that with applications of , one can transform the state into any target state , i.e. arbitrary quantum state transformation. Our method uses only the subtraction and identity components of note:coherentState and proceeds in a manner similar to Ref. Dakna1999 and generalizes the scheme presented in Ref. Fiurasek2005 . Specifically, by applying , where , to the state one can obtain provided that the set of coefficients is such that . Determining can be readily achieved via matrix inversion and a solution exists provided that note:Supplementary . For the initial state having , the probability of successful quantum state transformation is , where is the phonon number expectation prior to the th pulse. This probability may seem low, however, the experiment can readily be performed with a megahertz repetition rate using a 100 MHz mechanical oscillator and thus a practical number of heralding events can be attained in a reasonable time. Also, if the target state has a larger (smaller) dimension than the initial state one can apply creation (annihilation) as many times as necessary in order to make the dimensions the same prior to using .
An experimental approach.— There are numerous realizations of optomechanical systems and much progress has been made that can be built upon; the most pertinent being Refs. Lee2011 ; Lee2012 where phonon addition and subtraction were realized as separate operations. Combining these operations into a coherent superposition can be achieved with the setup in Fig. 1(c). Here we present an alternative route to fulfill the requirements of our proposed scheme using a mechanical element with a bulk acoustic wave vibration that forms an end mirror of a Fabry-Pérot cavity Kuhn2011 . This configuration has the advantage that the cavity decay can be controlled independently of the mechanical properties and such vibrational modes offer high mechanical resonance frequencies Borkje2012 . Moreover, simultaneous high reflectivity and high mechanical quality can be realized with multilayer crystalline reflectors Cole2008 . A 40 m diameter and several micrometer thick mirror has a mechanical resonance MHz with a 20 ng effective mass. With a finesse of , to achieve resolved sideband operation, i.e. , a 75 m cavity length can be used. For a drive laser with wavelength 1064 nm and a pulse duration of one hundred mechanical periods an optical power during the pulse of 1.3 mW is needed to achieve . During the interaction the mechanical resonator also interacts with its thermal environment. To neglect the effects of environmental coupling we require that , where is the mechanical phonon occupation in thermal equilibrium and is the mechanical quality factor. For and a mK bath, which can be readily achieved using dilution refrigeration, . Following the interaction the sideband needs to be separated from the drive field(s) prior to photon detection. For higher mechanical frequencies the filtering requirements simplify. However, it is possible to achieve sufficient filtering even for a 200 MHz mechanical frequency using an optical displacement and spectral filtering note:filtering . Realizing the displacement with optical-fiber-based components, which provide excellent spatial mode matching, one can achieve an interferometric visibility of that suppresses the drive by . The remaining drive can be further reduced by filtering with a cavity that has the same resonance frequency as the optomechanical cavity. To achieve a drive transmission times smaller than sideband transmission, a filter cavity amplitude decay rate of 2 kHz is required note:cavity . We would also like to emphasize that our scheme is robust against optical loss and inefficient detection as an optomechanically scattered photon that goes undetected does not trigger , hence, the primary effect of loss is to merely reduce the heralding probability note:twoPhoton . To characterize the mechanical motional state, as the parameter regime considered here is suited for the beam-splitter interaction, quantum state transfer of the mechanical motional state to the light Parkins1999 ; Zhang2003 can be performed followed by optical homodyne tomography. This interaction, following action(s) with to the stationary mechanical element, also provides a route to prepare optical continuous variable qubits or to synthesize arbitrary quantum states of a travelling optical field.
Conclusions.— A superposition of quanta addition and subtraction can orthogonalize any pure continuous-variable quantum state with known angle made by the mean of the state’s amplitude in phase-space. Such a superposition in combination with a controllable amount of the identity operation provides extensive control for quantum state engineering and quantum information applications. For stationary systems it is convenient to apply this tool multiple times, which we have utilized to illustrate how to perform arbitrary quantum state transformation. As the interactions we have used are available in many of the facets of quantum optics Hammerer2010 , the tools we introduce can be realized in numerous physical systems.
Acknowledgments.— We thank G. D. Cole, S. G. Hofer and K. Hammerer for useful discussion. M.R.V. is a member of the FWF Doctoral Programme CoQuS (W 1210), is a recipient of a DOC fellowship of the Austrian Academy of Sciences and gratefully acknowledges the Royal Society. We thank support provided by the European Comission (Q-ESSENCE), the European Research Council (ERC QOM), the Austrian Science Fund (FWF) (START, SFB FOQUS), the Foundational Questions Institute, the John Templeton Foundation and the Qatar National Research Fund (NPRP 4-554-1-084).
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Supplementary Material
.1 Determining the Effective Unitary Interaction
Central to our discussion in the main text is the measurement operator , which is used to describe the operation to the mechanical resonator via the optomechanical interaction and then single photon detection. , where the ket is the initial state of light at the cavity resonance for the two orthogonal polarizations used, the bra describes a -polarization photon detection with no photon detection, and is the effective optomechanical interaction including the manipulations to the optical field made after interaction with the mechanical resonator. In this supplementary we provide a discussion how is obtained.
The time evolutions described in Eq. (2) of the main text are generated by the beam-splitter and two-mode-squeezing effective interaction Hamiltonians. In the former case accumulates correlation with and in the latter case accumulates correlation with . For vacuum on the input of mode , the expectation of the number operator in the output of mode for the beam-splitter and two-mode-squeezing interactions are
respectively, where is the (intensity) reflectivity of the beam-splitter and is the squeezing parameter. In the optomechanical scheme we have considered, the mean photon number scattered by the optomechanical interaction for the beam-splitter and two-mode-squeezing interactions are
respectively. For small , , and we then have
for the effective optomechanical beam-splitter and two-mode-squeezing parameters, respectively. It is noted here that computing the mean number output in mode can also be performed to yield these parameters. As both the beam-splitter and two-mode-squeezing processes are driven simultaneously, we expect that the effective optomechanical unitary take the form , where and are the beam-splitter and two-mode-squeezing Hamiltonians respectively. To first order in the beam-splitter and squeezing parameters the effective unitary describing the cavity optomechanical interaction is then
Finally, to obtain the effective unitary used for the measurement operator, the polarization manipulations to the optical fields, as discussed in the main text, must be performed.
.2 Arbitrary Quantum State Transformation
In the main text we introduced a scheme for arbitrary quantum state transformation that generates a target state from a known input state. Here we further discuss our protocol and provide a specific quantum state transformation example.
The protocol works as follows. For a known initial state
which has no excitation beyond quanta (or has been approximated by truncation at this level), any target state of the form
can be generated by applying a controllably weighted superposition of identity and subtraction times, i.e.
(3) |
Applying this operation to the initial state we have
where we have used .
The operation is a non-unitary process and the un-normalized matrix elements of the state after application of are
(4) |
The target state is reached when . Provided that a set of coefficients fulfilling can be determined via matrix inversion. Once a set of coefficients is determined, a set of complex coefficients and that satisfy (3) can also readily be determined via matrix inversion.
We now provide a specific example of a quantum state transformation. Starting with an initial state we wish to reach the target state . This target state can be reached with three applications of identity and subtraction. Solving (4) we find that and . As identity has been used with each application we set and obtain
These equations can be readily solved exactly to provide the relative amplitudes between identity and subtraction to produce the target state. Numerical approximations to the solutions and the intermediate states during the quantum state transformation process are shown in Fig. 3.